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+Modeling communication asymmetry and algorithmic personalization in
+online social networks
+Franco Galantea,∗, Luca Vassioa, Michele Garettob, Emilio Leonardia
+aPolitecnico di Torino, Corso Duca Degli Abruzzi, 24, 10129 Torino
+bUniversit`a degli Studi di Torino, Corso Svizzera 185, 10149 Torino
+Abstract
+Modeling social interactions and their impact on opinion dynamics has attracted growing interest in
+recent decades, fuelled by the mounting popularity of online social networks (OSNs). On online social plat-
+forms, a few individuals, commonly referred to as influencers, produce the majority of content consumed by
+users. However, classic opinion models do not capture this communication asymmetry in OSNs. We develop
+an opinion model inspired by observations on leading social media platforms and tailored to the peculiarities
+of online interactions. Our work has two main objectives: first, to describe the inherent communication asym-
+metry in OSNs, where a tiny group of influencers hegemonizes the landscape of social debate, and second,
+to model the personalization of content by the social media platform. We derive a Fokker-Planck equation
+for the temporal evolution of users’ opinion distribution and analytically characterize the stationary system
+behavior. Analytical results, confirmed by Monte Carlo simulations, show how content personalization tends
+to radicalize user opinion and favor structurally advantaged influencers. These emerging behaviors suggest
+that algorithmic bias, inherently associated with platform filtering, can lead to undesirable outcomes. As an
+example application, we apply our model to Facebook during the Italian government crisis in the summer
+of 2019. Our work provides a flexible framework to assess the impact of algorithmic filtering on the opinion
+formation process and a fine-grained tool to study the complex interaction between influencers and social
+network users.
+Keywords:
+opinion dynamics, online social networks, algorithmic personalization, Fokker-Planck equation
+1. Introduction
+In recent years, the way people communicate has changed dramatically. With the advent of the Internet,
+new communication channels have emerged that allow people to transcend geographical and language barriers
+thanks to a global communication network and alternatives to text-only interaction (e.g., images and videos).
+Online social networks (OSNs) are probably the most notable example of such new interaction mechanisms
+and have greatly influenced our society by fostering discussions and disseminating information. The means
+that enable online social interactions are profoundly different from traditional interpersonal interactions and
+other mass media such as newspapers or television. The amount of content produced on such social media
+platforms is immense. Therefore, to keep users engaged within the social network, the platform performs
+filtering to select the posts offered to them. This filtering mechanism can reinforce the natural tendency
+(usually referred to as homophily in the literature) to interact with like-minded people. And, in turn, can
+lead to the formation of echo chambers [1], where people who share a similar point of view interact with
+each other but are isolated from the rest of the users. In addition, the reach of certain social media users can
+be extraordinary. Posts by very influential people can reach a large audience in virtually no time. Another
+∗Corresponding author
+Email addresses: franco.galante@polito.it (Franco Galante), luca.vassio@polito.it (Luca Vassio),
+michele.garetto@unito.it (Michele Garetto), emilio.leonardi@polito.it (Emilio Leonardi)
+Preprint submitted to Online Social Networks and Media (OSNEM)
+January 5, 2023
+arXiv:2301.01478v1  [cs.SI]  4 Jan 2023
+
+aspect worth mentioning is the diversity of topics discussed on the platform. It ranges from commentaries
+on the latest sporting events to debates on sensitive issues such as vaccinations. Last, communication on
+OSN takes place asymmetrically, i.e., few well-known individuals can exert influence on a large audience
+which, in turn, is composed of far less known people. These are some crucial aspects that characterize online
+social networks and distinguish them from ”offline” interactions. We believe that models seeking to capture
+the complexity of interactions occurring in online social networks must account for them.
+Trying to understand the mechanisms behind the opinion-forming process is a daunting challenge. The
+complexity driving this process is still poorly understood. Moreover, the individual reaction to external
+stimuli is utterly subjective and thus difficult to model. In a continuous framework, opinions can be in-
+terpreted as a person’s level of agreement with a statement or the interest they show in an issue (e.g.,
+adoption of technology, politics) and mapped to real-valued intervals. It is clear that measuring opinions in
+such intervals is arbitrary and can only lead to qualitative results. Indeed, most of the literature proposes
+theoretical models without the claim of accurately representing real-world scenarios. Very few works in the
+literature attempt to validate the emergent behavior of the model with physically observed phenomena (e.g.,
+[2] [1]). Some other works take the approach of supporting modeling decisions with real-world observations.
+Das et al. [3] did this by interviewing a group of people on specific topics, while Xiong and Liu [4] extracted
+information from Twitter networks. Following the above works, we present a model whose hypotheses are
+supported by data from social networks and whose outcomes are compared with emerging phenomena on
+two popular OSNs, i.e., Facebook and Instagram. As the limitations mentioned above also apply to our
+model, the results discussed here do not aim to be predictive. However, the proposed model provides a tool
+to study the emerging behavior on online social networks and the impact of algorithmic personalization.
+The main objective of this work is to develop an analytical framework tailored to online interactions,
+incorporating the following aspects:
+• The asymmetry typically found in OSNs. There exists a relatively small percentage of users of online
+social networks whose number of followers is orders of magnitude larger than that of other users.
+These individuals are commonly referred to in the literature as influencers or opinion leaders. They
+are particularly relevant, as their opinions can reach a vast fraction of the social network’s population.
+• The filtering performed on the content by the social media platform.
+Algorithmic personalization
+appears necessary in the context of OSNs, as the number of daily produced posts has become enormous.
+The aim is to increase engagement by showing users only the most relevant posts. The loop is then
+closed by taking into account user feedback on the posts received (e.g., likes).
+The proposed model:
+• Provides a tool for assessing the impact of different algorithmic personalization policies, focusing on
+the opinion leaders in the network. It can evaluate the extent to which these strategies might hinder
+diversity of opinion.
+• Exploits the observed characteristics of a large ensemble of Italian influencers from Facebook and
+Instagram social networks to ground its main hypotheses.
+• Allows for comparing its emergent behavior with observations on real online social networks. Further-
+more, the explanatory capabilities of the model are used in the study of the opposition of two Italian
+politicians during a government crisis by identifying a state of public opinion that can lead to the same
+behaviors observed in the collected data.
+One peculiar feature of our approach is the concept of reference direction, which is the individual’s main
+topic of interest and expertise. To our knowledge, the existing literature has not yet considered the impact
+of a reference topic for each influencer on the opinion formation process in multi-dimensional spaces.
+The influence exerted on non-reference directions depends heavily on platform personalization, which
+usually depends on how well-known an influencer is in its main field of expertise. For example, famous
+public figures (e.g., athletes, models) can express their point of view on potentially sensitive matters and
+2
+
+may resonate more than experts due to their popularity in their field. Therefore, since influencers discuss
+different topics, the reference direction loosely couples seemingly unrelated subjects brought up by the same
+person.
+Incorporating a personalization process into the dynamic behavior of the model is another crucial feature.
+The analytical results and the model simulation show that algorithmic personalization favors structurally
+advantaged individuals, resulting in less diversity of opinion. It is also interesting to observe that the model
+undergoes a phase transition in its behavior as a function of the degree of polarisation, at least in the case
+of two competing influencers. Below a certain threshold, there are diverse opinions in the population, and
+above this threshold, one of the influencers tends to polarise users’ attention.
+The paper is organized as follows. Section 2 discusses the relevant work in the literature and sets out
+the rationale for the need for a new opinion model tailored to OSNs. The Communication Asymmetry
+model is presented in Section 3, along with the notation used throughout the article. Section 4 presents
+some observations from real social networks supporting our modeling assumptions. Section 5 is devoted
+to the mean-field analysis of the model as the number of users grows large. The theoretical results on the
+steady-state behavior of the model are proved in the Appendix. Section 6 then investigates the impact of the
+model parameters on a reference scenario with two influencers. Analytical findings are validated in section
+7 by comparing the results of Monte-Carlo simulations with theoretical predictions specific to our reference
+scenario. Section 8 further validates our model with real data collected on Instagram and Facebook. At
+last, Section 9 concludes the article with a discussion of the implications and limitations of our work, setting
+the ground for future extensions.
+2. Related work
+The first steps in the field of opinion dynamics were taken in the late 1950s by a number of social
+psychologists, among which Solomon Ash [5], John R. P. French [6], and Leon Festinger [7] had great
+resonance in the field.
+Ash empirically observed that the individuals he studied engaged in conformist
+behavior because of the social pressure exerted by the rest of the social group. In short, Ash observed
+that an individual states a truth about something that is not true (e.g. “white is black” [3]) when the
+social group to which the individual belongs asserts it. French [6] developed a model to capture influence
+through interpersonal relationships within a group, focusing on leadership and using directed graphs to
+model interpersonal relationships. Festinger developed the theory of social comparison, according to which
+individuals tend to evaluate their position by comparing it with others. Moreover, the tendency to do so
+decreases the greater becomes the difference in opinion [7].
+Opinion models are divided customarily into two broad classes: those in which opinions are continuous
+variables and those in which opinions are discrete (often binary). In a recent review [8] examining agent-
+based opinion models, the authors show that more than 80% of the models considered are continuous.
+Much of the seminal work in the field of opinion dynamics is continuous in nature.
+For example, the
+DeGroot model [9] considers a networked social system in which individuals interact with their neighbors.
+Individuals average their current opinion with the opinion of their neighbors. The idea behind the model
+is to describe the process leading to consensus within a group. Subsequently, Friedkin and Johnsen [10]
+extended it by developing a flexible framework from which various opinion models (including French [6] and
+De Groot [9]) can be derived as particular cases. Their model of social influence encompasses both the
+processes of social conformity and social conflict that lead to behavior that goes beyond simple consensus
+and represents the persistent disagreement often observed in social networks. In the early 2000s, Hegselmann
+and Krause [11] and Deffuant and Weisbuch [12] proposed two similar models. In analogy with the DeGroot
+model, individuals interact by averaging opinions, but the authors introduced the central idea of bounded
+confidence. According to bounded confidence, individuals interact in a social network with other peers only
+if their beliefs are not too different. This mechanism implements the concept of homophily. Lorenz, in his
+review [13], provides the agent-based and density-based formulation of bounded confidence, distinguishing
+two main models: in the Hegselmann-Krause (HK) model, individuals modify their opinion as a result of
+interactions with all agents in their neighborhood, whereas, in the Deffuant-Weisbuch model, interactions
+are pairwise between connected individuals.
+3
+
+In addition to continuous models, discrete models have also appeared in the literature. The first and
+probably most prominent model of this kind is the voter model, independently introduced by Clifford and
+Sudbury [14] and Holley and Liggett [15]. Here, individuals are agents in a network of interactions, holding
+a binary opinion. At times dictated by a Poisson clock, an individual adopts the belief of a randomly chosen
+neighbor. This type of model has attracted a great deal of attention: several extensions have populated
+the recent literature, for example, taking evolving networks into account [16] [17] or allowing individuals to
+hold more than one opinion at a time [18], or introducing spontaneous changes of opinion [19] (noisy voter
+model).
+A consistent bulk of research on opinion dynamics comes from the physics literature, among which early
+contributions are Ben-Naim [20] and Toscani [21]. The idea underlying these models is that of describing
+interacting individuals using statistical mechanics by adequately defining the microscopic interactions be-
+tween the individuals, much like particles in a gas. Then, collective statistical phenomena are sought for
+the overall opinion of the population. In the papers mentioned above, Ben-Naim and Toscani consider two
+mechanisms of opinion formation: compromise, the human tendency to reach a reasonable trade-off on an
+issue to avoid conflict, and a process of introspection (in other models, e.g., [19], modeled as noise), which
+the authors believe represents the impact of external sources of information (e.g., media). A statistical
+approach is generally employed to study spin systems, and models such as the Ising model have also been
+applied to the opinion formation process. An extension of the Ising model is the Sznajd model [22], which
+implements social validation and for which Slanina and Lavicka [23] derived analytical results. In this model,
+the agreement of individual pairs leads to their neighbors agreeing with them, and a line graph is considered
+to capture the connection network. For a comprehensive review of opinion models, we refer to the survey
+by Castellano et al. [24].
+2.1. Models tailored to online platforms
+Most of the seminal literature on opinion dynamics is suited to describe the decision-making process in
+small groups of individuals, e.g., a board of directors, or to capture rather regular patterns determined by
+the daily personal interactions of individuals. Models such as the voter model have been studied extensively
+on regular lattices [25] [26]. The structure of interactions, especially those online, is far from homogeneous.
+As mentioned earlier, an inherent asymmetry in communication exists in OSNs where a limited number of
+individuals (influencers) monopolize the discussion. The voter model has been studied over heterogeneous
+networks (e.g., [27] [28]) to account for this diversity. On such networks, there can exist hubs (strongly
+connected nodes) playing a role similar to influencers in our framework, although the authors did not
+explicitly make such a distinction. Other works have divided the population into classes, e.g., [29] introduced
+stubborn agents, and if such individuals have opposing opinions, they hinder the possibility of the population
+converging to consensus.
+Recent work is drawing further attention to online platforms by adapting classical frameworks to the
+specificities of online interactions. Valensise et al. [30] have developed an opinion model that embodies al-
+gorithmic personalization, comparing its behavior to phenomena observed in social networks (e.g., Facebook
+and Twitter). Our work is different because we consider distinct classes of users, characterizing specifically
+influencers and closing the interaction loop between users and the platform by a feedback function. Other
+works that address content filtering bias in social media platforms include [31] [32]. Peralta et al. [31] develop
+a flexible framework to incorporate algorithmic bias into binary opinion dynamics by having agents interact
+at a lower rate with individuals who hold an opposing viewpoint. Considering both pairwise and group-wise
+interactions, the authors found that algorithmic bias either leads users to polarize their opinion (in the case
+of pairwise interactions) or results in the coexistence of beliefs (in the case of group-wise interactions).
+2.2. Validation of opinion formation models
+In [33], models of opinion dynamics are referred to as idealized because, in most cases, they assume basic
+underlying principles of interaction and observe emergent social behavior. There are two main approaches
+to validating opinion models in the literature: first, the use of observational data [2][1][30] and second, the
+use of controlled sociological experiments[34][35]. We will focus more on the first portion of the literature,
+4
+
+which is more relevant to our work. Attempts to validate opinion models are scarce for several reasons: i)
+the mapping of opinions into values, ii) an adequate definition of links between agents, and iv) the change
+in opinion after an interaction is hardly measurable.
+A notable exception is election and polling data,
+which make it possible to attribute a person’s opinion to the political orientation of the chosen candidate.
+Fortunato and Castellano [2] have shown that the distribution of vote counts is a universal scaling function
+and have derived a simple tree-like interaction structure with candidates as roots and an interaction that
+can turn the individuals reached into “activists” who can spread the idea and convince other individuals.
+The results of the model are in good agreement with empirical evidence. In [36], a noisy voter model could
+fit data from US elections. Other recent approaches [37] have used shared news on Facebook to assess the
+extent to which individuals are exposed to opposing views through their (online) friendship relationships,
+using users’ self-reported ideological affiliations to infer opinion. They found that individuals have access
+to cross-cutting content and that the degree of this exposure depends on the composition of one’s friends
+on social media. A more recent body of literature [1] [30] has directly employed data from online social
+networks, such as Gab, Facebook, Reddit, and Twitter, to observe the emergence of echo chambers [1] and
+to validate a model encompassing algorithmic personalization in the process of opinion formation [30].
+3. The Communication Asymmetry opinion model
+In this section, we first establish the notation used throughout the paper and then present the Commu-
+nication Asymmetry (CA) model in its most general formulation. We conclude the section with a discussion
+of the strengths and limitations of the proposed model.
+3.1. Notation
+In this work, we adopt the following vectorial notation. We denote vectors by bold symbols, whereas we
+denote their components with normal-font symbols whose subscript is the index in the vector, e.g., a = {ak}k.
+Lowercase letters denote parameters and dynamical variables associated with an individual. In general, index
+i runs over the set of influencers while index u runs over that of regular users. For those parameters/variables
+that can be associated with individuals of both classes (either influencers or regular users), the above indices
+are indicated between superscript parentheses, e.g., a(i), a(u), to immediately identify the class to which the
+individual belongs. If necessary, the dependence of variables on other system parameters is made explicit
+by specifying the independent variables between parentheses, e.g., α(·, ·). Italic capital letters denote sets,
+e.g., I is the set of all influencers in the population, while |I| is its cardinality.
+Capital letters represent outcomes of stochastic experiments whose characteristic parameters are low-
+ercase letters: e.g., Ω (ω(·, ·)). The operator E[·] represents an expected value, and a bar over a variable,
+e.g., ¯a, represents its average value. Whenever we need to express the probability of an event, we use the
+notation Pr[·]. We employ 1{·} for the indicator function. Lastly, time is denoted by t if it is considered
+continuous and by n if it is discrete.
+3.2. Description of the model
+We propose a continuous opinion model with two interacting classes of agents. Specifically, the population
+consists of Nu = |U| regular users and Ni = |I| influencers. This division mimics what happens in real
+social networks, where a small portion of the population, the influencers, has a much larger number of people
+following their posts on the online social network. We assume that the generation of new posts, i.e., messages
+in the OSN, is a Poisson Point Process (PPP) with intensity λ, where each event of the PPP corresponds
+to the creation of a new post from an influencer i ∈ I. The corresponding embedded discrete time will
+be denoted by the integer n ∈ N+, n = 1, 2, . . ., where n is the n-th post. Each post is sent1 to a subset
+of regular users, identified by the social platform according to an algorithmic personalization described by
+function ω (to be specified later). Regular users react to these posts through a feedback function θ (specified
+5
+
+User
+Influencer
+popularity update
+post generation
+post provision
+feedback
+Platform
+θ
+ω
+Figure 1: The picture depicts the relationship among the three players of the model: regular users, influencers, and the social
+media platform. Moreover, it highlights the role of the feedback function θ provided by the users and the filtering function ω
+used by the platform to propose the posts to the users.
+later).
+Figure 1 highlights this closed loop behavior brokered by the social media platform, positioned
+between regular users and influencers.
+The opinion space is X ⊂ Rd, where each dimension represents an uncorrelated topic on which users
+have a belief. Therefore, an opinion consists of a d-dimensional vector x(u)(n) ∈ Rd, which evolves as a
+result of the interaction between a regular user and every influencer on every possible topic. This model
+neglects interactions between regular users2. The prejudice of a user, denoted by z(u), is the other parameter
+that enters the opinion update rule alongside the user’s current opinion. It represents the user’s natural
+inclination toward different topics. Unless otherwise specified, we will assume that the user’s initial opinion
+is set equal to the prejudice: x(u)(0) = z(u).
+We will consider different distributions for the agents’ prejudice over the opinion space. In particular,
+delta, uniform, and Beta distributions are usually employed.
+Influencers are considered stubborn agents, which means that their opinions do not change over time,
+i.e., x(i)(n) = x(i)(0) = x(i) = z(i) ∈ Rd,
+∀n > 0 and i ∈ I. As we will show in Section 4, each influencer
+has a main topic of interest on which it publishes the majority of its posts and which typically coincides
+with the topic it is mainly known for on the OSN. It represents the reference direction r(i) ∈ {0, .., d − 1} of
+the influencer. Another parameter characterizing influencer i is its consistency c(i)(n), which indicates the
+probability that such an influencer publishes a post on its reference direction (it might change over time).
+Note that individuals with high consistency prefer to post in their reference topic. Moreover, we denote
+by f (i) the probability that a post is generated by influencer i at any time instant n, with �
+i∈I f (i) = 1.
+At last, we introduce the popularity vector p(n) := {pi(n)}i∈I, containing the current popularity of all
+influencers at time n, before the emission of the post at time n. We also introduce the normalized version
+of this vector π(n) = {πi(n)}i∈I where the components are the normalized probabilities πi =
+pi
+�
+j∈I pj .
+Dynamic variables of users (i.e., their opinion x(u)) and influencers (i.e., their popularity pi) are updated
+upon every post generation according to Algorithm 1. It provides a detailed description of the dynamics
+captured by our model.
+The model’s key features are further illustrated schematically in Figure 2: an
+influencer posts a message, the social media platform filters it according to ω, and users provide feedback
+via θ. These two features represent, respectively, an algorithmic effect (function ω): selective exposure,
+namely the tendency of a platform to suggest similar content to maximize time spent on the social platform,
+and an individual effect (function θ): confirmation bias, namely the tendency to value content that is close
+to one’s point of view, as discussed in [1] and the resources therein.
+These tendencies can explain the
+appearance of echo chambers in social networks.
+More specifically, in an elementary step of the dynamics, a post is generated by one of the influencers,
+selected according to the distribution f (i). The influencer i posts on its reference direction r(i) = j with a
+1In this paper, the verbs “send”, “suggest” and “reach” are used interchangeably referring to a post shown to a user by the
+platform.
+2From now on, we will refer to regular users simply as users. Moreover, the terms agent or individual are used to indicate
+a social network user of either class.
+6
+
+probability equal to its consistency c(i). Otherwise, it posts on one of the other directions in the opinion
+space j ∈ {0, 1, ..., d−1}\{r(i)} = Nr according to a given distribution, Pr[j = k] for k in Nr. In the rest of
+the paper, we assume for simplicity that this distribution is uniform over the set of non-reference directions.
+We suppose that each post contains exactly the influencer’s opinion on the topic (and that no noise in the
+user’s perception of the post is present). Then, note that the post contains a real-valued opinion that is the
+j-th component of the influencer’s opinion vector x(i). In principle, this generated post can reach any user:
+no explicit network structure is considered for the population3. Following the intuition that individuals
+with strongly divergent opinions are less likely to interact and therefore, as homophily suggests, like-minded
+individuals are more likely to interact, the subset of reachable users (i.e., users to whom the platform sends
+the post) is determined by considering the opinion distance in the reference direction between each user and
+the posting influencer. Adopting such distance as the central metric influencing the reachable group of users
+is of utmost importance as it couples the dynamics in different directions, which would otherwise evolve
+independently of each other. This posts-users matching process constitutes the content personalization we
+consider in this paper. Note that the social media platform suggests posts that might interest a user in
+addition to those that a user explicitly subscribes to (i.e., follows).
+Algorithm 1 Description of the Communication Asymmetry model
+Input:
+Ni influencers, Nu users, filtering function ω, feedback function θ
+Output: opinion of each regular user x(u)(n), ∀u
+Output: popularity of each influencer pi(n), ∀i
+1: loop
+2:
+select influencer i according to f (i)
+3:
+select a posting direction j, i.e., j = r(i) with probability c(i), other-
+wise j is selected uniformly on j ∈ {0, .., d − 1} \ {r(i)}
+4:
+pi(n + 1) = pi(n)
+5:
+for all regular user u in the population do
+6:
+x(u)
+j
+(n + 1) = x(u)
+j
+(n)
+7:
+if Ω
+�
+ω(|x(i)
+ri − x(u)
+ri |, πi(n))
+�
+= 1 then {post proposition}
+8:
+get feedback Θ
+�
+θ(|x(u)
+j
+− x(i)
+j |)
+�
+9:
+if Θ = 1 then {positive feedback}
+10:
+x(u)
+j
+(n + 1) = αz(u)
+j
++ βx(u)
+j
+(n) + (1 − α − β)x(i)
+j
+11:
+update popularity of i: pi(n + 1) += 1/Nu
+12:
+end if
+13:
+end if
+14:
+end for
+15: end loop
+To decide whether a given user is reached by a post (independently from other users), we extract a
+Bernoulli random variable Ω with parameter ω.
+The user receives the message when Ω(ω) = 1.
+The
+parameter ω can be interpreted as a visibility function from the influencer’s perspective, as it affects the
+subset of users reached by its posts. As already mentioned, ω should be a function of the opinion distance
+in the reference direction dr(n) = |x(u)
+r (n) − x(i)
+r (n)| and the popularity ratio πi of the posting influencer,
+so that the higher the popularity ratio, the more users an influencer can reach on average. Users express
+their feedback to a post on the platform through a Bernoulli random variable Θ
+�
+θ(|x(u)
+j
+− x(i)
+j |)
+�
+∈ {0, 1}
+3A complete bipartite graph, where I and U are the two sets of nodes and each link has a weight ω computed at each
+iteration of the dynamics, might represent the underlying network structure.
+7
+
+whose parameter θ depends on the difference in opinion on the actual direction j of the contribution. Only
+posts that receive positive feedback, i.e., Θ = 1, can influence the user’s opinion, reflecting the tendency to
+ignore unappreciated content. The social media platform collects feedback from all reached users to update
+the popularity pi of the posting influencer. Specifically, the update rule for the popularity of the posting
+influencer i reads as follows:
+pi(n + 1) = pi(n) + ΘT (θ, Upost)
+Nu
+(1)
+ΘT (θ, Upost) =
+�
+u∈Upost
+Θ
+�
+θ(|x(u)
+j
+(n) − x(i)
+j (n)|)
+�
+(2)
+where Upost is the subset of users who were made aware of the post by the platform, i.e., those for whom
+Ω(ω) takes the value one. The summation in the formula gives the aggregate feedback of all users who saw
+the post, which is normalized by the size of the population of regular users |U| = Nu to update the popularity.
+Note that this normalization is introduced only to avoid excessive growth of influencers’ popularity when
+the number of users becomes large. It does not affect the system dynamics, as these depend only on the
+normalized popularity values πi, which are not affected by the scaling factor 1/Nu.
+Influencer
+post generation 
+according to    
+Platform
+popularity update
+...
+...
+Ω=1
+Θ=1
+Ω=1
+Ω=1
+Θ=0
+Θ=1
+not reached
+reached
+Figure 2: Schematic representation of the model dynamics. The figure highlights the proportions of users who view a particular
+post Upost, i.e. those for which the random variable Ω takes the value 1. They react with their feedback Θ (e.g., likes), which
+depends on the opinion distance between them and the influencer i. Then the platform updates influencer’s i popularity.
+The core of the dynamic is represented by the opinion update rule, which dictates how the user’s opinion
+changes on the direction j of the post depending on the previous opinion x(u)(n), the prejudice z(u) and the
+opinion x(i) conveyed by the influencer through the post. The following system of equations characterizes
+the updating rule:
+x(u)
+j
+(n + 1) =
+�
+�
+�
+αz(u)
+j
++ βx(u)
+j
+(n) + γx(i)
+j
+if Ω (ω(dr, πi)) = 1 , Θ (θ(dj)) = 1
+x(u)
+j
+(n)
+otherwise
+(3)
+where γ = (1−α−β), being the updating rule a convex combination of x(u)(n), z(u) and x(i). Whenever
+the Bernoulli random variable Ω(ω) ∈ {0, 1} assumes the value 0, the post does not reach the user who keeps
+8
+
+its opinion. The individual’s opinion is also not affected by the post when the user receives it but does not
+appreciate it, i.e., the feedback variable Θ = 0. The actual opinion update is a convex (linear) combination
+of the user’s prejudice z(u)
+j
+, the current user’s opinion x(u)
+j
+(n) and the belief delivered by the influencer’s
+post x(i)
+j .
+Remark 1. The distance on the reference direction drives the filtering because we assume the platform is
+unaware of the specific topic associated with the post just created. Note the joint effect in the model of the
+distance between the user’s opinion and the influencer’s opinion on the reference direction and the distance
+along the direction defined by the post’s topic. Both contribute to determining the likelihood for the user to
+provide positive feedback to the message.
+Remark 2. In most OSNs, there are explicit subscriptions to influencers (i.e., the follow mechanism). Our
+approach does not consider this type of relationship, as we only account for homophilic contacts. Since the
+number of influencers is considerable in practice, homophily is not the only mechanism driving interaction. A
+regular user does not follow all its homophilic influencers. However, nowadays, most social media platforms
+(e.g., Facebook, Instagram, Twitter) not only offer their users content they explicitly subscribe to but also
+content that users might like based on their activity on the platform. This resembles the mechanism we are
+considering in our model.
+Remark 3. In our framework, regular users are passive, as they merely consume content produced by influ-
+encers: this constitutes a rather simplistic assumption. First, users can share the posts they receive, which
+increases their reach. Secondly, users themselves write posts that reflect their opinion, influencing other
+users. The impact of active users is beyond the scope of this article and will be considered in future work.
+4. Observations from Online Social Networks
+This section presents data from real-world social networks to motivate some of our modeling choices. For
+a detailed description of the dataset used, see Appendix A. One of the most important features introduced
+in this paper is the concept of reference direction, i.e., the main topic an influencer is interested in and
+on which they publish most of their posts. While this is a reasonable assumption, this claim needs to be
+supported by evidence from real social networks. Moreover, we examine the post-generation process to
+justify the choice of a Poisson Point Process to describe it.
+Music
+Politics
+Sports
+Food Pandemic
+Topics
+0.0
+0.2
+0.4
+0.6
+0.8
+Fraction of posts
+Luca Zaia
+AC Milan
+(a)
+0
+10
+20
+30
+40
+lag
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+autocorrelation
+music
+Luca Zaia
+AC Milan
+(b)
+Figure 3: (3a) Percentage of labeled posts on each of the considered topics for Luca Zaia, an Italian politician, and AC Milan,
+an Italian football club. (3b) Autocorrelation function on a secondary topic, i.e., music, for both influencers.
+9
+
+4.1. The reference direction
+This section shows that influencers prefer to post about a specific topic rather than discuss multiple
+ones. We have developed a post classifier that flags posts based on their topic. We should point out that
+classifying posts on OSNs into topics is not straightforward, and interpreting the results should be done
+with caution. First, the range of possible subjects discussed in a social network is practically countless. For
+practical reasons, we will only focus on a subset of five topics: Sports, Politics, Food and Cooking, Music,
+and Pandemics.
+These can be considered popular and general enough to cover a substantial fraction of the influencer
+discussions on OSN. We took a subset of the influencers in the dataset, namely those with the highest
+number of classified posts (see Appendix
+B for details on the classification and filtering process on the
+data). Note that even if the selected topics can be assumed uncorrelated, they are sometimes discussed
+jointly in one post. In such cases, it is not always clear which is the main topic of the post.
+After classification, we examined the distribution of posts on the topics for each influencer. In Figure 3a,
+we show two example influencers. In these two cases, the influencers have one topic on which they write most
+of their posts. Luca Zaia, an Italian politician, posts mainly about politics, and AC Milan, a soccer club,
+discusses sports predominantly. This behavior supports the existence of a reference direction for influencers.
+Figure 4a shows the distribution of the proportion of posts dealing with the main topic of each influencer.
+Recall that this proportion was called consistency in the jargon of our model.
+Most influencers have a clear reference topic on which they write more than half of their posts. Figure
+4b shows the average per-topic percentage of all influencers in the dataset in descending order, regardless
+of the specific topic. On average, almost ninety percent of the posts are in the reference direction. We
+discovered that influencers with low consistency values are affected by the presence of news outlets in the
+considered profiles, for which the lack of a sharp main topic is sensible.
+0.4
+0.6
+0.8
+1.0
+Consistency
+0
+20
+40
+60
+Number of Influencers
+(a)
+main
+2nd
+3rd
+4th
+5th
+Topics
+0.0
+0.2
+0.4
+0.6
+0.8
+Fraction of posts
+(b)
+Figure 4: (4a) Distribution of the fraction of posts published on the main topic of interest by the subset of influencers considered
+in this experiment, i.e., their consistency. (4b) The average percentage of labeled posts on each topic in decreasing order for
+all the influencers considered. The 95% confidence interval for each average value is reported in the figure.
+4.2. Independence of posts’ generation process on secondary directions
+The way users interact in an OSN is by posting content (i.e., text, images, videos) and receiving sugges-
+tions about what other users of the OSN posted, according to the filtering process set up by the social media
+platform. Since influencers’ posts have a much greater reach than those of regular users, they are the focus of
+our study. Namely, we examine the correlation between posts on each topic by looking at the chronological
+sequence of the messages of the individual influencers. Our primary focus now is on secondary topics, i.e.,
+the topics that are not the reference for the influencer, as they post less frequently in these topics, and one
+might expect to observe a bursty posting behavior, not well captured by the Poisson process.
+10
+
+In the previous section, we were able to assign a reference direction r(i) to each influencer. Here we look
+at the time series of the Influencers’ labeled posts. For each secondary direction s(i)
+j , we define an indicator
+function 1{postlabel=s(i)
+j
+} that takes the value 1 if the post was labelled as s(i)
+j
+and 0 otherwise. For each
+influencer, we thus obtain four sequences (recall that we consider five topics in total) of Bernoulli random
+variables indicating whether a post belongs to that particular direction. For these sequences, we calculated
+the autocorrelation function a(t). Figure 3b shows two examples of such autocorrelation functions, limited
+to 40-time lags, for the profiles of Luca Zaia and AC Milan. The time is discretized, i.e., the actual time
+between postings is not taken into account: only the posting events matter. An autocorrelation that equals
+zero everywhere except at τ = 0 would represent uncorrelated samples. In our case, the autocorrelation
+takes moderate values in most cases (≪ 1). Therefore, it is reasonable to assume that the post-generation
+is independent, and a Poisson Point Process is an appropriate choice. Lastly, note that the autocorrelation
+function for the pandemic topic takes larger values than for the other topics (see Figure 5), suggesting that
+the samples are weakly correlated. This fact is due to the exceptional public interest in the topic and because
+the outbreak of the epidemic only interested the last part of the considered time horizon.
+0.00
+0.25
+0.50
+0.75
+1.00
+sports
+politics
+music
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+food
+0
+10
+20
+30
+40
+pandemic
+0
+10
+20
+30
+40
+average
+lag
+autocorrelation
+Figure 5: Mean autocorrelation values of the post-generation process for each secondary topic of all the influencers. The last
+plot represents the average value over all topics. The 95% confidence interval is shown in each plot.
+5. Asymptotic Analysis of the Model
+This section is devoted to the analytical study of the model. In particular, results are obtained using a
+mean-field approach, considering Nu → ∞. In this situation, the equilibrium value for the influencers’ mean-
+popularity ratios ¯πi and users’ mean opinion value ¯x(z) (which depend on prejudice z) can be analytically
+determined. Furthermore, transient analysis of the system can be carried out by describing the dynamics
+of the users through a Fokker-Plank equation. For simplicity, we restrict our investigation to the situation
+where the opinion space is one-dimensional. However, we remark that it is possible to extend the analysis
+to the more general case by following the same approach.
+11
+
+5.1. Mean field approach
+When the number of users grows large, it is convenient to characterize the system state by the users’
+opinion distribution over the space. Moreover, hereinafter we will refer to system dynamics over continuous
+time t.
+Let (X(t), Z(t)) = (X(t), Z) be the current position (opinion) and prejudice of a randomly selected user.
+We introduce the cumulative distribution function F(x, z, t) = Pr[X(t) < x, Z < z]. The corresponding
+probability density function is f(x, z, t) =
+∂2
+∂x∂zF(x, z, t). Note that, by hypothesis, there are no dynamics
+along the z-axes, thus h(z) =
+�
+x f(x, z, t)dx does not depend on t and corresponds to the initial distribution
+of users’ prejudice. In Section 5.2, we will derive a Fokker-Plank equation for the evolution of the opinion
+distribution over time and space.
+For what concerns the evolution of the popularity of a generic influencer i, recall that we distinguish
+between its absolute popularity value pi(t) and the normalized value πi =
+pi(t)
+�
+j pj(t). We can already write
+down the equation for the evolution of the mean popularity ¯pi(t) (we remark that influencer’s popularities
+concentrate around their average as Nu grows large, as it can be easily shown):
+d¯pi(t)
+dt
+=
+1
+Nu
+λf (i)
+�
+x
+�
+z
+f(x, z, t) θ
+�
+|x − x(i)|
+�
+ω
+�
+¯πi, |x − x(i)|
+�
+dz dx
+(4)
+Indeed, the rate at which the popularity of influencer i grows is proportional to its posting rate (term
+λf (i)) times the probability that a generic user at (x, z) provides positive feedback to the post generated at
+time t (integral term). Moreover, recall from Algorithm 1 that each positive feedback increases the absolute
+popularity of the influencer by 1/Nu.
+5.2. Fokker-Planck equation for the opinion distribution
+In this section, we derive a mean-field Fokker-Planck (FP) equation for the population’s opinion distri-
+bution, assuming that the number of users grows large.
+To be specific, in the continuous-time FP approximation, we assume that for the effect of a post,
+“users/particles” reach their new position by moving at a constant speed during the interval ∆T equal
+to the average time 1/λ that elapses between the generation of two successive posts. Therefore, assuming
+that at time t a post is generated by user i, the following equation describes how the opinion of a user with
+prejudice z evolves from t to t + ∆T:
+x(t + ∆T) = αz + βx(t) + γx(i)(t)
+Thus, the increment is:
+∆x(i) = x(t + ∆T) − x(t) = α(z − x(i)(t)) + (1 − β)(x(i)(t) − x(t))
+(5)
+where we remark that ∆x(i) here represents the change in position of a user in position x, providing positive
+feedback to a post of influencer i.
+We can then compute its average velocity as:
+E[vx(x, z, t) | X(t) = x, Z = z] = E
+�[X(t + ∆T) − X(t) | X(t) = x, Z = z]
+∆T
+�
+=
+�
+i
+λf (i)∆T θ
+�
+|x − x(i)|
+�
+ω
+�
+¯πi(t), |x − x(i)|
+� ∆x(i)
+∆T
+=
+�
+i
+λf (i)θ
+�
+|x − x(i)|
+�
+ω
+�
+¯πi(t), |x − x(i)|
+�
+∆x(i)
+(6)
+where θ
+�
+|x − x(i)|
+�
+is the probability of providing positive feedback (users move only in this case), while
+ω
+�
+¯πi(t), |x − x(i)|
+�
+is the probability with which a user in x is exposed to a post created by influencer i at
+12
+
+time t. Indeed, users only move if they are exposed to the post and provide positive feedback. Note that, to
+avoid a cumbersome notation, we have omitted the dependency on the time of the distance term |x − x(i)|.
+The variance of the velocity is given by the relation:
+σ2
+x(x, z, t) =
+�
+i
+λf (i)∆Tθ
+�
+|x − x(i)|
+�
+ω
+�
+¯πi(t), |x − x(i)|
+� (∆x(i) − E[vx(x, z, t)]∆T)2
+(∆T)2
+=
+1
+(∆T)2
+�
+i
+f (i)θ
+�
+|x − x(i)|
+�
+ω
+�
+¯πi(t), |x − x(i)|
+�
+(∆x(i) − E[vx(x, z, t)]∆T)2
+This allows us to write down a Fokker-Plank equation [38] for the probability density function f(x, z, t)
+where x, z ∈ [a, b]:
+∂f(x, z, t)
+∂t
+= −∂vx(x, z, t)f(x, z, t)
+∂x
++ 1
+2
+∂2σ2
+x(x, z, t)f(x, z, t)
+∂x2
+(7)
+5.3. Steady state analysis
+Now we direct our attention to the existence of stationary solutions for the system. Stationary solutions
+of (7) necessarily satisfy:
+∂
+∂x
+�
+−vx(x, z)f(x, z) + 1
+2
+∂σ2
+x(x, z)f(x, z)
+∂x
+�
+= 0
+where vx(x, z) and σ2
+x(x, z) must be constant over time. This requires normalized popularities to be static
+(i,e. ω(·) to be constant over time). From previous equation, integrating both sides with respect to x, we
+get:
+�
+−vx(x, z)f(x, z) + 1
+2
+∂σ2
+x(x, z)f(x, z)
+∂x
+�
+= c0(z)
+(8)
+where c0(z) is a uni-dimensional arbitrary in z. Now, observe that, for every z, previous equation is a first
+order linear ODE in x, and therefore an explicitly solution for f(x, z) can be obtained:
+f(x, z) =
+�
+c1(z) exp(A(x, z) − A(a, z)) + c0(z) exp(−A(x, z))
+� x
+a
+exp(A(θ, z))dθ
+�
+h(z)
+(9)
+where
+A(x, z) =
+� x
+a
+η(u, z)du
+η(x, z) = −2vx(x, z) − 1
+2
+∂σ2
+x(x,z)
+∂x
+σ2x(x, z)
+.
+Function c0(z) can be obtained by imposing boundary conditions:
+�
+−vx(x, z)f(x, z) + 1
+2
+∂
+∂xσ2
+x(x, z)f(x, z)
+� ���
+x=a,b= 0.
+∀z
+which leads to c0(z) = 0, while function c1(z) is determined by imposing the normalization condition:
+�
+f(x, z)dx = h(z).
+Observe that when σ2
+x(x, z) → 0 and ∂σ2
+x(x,z)
+∂x
+→ 0, from (8), with x0(z) = 0, we obtain that necessarily
+the mass concentrates around points for which vx(x, z) = 0. Such points, improperly referred to in the
+following as equilibrium points, will be characterized analytically later on.
+Turning our attention to popularity dynamics, recall that stationary conditions necessarily imply nor-
+13
+
+malized popularities to be constant over time:
+¯πi(t) = ¯πi
+∀i
+On the other hand, absolute popularities naturally grow over time, but the ratio between any two of them (say
+i, j) must converge to a constant value cij equal to the ratio of their corresponding normalized popularities:
+¯pi(t)
+¯pj(t) = cij = ¯πi
+¯πj
+∀i, j ∈ I, i ̸= j
+(10)
+Now observe that in stationary conditions the r.h.s. of (4) does not depend on time, therefore (4) admits
+the following trivial solution:
+¯pi(t) =
+�
+λf (i)
+�
+x
+�
+z
+θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z)
+� t
+Nu
++ ¯pi(0)
+(11)
+Therefore, we meet conditions (10) for any t ≥ 0, iff normalized popularities of influencers {¯πi}i satisfy
+the following system of equations:
+λf (i)
+�
+x
+�
+z
+θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z) = c¯πi
+∀i, for some c ∈ R+
+s.t. ¯πi ≥ 0 and
+�
+i
+¯πi = 1.
+(12)
+and the initial condition {pi(0)}i satisfies (10) (i.e. pi(0) = k¯πi for some k > 0).
+Let
+ki(¯πi) := λf (i)
+�
+x
+�
+z
+θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z)
+¯πi ∈ [0, 1]
+(13)
+We can show that:
+Theorem 5.1. Solutions of (12) always exist whenever ki(·) ∈ C1[0, 1], ki(·) is increasing, continuous and
+strictly concave.
+The proof is reported in Appendix C.
+We remark that when ki(0) > 0 ∀i, the solution is always unique with ¯πi ∈ (0, 1). Instead when ki(0) = 0
+for some i, the solution is not guaranteed to be unique.
+Now, the problem is how to jointly solve for stationary solutions of {¯πi}i and F(x, z). In a schematic
+way, on the one hand, we have shown that given ¯π = {¯πi}i, and h(z), we can uniquely determine a
+F¯π(x, z) = H(¯π), where F¯π(x, z) =
+� x
+−∞
+� z
+−∞ f¯π(y, w) dy dw is the opinion distribution of users resulting
+from fixed influencers’ popularities ¯π (by (9)).
+On the other hand, under the conditions: ki(·) ∈ C1[0, 1], ki(·) is increasing and strictly concave,
+ki(0) > 0 ∀i, given F(x, z), we can obtain a ¯πF = G(F(x, z)) that uniquely corresponds to i (Theorem 5.1).
+The existence of a unique fixed point for the joint system of (stationary) users’ opinions and influencers’
+popularities is guaranteed under the condition that the operator H ◦ G(·) is a contraction over a complete
+space.
+Theorem 5.2. Under the assumption that both ω(·, ·) and θ(·) exhibit a sufficiently weak dependence on their
+variables, the operator H ◦ G(·) is a contraction over a complete space, and therefore a unique stationary
+solution exists.
+The proof is reported in Appendix C.
+5.4. Asymptotic analysis of the fluid limit
+Previous theoretical analysis is, unfortunately, non-constructive, meaning that it does not allow for direct
+computation of stationary solutions of our dynamical system. To complement the previous analysis, in this
+14
+
+section we propose a methodology to compute numerically stationary solutions, even in multi-dimensional
+scenarios, under the assumption that Nu → ∞, β → 1, σ2
+x(x, z) → 0 and ∂σ2
+x(x,z)
+∂x
+→ 0. In the following, we
+will refer to the such regime as fluid limit.
+5.4.1. Mean opinion assuming that normalized popularities converge
+As already observed in Section 5.3, recall that, given ¯π = {¯πi}i, the distribution of users with a given
+prejudice z concentrates around equilibrium points, i.e., points ¯x(z) at which v(x, z), as given in (6), is null
+(i.e. v(¯x(z), z) = 0). Therefore, points ¯x(z) must satisfy equation:
+0 =
+�
+i
+f (i)ω
+�
+¯πi, |¯x − x(i)|
+�
+θ
+�
+|¯x − x(i)|
+� �
+α(z − x(i)) + (1 − β)(x(i) − ¯x)
+�
+(14)
+Defining for compactness di,¯x =
+��¯x − x(i)�� and recalling γ = 1 − α − β, from (14) we get:
+¯x(z) =
+α
+1 − β z +
+γ
+1 − β
+�
+i∈I f (i)ω
+�
+¯πi, di,¯x�
+θ
+�
+di,¯x�
+x(i)
+�
+i∈I f (i)ω (¯πi, di,¯x) θ (di,¯x)
+(15)
+The assumption β → 1 is required to avoid too large oscillations of users’ opinions in response to a single
+post generated by an influencer, which may reduce the accuracy of our mean-field approximation.
+This hypothesis is not restrictive: since β represents the weight individuals give to their current opinion,
+we can reasonably assume that users do not dramatically change their opinion in response to single post
+events.
+5.4.2. Normalized popularities assuming opinion convergence
+Here we assume that users with prejudice z are concentrated in opinion point ¯x(z), and we look for
+the stationary popularity ratios ¯πi.
+To simplify the expressions, we introduce the quantity Fi(¯πi) ≜
+�
+z f (i)ω(¯πi, di,¯x(z))θ(di,¯x(z))h(z)dz.
+Observe that solutions of (12) are necessarily in the form:
+¯πi =
+Fi(¯πi)
+�
+j∈I Fj(¯πj)
+(16)
+where c appearing in (12) is given by c =
+1
+�
+j∈I Fj(¯πj). Under the assumption that ω(·, ·) is concave in its
+first argument (for any choice of the second), Theorem 5.1, guarantees the existence of such solutions for
+every choice of function ¯x(z). Moreover, even in the more general case, i.e., when ω(·, ·) is non-concave in
+its first argument, solutions of (16) can be found numerically in many cases, through a fixed point iteration
+method.
+To conclude, observe that a pair (¯x(z), {¯πi}i) represents a stationary solution if it jointly satisfies (15)
+and (16). The existence of such solution can be, again, only verified numerically through a fixed point
+approach.
+At last, note that, in the special case in which all users have the same prejudice z we can rewrite (15)
+as:
+¯x =
+α
+1 − β z +
+γ
+1 − β
+�
+i∈I Fi(¯πi, ¯x)x(i)
+�
+i∈I Fi(¯πi, ¯x)
+=
+α
+1 − β z +
+γ
+1 − β
+�
+i∈I
+¯πix(i)
+(17)
+which provides a direct formula for the mean opinion ¯x in terms of the normalized popularities ¯πi and the
+influencers’ opinions x(i).
+6. Model predictions
+In this section, we present a selection of results obtained while varying the model parameters, providing
+valuable insights into the impact of algorithmic personalization. Results are obtained through a Monte Carlo
+15
+
+Table 1: Parameters and functions shared across experiments
+Symbol
+Value - Form
+Description
+Ni
+2
+Number of influencers
+x(0)
+j
+0
+Opinion of influencer 0 on direction j
+x(1)
+j
+1
+Opinion of influencer 1 on direction j
+r(0)
+0
+Reference direction of influencer 0
+r(1)
+1
+Reference direction of influencer 1
+p0,1(0)
+100
+Initial absolute popularity of both influencers
+Nu
+10000
+Number of regular users
+Niter
+100000
+Number of iterations for each simulation
+α
+0.05
+First weight in the updating rule in Eq. 3
+β
+0.93
+Second weight in the updating rule in Eq. 3
+θ(·)
+1 −
+���x(i)
+j
+− x(u)
+j
+���
+Functional form of the feedback function
+ω(·)
+e−ρ(x(u)
+r
+−x(i)
+r )
+2
+πi
+Functional form of the visibility function
+0.00
+0.25
+0.50
+0.75
+1.00
+x0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x1
+Figure 6: Table 1 summarizes some of the parameters of the system shared across the different experiments. On the right-hand
+side, a realization of the initial opinion distribution of the regular users (being also the prejudice since z(u) = x(u)(0)).
+approach. We focus on the two main dynamic variables of the system: the average opinion ¯x of regular
+users and the normalized popularities {¯πi}i for influencers.
+Note that the quantities shown in this section, i.e., the pair (¯x, ¯π), are obtained as empirical averages
+over multiple runs, and over all the regular users, as far as ¯x is concerned. Hence, they are different from
+the values presented in the previous section, in principle, which pertains to the limiting case of an infinite
+population of users with the same prejudice z, and where β approaches 1. Moreover, in some cases, to
+save space, we omit the results on average user opinion because it is tightly coupled with the normalized
+popularities, as observed in the previous section. Lastly, to facilitate the interpretation of results, we restrict
+ourselves to the case of two “competing” influencers. We provide further details on the scenario considered in
+section 6.1. The model can clearly be applied to scenarios with an arbitrary number of influencers occupying
+any position in the opinion space. In section 6.2, we present the behavior as function of publication frequency
+f (i), and in section 6.3 as function of consistency c(i). Then we show examples of final opinion distributions
+of the regular users in a few paradigmatic cases in section 6.4. Finally, in section 6.5, we consider the degree
+of stubbornness, defined as δ = α
+γ , which governs the opinion update of the users.
+6.1. Description of the scenario
+The default parameters of our reference scenario are reported in Table 1 unless otherwise explicitly
+stated. As mentioned earlier, we restrict ourselves to the case of two “competing” influencers, i.e., Ni = 2.
+We assume that x(0)
+j
+= 0 and x(1)
+j
+= 1 ∀j. We consider the case of different reference directions r(0) ̸= r(1). In
+this section, we consider a two-dimensional opinion space and assume that the vast majority of the regular
+users’ population initially takes a moderate position on both topics. More precisely, initial opinions are
+distributed according to a Beta distribution, independently on each axis, with shape parameters a = b = 10,
+as shown in Figure 6. Recall that we assume for simplicity that the prejudice of the user z(u) corresponds
+to the initial opinion x(u), hence Figure 6 also provides the prejudice distribution of users.
+The functional form of visibility ω and feedback θ is also reported in Table 1. We take as ω(·) a Gaussian
+function similar to the trust function in [39], but modulated by ¯πi. Here, the coefficient ρ is a parameter that
+controls the extent to which the social media platform filters content. Small values of ρ correspond to smooth
+personalization, i.e., influencers can reach users whose opinion strongly differs from theirs. Conversely, high
+values of ρ correspond to sharp personalization: only close users (in the opinion space) are reachable with
+non-negligible probability. The function θ(·) is assumed to be a decreasing, linear function of the opinion
+difference dj = |x(u)
+j
+− x(i)
+j |.
+16
+
+6.2. Behaviour as function of the frequency of publication
+The frequency of publication f (i) is one of the basic parameters that characterize influencers.
+The
+higher f (i), the higher the structural advantage of the influencer because it more frequently reaches users
+through posts, attracting them to its own opinion. In this section, we examine the value of mean normalized
+popularity ¯π0 as a function of f (0). Note that in the case of two influencers, f (1) = 1 − f (0). We performed
+this experiment by fixing the consistency of the two influencers: c(0) = c(1) = 0.8, which is approximately
+the average consistency observed on real-world data (Figure 4a).
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+f(0)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+Values of 
+0.0
+0.3
+0.5
+1.0
+Figure 7: Popularity ratio ¯π0 of influencer 0 as function of the publication rate f(0). Each point is obtained by averaging
+over 100 time samples and 10 different process’ realizations. Different levels of personalization are considered by varying the
+parameter ρ. The two influencers have the same consistency c(0) = c(1) = 0.8. Note that, in the considered scenario, the curves
+are symmetric for values of f(0) in [0.5, 1.0].
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+f(0)
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+Distance from i = 0 on r(0)
+Values of 
+0.0
+0.3
+0.5
+1.0
+(a)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+f(0)
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+Distance from i = 1 on r(1)
+Values of 
+0.0
+0.3
+0.5
+1.0
+(b)
+Figure 8: Opinion distance of the influencers’ opinion on their reference direction x(i)
+r(i) and the average opinion of the regular
+users’ population on the same direction ¯xr(i). Various degrees of personalization are considered, tuning the parameter ρ, the
+setting is the same as that of Figure 7.
+In Figure 7, we consider different levels of personalization by varying the parameter ρ in the exponent
+of the visibility function ω. We see that the higher the degree of personalization (i.e., the higher the value
+of ρ), the lower the normalized popularity of influencer i = 0, for any given f (0). This result suggests
+that algorithmic personalization favors the structurally advantaged individual, i.e., the one with higher f (i).
+This mechanism, in turn, leads to more radical positions in the population of regular users, as the platform
+preferentially exposes them to the belief of the advantaged influencer. Figure 8 clearly shows this behavior.
+17
+
+Note that for high values of ρ, the average user opinion exhibits a significant bias toward the structurally
+advantaged influencer. Such bias persists up to a critical value of posting frequency. For example, when
+the personalization parameter is ρ = 1.0, the critical posting frequency value is roughly 0.35; when the
+personalization parameter is ρ = 0.5, the critical posting frequency value is approximately 0.25.
+We argue that content filtering in OSN poses a potential threat to opinion diversity.
+This premise
+is inextricably linked to the goal of usage maximization [32] pursued by the social media platform. Many
+platforms indeed prefer to suggest just similar content rather than exposing individuals to radically different
+opinions, allowing for so-called serendipity.
+6.3. Behaviour as a function of the consistency
+In section 4.1, we showed the existence of a reference direction for real influencers. Here, we investigate
+the impact on dynamics of the extent to which an influencer publishes on its reference direction, i.e., its
+consistency c(i). In this experiment, we consider two influencers with the same posting frequency f (0) =
+f (1) = 0.5, and we let c(0) vary while keeping c(1) fixed. We then consider different choices of c(1) to grasp its
+impact on the dynamics. From Figure 9, we see that consistency does not significantly affect the normalized
+popularities when personalization is smooth (ρ = 0.0001), while it becomes relevant when the platform
+applies sharp personalization to the content (ρ = 1).
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+c(0)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+Values of 
+0.0001
+1.0
+(a) c(1) = 0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+c(0)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+Values of 
+0.0001
+1.0
+(b) c(1) = 0.8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+c(0)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+Values of 
+0.0001
+1.0
+(c) c(1) = 1.0
+Figure 9: Popularity ratio ¯π0 of i0 as function of its consistency c(0), while considering f(0) = f(1) = 0.5 and keeping fixed
+the consistency of the second influencer at (9a) c(1) = 0.5, (9b) c(1) = 0.8 and (9c) c(1) = 1.0. The two colors represent two
+different levels of personalization (i.e., smooth and sharp). Each point is obtained by averaging over 100 time samples and 10
+different realizations of the process. It is interesting to observe that the maximum of ¯π0 moves to the left (i.e., is achieved for
+a lower value of consistency c(0)) as the consistency of i = 1 increases.
+18
+
+Before discussing the results concerning sharp personalization shown in Figure 9, it is crucial to note
+that when considering a bi-dimensional opinion space, a consistency c(i) < 0.5 implies that the choice of the
+reference direction for influencer i is somehow unnatural since it produces the majority of posts on the other
+direction. This choice is in contradiction with the definition of reference direction itself. Nonetheless, we
+leave this situation as a possibility: let us imagine an influencer can adopt consistency values of less than
+0.5 while undertaking a transition phase during which it changes its main topic for its posts. In this case,
+the platform would still perform personalization on the given reference direction r(i), but the consistency
+would be less than 0.5 due to the change in posting pattern. It represents a scenario of interest, and as such,
+we allow for c(i) < 0.5.
+First, we note that the shape of the curves in Figure 9 depends strongly on the value of the consistency
+of the “opposing” influencer c(1).
+Second, a perfectly balanced condition is achieved whenever the two
+influencers have the same consistency since all parameters are symmetric (even the curves associated with
+ρ = 0.0001 and ρ = 1.0 coincide on this point), see Figure 9c at c(0) = 1 for example. The observed pattern
+is consistent in all three figures with ¯π0 being first increasing and then decreasing, exhibiting a unique
+maximum in all three diagrams.
+Let us start the discussion by considering Figure 9c because its interpretation is instrumental to better
+understanding the other scenarios. It represents a rather degenerate situation since the influencer i = 1
+posts exclusively in its reference direction r(1) = 1. However, the simplicity of the scenarios allows us to
+interpret the results straightforwardly. In this case, the influencer i = 0 has r(0) = 0 and for 0 < c(0) < 1
+4 posts in both directions, with the social media platform filtering according to distance in the reference
+direction. In the direction r(0) = 0, the influencer has no competition at all, since c(1) = 1, so it is able
+to attract the user population to its “reference opinion” while competing with the other influencer in the
+non-reference direction. The lower the consistency c(0), the greater the competition on r(1) = 1, i.e., for
+values of c(0) close to 0, the influencer i = 0 posts the vast majority of its messages on r(1) = 1. Therefore,
+the final value of ¯π0 reaches higher values, as demonstrated in Figure 9c. It happens because the influencer
+i = 0 has a stable feedback stream from the posts in its reference direction, where it does not face any
+competition, and it competes with i = 1 in the other direction, being at an advantage since the visibility
+of its posts is determined by the opinion distance in its reference direction r(0). From these observations,
+we can conclude that influencer i = 1, i.e., the influencer with the higher consistency, is disadvantaged for
+virtually all values of the other influencer’s consistency c(0). Here we are considering the extreme case, where
+the influencer i = 1 has the maximum attainable consistency c(1) = 1. Thus, we can easily conclude that
+the individual with lower consistency holds the structural advantage. This is consistent with the results in
+Figure 9a and 9b, for which we observe a decrease in normalized popularity ¯π0 on the right of the point at
+which c(0) = c(1). Namely, the influencer with higher consistency is penalized and eventually reaches lower
+values of ¯πi. This assertion is not true in the first part of all plots in Figure 9 where ¯π0 is less than 0.5,
+i.e., the influencer i = 1 is favored, even though c(0) < c(1). It is important to note that this is only true
+for values of c(i) < 0.5, which, as discussed above, are only relevant in certain situations. This behavior
+depends on the interplay between the actual main posting direction (which is different from the reference
+direction when c(i) < 0.5) and the algorithmic personalization performed on r(i), see also the discussion in
+the footnote. Interestingly, the curves intersect for the first time when c(0) ≈ 1 − c(1).
+From our results, choosing a consistency c(i) around 0.5 for i = 1 seems to be a successful choice:
+from Figure 9a it is clear that the influencer i = 0 can only reach and never exceed the value of the
+normalized popularity of its “opposing” influencer. In Figure 9b we assign a higher consistency c(1) for the
+competitor. The discussion developed above applies, and for values of c(0) ≥ 0.5, the influencer with the
+lower consistency always has an advantage. Indeed, we find that for consistency values c(0) < 0.8 = c(1)
+(and c(0) > 0.2) influencer i = 0 achieves higher values of ¯πi, see Figure 9b.
+In summary, the results of this section suggest that a given influencer can gain an advantage over its
+4The point c(0) = 0 corresponds to a rather peculiar situation where the influencer i = 0 does not post on its reference
+direction, but only on the other direction. Then the filtering depends on the initial configuration of the users in this direction,
+where no dynamics occur. It can be concluded that the scenario is fairly balanced, with the influencer i = 1 having a slight
+advantage since the filtering occurs in the direction where the dynamics take place.
+19
+
+competitors if it has a lower consistency c(i). This observation also reflects the natural tendency of people
+to seek varied content.
+6.4. Opinion configuration considering combinations of reference directions
+In previous sections, we have focused primarily on the influencer perspective, looking at normalized
+popularity values ¯πi. Here we present possible final opinion configurations in scenarios in which the reference
+directions of influencers (r(0), r(1)) are either coincident or different. It was observed above that when an
+influencer has a structural advantage, it achieves a higher πi and, in turn, can exert a higher attracting force
+to the regular users towards its opinion. We then argue that it is interesting to examine what can happen
+in a symmetric scenario in terms of frequency of publication (f (0) = f (1)) and consistency (c(0) = c(1)). As
+before, the influencers hold opinions x(0) = (0, 0) and x(1) = (1, 1). We consider the two possible cases where
+the two opinion leaders have either the same or different reference directions and the impact of algorithmic
+personalization.
+This symmetrical scenario is interesting because, in most cases, it guarantees the coexistence of both
+influencers (neither of them ‘wins’), see the normalized popularity plots in Figure 10. Therefore, the final
+opinion distribution is the result of the joint influence of both agents.
+In Figure 10a, we observe only a negligible perturbation with respect to the initial distribution shown in
+Figure 6. In this case, the platform practically does not filter the content, so every post reaches all users.
+From a regular user perspective, individuals are exposed to nearly identical forces, i.e., “opposite” stimuli
+from the two influencers, which almost perfectly cancel each other. In Figure 10b, the impact of strong
+personalization is clear: the filtering effect introduced by the platform leads to the emergence of two echo
+chambers, whose membership is determined mainly by user’s prejudice. Each user reaches an equilibrium
+point at which the resultant attraction induced by the two influencers is balanced by the attraction exerted
+by its own prejudice. Interestingly, users also tend to cluster in the non-reference direction (x1 in Fig. 10b)
+and align their opinion with that of the influencer associated with the echo chamber they end up in. We
+remark that this is a metastable condition, as the πi diagram indicates. By extending the time horizon,
+we may observe a different final situation in which one of the two influencers “wins” (exhibiting behavior
+similar to Fig. 10d, but just taking place at a different time scale.)
+Figures 10c and 10d refer to the case of different reference directions: the two influencers do not compete
+on the same topic. In Figure 10c, it is clear that there is no competition as the two influencers are able to
+attract users to their reference opinion, i.e., x0 = 0 the reference opinion of i = 0 and x1 = 0 that of i = 1.
+It constitutes a particularly relevant case, whose occurrence is linked indissolubly to the newly introduced
+concept of reference direction. In the last scenario, shown in Figure 10d, the influencer i = 1 “wins ”, i.e.,
+¯π1 → 1, which brings public opinion closer to their belief on both issues. The users’ opinion does not overlap
+with that of the winning influencer because they are anchored by their prejudice. In this case, an unstable
+behavior is observed since the identity of the winner influencer (as expected, as a result of perfect symmetry)
+depends on random factors, and different sample-paths lead to diverse winners. It should also be noted that
+sharp personalization leads to a situation where the public scene is monopolized by only one individual.
+6.5. Behaviour as function of the updating weights
+The behavior of the system depends not only on the characteristics of the influencers and the composition
+of public opinion, but also on the parameters controlling the opinion update rule in equation (3). The update
+is a convex combination of the prejudice, the current opinion, and the opinion conveyed by the post. We
+chose to hold fixed the weight β associated with the current opinion and consider the ratio of the other two
+weights α
+γ , which we termed degree of stubbornness, as it gives an indication of the extent to which users
+change their opinions.
+We considered an unbalanced scenario in which influencer i = 1 has a structural advantage, i.e.,
+f (1) = 0.7 > f (0).
+Figure 11 again shows that personalization favors the structurally advantaged indi-
+vidual (consistent with section 6.2). Note that the x-scale is logarithmic to highlight the sudden drop of ¯π0
+for α
+γ ≈ 10−3 (corresponding to modest values of α) when sharp personalization is applied. The shape of the
+two curves is quite similar, only the decrease is observed at different values of α
+γ . Smooth personalization
+20
+
+0.00
+0.25
+0.50
+0.75
+1.00
+x0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x1
+0
+20000
+40000
+60000
+80000
+100000
+n
+0.0
+0.5
+1.0
+i
+0
+1
+(a)
+0.00
+0.25
+0.50
+0.75
+1.00
+x0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x1
+0
+20000
+40000
+60000
+80000
+100000
+n
+0.0
+0.5
+1.0
+i
+0
+1
+(b)
+0.00
+0.25
+0.50
+0.75
+1.00
+x0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x1
+0
+20000
+40000
+60000
+80000
+100000
+n
+0.0
+0.5
+1.0
+i
+0
+1
+(c)
+0.00
+0.25
+0.50
+0.75
+1.00
+x0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x1
+0
+20000
+40000
+60000
+80000
+100000
+n
+0.0
+0.5
+1.0
+i
+0
+1
+(d)
+Figure 10: (10a,10b) Influencers with the same reference direction, i.e., r(0) = r(1) = 0, while (10c,10d) influencers have
+r(0) = 0 and r(1) = 1. In both cases, two different degrees of personalization are considered: smooth (left column, ρ = 0.0001)
+and sharp (right column, ρ = 5.0). In all cases, the influencers have consistency c(0) = c(1) = 0.8. The distributions were
+obtained as the time average of the opinion distribution in one realization of the process. The normalized popularities of the
+two influencers in the given realization are shown along with the distributions; it is clear that the influencers coexist except in
+10d.
+21
+
+allows the coexistence of influencers on the whole domain, while with sharp personalization, for a wide range
+of parameters, influencer i = 1 “wins.”
+In both cases, there is an initial phase (for low values of α
+γ ) in which the two influencers coexist, and this
+is followed by a drop of the normalized popularity of the disadvantaged influencer. This can be explained
+by the fact that small values of α
+γ imply that a negligible weight is given to the prejudice, and therefore
+regular users concentrate around the two influencers’ opinions on their reference direction. This can be easily
+confirmed by looking at the final opinion configuration of users, who concentrate in the upper corners of the
+opinion space (around [0, 1] and [1, 1]). This is because the influencer i = 1, whose opinion is x(1) = [1, 1]
+is stronger than the other in terms of popularity and is able to pull users along its non-reference direction
+as well. We remark that when users are very close in opinion to a particular influencer, it is difficult for
+the other to persuade them, as the probability of this happening is proportional to the product ω · θ, both
+of which are a function of opinion distance. In these scenarios, the distance from the “further ” influencer
+is dj ≈ 1, which drastically reduces the probability of reaching the users.
+Thus, as long as
+α
+γ is small
+enough, both influencers can build their user base. These situations represent rather degenerate cases where
+the population almost disregards their prejudice in favor of the opinion conveyed by the post. It might be
+interesting to consider users with varying degrees of “volatility” who are able to pull along the opinion of
+their neighborhood.
+As the degree of stubborness increases, so does the inertia of the users. They are more entrenched in
+their prejudice and therefore no longer concentrate in a small neighborhood of the influencer’s opinions. This
+favors the structurally advantaged influencer, as the other (i.e., i = 0) is unable to build its user base because
+users do not get close enough to it (see Figure 11 for i = 1 and ρ = 0.0001 we have ¯π0 → 0). The subsequent
+rise in ¯π0 depends on the fact that when α approaches the maximum value αmax = 1 − β, users give
+importance only to their prejudice, and therefore they do not deviate too much from their initial position.
+As a consequence, it can not be triggered the positive feedback between users’ opinion and influencers’
+popularity that leads to the complete victory of one influencer.
+10
+4
+10
+2
+10
+0
+degree of stubborness 
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0
+Values of 
+0.0001
+1.0
+Figure 11: Normalized popularity ¯π0 as a function of the degree of stubbornness α
+γ , the points are obtained considering 50
+realizations of the process and averaging over 100 discrete time instants. Again, two levels of algorithmic personalization are
+considered.
+7. Analysis of the fluid limit
+In this section, we compare predictions of the simplified fluid limit against simulation results of the full
+stochastic model described by algorithm 1 (obtained through a Monte-Carlo approach) . We restrict ourselves
+to a one-dimensional opinion space, as in section 5, and assume that all users share the same prejudice z.
+Again, we consider two “competing” influencers. A similar analysis could be performed in scenarios with any
+number of influencers at any point in the opinion space, but this would be computationally more challenging
+since multiple stationary points may exist, each with its own attraction basin.
+22
+
+First, we derive in section 7.1 some preliminary analytical results for the case of two influencers, using
+the results of the fluid limit introduced in section 5.4. Then in Section 7.2 two extreme instances of the
+model are solved in closed form. Section 7.3 is devoted to comparing the analytical results of the fluid model
+with simulations. Finally, we discuss the impact of content personalization.
+7.1. Two competing influencers
+Let us specialize the equations presented in section 5.4 for the mean opinion ¯x(z) (Eq. (17)) and the
+normalized popularities ¯πi (Eq. (16)). Note that for Ni = 2, ¯π0 = 1 − ¯π1, so it is sufficient to study ¯π1.
+As for the mean user opinion ¯x(z), equation (17) allows us to write the asymptotic mean directly as a
+function of ¯π1 and the opinions of the two influencers x(0), x(1):
+¯x(z) =
+α
+1 − β z +
+γ
+1 − β
+�
+(1 − ¯π1) x(0) + ¯π1x(1)�
+(18)
+Substituting the functional forms of the visibility ω and feedback θ into equation (16), we obtain the
+following expression for the normalized popularity ¯π1:
+¯π1 =
+f (1)e−ρ(x(1)−¯x)
+2
+¯π1
+�
+1 − |x(1) − ¯x|
+�
+�
+i∈{0,1} f (i)e−ρ (xi−¯x)2
+¯π1
+�
+1 − |x(i) − ¯x|
+� = f(¯π1, ¯x)
+(19)
+Moreover, if we combine the above expression with equation (18) for ¯x, we get ¯π1 = f(¯π1), which can
+be solved numerically through a fixed-point approximation (FPA) (a graphical representation is shown on
+Fig. 13). The outcome of this FPA and the corresponding simulation results are compared in Figure 12.
+7.2. Closed form computations in extremal cases
+The combination of equations (18) and (19) cannot be solved in closed form in the general case. However,
+there are at least two scenarios in which this is possible, separately considered in the following subsections.
+7.2.1. When an influencer “wins”
+We consider an influencer a “winner” if its normalized popularity ¯πi approaches 1. Suppose that the
+influencer whose opinion is x(1) = 1 wins, then ¯π1 → 1. This implies ¯π0 → 0 and thus ω → 0+: the influencer
+with x(0) = 0 is seen by a negligible fraction of users and in practice, only influencer i = 1 remains visible.
+Note that in the extreme case in which influencer 1 wins, users see only x(1), and asymptotically all users
+move towards it. In this case, the final opinion ¯x(z) can be easily calculated with a recursion of the update
+rule (3):
+x(u)(n) =
+n
+�
+i=0
+βi �
+αz + γx(1)�
++ βnx(0)
+For n → ∞ and considering β < 1 (the case β = 1 coincides with the trivial case where users remain
+fixed at their initial opinion) we get:
+x(w) =
+α
+1 − β z +
+γ
+1 − β x(1),
+(20)
+which is in agreement with (18) if one sets ¯π1 = 1. This corresponds to one of the extreme cases that we
+will use later to examine the model behavior as a function of the personalization parameter ρ. It should be
+noted that this construction relies on the knowledge of the winning influencer, which is unknown in advance.
+However in the fluid limit, we expect that the winning influencer, if any, is the one that has a structural
+advantage over the others at the beginning (e.g., a higher posting rate f (i), see Figure 7).
+23
+
+7.2.2. Constant personalization function
+The other extreme case we consider is the one in which ρ = 0. In this case, the personalization function
+ω no longer depends on ¯πi, and it is easy to see from Table 6a that it returns ω ≡ 1. Moreover, we consider
+x(1) = 1, x(0) = 0, which further simplifies (18). The above formulas (Eq. 19 and Eq. 18) can then be solved
+in closed form. In particular, equation (19) for the normalized popularity ¯π1 becomes:
+¯π1 =
+f (1) (q + m ¯π1)
+f (0) (1 − (q + m ¯π1)) + f (1) (q + m ¯π1)
+where m ≜
+γ
+1−β and q ≜
+α
+1−β z for compactness. This leads to a second order equation which can be easily
+solved for ¯π1:
+¯π2
+1 m(f (1) − f (0)) + ¯π1
+�
+f (0)(1 − q) + f (1)(q − m)
+�
+− f (1)q = 0
+(21)
+7.3. Comparison between analytical prediction and Monte Carlo simulations
+This section is devoted to comparing the analytical results derived in section 5 with simulations of the
+model. Numerical and graphical solutions of equation (19) are also provided, shedding light on the impact
+of the algorithmic personalization performed by the platform.
+7.3.1. Description of the scenario
+The scenario setting is analogous to that described in section 6.1 and Table 1.However, here, we consider
+a one-dimensional opinion space [0, 1] and we assume all users to have the same prejudice, i.e., z(u) =
+z = 0.4, ∀u ∈ U matching their initial opinion x(u)(0).
+The “competing ” influencers have opinions at
+the extremes of the domain, and their posting frequencies are f (1) = 0.7 and f (0) = 0.3, i.e., influencer
+i = 1 has a structural advantage over influencer i = 0. Note that in a one-dimensional space, the reference
+direction r(i), and hence the consistency c(i), lose their significance. To avoid obtaining trivial results in
+which influencer 1 obviously wins, regular users are initially placed closer to the disadvantaged influencer
+i = 0.
+7.3.2. Simulation, fluid limit and fixed-point approximation
+Comprehensive validation and comparison of the approaches used to obtain the system equilibria are
+shown in Figure 12. First, the stochastic model described by Algorithm 1 is “simulated” by obtaining 100
+different sample whose length is 500000 elementary steps. The variables of interest ¯x(z) and ¯π1 are obtained
+by averaging the process over both discrete times steps n and sample paths and are represented by circle
+marks. Second, equation (18), which is a specialization of (17) obtained from the fluid limit, indicates that
+the state of the system lies on a line in the plane ¯π1,¯x (dashed line in Figure 12). Third, the extreme cases of
+the model analyzed in section 7.2, for which we derived a closed-form solution, are represented by star-like
+marks. Lastly, diamonds are solutions of (19) employing the fixed-point approximation.
+We observe that, for given ρ, simulation marks match well with analytical marks. The only exception
+is for ρ = 0.5, for which simulations provide ¯π1 ≈ 0.79, whereas the analysis provides ¯π1 ≈ 1 (see also
+the table on Fig. 13). This mismatch is due to the fact that ρ = 0.5 is close to a ‘phase transition’, at
+which the system switches from a regime in which two stable solutions exist (in particular, one in which
+both influencers survive) to a regime in which influencer i = 1 wins. This behavior is better illustrated in
+Fig. 13, where the curve corresponding to ρ = 0.5 is almost tangent to the bisector. It should be noted that
+the “empty” region in Figure 12 is directly related to this behavior since no stable solutions can exist for
+that values of ¯πi. In fact, there is no stable intersection with the bisector in Figure 13 in the corresponding
+interval.
+7.4. Implications of algorithmic personalization
+We summarize here the insights into algorithmic personalization suggested by the emergent behavior
+of our model. We already mentioned how content filtering favors influencers with a structural advantage.
+24
+
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+1
+0.48
+0.50
+0.52
+0.54
+0.56
+x
+simulation
+fixed-point
+1
+z + 1
+1
+closed-form 
+= 0
+closed-form 
+1
+1
+10
+3
+10
+2
+10
+1
+10
+0
+ in simulation
+Figure 12: Comparison between analytical results, including the exact extreme points calculated in 7.2.2 and 7.2.1, and the
+linear relationship between ¯x and ¯π1 according to Equation (18). Diamonds represent the fixed-point approximation for the
+solution of equation (19). Simulation results of the stochastic dynamics, represented by circles, were obtained by averaging
+100 realizations of the process as described by Algorithm 1. We consider a scenario in which α = 0.05, β = 0.93, with two
+influencers at the extremes of the domain, with f(0) = 0.3, f(1) = 0.7 and the same initial absolute popularity p0 = p1 = 100.
+Numerical values from simulation and fixed-point approximation are reported in the table alongside the plot in Fig. 13.
+Table 2: Simulation and FPA
+ρ
+¯π1
+¯π1
+SIM
+FPA
+0.0
+0.682
+0.684
+0.001
+0.683
+0.684
+0.01
+0.684
+0.685
+0.1
+0.693
+0.695
+0.3
+0.725
+0.728
+0.4
+0.749
+0.758
+0.5
+0.789
+0.999
+0.8
+0.986
+0.999
+1.0
+0.995
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+f(
+1)
+= 0
+= 0.001
+= 0.01
+= 0.1
+= 0.3
+= 0.4
+= 0.5
+= 0.8
+= 1.0
+1
+Figure 13: Graphical solution of ¯πi = f(¯πi), (19). Stable solutions corresponds to intercepts between f(¯πi) and the bisector,
+such that f′(¯πi) < 1. We observe that non-trivial solutions (i.e., solutions in which both influencers survive) exist, roughly in
+the interval [0.7, 0.8], provided that ρ is not too large (i.e., ρ < 0.5). For ρ > 0.5, the only stable solution is ¯π = 1. This explains
+the results in Figure 12. Simulation results reported on the alongside table confirm the validity of the analytical predictions.
+For instance, in section 6.2, we showed that personalization promotes the influencer with higher posting
+frequency, and, in section 6.3, the one with lower consistency. In addition, in 7.3.2, we presented a case
+25
+
+where a ‘phase transition’ is observed as a function of the filtering strength (i.e., ρ). Indeed, after a certain
+threshold, the favored influencer (say influencer 1) is the only one that survives (¯π0 → 0). In such a situation,
+the population is exposed to the opinions of a single individual, hindering diversity on the social platform.
+It is also interesting to discuss the results in section 6.4 concerning the effects of personalization on the
+opinion distribution of regular users. The two possible outcomes when sharp personalization is applied are:
+either the emergence of two echo chambers with users holding more radical positions in both directions or
+the onset of an unstable situation in which the two influencers coexist for a limited amount of time, after
+which ¯πi → 1 for an i dependent on the specific sample path.
+8. Online social network data
+This section examines data collected from Facebook and Instagram social networks and compares the
+observed behavior with some of the findings of our Communication Asymmetry model.
+8.1. Correlation between frequency of publication and popularity
+In previous sections, especially in 6 and 7.3, we discussed structural advantage from the influencer’s
+point of view. One of the key advantage parameters, as observed across all experiments, is the publication
+frequency f (i): the higher f (i), the greater the advantage (see Figure 7). In this section, we attempt to
+validate this finding by correlating the frequency of publication of influencers with their popularity growth,
+using the total number of followers, i.e., the number of people subscribed to the profile, as a proxy for
+popularity. We consider temporal sequences from Instagram on a sample set of 110 influencers.
+0.5
+0.0
+0.5
+Pearson correlation coefficient
+0
+5
+10
+15
+20
+25
+Number of influencers
+Figure 14: Distribution of the correlation coefficient between monthly number of posts and popularity growth (in terms of
+number of followers).
+For each influencer, we considered a temporal granularity of one month, determined the number of posts
+during this period, and calculated the relative change in the number of followers considering the values at
+the beginning and end of the interval. Then for each user, we calculated the Pearson correlation coefficient
+between the number of posts and the relative variation of followers in the month. In Figure 14, we show
+the distribution of these correlation coefficients. Results suggest that there exists, in general, a positive
+correlation between the two quantities, i.e., influencers with aggressive posting habits tend (but not always)
+to get more followers, which likely favors them when in competition with other influencers on social media
+platforms. This is consistent with the model predictions shown in section 6.2.
+8.2. Case Study: Italian government crisis in August 2019
+In June 2018, a few months after the general elections, Giuseppe Conte was appointed Italian Prime
+Minister. Two parties formed his supporting coalition: Movimento 5 Stelle (his own party, holding the
+26
+
+relative majority of the Italian Parliament) and Lega, whose leader was Matteo Salvini. In August 2019,
+Salvini decided to withdraw Lega’s support to the government, starting a crisis aimed at driving Italians
+to new elections and gaining more votes. However, Movimento 5 Stelle managed to reach an agreement
+with various parties to form a new government, and on September 5, 2019, Giuseppe Conte became Prime
+Minister for the second time. The coalition that supported this new administration clearly excluded Lega.
+In this section, we apply the proposed model to reproduce the sudden rise of Giuseppe Conte’s popularity
+in social networks during the government crisis in August 2019. We exploit the multidimensional capability
+of the model considering two directions: Politics, reference topic for Salvini and Conte, and attitude toward
+government fall (End government, see Figure 15).
+(a) Initial distribution
+(b) After the transient
+Figure 15: Initial distribution density of the population along the Politics direction and End government direction. The opinion
+position of the two leaders in the space is depicted with a green (Salvini) and a yellow (Conte) point.
+In the opinion space, we assume Salvini has a more radical political viewpoint, while Conte has an
+opposing and more moderate position (described somehow arbitrarily by putting Salvini at xP olitics = 0,
+Conte at xP olitics = 0.76). We assume that the population has a moderate initial opinion (centered at
+xP olitics = 0.5, see Figure 15a). Regarding the attitude toward government fall, the two politicians obviously
+have a completely different opinion (Salvini has xEndgovernment = 1 while Conte has xEndgovernment = 0). We
+assume that the population is strongly polarized towards Conte’s opinion in this latter direction (Figure 15).
+This is an a posteriori assumption made knowing the outcome of the social confrontation.
+A period of eleven weeks is considered, from July 7 to September 22, during which data was collected
+weekly from Facebook. A total of 1162 posts were published, of which 125 were by Conte. The rate f (i) is
+calculated as the number of posts by an influencer relative to the total number of posts (f (Conte) = 0.108,
+f (Salvini) = 0.892).
+Some simplifying assumptions are necessary to apply the model. We assume that the two politicians
+have a consistency c(i) of exactly one (real values are often close to this value, see Figure 4a). Moreover,
+Giuseppe Conte and Matteo Salvini are the only influencers. Although this hypothesis is restrictive, in the
+scenario studied, the two influencers were the main (active and popular) protagonists during the government
+crisis. Moreover, we consider the simplest scenario in which personalization is not employed: ρ = 0 and thus
+ω ≡ 1. We consider a feedback function of the form θ = e−8.25(x(u)−x(i))2 for both opinion directions. For an
+exhaustive list of the parameters, we refer the reader to Table 3.
+27
+
+transient
+phase 
+7th July
+22nd Sept.
+switch
+posts along 
+politics
+government crisis
+28th July
+1st Sept.
+switch
+posts along 
+politics
+posts along 
+end government
+0
+Figure 16: Timeline of modelled scenario from July 7, to September 22. From July 28 to September 1 we have a consistency
+switch, with posts along End government direction.
+Figure 16 shows the timeline of the experiment. The two influencers start with the same initial popularity.
+We consider a transient of Nt = 10000 discrete time-units, after which the stationary normalized popularities
+πi roughly correspond to the empirical normalized popularities obtained by dividing the number of followers
+of each influencer by the total number of the two.
+After the transient, we can see in Figure 15b that
+the distribution of public opinion is skewed towards Salvini, who, in turn, has a higher popularity ratio
+due to his higher publication frequency. After the transient, the crisis happens and both influencers start
+posting in the Endgoverment direction (i.e., we observe a consistency shift for both influencers), during a
+time window of five weeks that approximates the duration of the government crisis, after which the two
+politicians switch back to posting on the Politics direction. Note that the initial users’ opinion distribution
+along the Endgoverment axes is concentrated around Conte’s point of view.
+Even with these limitations, it is still possible to reproduce the observed social behavior as a whole: it
+corresponds to a situation where an influencer is in stark contrast to the opinions of its user base and loses
+ground with respect to the other influencer. Note that, in the model, only a very unbalanced distribution
+of the population towards Conte’s opinion (against the government fall) can explain the sudden increase
+in Conte’s popularity, despite the remarkable differences in popularity ratios in favor of Salvini. Figure 17
+compares the simulation results of the described setting and Facebook’s measurements. Note how the model
+can explain the sudden rise in Giuseppe Conte’s popularity, precisely in the weeks of the government crisis.
+Clearly, our model does not exactly fit empirical observations but simply provides qualitative insights
+into the possible causes of the rather sudden popularity shift that was observed.
+Many of the model’s
+parameters are unknown, such as the opinion distribution, the weights of the updating rule, or the feedback
+function. However, by making reasonable assumptions about some of the parameters, one can obtain a
+reasonably good fit, and exploit the explanatory capability of the model to acquire better confidence in the
+hidden mechanisms beneath observed dynamics. In this experiment, we followed exactly this approach and
+we looked for some mechanisms that could justify the same sudden surge in popularity that occurred during
+the government crisis.
+As the main outcome of our analysis, we conclude that the observed popularity trends of the two
+considered influencers can be largely explained by considering the fear of political instability in the user
+base.
+9. Conclusions
+In recent times, online social interactions appear essential to human relationships and play an increasingly
+important role in opinion formation. To understand the mechanisms underlying this novel communication
+paradigm, it is of utmost importance to develop flexible frameworks suitable for describing interactions on
+social media platforms. In this work, we have developed an opinion model tailored to online interactions, with
+28
+
+07 Jul
+14 Jul
+21 Jul
+28 Jul
+04 Aug
+11 Aug
+18 Aug
+25 Aug
+01 Sep
+08 Sep
+15 Sep
+22 Sep
+0.215
+0.220
+0.225
+0.230
+0.235
+0.240
+Conte
+Model outcome
+Facebook data
+Figure 17: The popularity ratio πConte for Conte, the one obtained from Facebbok data and the one from the model along with
+its 95% confidence interval, computed over 10 realizations of the process. It can be seen how the model follows the increase in
+popularity during August 2019.
+Table 3: Parameters and functions for the Case Study
+Symbol
+Value - Form
+Description
+Ni
+2
+Number of influencers
+x(Conte)
+0
+0.76
+Opinion of Giuseppe Conte on direction j
+x(Salvini)
+0
+0.0
+Opinion of influencer 1 on direction j
+f(Conte)
+0.108
+Opinion of Giuseppe Conte on direction j
+f(Salvini)
+0.892
+Opinion of influencer 1 on direction j
+r(Conte),(Salvini)
+0
+Refrence direction of both influencers
+pConte,Salvini(0)
+20
+Initial absolute popularity of both influencers
+Nu
+10000
+Number of regular users
+Niter
+15000
+Number of iterations for each simulation
+Nt
+10000
+Duration of the transient phase
+w
+550
+Length of the government crisis
+α
+0.3
+First weight in the updating rule in Eq. 3
+β
+0.65
+Second weight in the updating rule in Eq. 3
+θ(·)
+e−8.25(x(u)−x(i))2
+Functional form of the feedback function
+ω(·)
+ρ = 0 =⇒ ω ≡ 1
+Functional form of the visibility function
+particular attention to distinguishing between two classes of users, namely regular users and influencers.
+We characterized the influencers by introducing the concept of reference direction, which links unrelated
+topics discussed by the same influencer. Measurements collected from real online social networks support
+our modeling assumptions.
+Similarly to other works in the recent literature, we integrated algorithmic
+personalization in a flexible and tunable manner. We have shown how content filtering reinforces inequality
+by favoring the structurally advantaged influencer and, in most cases, preventing the “competing” influencer
+from remaining visible to the population. Moreover, even in structurally balanced conditions, personalization
+can lead to the emergence of echo-chambers, in which users’ opinion also radicalizes along non-reference
+directions.
+The proposed model is a preliminary attempt to describe the complexity of online interactions and
+comes with some limitations. In our model, users are passive entities, and influencers are stubborn agents.
+Moreover, homophily is the only driver of individuals’ interaction, as no other relationship structure was
+considered. Nonetheless, despite the simplifying assumptions, the emergent behavior of the model proved
+29
+
+rich enough to reveal the effects of content personalization and shed light on influencer popularity dynamics.
+Our work points to several research directions, such as viewing users as active agents capable of publishing
+their own posts and forwarding (i.e., sharing) posts from influencers. This can pose significant challenges
+in terms of analytic tractability. Another promising direction could be to look at influencers as “strategic”
+players aiming at maximizing their popularity on the platform by exploiting the internal mechanisms of the
+platform itself (such as algorithmic content filtering).
+Declaration of Competing Interest
+The authors declare that they have no known competing financial interests or personal relationships that
+could have appeared to influence the work reported in this paper.
+Appendix A. Description of the dataset
+We collected data from real online social networks to support the hypotheses of our model and compare
+emergent behaviors. We focus on two popular social networks: Facebook (FB) and Instagram (IG). Facebook
+has long been the most popular social media application, while Instagram has undergone a surge in popularity
+in recent years.
+In Facebook and Instagram, a profile, i.e., a social network user, can be followed by other profiles, i.e.,
+its followers. A profile with a large number of followers is also called an influencer - we consider profiles
+with more than ten thousand followers as influencers. Influencers post content (i.e., posts) consisting of a
+photo, a video, plain text, or a combination of these. The profile’s followers and anyone registered on the
+platform can see, like and comment on the influencer’s posts. Note that when we use the term influencer,
+we do not only mean individuals but also groups, soccer teams, newspapers, or companies.
+In this work, we are interested in the plain-text messages of influencers, their temporal sequence,
+and metadata describing the features of the influencers.
+To get the list of such popular profiles, we
+exploited the online analytics platform hypeauditor.com for IG, and www.socialbakers.com and www.
+pubblicodelirio.it for FB. We restricted the analysis to influencers with at least 10, 000 followers on
+June 1, 2021.
+The lists of 649 influencers we used are publicly available.5
+For each monitored profile,
+we downloaded the corresponding metadata, i.e., the profile information and all generated posts, using the
+CrowdTangle tool and its API6. CrowdTangle is a content discovery and social analytics tool owned by Meta
+and available to researchers and analysts worldwide to support research, subject to a partnership agreement.
+For each influencer, we downloaded all the data related to the posts published between January 1, 2016,
+and June 1, 2021. Finally, we stored the data, which takes around 110 GB of disk space, on a Hadoop-based
+cluster, and we used PySpark for scalable processing.
+Appendix B. Details on post classification
+One of the novelties introduced in this work is the concept of
+reference direction, which states that
+influencers have a preferred topic of discussion. To confirm this hypothesis, we developed a classifier that
+can categorize posts according to a particular set of subjects. First, we arbitrarily identified a subset of topics
+that sufficiently characterize the discussions on the monitored profiles. Specifically, these topics are sports,
+politics, food and cooking, music, and pandemics, which are intentionally loose and relatively uncorrelated
+to each other. We developed a keyword classifier to classify the posts. For each topic, we manually defined
+a list of representative keywords. For example, if we consider pandemic, we search for words like COVID,
+pandemic, and coronavirus in Italian (and commonly used terms in other languages). We search for the
+topic-specific terms in the text corpus of the post, and if we find a match, we mark the post as belonging
+to the topic. Notice that since keywords of various topics may be present in the same corpus, we can flag a
+5https://mplanestore.polito.it:5001/sharing/P4WnRClQn
+6https://github.com/CrowdTangle/API
+30
+
+message as discussing multiple topics. In this work, we discard posts marked as multiple and only consider
+posts associated with a single topic.
+We are not interested in classifying all posts by an influencer, first because our list of topics does not
+cover all possible ones, and second because we only need a large enough subsample of posts to make some
+statistical considerations. Conversely, it is of utmost importance that the accuracy of the classifier is high
+since misclassified posts could lead to wrong conclusions about the distribution among the available topics.
+Therefore, we manually validate the accuracy of our methodology for topic detection, as described in the
+following paragraph.
+Appendix B.1. Classifier Precision Evaluation
+We empirically evaluated the accuracy of the classifier by taking a random subsample of the labeled
+posts, i.e., 100 posts for each topic for a total of 500 messages, and manually classifying them. To this
+end, we defined a lower and upper bound for accuracy. Indeed, even for a human being, it is challenging
+to univocally classify posts based on their content. Therefore, we defined three possible states for each
+classification decision: “t” correct classification, “f” incorrect classification, and “ncc” standing for not
+completely correct (indicating that the assigned topic is related to the post but may not be the main topic
+of the post or the classification of the post is difficult). Given this states subdivision, the precision bounds
+are as follows:
+PL =
+Nt
+Nt + Nf + Nncc
+(B.1)
+PU =
+Nt + Nncc
+Nt + Nf + Nncc
+(B.2)
+We refined our term selection for each topic to improve precision based on this analysis .7 The classifier’s
+precision is subject-dependent but was consistently above 80% considering the upper bound defined in
+(B.1). The classification is particularly effective in the case of politics and pandemic, where the precision
+goes above 90%. Table B.1 summarises the bounds on precision achieved by the procedure described above.
+These results are sufficient to use the classification to support our modelling assumptions.
+Table B.1: Per-topic Precision
+Topic
+Precision l.b.
+Precision u.b.
+Sports
+76.9
+83.2
+Politics
+87.0
+94.4
+Music
+53.4
+84.5
+Food
+65.5
+82.4
+Pandemic
+76.6
+93.1
+The average percentage of messages classified is 27.8% for all influencers in the dataset. Considering
+the final classifier and the analysed dataset, we automatically flagged about one million posts8 with at least
+one topic. Of these, only 6.7% of the posts were flagged with multiple labels, indicating the message dealt
+with more than one topic. We decided to consider in the rest of the work only influencers for whom it was
+possible to classify more than a thousand posts in the observed period. At the end of this filtering process,
+we could keep 237 influencers for whom the average posts’ classification percentage is 53.2%.
+The dataset used contains a subset of Italian politicians.
+To check the correctness of the labelling
+procedure, we checked whether the derived reference topic for all politicians was politics. It turned out that
+7We make the final list of terms available at https://mplanestore.polito.it:5001/sharing/0wD5oU6xr.
+81167963 posts were tagged with at least one label.
+31
+
+two politicians did not have politics as reference: Vincenzo De Luca had pandemic, and Renata Briano had
+food. However, this is entirely understandable as the latter runs a food blog and the former was known for
+his firm and frequent statements on the pandemic situation during the COVID -19 pandemic.
+Appendix C. Proofs of Theorems (5.1) and (5.2)
+Appendix C.1.
+Proof of Theorem (5.1)
+Let us start assuming ki(0) > 0 ∀i.
+In such a case we denote with i0 = arg maxi ki(1) and with
+K := ki0(1). First we show that the problem:
+ki(yi) − cyi = 0,
+with yi ∈ [0, 1] ∀i
+(C.1)
+admits a solution for any c ≥ K. Indeed by choosing c ≥ K we have that necessarily ki(1) ≤ ki0(1) ≤ c·1 ∀i
+while ki(0) > c · 0 = 0; therefore a zero zi(c) must exist for every i. This zero is unique as a consequence
+of the concavity of ki(·). The set of zeros zi(c)i provides a solution of (C.1). Now to get a solution of the
+original problem (12) we need to show that there exist a c such that {zi(c)}i are normalized. Observe that
+for c = K by construction zi0(K) = 1 while 0 < zi(K) ≤ 1 for i ̸= i0, therefore �
+i zi(K) > 1. Now, due
+to the monotonicity and concavity of ki(·), zi(c) is by construction decreasing with respect to c, moreover
+zi(c) → 0 as c → ∞ ∀i, therefore since �
+i zi(·) is a continuous function of its argument, there will necessarily
+be a c0 in correspondence of which �
+i zi(c0) = 1. In the case in which ki(0) = 0, observe that 0 is a solution
+of (C.1) for any c, i.e. zi(c) = 0. Moreover for any c ≥ K a second zero may exist. For example, by
+construction, zi0(K) = {0, 1}. Therefore for c = K, as before, we can always choose as set of zeros {zi(K)}i,
+such that zi(K) = 0 if ki(0) = 0, and i ̸= i0, zi0(K) = 1. By construction �
+i zi(K) ≥ 1. In particular
+�
+i zi(K) > 1 is there exists a i such that ki(0) > 0. In this latter case, by increasing c all the non null zeros
+decrease, therefore, as before, there will necessarily be a c0 in correspondence of which �
+i zi(c0) = 1.
+□
+Appendix C.2.
+Proof of Theorem (5.2)
+We first show that ||¯π(1) − ¯π(2)||L∞ = maxi |¯π(1)
+i
+− ¯π(2)
+i
+| = ||G(F1(x, z)) − G(F2(x, z))||L∞ ≤ M||F1(x) −
+F2(x)||L∞; then we show that we can always enforce: ||F1(x, z) − F2(x, z)||L∞ = ||H(¯π(1)) − H(¯π(2))|| ≤
+1/(2M)||¯π(1) − ¯π(2)||L∞ by properly choosing ω(·, ·) and θ(·).
+Therefore, we can conclude that ||H ◦
+G(F1(x, z)) − H ◦ G(F2(x, z))|| ≤ 1/(2M)||(G(F1(x)) − (G(F2(x))|| ≤ M/(2M)||F1(x, z) − F2(x, z)|| =
+1/2||F1(x, z) − F2(x, z)||.
+First note that ||F1(x, z) − F2(x, z)||L∞ = supx |F1(x, z) − F2(x, z)| coincides with the Kolmogorov
+distance between the two distributions.
+Let us denote with
+ki(y, F1(x, z)) = λf (i)
+� �
+θ(|x − xi|)ρ(¯πi, |x − xi|)dF1(x, z),
+and similarly for ki(y, F2(x, z)) we assume that:
+sup
+y∈[0,1],i
+|ki(y, F1(x, z)) − ki(y, F2(x, z))| := ∆K(F1, F2) ≤ a||F1(x, z) − F2(x, z)||L∞
+a ∈ R+
+and
+dki(y, F1(x, z))
+dy
+|y=0< max
+i
+ki(1, F1(x, z))
+dki(y, F2(x, z))
+dy
+|y=0< max
+i
+ki(1, F2(x, z))
+∀i.
+Without lack of generality we assume maxi ki(1, F1(x)) ≥ maxi ki(1, F2(x)). Let the pair (¯π(1) = {¯π(1)
+i
+}i, c1)
+be the solution of
+ki(yi, F1(x, z)) − cyi = 0
+s.t
+�
+i
+yi = 1, yi ≥ 0, ∀i
+32
+
+now let ({�p(2)
+i }i) the non necessarily normalized solution of
+ki(yi, F2(x, z)) − c1yi = 0
+s.t yi ≥ 0, ∀i.
+by means of elementary geometric considerations we can bound:
+|¯π(1)
+i
+− �p(2)
+i | ≤ ∆K(F1, F2)
+c1 − h1
+where h1 =
+dki(y,F1(x))
+dy
+|y=min(¯π(2)
+i
+,�p(2)
+i
+)≤
+dki(y,F1(x))
+dy
+|y=0.
+We recall that by construction (see proof of
+Theorem 5.1) we have c1 > maxi ki(1, F1(x))).
+Denoting with |�p(2)| = �
+i �p(2)
+i
+, we have
+1 −
+�
+i
+|�p(2)
+i
+− ¯π(1)
+i
+| ≤ |�p(2)| ≤ 1 +
+�
+i
+|�p(2)
+i
+− ¯π(1)
+i
+|
+Now denoted with ({¯π(2)
+i
+}i, c2) the solution of
+ki(yi, F2(x, z)) − cyi = 0
+s.t
+�
+i
+yi = 1, yi ≥ 0, ∀i
+we have, by construction, that:
+1
+max(1, |�p(2)|) < c1
+c2
+<
+1
+min(1, |�p(2)|)
+and therefore, exploiting again elementary geometrical arguments, we can bound:
+|�p(2)
+i
+− ¯π(2)
+i
+| ≤
+����
+�c1 − h2
+c2 − h2
+− 1
+�
+�p(2)
+i
+����
+where h2 = dki(y,F2(x,z))
+dy
+|y=min(�p(2)
+i
+,¯π(2)
+i
+)= dki(y,F2(x,z))
+dy
+|y=0. Putting everything together, we have proved
+that:
+max
+i
+|||¯π(1)
+i
+− ¯π(2)
+i
+|| = ||G(F1(x, z)) − G(F2(x, z))||L∞ ≤ M||F1(x, z) − F2(x, z)||L∞
+To conclude the proof, first note that by properly choosing ρ(·, ·) and θ(·) we can assume vx(x, z) and σ2
+x(x, z)
+to depend sufficiently smoothly on ¯π, i.e. ∀ε > 0 we can assume:
+sup
+x
+���
+���v(1)
+x (x, z) − v(2)
+x (x, z)
+���
+���
+L∞
+≤ ε||¯π(1) − ¯π(2)||L∞
+∀z,
+sup
+x
+���
+���σ2,(1)
+x
+(x, z) − σ2,(2)
+x
+(x, z)
+���
+���
+L∞ ≤ ε||¯π(1) − ¯π(2)||L∞
+∀z,
+and
+sup
+x
+�����
+�����
+∂σ2,(1)
+x
+(x, z)
+∂x
+− ∂σ2,(2)
+x
+(x, z)
+∂x
+�����
+�����
+L∞
+≤ ε||¯π(1) − ¯π(2)||L∞
+∀z.
+Then observe that the solution of the Fokker-Planck equation given in (9) on a compact interval (and so
+also its primitive) depends smoothly on function vx(x, z), function σ2
+x(x, z) and its first derivative, as long as
+infx,z σ2
+x(x, z) is bounded away from zero. As final remark note that the set of weakly-increasing functions
+F(x), such that F(a) = 0 and F(b) = 1 equipped with the L∞-norm forms a closed set in a complete metric
+space. □
+33
+
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+
diff --git a/-NAzT4oBgHgl3EQfg_yi/content/tmp_files/load_file.txt b/-NAzT4oBgHgl3EQfg_yi/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..00fcd473704080aef8dd32d31ce56ef841510eb5
--- /dev/null
+++ b/-NAzT4oBgHgl3EQfg_yi/content/tmp_files/load_file.txt
@@ -0,0 +1,1843 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf,len=1842
+page_content='Modeling communication asymmetry and algorithmic personalization in  online social networks  Franco Galantea,∗, Luca Vassioa, Michele Garettob, Emilio Leonardia  aPolitecnico di Torino, Corso Duca Degli Abruzzi, 24, 10129 Torino  bUniversit`a degli Studi di Torino, Corso Svizzera 185, 10149 Torino  Abstract  Modeling social interactions and their impact on opinion dynamics has attracted growing interest in  recent decades, fuelled by the mounting popularity of online social networks (OSNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' On online social plat-  forms, a few individuals, commonly referred to as influencers, produce the majority of content consumed by  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, classic opinion models do not capture this communication asymmetry in OSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We develop  an opinion model inspired by observations on leading social media platforms and tailored to the peculiarities  of online interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Our work has two main objectives: first, to describe the inherent communication asym-  metry in OSNs, where a tiny group of influencers hegemonizes the landscape of social debate, and second,  to model the personalization of content by the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We derive a Fokker-Planck equation  for the temporal evolution of users’ opinion distribution and analytically characterize the stationary system  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Analytical results, confirmed by Monte Carlo simulations, show how content personalization tends  to radicalize user opinion and favor structurally advantaged influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' These emerging behaviors suggest  that algorithmic bias, inherently associated with platform filtering, can lead to undesirable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' As an  example application, we apply our model to Facebook during the Italian government crisis in the summer  of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Our work provides a flexible framework to assess the impact of algorithmic filtering on the opinion  formation process and a fine-grained tool to study the complex interaction between influencers and social  network users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Keywords:  opinion dynamics, online social networks, algorithmic personalization, Fokker-Planck equation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Introduction  In recent years, the way people communicate has changed dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' With the advent of the Internet,  new communication channels have emerged that allow people to transcend geographical and language barriers  thanks to a global communication network and alternatives to text-only interaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', images and videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Online social networks (OSNs) are probably the most notable example of such new interaction mechanisms  and have greatly influenced our society by fostering discussions and disseminating information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The means  that enable online social interactions are profoundly different from traditional interpersonal interactions and  other mass media such as newspapers or television.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The amount of content produced on such social media  platforms is immense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore, to keep users engaged within the social network, the platform performs  filtering to select the posts offered to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This filtering mechanism can reinforce the natural tendency  (usually referred to as homophily in the literature) to interact with like-minded people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' And, in turn, can  lead to the formation of echo chambers [1], where people who share a similar point of view interact with  each other but are isolated from the rest of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In addition, the reach of certain social media users can  be extraordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Posts by very influential people can reach a large audience in virtually no time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Another  ∗Corresponding author  Email addresses: franco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='galante@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='it (Franco Galante), luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='vassio@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='it (Luca Vassio),  michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='garetto@unito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='it (Michele Garetto), emilio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='leonardi@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='it (Emilio Leonardi)  Preprint submitted to Online Social Networks and Media (OSNEM)  January 5, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='01478v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='SI]  4 Jan 2023  aspect worth mentioning is the diversity of topics discussed on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It ranges from commentaries  on the latest sporting events to debates on sensitive issues such as vaccinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Last, communication on  OSN takes place asymmetrically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', few well-known individuals can exert influence on a large audience  which, in turn, is composed of far less known people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' These are some crucial aspects that characterize online  social networks and distinguish them from ”offline” interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We believe that models seeking to capture  the complexity of interactions occurring in online social networks must account for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Trying to understand the mechanisms behind the opinion-forming process is a daunting challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The  complexity driving this process is still poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, the individual reaction to external  stimuli is utterly subjective and thus difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In a continuous framework, opinions can be in-  terpreted as a person’s level of agreement with a statement or the interest they show in an issue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=',  adoption of technology, politics) and mapped to real-valued intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It is clear that measuring opinions in  such intervals is arbitrary and can only lead to qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Indeed, most of the literature proposes  theoretical models without the claim of accurately representing real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Very few works in the  literature attempt to validate the emergent behavior of the model with physically observed phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=',  [2] [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Some other works take the approach of supporting modeling decisions with real-world observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' [3] did this by interviewing a group of people on specific topics, while Xiong and Liu [4] extracted  information from Twitter networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Following the above works, we present a model whose hypotheses are  supported by data from social networks and whose outcomes are compared with emerging phenomena on  two popular OSNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', Facebook and Instagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' As the limitations mentioned above also apply to our  model, the results discussed here do not aim to be predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, the proposed model provides a tool  to study the emerging behavior on online social networks and the impact of algorithmic personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The main objective of this work is to develop an analytical framework tailored to online interactions,  incorporating the following aspects:  The asymmetry typically found in OSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' There exists a relatively small percentage of users of online  social networks whose number of followers is orders of magnitude larger than that of other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  These individuals are commonly referred to in the literature as influencers or opinion leaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' They  are particularly relevant, as their opinions can reach a vast fraction of the social network’s population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The filtering performed on the content by the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Algorithmic personalization  appears necessary in the context of OSNs, as the number of daily produced posts has become enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The aim is to increase engagement by showing users only the most relevant posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The loop is then  closed by taking into account user feedback on the posts received (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', likes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The proposed model:  Provides a tool for assessing the impact of different algorithmic personalization policies, focusing on  the opinion leaders in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It can evaluate the extent to which these strategies might hinder  diversity of opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Exploits the observed characteristics of a large ensemble of Italian influencers from Facebook and  Instagram social networks to ground its main hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Allows for comparing its emergent behavior with observations on real online social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Further-  more, the explanatory capabilities of the model are used in the study of the opposition of two Italian  politicians during a government crisis by identifying a state of public opinion that can lead to the same  behaviors observed in the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  One peculiar feature of our approach is the concept of reference direction, which is the individual’s main  topic of interest and expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' To our knowledge, the existing literature has not yet considered the impact  of a reference topic for each influencer on the opinion formation process in multi-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The influence exerted on non-reference directions depends heavily on platform personalization, which  usually depends on how well-known an influencer is in its main field of expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For example, famous  public figures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', athletes, models) can express their point of view on potentially sensitive matters and  2  may resonate more than experts due to their popularity in their field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore, since influencers discuss  different topics, the reference direction loosely couples seemingly unrelated subjects brought up by the same  person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Incorporating a personalization process into the dynamic behavior of the model is another crucial feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The analytical results and the model simulation show that algorithmic personalization favors structurally  advantaged individuals, resulting in less diversity of opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It is also interesting to observe that the model  undergoes a phase transition in its behavior as a function of the degree of polarisation, at least in the case  of two competing influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Below a certain threshold, there are diverse opinions in the population, and  above this threshold, one of the influencers tends to polarise users’ attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Section 2 discusses the relevant work in the literature and sets out  the rationale for the need for a new opinion model tailored to OSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The Communication Asymmetry  model is presented in Section 3, along with the notation used throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Section 4 presents  some observations from real social networks supporting our modeling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Section 5 is devoted  to the mean-field analysis of the model as the number of users grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The theoretical results on the  steady-state behavior of the model are proved in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Section 6 then investigates the impact of the  model parameters on a reference scenario with two influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Analytical findings are validated in section  7 by comparing the results of Monte-Carlo simulations with theoretical predictions specific to our reference  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Section 8 further validates our model with real data collected on Instagram and Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' At  last, Section 9 concludes the article with a discussion of the implications and limitations of our work, setting  the ground for future extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Related work  The first steps in the field of opinion dynamics were taken in the late 1950s by a number of social  psychologists, among which Solomon Ash [5], John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' French [6], and Leon Festinger [7] had great  resonance in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Ash empirically observed that the individuals he studied engaged in conformist  behavior because of the social pressure exerted by the rest of the social group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In short, Ash observed  that an individual states a truth about something that is not true (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' “white is black” [3]) when the  social group to which the individual belongs asserts it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' French [6] developed a model to capture influence  through interpersonal relationships within a group, focusing on leadership and using directed graphs to  model interpersonal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Festinger developed the theory of social comparison, according to which  individuals tend to evaluate their position by comparing it with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, the tendency to do so  decreases the greater becomes the difference in opinion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Opinion models are divided customarily into two broad classes: those in which opinions are continuous  variables and those in which opinions are discrete (often binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In a recent review [8] examining agent-  based opinion models, the authors show that more than 80% of the models considered are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Much of the seminal work in the field of opinion dynamics is continuous in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  For example, the  DeGroot model [9] considers a networked social system in which individuals interact with their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Individuals average their current opinion with the opinion of their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The idea behind the model  is to describe the process leading to consensus within a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Subsequently, Friedkin and Johnsen [10]  extended it by developing a flexible framework from which various opinion models (including French [6] and  De Groot [9]) can be derived as particular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Their model of social influence encompasses both the  processes of social conformity and social conflict that lead to behavior that goes beyond simple consensus  and represents the persistent disagreement often observed in social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In the early 2000s, Hegselmann  and Krause [11] and Deffuant and Weisbuch [12] proposed two similar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In analogy with the DeGroot  model, individuals interact by averaging opinions, but the authors introduced the central idea of bounded  confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' According to bounded confidence, individuals interact in a social network with other peers only  if their beliefs are not too different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This mechanism implements the concept of homophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Lorenz, in his  review [13], provides the agent-based and density-based formulation of bounded confidence, distinguishing  two main models: in the Hegselmann-Krause (HK) model, individuals modify their opinion as a result of  interactions with all agents in their neighborhood, whereas, in the Deffuant-Weisbuch model, interactions  are pairwise between connected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  3  In addition to continuous models, discrete models have also appeared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The first and  probably most prominent model of this kind is the voter model, independently introduced by Clifford and  Sudbury [14] and Holley and Liggett [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Here, individuals are agents in a network of interactions, holding  a binary opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' At times dictated by a Poisson clock, an individual adopts the belief of a randomly chosen  neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This type of model has attracted a great deal of attention: several extensions have populated  the recent literature, for example, taking evolving networks into account [16] [17] or allowing individuals to  hold more than one opinion at a time [18], or introducing spontaneous changes of opinion [19] (noisy voter  model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  A consistent bulk of research on opinion dynamics comes from the physics literature, among which early  contributions are Ben-Naim [20] and Toscani [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The idea underlying these models is that of describing  interacting individuals using statistical mechanics by adequately defining the microscopic interactions be-  tween the individuals, much like particles in a gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Then, collective statistical phenomena are sought for  the overall opinion of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In the papers mentioned above, Ben-Naim and Toscani consider two  mechanisms of opinion formation: compromise, the human tendency to reach a reasonable trade-off on an  issue to avoid conflict, and a process of introspection (in other models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', [19], modeled as noise), which  the authors believe represents the impact of external sources of information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' A statistical  approach is generally employed to study spin systems, and models such as the Ising model have also been  applied to the opinion formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' An extension of the Ising model is the Sznajd model [22], which  implements social validation and for which Slanina and Lavicka [23] derived analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this model,  the agreement of individual pairs leads to their neighbors agreeing with them, and a line graph is considered  to capture the connection network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For a comprehensive review of opinion models, we refer to the survey  by Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Models tailored to online platforms  Most of the seminal literature on opinion dynamics is suited to describe the decision-making process in  small groups of individuals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', a board of directors, or to capture rather regular patterns determined by  the daily personal interactions of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Models such as the voter model have been studied extensively  on regular lattices [25] [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The structure of interactions, especially those online, is far from homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  As mentioned earlier, an inherent asymmetry in communication exists in OSNs where a limited number of  individuals (influencers) monopolize the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The voter model has been studied over heterogeneous  networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', [27] [28]) to account for this diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' On such networks, there can exist hubs (strongly  connected nodes) playing a role similar to influencers in our framework, although the authors did not  explicitly make such a distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Other works have divided the population into classes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', [29] introduced  stubborn agents, and if such individuals have opposing opinions, they hinder the possibility of the population  converging to consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Recent work is drawing further attention to online platforms by adapting classical frameworks to the  specificities of online interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Valensise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' [30] have developed an opinion model that embodies al-  gorithmic personalization, comparing its behavior to phenomena observed in social networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', Facebook  and Twitter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Our work is different because we consider distinct classes of users, characterizing specifically  influencers and closing the interaction loop between users and the platform by a feedback function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Other  works that address content filtering bias in social media platforms include [31] [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Peralta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' [31] develop  a flexible framework to incorporate algorithmic bias into binary opinion dynamics by having agents interact  at a lower rate with individuals who hold an opposing viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Considering both pairwise and group-wise  interactions, the authors found that algorithmic bias either leads users to polarize their opinion (in the case  of pairwise interactions) or results in the coexistence of beliefs (in the case of group-wise interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Validation of opinion formation models  In [33], models of opinion dynamics are referred to as idealized because, in most cases, they assume basic  underlying principles of interaction and observe emergent social behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' There are two main approaches  to validating opinion models in the literature: first, the use of observational data [2][1][30] and second, the  use of controlled sociological experiments[34][35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We will focus more on the first portion of the literature,  4  which is more relevant to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Attempts to validate opinion models are scarce for several reasons: i)  the mapping of opinions into values, ii) an adequate definition of links between agents, and iv) the change  in opinion after an interaction is hardly measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  A notable exception is election and polling data,  which make it possible to attribute a person’s opinion to the political orientation of the chosen candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Fortunato and Castellano [2] have shown that the distribution of vote counts is a universal scaling function  and have derived a simple tree-like interaction structure with candidates as roots and an interaction that  can turn the individuals reached into “activists” who can spread the idea and convince other individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The results of the model are in good agreement with empirical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In [36], a noisy voter model could  fit data from US elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Other recent approaches [37] have used shared news on Facebook to assess the  extent to which individuals are exposed to opposing views through their (online) friendship relationships,  using users’ self-reported ideological affiliations to infer opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' They found that individuals have access  to cross-cutting content and that the degree of this exposure depends on the composition of one’s friends  on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' A more recent body of literature [1] [30] has directly employed data from online social  networks, such as Gab, Facebook, Reddit, and Twitter, to observe the emergence of echo chambers [1] and  to validate a model encompassing algorithmic personalization in the process of opinion formation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The Communication Asymmetry opinion model  In this section, we first establish the notation used throughout the paper and then present the Commu-  nication Asymmetry (CA) model in its most general formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We conclude the section with a discussion  of the strengths and limitations of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Notation  In this work, we adopt the following vectorial notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We denote vectors by bold symbols, whereas we  denote their components with normal-font symbols whose subscript is the index in the vector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', a = {ak}k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Lowercase letters denote parameters and dynamical variables associated with an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In general, index  i runs over the set of influencers while index u runs over that of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For those parameters/variables  that can be associated with individuals of both classes (either influencers or regular users), the above indices  are indicated between superscript parentheses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', a(i), a(u), to immediately identify the class to which the  individual belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' If necessary, the dependence of variables on other system parameters is made explicit  by specifying the independent variables between parentheses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', α(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Italic capital letters denote sets,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', I is the set of all influencers in the population, while |I| is its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Capital letters represent outcomes of stochastic experiments whose characteristic parameters are low-  ercase letters: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', Ω (ω(·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The operator E[·] represents an expected value, and a bar over a variable,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', ¯a, represents its average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Whenever we need to express the probability of an event, we use the  notation Pr[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We employ 1{·} for the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Lastly, time is denoted by t if it is considered  continuous and by n if it is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Description of the model  We propose a continuous opinion model with two interacting classes of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Specifically, the population  consists of Nu = |U| regular users and Ni = |I| influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This division mimics what happens in real  social networks, where a small portion of the population, the influencers, has a much larger number of people  following their posts on the online social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We assume that the generation of new posts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', messages  in the OSN, is a Poisson Point Process (PPP) with intensity λ, where each event of the PPP corresponds  to the creation of a new post from an influencer i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The corresponding embedded discrete time will  be denoted by the integer n ∈ N+, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', where n is the n-th post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Each post is sent1 to a subset  of regular users, identified by the social platform according to an algorithmic personalization described by  function ω (to be specified later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Regular users react to these posts through a feedback function θ (specified  5  User  Influencer  popularity update  post generation  post provision  feedback  Platform  θ  ω  Figure 1: The picture depicts the relationship among the three players of the model: regular users, influencers, and the social  media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, it highlights the role of the feedback function θ provided by the users and the filtering function ω  used by the platform to propose the posts to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Figure 1 highlights this closed loop behavior brokered by the social media platform, positioned  between regular users and influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The opinion space is X ⊂ Rd, where each dimension represents an uncorrelated topic on which users  have a belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore, an opinion consists of a d-dimensional vector x(u)(n) ∈ Rd, which evolves as a  result of the interaction between a regular user and every influencer on every possible topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This model  neglects interactions between regular users2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The prejudice of a user, denoted by z(u), is the other parameter  that enters the opinion update rule alongside the user’s current opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It represents the user’s natural  inclination toward different topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Unless otherwise specified, we will assume that the user’s initial opinion  is set equal to the prejudice: x(u)(0) = z(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We will consider different distributions for the agents’ prejudice over the opinion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In particular,  delta, uniform, and Beta distributions are usually employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Influencers are considered stubborn agents, which means that their opinions do not change over time,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', x(i)(n) = x(i)(0) = x(i) = z(i) ∈ Rd,  ∀n > 0 and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' As we will show in Section 4, each influencer  has a main topic of interest on which it publishes the majority of its posts and which typically coincides  with the topic it is mainly known for on the OSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It represents the reference direction r(i) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='., d − 1} of  the influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Another parameter characterizing influencer i is its consistency c(i)(n), which indicates the  probability that such an influencer publishes a post on its reference direction (it might change over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Note that individuals with high consistency prefer to post in their reference topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, we denote  by f (i) the probability that a post is generated by influencer i at any time instant n, with �  i∈I f (i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  At last, we introduce the popularity vector p(n) := {pi(n)}i∈I, containing the current popularity of all  influencers at time n, before the emission of the post at time n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We also introduce the normalized version  of this vector π(n) = {πi(n)}i∈I where the components are the normalized probabilities πi =  pi  �  j∈I pj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Dynamic variables of users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', their opinion x(u)) and influencers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', their popularity pi) are updated  upon every post generation according to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It provides a detailed description of the dynamics  captured by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The model’s key features are further illustrated schematically in Figure 2: an  influencer posts a message, the social media platform filters it according to ω, and users provide feedback  via θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' These two features represent, respectively, an algorithmic effect (function ω): selective exposure,  namely the tendency of a platform to suggest similar content to maximize time spent on the social platform,  and an individual effect (function θ): confirmation bias, namely the tendency to value content that is close  to one’s point of view, as discussed in [1] and the resources therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  These tendencies can explain the  appearance of echo chambers in social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  More specifically, in an elementary step of the dynamics, a post is generated by one of the influencers,  selected according to the distribution f (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The influencer i posts on its reference direction r(i) = j with a  1In this paper, the verbs “send”, “suggest” and “reach” are used interchangeably referring to a post shown to a user by the  platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  2From now on, we will refer to regular users simply as users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, the terms agent or individual are used to indicate  a social network user of either class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  6  probability equal to its consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Otherwise, it posts on one of the other directions in the opinion  space j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', d−1}\\{r(i)} = Nr according to a given distribution, Pr[j = k] for k in Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In the rest of  the paper, we assume for simplicity that this distribution is uniform over the set of non-reference directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We suppose that each post contains exactly the influencer’s opinion on the topic (and that no noise in the  user’s perception of the post is present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Then, note that the post contains a real-valued opinion that is the  j-th component of the influencer’s opinion vector x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In principle, this generated post can reach any user:  no explicit network structure is considered for the population3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Following the intuition that individuals  with strongly divergent opinions are less likely to interact and therefore, as homophily suggests, like-minded  individuals are more likely to interact, the subset of reachable users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', users to whom the platform sends  the post) is determined by considering the opinion distance in the reference direction between each user and  the posting influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Adopting such distance as the central metric influencing the reachable group of users  is of utmost importance as it couples the dynamics in different directions, which would otherwise evolve  independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This posts-users matching process constitutes the content personalization we  consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that the social media platform suggests posts that might interest a user in  addition to those that a user explicitly subscribes to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', follows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Algorithm 1 Description of the Communication Asymmetry model  Input:  Ni influencers, Nu users, filtering function ω, feedback function θ  Output: opinion of each regular user x(u)(n), ∀u  Output: popularity of each influencer pi(n), ∀i  1: loop  2:  select influencer i according to f (i)  3:  select a posting direction j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', j = r(i) with probability c(i), other-  wise j is selected uniformly on j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='.,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' d − 1} \\ {r(i)}  4:  pi(n + 1) = pi(n)  5:  for all regular user u in the population do  6:  x(u)  j  (n + 1) = x(u)  j  (n)  7:  if Ω  �  ω(|x(i)  ri − x(u)  ri |,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' πi(n))  �  = 1 then {post proposition}  8:  get feedback Θ  �  θ(|x(u)  j  − x(i)  j |)  �  9:  if Θ = 1 then {positive feedback}  10:  x(u)  j  (n + 1) = αz(u)  j  + βx(u)  j  (n) + (1 − α − β)x(i)  j  11:  update popularity of i: pi(n + 1) += 1/Nu  12:  end if  13:  end if  14:  end for  15: end loop  To decide whether a given user is reached by a post (independently from other users),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' we extract a  Bernoulli random variable Ω with parameter ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The user receives the message when Ω(ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The  parameter ω can be interpreted as a visibility function from the influencer’s perspective, as it affects the  subset of users reached by its posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' As already mentioned, ω should be a function of the opinion distance  in the reference direction dr(n) = |x(u)  r (n) − x(i)  r (n)| and the popularity ratio πi of the posting influencer,  so that the higher the popularity ratio, the more users an influencer can reach on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Users express  their feedback to a post on the platform through a Bernoulli random variable Θ  �  θ(|x(u)  j  − x(i)  j |)  �  ∈ {0, 1}  3A complete bipartite graph, where I and U are the two sets of nodes and each link has a weight ω computed at each  iteration of the dynamics, might represent the underlying network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7  whose parameter θ depends on the difference in opinion on the actual direction j of the contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Only  posts that receive positive feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', Θ = 1, can influence the user’s opinion, reflecting the tendency to  ignore unappreciated content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The social media platform collects feedback from all reached users to update  the popularity pi of the posting influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Specifically, the update rule for the popularity of the posting  influencer i reads as follows:  pi(n + 1) = pi(n) + ΘT (θ, Upost)  Nu  (1)  ΘT (θ, Upost) =  �  u∈Upost  Θ  �  θ(|x(u)  j  (n) − x(i)  j (n)|)  �  (2)  where Upost is the subset of users who were made aware of the post by the platform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', those for whom  Ω(ω) takes the value one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The summation in the formula gives the aggregate feedback of all users who saw  the post, which is normalized by the size of the population of regular users |U| = Nu to update the popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Note that this normalization is introduced only to avoid excessive growth of influencers’ popularity when  the number of users becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It does not affect the system dynamics, as these depend only on the  normalized popularity values πi, which are not affected by the scaling factor 1/Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Influencer  post generation  according to  Platform  popularity update  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Ω=1  Θ=1  Ω=1  Ω=1  Θ=0  Θ=1  not reached  reached  Figure 2: Schematic representation of the model dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The figure highlights the proportions of users who view a particular  post Upost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' those for which the random variable Ω takes the value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' They react with their feedback Θ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', likes), which  depends on the opinion distance between them and the influencer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Then the platform updates influencer’s i popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The core of the dynamic is represented by the opinion update rule, which dictates how the user’s opinion  changes on the direction j of the post depending on the previous opinion x(u)(n), the prejudice z(u) and the  opinion x(i) conveyed by the influencer through the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The following system of equations characterizes  the updating rule:  x(u)  j  (n + 1) =  �  �  �  αz(u)  j  + βx(u)  j  (n) + γx(i)  j  if Ω (ω(dr, πi)) = 1 , Θ (θ(dj)) = 1  x(u)  j  (n)  otherwise  (3)  where γ = (1−α−β), being the updating rule a convex combination of x(u)(n), z(u) and x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Whenever  the Bernoulli random variable Ω(ω) ∈ {0, 1} assumes the value 0, the post does not reach the user who keeps  8  its opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The individual’s opinion is also not affected by the post when the user receives it but does not  appreciate it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the feedback variable Θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The actual opinion update is a convex (linear) combination  of the user’s prejudice z(u)  j  , the current user’s opinion x(u)  j  (n) and the belief delivered by the influencer’s  post x(i)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The distance on the reference direction drives the filtering because we assume the platform is  unaware of the specific topic associated with the post just created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note the joint effect in the model of the  distance between the user’s opinion and the influencer’s opinion on the reference direction and the distance  along the direction defined by the post’s topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Both contribute to determining the likelihood for the user to  provide positive feedback to the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In most OSNs, there are explicit subscriptions to influencers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the follow mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Our  approach does not consider this type of relationship, as we only account for homophilic contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Since the  number of influencers is considerable in practice, homophily is not the only mechanism driving interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' A  regular user does not follow all its homophilic influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, nowadays, most social media platforms  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', Facebook, Instagram, Twitter) not only offer their users content they explicitly subscribe to but also  content that users might like based on their activity on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This resembles the mechanism we are  considering in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In our framework, regular users are passive, as they merely consume content produced by influ-  encers: this constitutes a rather simplistic assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' First, users can share the posts they receive, which  increases their reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Secondly, users themselves write posts that reflect their opinion, influencing other  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The impact of active users is beyond the scope of this article and will be considered in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Observations from Online Social Networks  This section presents data from real-world social networks to motivate some of our modeling choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For  a detailed description of the dataset used, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' One of the most important features introduced  in this paper is the concept of reference direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the main topic an influencer is interested in and  on which they publish most of their posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' While this is a reasonable assumption, this claim needs to be  supported by evidence from real social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, we examine the post-generation process to  justify the choice of a Poisson Point Process to describe it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Music  Politics  Sports  Food Pandemic  Topics  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  Fraction of posts  Luca Zaia  AC Milan  (a)  0  10  20  30  40  lag  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  autocorrelation  music  Luca Zaia  AC Milan  (b)  Figure 3: (3a) Percentage of labeled posts on each of the considered topics for Luca Zaia, an Italian politician, and AC Milan,  an Italian football club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' (3b) Autocorrelation function on a secondary topic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', music, for both influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The reference direction  This section shows that influencers prefer to post about a specific topic rather than discuss multiple  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We have developed a post classifier that flags posts based on their topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We should point out that  classifying posts on OSNs into topics is not straightforward, and interpreting the results should be done  with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' First, the range of possible subjects discussed in a social network is practically countless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For  practical reasons, we will only focus on a subset of five topics: Sports, Politics, Food and Cooking, Music,  and Pandemics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  These can be considered popular and general enough to cover a substantial fraction of the influencer  discussions on OSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We took a subset of the influencers in the dataset, namely those with the highest  number of classified posts (see Appendix  B for details on the classification and filtering process on the  data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that even if the selected topics can be assumed uncorrelated, they are sometimes discussed  jointly in one post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In such cases, it is not always clear which is the main topic of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  After classification, we examined the distribution of posts on the topics for each influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In Figure 3a,  we show two example influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In these two cases, the influencers have one topic on which they write most  of their posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Luca Zaia, an Italian politician, posts mainly about politics, and AC Milan, a soccer club,  discusses sports predominantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This behavior supports the existence of a reference direction for influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Figure 4a shows the distribution of the proportion of posts dealing with the main topic of each influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Recall that this proportion was called consistency in the jargon of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Most influencers have a clear reference topic on which they write more than half of their posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Figure  4b shows the average per-topic percentage of all influencers in the dataset in descending order, regardless  of the specific topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' On average, almost ninety percent of the posts are in the reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We  discovered that influencers with low consistency values are affected by the presence of news outlets in the  considered profiles, for which the lack of a sharp main topic is sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  Consistency  0  20  40  60  Number of Influencers  (a)  main  2nd  3rd  4th  5th  Topics  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  Fraction of posts  (b)  Figure 4: (4a) Distribution of the fraction of posts published on the main topic of interest by the subset of influencers considered  in this experiment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', their consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' (4b) The average percentage of labeled posts on each topic in decreasing order for  all the influencers considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The 95% confidence interval for each average value is reported in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Independence of posts’ generation process on secondary directions  The way users interact in an OSN is by posting content (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', text, images, videos) and receiving sugges-  tions about what other users of the OSN posted, according to the filtering process set up by the social media  platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Since influencers’ posts have a much greater reach than those of regular users, they are the focus of  our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Namely, we examine the correlation between posts on each topic by looking at the chronological  sequence of the messages of the individual influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Our primary focus now is on secondary topics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=',  the topics that are not the reference for the influencer, as they post less frequently in these topics, and one  might expect to observe a bursty posting behavior, not well captured by the Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  10  In the previous section, we were able to assign a reference direction r(i) to each influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Here we look  at the time series of the Influencers’ labeled posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For each secondary direction s(i)  j , we define an indicator  function 1{postlabel=s(i)  j  } that takes the value 1 if the post was labelled as s(i)  j  and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For each  influencer, we thus obtain four sequences (recall that we consider five topics in total) of Bernoulli random  variables indicating whether a post belongs to that particular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For these sequences, we calculated  the autocorrelation function a(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Figure 3b shows two examples of such autocorrelation functions, limited  to 40-time lags, for the profiles of Luca Zaia and AC Milan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The time is discretized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the actual time  between postings is not taken into account: only the posting events matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' An autocorrelation that equals  zero everywhere except at τ = 0 would represent uncorrelated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In our case, the autocorrelation  takes moderate values in most cases (≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore, it is reasonable to assume that the post-generation  is independent, and a Poisson Point Process is an appropriate choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Lastly, note that the autocorrelation  function for the pandemic topic takes larger values than for the other topics (see Figure 5), suggesting that  the samples are weakly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This fact is due to the exceptional public interest in the topic and because  the outbreak of the epidemic only interested the last part of the considered time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  sports  politics  music  0  10  20  30  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  food  0  10  20  30  40  pandemic  0  10  20  30  40  average  lag  autocorrelation  Figure 5: Mean autocorrelation values of the post-generation process for each secondary topic of all the influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The last  plot represents the average value over all topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The 95% confidence interval is shown in each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Asymptotic Analysis of the Model  This section is devoted to the analytical study of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In particular, results are obtained using a  mean-field approach, considering Nu → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this situation, the equilibrium value for the influencers’ mean-  popularity ratios ¯πi and users’ mean opinion value ¯x(z) (which depend on prejudice z) can be analytically  determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Furthermore, transient analysis of the system can be carried out by describing the dynamics  of the users through a Fokker-Plank equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For simplicity, we restrict our investigation to the situation  where the opinion space is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, we remark that it is possible to extend the analysis  to the more general case by following the same approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  11  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Mean field approach  When the number of users grows large, it is convenient to characterize the system state by the users’  opinion distribution over the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, hereinafter we will refer to system dynamics over continuous  time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Let (X(t), Z(t)) = (X(t), Z) be the current position (opinion) and prejudice of a randomly selected user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We introduce the cumulative distribution function F(x, z, t) = Pr[X(t) < x, Z < z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The corresponding  probability density function is f(x, z, t) =  ∂2  ∂x∂zF(x, z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that, by hypothesis, there are no dynamics  along the z-axes, thus h(z) =  �  x f(x, z, t)dx does not depend on t and corresponds to the initial distribution  of users’ prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2, we will derive a Fokker-Plank equation for the evolution of the opinion  distribution over time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  For what concerns the evolution of the popularity of a generic influencer i, recall that we distinguish  between its absolute popularity value pi(t) and the normalized value πi =  pi(t)  �  j pj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We can already write  down the equation for the evolution of the mean popularity ¯pi(t) (we remark that influencer’s popularities  concentrate around their average as Nu grows large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' as it can be easily shown):  d¯pi(t)  dt  =  1  Nu  λf (i)  �  x  �  z  f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t) θ  �  |x − x(i)|  �  ω  �  ¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' |x − x(i)|  �  dz dx  (4)  Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' the rate at which the popularity of influencer i grows is proportional to its posting rate (term  λf (i)) times the probability that a generic user at (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z) provides positive feedback to the post generated at  time t (integral term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, recall from Algorithm 1 that each positive feedback increases the absolute  popularity of the influencer by 1/Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Fokker-Planck equation for the opinion distribution  In this section, we derive a mean-field Fokker-Planck (FP) equation for the population’s opinion distri-  bution, assuming that the number of users grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  To be specific, in the continuous-time FP approximation, we assume that for the effect of a post,  “users/particles” reach their new position by moving at a constant speed during the interval ∆T equal  to the average time 1/λ that elapses between the generation of two successive posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore, assuming  that at time t a post is generated by user i, the following equation describes how the opinion of a user with  prejudice z evolves from t to t + ∆T:  x(t + ∆T) = αz + βx(t) + γx(i)(t)  Thus, the increment is:  ∆x(i) = x(t + ∆T) − x(t) = α(z − x(i)(t)) + (1 − β)(x(i)(t) − x(t))  (5)  where we remark that ∆x(i) here represents the change in position of a user in position x, providing positive  feedback to a post of influencer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We can then compute its average velocity as:  E[vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t) | X(t) = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Z = z] = E  �[X(t + ∆T) − X(t) | X(t) = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Z = z]  ∆T  �  =  �  i  λf (i)∆T θ  �  |x − x(i)|  �  ω  �  ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' |x − x(i)|  � ∆x(i)  ∆T  =  �  i  λf (i)θ  �  |x − x(i)|  �  ω  �  ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' |x − x(i)|  �  ∆x(i)  (6)  where θ  �  |x − x(i)|  �  is the probability of providing positive feedback (users move only in this case),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' while  ω  �  ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' |x − x(i)|  �  is the probability with which a user in x is exposed to a post created by influencer i at  12  time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Indeed, users only move if they are exposed to the post and provide positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that, to  avoid a cumbersome notation, we have omitted the dependency on the time of the distance term |x − x(i)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The variance of the velocity is given by the relation:  σ2  x(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t) =  �  i  λf (i)∆Tθ  �  |x − x(i)|  �  ω  �  ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' |x − x(i)|  � (∆x(i) − E[vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)]∆T)2  (∆T)2  =  1  (∆T)2  �  i  f (i)θ  �  |x − x(i)|  �  ω  �  ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' |x − x(i)|  �  (∆x(i) − E[vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)]∆T)2  This allows us to write down a Fokker-Plank equation [38] for the probability density function f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)  where x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z ∈ [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' b]:  ∂f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)  ∂t  = −∂vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)  ∂x  + 1  2  ∂2σ2  x(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' t)  ∂x2  (7)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Steady state analysis  Now we direct our attention to the existence of stationary solutions for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Stationary solutions  of (7) necessarily satisfy:  ∂  ∂x  �  −vx(x, z)f(x, z) + 1  2  ∂σ2  x(x, z)f(x, z)  ∂x  �  = 0  where vx(x, z) and σ2  x(x, z) must be constant over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This requires normalized popularities to be static  (i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' ω(·) to be constant over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' From previous equation, integrating both sides with respect to x, we  get:  �  −vx(x, z)f(x, z) + 1  2  ∂σ2  x(x, z)f(x, z)  ∂x  �  = c0(z)  (8)  where c0(z) is a uni-dimensional arbitrary in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Now, observe that, for every z, previous equation is a first  order linear ODE in x, and therefore an explicitly solution for f(x, z) can be obtained:  f(x, z) =  �  c1(z) exp(A(x, z) − A(a, z)) + c0(z) exp(−A(x, z))  � x  a  exp(A(θ, z))dθ  �  h(z)  (9)  where  A(x, z) =  � x  a  η(u, z)du  η(x, z) = −2vx(x, z) − 1  2  ∂σ2  x(x,z)  ∂x  σ2x(x, z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Function c0(z) can be obtained by imposing boundary conditions:  �  −vx(x, z)f(x, z) + 1  2  ∂  ∂xσ2  x(x, z)f(x, z)  � ���  x=a,b= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  ∀z  which leads to c0(z) = 0, while function c1(z) is determined by imposing the normalization condition:  �  f(x, z)dx = h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Observe that when σ2  x(x, z) → 0 and ∂σ2  x(x,z)  ∂x  → 0, from (8), with x0(z) = 0, we obtain that necessarily  the mass concentrates around points for which vx(x, z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Such points, improperly referred to in the  following as equilibrium points, will be characterized analytically later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Turning our attention to popularity dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' recall that stationary conditions necessarily imply nor-  13  malized popularities to be constant over time:  ¯πi(t) = ¯πi  ∀i  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' absolute popularities naturally grow over time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' but the ratio between any two of them (say  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' j) must converge to a constant value cij equal to the ratio of their corresponding normalized popularities:  ¯pi(t)  ¯pj(t) = cij = ¯πi  ¯πj  ∀i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' j ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' i ̸= j  (10)  Now observe that in stationary conditions the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' of (4) does not depend on time, therefore (4) admits  the following trivial solution:  ¯pi(t) =  �  λf (i)  �  x  �  z  θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z)  � t  Nu  + ¯pi(0)  (11)  Therefore, we meet conditions (10) for any t ≥ 0, iff normalized popularities of influencers {¯πi}i satisfy  the following system of equations:  λf (i)  �  x  �  z  θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z) = c¯πi  ∀i, for some c ∈ R+  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' ¯πi ≥ 0 and  �  i  ¯πi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  (12)  and the initial condition {pi(0)}i satisfies (10) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' pi(0) = k¯πi for some k > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Let  ki(¯πi) := λf (i)  �  x  �  z  θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z)  ¯πi ∈ [0, 1]  (13)  We can show that:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Solutions of (12) always exist whenever ki(·) ∈ C1[0, 1], ki(·) is increasing, continuous and  strictly concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The proof is reported in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We remark that when ki(0) > 0 ∀i, the solution is always unique with ¯πi ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Instead when ki(0) = 0  for some i, the solution is not guaranteed to be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Now, the problem is how to jointly solve for stationary solutions of {¯πi}i and F(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In a schematic  way, on the one hand, we have shown that given ¯π = {¯πi}i, and h(z), we can uniquely determine a  F¯π(x, z) = H(¯π), where F¯π(x, z) =  � x  −∞  � z  −∞ f¯π(y, w) dy dw is the opinion distribution of users resulting  from fixed influencers’ popularities ¯π (by (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  On the other hand, under the conditions: ki(·) ∈ C1[0, 1], ki(·) is increasing and strictly concave,  ki(0) > 0 ∀i, given F(x, z), we can obtain a ¯πF = G(F(x, z)) that uniquely corresponds to i (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The existence of a unique fixed point for the joint system of (stationary) users’ opinions and influencers’  popularities is guaranteed under the condition that the operator H ◦ G(·) is a contraction over a complete  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Under the assumption that both ω(·, ·) and θ(·) exhibit a sufficiently weak dependence on their  variables, the operator H ◦ G(·) is a contraction over a complete space, and therefore a unique stationary  solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The proof is reported in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Asymptotic analysis of the fluid limit  Previous theoretical analysis is, unfortunately, non-constructive, meaning that it does not allow for direct  computation of stationary solutions of our dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' To complement the previous analysis, in this  14  section we propose a methodology to compute numerically stationary solutions, even in multi-dimensional  scenarios, under the assumption that Nu → ∞, β → 1, σ2  x(x, z) → 0 and ∂σ2  x(x,z)  ∂x  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In the following, we  will refer to the such regime as fluid limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Mean opinion assuming that normalized popularities converge  As already observed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3, recall that, given ¯π = {¯πi}i, the distribution of users with a given  prejudice z concentrates around equilibrium points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', points ¯x(z) at which v(x, z), as given in (6), is null  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' v(¯x(z), z) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' points ¯x(z) must satisfy equation:  0 =  �  i  f (i)ω  �  ¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' |¯x − x(i)|  �  θ  �  |¯x − x(i)|  � �  α(z − x(i)) + (1 − β)(x(i) − ¯x)  �  (14)  Defining for compactness di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='¯x =  ��¯x − x(i)�� and recalling γ = 1 − α − β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' from (14) we get:  ¯x(z) =  α  1 − β z +  γ  1 − β  �  i∈I f (i)ω  �  ¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='¯x�  θ  �  di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='¯x�  x(i)  �  i∈I f (i)ω (¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='¯x) θ (di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='¯x)  (15)  The assumption β → 1 is required to avoid too large oscillations of users’ opinions in response to a single  post generated by an influencer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' which may reduce the accuracy of our mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  This hypothesis is not restrictive: since β represents the weight individuals give to their current opinion,  we can reasonably assume that users do not dramatically change their opinion in response to single post  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Normalized popularities assuming opinion convergence  Here we assume that users with prejudice z are concentrated in opinion point ¯x(z), and we look for  the stationary popularity ratios ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  To simplify the expressions, we introduce the quantity Fi(¯πi) ≜  �  z f (i)ω(¯πi, di,¯x(z))θ(di,¯x(z))h(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Observe that solutions of (12) are necessarily in the form:  ¯πi =  Fi(¯πi)  �  j∈I Fj(¯πj)  (16)  where c appearing in (12) is given by c =  1  �  j∈I Fj(¯πj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Under the assumption that ω(·, ·) is concave in its  first argument (for any choice of the second), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1, guarantees the existence of such solutions for  every choice of function ¯x(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, even in the more general case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', when ω(·, ·) is non-concave in  its first argument, solutions of (16) can be found numerically in many cases, through a fixed point iteration  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  To conclude, observe that a pair (¯x(z), {¯πi}i) represents a stationary solution if it jointly satisfies (15)  and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The existence of such solution can be, again, only verified numerically through a fixed point  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  At last, note that, in the special case in which all users have the same prejudice z we can rewrite (15)  as:  ¯x =  α  1 − β z +  γ  1 − β  �  i∈I Fi(¯πi, ¯x)x(i)  �  i∈I Fi(¯πi, ¯x)  =  α  1 − β z +  γ  1 − β  �  i∈I  ¯πix(i)  (17)  which provides a direct formula for the mean opinion ¯x in terms of the normalized popularities ¯πi and the  influencers’ opinions x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Model predictions  In this section, we present a selection of results obtained while varying the model parameters, providing  valuable insights into the impact of algorithmic personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Results are obtained through a Monte Carlo  15  Table 1: Parameters and functions shared across experiments  Symbol  Value - Form  Description  Ni  2  Number of influencers  x(0)  j  0  Opinion of influencer 0 on direction j  x(1)  j  1  Opinion of influencer 1 on direction j  r(0)  0  Reference direction of influencer 0  r(1)  1  Reference direction of influencer 1  p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1(0)  100  Initial absolute popularity of both influencers  Nu  10000  Number of regular users  Niter  100000  Number of iterations for each simulation  α  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='05  First weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 3  β  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='93  Second weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 3  θ(·)  1 −  ���x(i)  j  − x(u)  j  ���  Functional form of the feedback function  ω(·)  e−ρ(x(u)  r  −x(i)  r )  2  πi  Functional form of the visibility function  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  x0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  x1  Figure 6: Table 1 summarizes some of the parameters of the system shared across the different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' On the right-hand  side, a realization of the initial opinion distribution of the regular users (being also the prejudice since z(u) = x(u)(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We focus on the two main dynamic variables of the system: the average opinion ¯x of regular  users and the normalized popularities {¯πi}i for influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Note that the quantities shown in this section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the pair (¯x, ¯π), are obtained as empirical averages  over multiple runs, and over all the regular users, as far as ¯x is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Hence, they are different from  the values presented in the previous section, in principle, which pertains to the limiting case of an infinite  population of users with the same prejudice z, and where β approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, in some cases, to  save space, we omit the results on average user opinion because it is tightly coupled with the normalized  popularities, as observed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Lastly, to facilitate the interpretation of results, we restrict  ourselves to the case of two “competing” influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We provide further details on the scenario considered in  section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The model can clearly be applied to scenarios with an arbitrary number of influencers occupying  any position in the opinion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2, we present the behavior as function of publication frequency  f (i), and in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3 as function of consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Then we show examples of final opinion distributions  of the regular users in a few paradigmatic cases in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Finally, in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, we consider the degree  of stubbornness, defined as δ = α  γ , which governs the opinion update of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Description of the scenario  The default parameters of our reference scenario are reported in Table 1 unless otherwise explicitly  stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' As mentioned earlier, we restrict ourselves to the case of two “competing” influencers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', Ni = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We assume that x(0)  j  = 0 and x(1)  j  = 1 ∀j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We consider the case of different reference directions r(0) ̸= r(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In  this section, we consider a two-dimensional opinion space and assume that the vast majority of the regular  users’ population initially takes a moderate position on both topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' More precisely, initial opinions are  distributed according to a Beta distribution, independently on each axis, with shape parameters a = b = 10,  as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Recall that we assume for simplicity that the prejudice of the user z(u) corresponds  to the initial opinion x(u), hence Figure 6 also provides the prejudice distribution of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The functional form of visibility ω and feedback θ is also reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We take as ω(·) a Gaussian  function similar to the trust function in [39], but modulated by ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Here, the coefficient ρ is a parameter that  controls the extent to which the social media platform filters content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Small values of ρ correspond to smooth  personalization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', influencers can reach users whose opinion strongly differs from theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Conversely, high  values of ρ correspond to sharp personalization: only close users (in the opinion space) are reachable with  non-negligible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The function θ(·) is assumed to be a decreasing, linear function of the opinion  difference dj = |x(u)  j  − x(i)  j |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  16  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Behaviour as function of the frequency of publication  The frequency of publication f (i) is one of the basic parameters that characterize influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The  higher f (i), the higher the structural advantage of the influencer because it more frequently reaches users  through posts, attracting them to its own opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this section, we examine the value of mean normalized  popularity ¯π0 as a function of f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that in the case of two influencers, f (1) = 1 − f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We performed  this experiment by fixing the consistency of the two influencers: c(0) = c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8, which is approximately  the average consistency observed on real-world data (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  f(0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  0  Values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  Figure 7: Popularity ratio ¯π0 of influencer 0 as function of the publication rate f(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Each point is obtained by averaging  over 100 time samples and 10 different process’ realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Different levels of personalization are considered by varying the  parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The two influencers have the same consistency c(0) = c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that, in the considered scenario, the curves  are symmetric for values of f(0) in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  f(0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='65  Distance from i = 0 on r(0)  Values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  f(0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='65  Distance from i = 1 on r(1)  Values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  (b)  Figure 8: Opinion distance of the influencers’ opinion on their reference direction x(i)  r(i) and the average opinion of the regular  users’ population on the same direction ¯xr(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Various degrees of personalization are considered, tuning the parameter ρ, the  setting is the same as that of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In Figure 7, we consider different levels of personalization by varying the parameter ρ in the exponent  of the visibility function ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We see that the higher the degree of personalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the higher the value  of ρ), the lower the normalized popularity of influencer i = 0, for any given f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This result suggests  that algorithmic personalization favors the structurally advantaged individual, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the one with higher f (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  This mechanism, in turn, leads to more radical positions in the population of regular users, as the platform  preferentially exposes them to the belief of the advantaged influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Figure 8 clearly shows this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  17  Note that for high values of ρ, the average user opinion exhibits a significant bias toward the structurally  advantaged influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Such bias persists up to a critical value of posting frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For example, when  the personalization parameter is ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0, the critical posting frequency value is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' when the  personalization parameter is ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, the critical posting frequency value is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We argue that content filtering in OSN poses a potential threat to opinion diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  This premise  is inextricably linked to the goal of usage maximization [32] pursued by the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Many  platforms indeed prefer to suggest just similar content rather than exposing individuals to radically different  opinions, allowing for so-called serendipity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Behaviour as a function of the consistency  In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1, we showed the existence of a reference direction for real influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Here, we investigate  the impact on dynamics of the extent to which an influencer publishes on its reference direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', its  consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this experiment, we consider two influencers with the same posting frequency f (0) =  f (1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, and we let c(0) vary while keeping c(1) fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We then consider different choices of c(1) to grasp its  impact on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' From Figure 9, we see that consistency does not significantly affect the normalized  popularities when personalization is smooth (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0001), while it becomes relevant when the platform  applies sharp personalization to the content (ρ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  c(0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7  0  Values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='0  (a) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  c(0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7  0  Values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0001  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  (b) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  c(0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7  0  Values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0001  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  (c) c(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  Figure 9: Popularity ratio ¯π0 of i0 as function of its consistency c(0), while considering f(0) = f(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5 and keeping fixed  the consistency of the second influencer at (9a) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, (9b) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8 and (9c) c(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The two colors represent two  different levels of personalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', smooth and sharp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Each point is obtained by averaging over 100 time samples and 10  different realizations of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It is interesting to observe that the maximum of ¯π0 moves to the left (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', is achieved for  a lower value of consistency c(0)) as the consistency of i = 1 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  18  Before discussing the results concerning sharp personalization shown in Figure 9, it is crucial to note  that when considering a bi-dimensional opinion space, a consistency c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5 implies that the choice of the  reference direction for influencer i is somehow unnatural since it produces the majority of posts on the other  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This choice is in contradiction with the definition of reference direction itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Nonetheless, we  leave this situation as a possibility: let us imagine an influencer can adopt consistency values of less than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5 while undertaking a transition phase during which it changes its main topic for its posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this case,  the platform would still perform personalization on the given reference direction r(i), but the consistency  would be less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5 due to the change in posting pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It represents a scenario of interest, and as such,  we allow for c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  First, we note that the shape of the curves in Figure 9 depends strongly on the value of the consistency  of the “opposing” influencer c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Second, a perfectly balanced condition is achieved whenever the two  influencers have the same consistency since all parameters are symmetric (even the curves associated with  ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0001 and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0 coincide on this point), see Figure 9c at c(0) = 1 for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The observed pattern  is consistent in all three figures with ¯π0 being first increasing and then decreasing, exhibiting a unique  maximum in all three diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Let us start the discussion by considering Figure 9c because its interpretation is instrumental to better  understanding the other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It represents a rather degenerate situation since the influencer i = 1  posts exclusively in its reference direction r(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, the simplicity of the scenarios allows us to  interpret the results straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this case, the influencer i = 0 has r(0) = 0 and for 0 < c(0) < 1  4 posts in both directions, with the social media platform filtering according to distance in the reference  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In the direction r(0) = 0, the influencer has no competition at all, since c(1) = 1, so it is able  to attract the user population to its “reference opinion” while competing with the other influencer in the  non-reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The lower the consistency c(0), the greater the competition on r(1) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', for  values of c(0) close to 0, the influencer i = 0 posts the vast majority of its messages on r(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore,  the final value of ¯π0 reaches higher values, as demonstrated in Figure 9c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It happens because the influencer  i = 0 has a stable feedback stream from the posts in its reference direction, where it does not face any  competition, and it competes with i = 1 in the other direction, being at an advantage since the visibility  of its posts is determined by the opinion distance in its reference direction r(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' From these observations,  we can conclude that influencer i = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the influencer with the higher consistency, is disadvantaged for  virtually all values of the other influencer’s consistency c(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Here we are considering the extreme case, where  the influencer i = 1 has the maximum attainable consistency c(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Thus, we can easily conclude that  the individual with lower consistency holds the structural advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This is consistent with the results in  Figure 9a and 9b, for which we observe a decrease in normalized popularity ¯π0 on the right of the point at  which c(0) = c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Namely, the influencer with higher consistency is penalized and eventually reaches lower  values of ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This assertion is not true in the first part of all plots in Figure 9 where ¯π0 is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the influencer i = 1 is favored, even though c(0) < c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It is important to note that this is only true  for values of c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, which, as discussed above, are only relevant in certain situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This behavior  depends on the interplay between the actual main posting direction (which is different from the reference  direction when c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5) and the algorithmic personalization performed on r(i), see also the discussion in  the footnote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Interestingly, the curves intersect for the first time when c(0) ≈ 1 − c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  From our results, choosing a consistency c(i) around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5 for i = 1 seems to be a successful choice:  from Figure 9a it is clear that the influencer i = 0 can only reach and never exceed the value of the  normalized popularity of its “opposing” influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In Figure 9b we assign a higher consistency c(1) for the  competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The discussion developed above applies, and for values of c(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, the influencer with the  lower consistency always has an advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Indeed, we find that for consistency values c(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8 = c(1)  (and c(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2) influencer i = 0 achieves higher values of ¯πi, see Figure 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In summary, the results of this section suggest that a given influencer can gain an advantage over its  4The point c(0) = 0 corresponds to a rather peculiar situation where the influencer i = 0 does not post on its reference  direction, but only on the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Then the filtering depends on the initial configuration of the users in this direction,  where no dynamics occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It can be concluded that the scenario is fairly balanced, with the influencer i = 1 having a slight  advantage since the filtering occurs in the direction where the dynamics take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  19  competitors if it has a lower consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This observation also reflects the natural tendency of people  to seek varied content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Opinion configuration considering combinations of reference directions  In previous sections, we have focused primarily on the influencer perspective, looking at normalized  popularity values ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Here we present possible final opinion configurations in scenarios in which the reference  directions of influencers (r(0), r(1)) are either coincident or different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It was observed above that when an  influencer has a structural advantage, it achieves a higher πi and, in turn, can exert a higher attracting force  to the regular users towards its opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We then argue that it is interesting to examine what can happen  in a symmetric scenario in terms of frequency of publication (f (0) = f (1)) and consistency (c(0) = c(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' As  before, the influencers hold opinions x(0) = (0, 0) and x(1) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We consider the two possible cases where  the two opinion leaders have either the same or different reference directions and the impact of algorithmic  personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  This symmetrical scenario is interesting because, in most cases, it guarantees the coexistence of both  influencers (neither of them ‘wins’), see the normalized popularity plots in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore, the final  opinion distribution is the result of the joint influence of both agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In Figure 10a, we observe only a negligible perturbation with respect to the initial distribution shown in  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this case, the platform practically does not filter the content, so every post reaches all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  From a regular user perspective, individuals are exposed to nearly identical forces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', “opposite” stimuli  from the two influencers, which almost perfectly cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In Figure 10b, the impact of strong  personalization is clear: the filtering effect introduced by the platform leads to the emergence of two echo  chambers, whose membership is determined mainly by user’s prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Each user reaches an equilibrium  point at which the resultant attraction induced by the two influencers is balanced by the attraction exerted  by its own prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Interestingly, users also tend to cluster in the non-reference direction (x1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 10b)  and align their opinion with that of the influencer associated with the echo chamber they end up in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We  remark that this is a metastable condition, as the πi diagram indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' By extending the time horizon,  we may observe a different final situation in which one of the two influencers “wins” (exhibiting behavior  similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 10d, but just taking place at a different time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=')  Figures 10c and 10d refer to the case of different reference directions: the two influencers do not compete  on the same topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In Figure 10c, it is clear that there is no competition as the two influencers are able to  attract users to their reference opinion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', x0 = 0 the reference opinion of i = 0 and x1 = 0 that of i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  It constitutes a particularly relevant case, whose occurrence is linked indissolubly to the newly introduced  concept of reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In the last scenario, shown in Figure 10d, the influencer i = 1 “wins ”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=',  ¯π1 → 1, which brings public opinion closer to their belief on both issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The users’ opinion does not overlap  with that of the winning influencer because they are anchored by their prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this case, an unstable  behavior is observed since the identity of the winner influencer (as expected, as a result of perfect symmetry)  depends on random factors, and different sample-paths lead to diverse winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It should also be noted that  sharp personalization leads to a situation where the public scene is monopolized by only one individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Behaviour as function of the updating weights  The behavior of the system depends not only on the characteristics of the influencers and the composition  of public opinion, but also on the parameters controlling the opinion update rule in equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The update  is a convex combination of the prejudice, the current opinion, and the opinion conveyed by the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We  chose to hold fixed the weight β associated with the current opinion and consider the ratio of the other two  weights α  γ , which we termed degree of stubbornness, as it gives an indication of the extent to which users  change their opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We considered an unbalanced scenario in which influencer i = 1 has a structural advantage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=',  f (1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7 > f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Figure 11 again shows that personalization favors the structurally advantaged indi-  vidual (consistent with section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that the x-scale is logarithmic to highlight the sudden drop of ¯π0  for α  γ ≈ 10−3 (corresponding to modest values of α) when sharp personalization is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The shape of the  two curves is quite similar, only the decrease is observed at different values of α  γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Smooth personalization  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='0  i  0  1  (d)  Figure 10: (10a,10b) Influencers with the same reference direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', r(0) = r(1) = 0, while (10c,10d) influencers have  r(0) = 0 and r(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In both cases, two different degrees of personalization are considered: smooth (left column, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0001)  and sharp (right column, ρ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In all cases, the influencers have consistency c(0) = c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The distributions were  obtained as the time average of the opinion distribution in one realization of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The normalized popularities of the  two influencers in the given realization are shown along with the distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' it is clear that the influencers coexist except in  10d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  21  allows the coexistence of influencers on the whole domain, while with sharp personalization, for a wide range  of parameters, influencer i = 1 “wins.”  In both cases, there is an initial phase (for low values of α  γ ) in which the two influencers coexist, and this  is followed by a drop of the normalized popularity of the disadvantaged influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This can be explained  by the fact that small values of α  γ imply that a negligible weight is given to the prejudice, and therefore  regular users concentrate around the two influencers’ opinions on their reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This can be easily  confirmed by looking at the final opinion configuration of users, who concentrate in the upper corners of the  opinion space (around [0, 1] and [1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This is because the influencer i = 1, whose opinion is x(1) = [1, 1]  is stronger than the other in terms of popularity and is able to pull users along its non-reference direction  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We remark that when users are very close in opinion to a particular influencer, it is difficult for  the other to persuade them, as the probability of this happening is proportional to the product ω · θ, both  of which are a function of opinion distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In these scenarios, the distance from the “further ” influencer  is dj ≈ 1, which drastically reduces the probability of reaching the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Thus, as long as  α  γ is small  enough, both influencers can build their user base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' These situations represent rather degenerate cases where  the population almost disregards their prejudice in favor of the opinion conveyed by the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It might be  interesting to consider users with varying degrees of “volatility” who are able to pull along the opinion of  their neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  As the degree of stubborness increases, so does the inertia of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' They are more entrenched in  their prejudice and therefore no longer concentrate in a small neighborhood of the influencer’s opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This  favors the structurally advantaged influencer, as the other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', i = 0) is unable to build its user base because  users do not get close enough to it (see Figure 11 for i = 1 and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0001 we have ¯π0 → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The subsequent  rise in ¯π0 depends on the fact that when α approaches the maximum value αmax = 1 − β, users give  importance only to their prejudice, and therefore they do not deviate too much from their initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  As a consequence, it can not be triggered the positive feedback between users’ opinion and influencers’  popularity that leads to the complete victory of one influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  10  4  10  2  10  0  degree of stubborness  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='30  0  Values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0001  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  Figure 11: Normalized popularity ¯π0 as a function of the degree of stubbornness α  γ , the points are obtained considering 50  realizations of the process and averaging over 100 discrete time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Again, two levels of algorithmic personalization are  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Analysis of the fluid limit  In this section, we compare predictions of the simplified fluid limit against simulation results of the full  stochastic model described by algorithm 1 (obtained through a Monte-Carlo approach) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We restrict ourselves  to a one-dimensional opinion space, as in section 5, and assume that all users share the same prejudice z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Again, we consider two “competing” influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' A similar analysis could be performed in scenarios with any  number of influencers at any point in the opinion space, but this would be computationally more challenging  since multiple stationary points may exist, each with its own attraction basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  22  First, we derive in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1 some preliminary analytical results for the case of two influencers, using  the results of the fluid limit introduced in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Then in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2 two extreme instances of the  model are solved in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3 is devoted to comparing the analytical results of the fluid model  with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Finally, we discuss the impact of content personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Two competing influencers  Let us specialize the equations presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4 for the mean opinion ¯x(z) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' (17)) and the  normalized popularities ¯πi (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that for Ni = 2, ¯π0 = 1 − ¯π1, so it is sufficient to study ¯π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  As for the mean user opinion ¯x(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' equation (17) allows us to write the asymptotic mean directly as a  function of ¯π1 and the opinions of the two influencers x(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' x(1):  ¯x(z) =  α  1 − β z +  γ  1 − β  �  (1 − ¯π1) x(0) + ¯π1x(1)�  (18)  Substituting the functional forms of the visibility ω and feedback θ into equation (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' we obtain the  following expression for the normalized popularity ¯π1:  ¯π1 =  f (1)e−ρ(x(1)−¯x)  2  ¯π1  �  1 − |x(1) − ¯x|  �  �  i∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1} f (i)e−ρ (xi−¯x)2  ¯π1  �  1 − |x(i) − ¯x|  � = f(¯π1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' ¯x)  (19)  Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' if we combine the above expression with equation (18) for ¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' we get ¯π1 = f(¯π1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' which can  be solved numerically through a fixed-point approximation (FPA) (a graphical representation is shown on  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The outcome of this FPA and the corresponding simulation results are compared in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Closed form computations in extremal cases  The combination of equations (18) and (19) cannot be solved in closed form in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However,  there are at least two scenarios in which this is possible, separately considered in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' When an influencer “wins”  We consider an influencer a “winner” if its normalized popularity ¯πi approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Suppose that the  influencer whose opinion is x(1) = 1 wins, then ¯π1 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This implies ¯π0 → 0 and thus ω → 0+: the influencer  with x(0) = 0 is seen by a negligible fraction of users and in practice, only influencer i = 1 remains visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Note that in the extreme case in which influencer 1 wins, users see only x(1), and asymptotically all users  move towards it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this case, the final opinion ¯x(z) can be easily calculated with a recursion of the update  rule (3):  x(u)(n) =  n  �  i=0  βi �  αz + γx(1)�  + βnx(0)  For n → ∞ and considering β < 1 (the case β = 1 coincides with the trivial case where users remain  fixed at their initial opinion) we get:  x(w) =  α  1 − β z +  γ  1 − β x(1),  (20)  which is in agreement with (18) if one sets ¯π1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This corresponds to one of the extreme cases that we  will use later to examine the model behavior as a function of the personalization parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It should be  noted that this construction relies on the knowledge of the winning influencer, which is unknown in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  However in the fluid limit, we expect that the winning influencer, if any, is the one that has a structural  advantage over the others at the beginning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', a higher posting rate f (i), see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  23  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Constant personalization function  The other extreme case we consider is the one in which ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this case, the personalization function  ω no longer depends on ¯πi, and it is easy to see from Table 6a that it returns ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, we consider  x(1) = 1, x(0) = 0, which further simplifies (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The above formulas (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 19 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 18) can then be solved  in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In particular, equation (19) for the normalized popularity ¯π1 becomes:  ¯π1 =  f (1) (q + m ¯π1)  f (0) (1 − (q + m ¯π1)) + f (1) (q + m ¯π1)  where m ≜  γ  1−β and q ≜  α  1−β z for compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This leads to a second order equation which can be easily  solved for ¯π1:  ¯π2  1 m(f (1) − f (0)) + ¯π1  �  f (0)(1 − q) + f (1)(q − m)  �  − f (1)q = 0  (21)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Comparison between analytical prediction and Monte Carlo simulations  This section is devoted to comparing the analytical results derived in section 5 with simulations of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Numerical and graphical solutions of equation (19) are also provided, shedding light on the impact  of the algorithmic personalization performed by the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Description of the scenario  The scenario setting is analogous to that described in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1 and Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='However, here, we consider  a one-dimensional opinion space [0, 1] and we assume all users to have the same prejudice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', z(u) =  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4, ∀u ∈ U matching their initial opinion x(u)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The “competing ” influencers have opinions at  the extremes of the domain, and their posting frequencies are f (1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7 and f (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', influencer  i = 1 has a structural advantage over influencer i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that in a one-dimensional space, the reference  direction r(i), and hence the consistency c(i), lose their significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' To avoid obtaining trivial results in  which influencer 1 obviously wins, regular users are initially placed closer to the disadvantaged influencer  i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Simulation, fluid limit and fixed-point approximation  Comprehensive validation and comparison of the approaches used to obtain the system equilibria are  shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' First, the stochastic model described by Algorithm 1 is “simulated” by obtaining 100  different sample whose length is 500000 elementary steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The variables of interest ¯x(z) and ¯π1 are obtained  by averaging the process over both discrete times steps n and sample paths and are represented by circle  marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Second, equation (18), which is a specialization of (17) obtained from the fluid limit, indicates that  the state of the system lies on a line in the plane ¯π1,¯x (dashed line in Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Third, the extreme cases of  the model analyzed in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2, for which we derived a closed-form solution, are represented by star-like  marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Lastly, diamonds are solutions of (19) employing the fixed-point approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We observe that, for given ρ, simulation marks match well with analytical marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The only exception  is for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, for which simulations provide ¯π1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='79, whereas the analysis provides ¯π1 ≈ 1 (see also  the table on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This mismatch is due to the fact that ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5 is close to a ‘phase transition’, at  which the system switches from a regime in which two stable solutions exist (in particular, one in which  both influencers survive) to a regime in which influencer i = 1 wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This behavior is better illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 13, where the curve corresponding to ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5 is almost tangent to the bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It should be noted that  the “empty” region in Figure 12 is directly related to this behavior since no stable solutions can exist for  that values of ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In fact, there is no stable intersection with the bisector in Figure 13 in the corresponding  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Implications of algorithmic personalization  We summarize here the insights into algorithmic personalization suggested by the emergent behavior  of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We already mentioned how content filtering favors influencers with a structural advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='00  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='56  x  simulation  fixed-point  1  z + 1  1  closed-form  = 0  closed-form  1  1  10  3  10  2  10  1  10  0  in simulation  Figure 12: Comparison between analytical results, including the exact extreme points calculated in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1, and the  linear relationship between ¯x and ¯π1 according to Equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Diamonds represent the fixed-point approximation for the  solution of equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Simulation results of the stochastic dynamics, represented by circles, were obtained by averaging  100 realizations of the process as described by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We consider a scenario in which α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='05, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='93, with two  influencers at the extremes of the domain, with f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3, f(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7 and the same initial absolute popularity p0 = p1 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Numerical values from simulation and fixed-point approximation are reported in the table alongside the plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Table 2: Simulation and FPA  ρ  ¯π1  ¯π1  SIM  FPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  f(  1)  = 0  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='001  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='01  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  1  Figure 13: Graphical solution of ¯πi = f(¯πi), (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Stable solutions corresponds to intercepts between f(¯πi) and the bisector,  such that f′(¯πi) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We observe that non-trivial solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', solutions in which both influencers survive) exist, roughly in  the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8], provided that ρ is not too large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, the only stable solution is ¯π = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This explains  the results in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Simulation results reported on the alongside table confirm the validity of the analytical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  For instance, in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2, we showed that personalization promotes the influencer with higher posting  frequency, and, in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3, the one with lower consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In addition, in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2, we presented a case  25  where a ‘phase transition’ is observed as a function of the filtering strength (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Indeed, after a certain  threshold, the favored influencer (say influencer 1) is the only one that survives (¯π0 → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In such a situation,  the population is exposed to the opinions of a single individual, hindering diversity on the social platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  It is also interesting to discuss the results in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4 concerning the effects of personalization on the  opinion distribution of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The two possible outcomes when sharp personalization is applied are:  either the emergence of two echo chambers with users holding more radical positions in both directions or  the onset of an unstable situation in which the two influencers coexist for a limited amount of time, after  which ¯πi → 1 for an i dependent on the specific sample path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Online social network data  This section examines data collected from Facebook and Instagram social networks and compares the  observed behavior with some of the findings of our Communication Asymmetry model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Correlation between frequency of publication and popularity  In previous sections, especially in 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3, we discussed structural advantage from the influencer’s  point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' One of the key advantage parameters, as observed across all experiments, is the publication  frequency f (i): the higher f (i), the greater the advantage (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this section, we attempt to  validate this finding by correlating the frequency of publication of influencers with their popularity growth,  using the total number of followers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the number of people subscribed to the profile, as a proxy for  popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We consider temporal sequences from Instagram on a sample set of 110 influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  Pearson correlation coefficient  0  5  10  15  20  25  Number of influencers  Figure 14: Distribution of the correlation coefficient between monthly number of posts and popularity growth (in terms of  number of followers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  For each influencer, we considered a temporal granularity of one month, determined the number of posts  during this period, and calculated the relative change in the number of followers considering the values at  the beginning and end of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Then for each user, we calculated the Pearson correlation coefficient  between the number of posts and the relative variation of followers in the month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In Figure 14, we show  the distribution of these correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Results suggest that there exists, in general, a positive  correlation between the two quantities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', influencers with aggressive posting habits tend (but not always)  to get more followers, which likely favors them when in competition with other influencers on social media  platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This is consistent with the model predictions shown in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Case Study: Italian government crisis in August 2019  In June 2018, a few months after the general elections, Giuseppe Conte was appointed Italian Prime  Minister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Two parties formed his supporting coalition: Movimento 5 Stelle (his own party, holding the  26  relative majority of the Italian Parliament) and Lega, whose leader was Matteo Salvini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In August 2019,  Salvini decided to withdraw Lega’s support to the government, starting a crisis aimed at driving Italians  to new elections and gaining more votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, Movimento 5 Stelle managed to reach an agreement  with various parties to form a new government, and on September 5, 2019, Giuseppe Conte became Prime  Minister for the second time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The coalition that supported this new administration clearly excluded Lega.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In this section, we apply the proposed model to reproduce the sudden rise of Giuseppe Conte’s popularity  in social networks during the government crisis in August 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We exploit the multidimensional capability  of the model considering two directions: Politics, reference topic for Salvini and Conte, and attitude toward  government fall (End government, see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  (a) Initial distribution  (b) After the transient  Figure 15: Initial distribution density of the population along the Politics direction and End government direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The opinion  position of the two leaders in the space is depicted with a green (Salvini) and a yellow (Conte) point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In the opinion space, we assume Salvini has a more radical political viewpoint, while Conte has an  opposing and more moderate position (described somehow arbitrarily by putting Salvini at xP olitics = 0,  Conte at xP olitics = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We assume that the population has a moderate initial opinion (centered at  xP olitics = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5, see Figure 15a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Regarding the attitude toward government fall, the two politicians obviously  have a completely different opinion (Salvini has xEndgovernment = 1 while Conte has xEndgovernment = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We  assume that the population is strongly polarized towards Conte’s opinion in this latter direction (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  This is an a posteriori assumption made knowing the outcome of the social confrontation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  A period of eleven weeks is considered, from July 7 to September 22, during which data was collected  weekly from Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' A total of 1162 posts were published, of which 125 were by Conte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The rate f (i) is  calculated as the number of posts by an influencer relative to the total number of posts (f (Conte) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='108,  f (Salvini) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='892).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Some simplifying assumptions are necessary to apply the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We assume that the two politicians  have a consistency c(i) of exactly one (real values are often close to this value, see Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover,  Giuseppe Conte and Matteo Salvini are the only influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Although this hypothesis is restrictive, in the  scenario studied, the two influencers were the main (active and popular) protagonists during the government  crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, we consider the simplest scenario in which personalization is not employed: ρ = 0 and thus  ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We consider a feedback function of the form θ = e−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='25(x(u)−x(i))2 for both opinion directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For an  exhaustive list of the parameters, we refer the reader to Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  27  transient  phase  7th July  22nd Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  switch  posts along  politics  government crisis  28th July  1st Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  switch  posts along  politics  posts along  end government  0  Figure 16: Timeline of modelled scenario from July 7, to September 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' From July 28 to September 1 we have a consistency  switch, with posts along End government direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Figure 16 shows the timeline of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The two influencers start with the same initial popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We consider a transient of Nt = 10000 discrete time-units, after which the stationary normalized popularities  πi roughly correspond to the empirical normalized popularities obtained by dividing the number of followers  of each influencer by the total number of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  After the transient, we can see in Figure 15b that  the distribution of public opinion is skewed towards Salvini, who, in turn, has a higher popularity ratio  due to his higher publication frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' After the transient, the crisis happens and both influencers start  posting in the Endgoverment direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', we observe a consistency shift for both influencers), during a  time window of five weeks that approximates the duration of the government crisis, after which the two  politicians switch back to posting on the Politics direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that the initial users’ opinion distribution  along the Endgoverment axes is concentrated around Conte’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Even with these limitations, it is still possible to reproduce the observed social behavior as a whole: it  corresponds to a situation where an influencer is in stark contrast to the opinions of its user base and loses  ground with respect to the other influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that, in the model, only a very unbalanced distribution  of the population towards Conte’s opinion (against the government fall) can explain the sudden increase  in Conte’s popularity, despite the remarkable differences in popularity ratios in favor of Salvini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Figure 17  compares the simulation results of the described setting and Facebook’s measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note how the model  can explain the sudden rise in Giuseppe Conte’s popularity, precisely in the weeks of the government crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Clearly, our model does not exactly fit empirical observations but simply provides qualitative insights  into the possible causes of the rather sudden popularity shift that was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Many of the model’s  parameters are unknown, such as the opinion distribution, the weights of the updating rule, or the feedback  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, by making reasonable assumptions about some of the parameters, one can obtain a  reasonably good fit, and exploit the explanatory capability of the model to acquire better confidence in the  hidden mechanisms beneath observed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this experiment, we followed exactly this approach and  we looked for some mechanisms that could justify the same sudden surge in popularity that occurred during  the government crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  As the main outcome of our analysis, we conclude that the observed popularity trends of the two  considered influencers can be largely explained by considering the fear of political instability in the user  base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Conclusions  In recent times, online social interactions appear essential to human relationships and play an increasingly  important role in opinion formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' To understand the mechanisms underlying this novel communication  paradigm, it is of utmost importance to develop flexible frameworks suitable for describing interactions on  social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this work, we have developed an opinion model tailored to online interactions, with  28  07 Jul  14 Jul  21 Jul  28 Jul  04 Aug  11 Aug  18 Aug  25 Aug  01 Sep  08 Sep  15 Sep  22 Sep  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='215  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='220  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='225  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='230  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='235  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='240  Conte  Model outcome  Facebook data  Figure 17: The popularity ratio πConte for Conte, the one obtained from Facebbok data and the one from the model along with  its 95% confidence interval, computed over 10 realizations of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It can be seen how the model follows the increase in  popularity during August 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Table 3: Parameters and functions for the Case Study  Symbol  Value - Form  Description  Ni  2  Number of influencers  x(Conte)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='76  Opinion of Giuseppe Conte on direction j  x(Salvini)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  Opinion of influencer 1 on direction j  f(Conte)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='108  Opinion of Giuseppe Conte on direction j  f(Salvini)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='892  Opinion of influencer 1 on direction j  r(Conte),(Salvini)  0  Refrence direction of both influencers  pConte,Salvini(0)  20  Initial absolute popularity of both influencers  Nu  10000  Number of regular users  Niter  15000  Number of iterations for each simulation  Nt  10000  Duration of the transient phase  w  550  Length of the government crisis  α  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='3  First weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 3  β  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='65  Second weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 3  θ(·)  e−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='25(x(u)−x(i))2  Functional form of the feedback function  ω(·)  ρ = 0 =⇒ ω ≡ 1  Functional form of the visibility function  particular attention to distinguishing between two classes of users, namely regular users and influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We characterized the influencers by introducing the concept of reference direction, which links unrelated  topics discussed by the same influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Measurements collected from real online social networks support  our modeling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Similarly to other works in the recent literature, we integrated algorithmic  personalization in a flexible and tunable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We have shown how content filtering reinforces inequality  by favoring the structurally advantaged influencer and, in most cases, preventing the “competing” influencer  from remaining visible to the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover, even in structurally balanced conditions, personalization  can lead to the emergence of echo-chambers, in which users’ opinion also radicalizes along non-reference  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The proposed model is a preliminary attempt to describe the complexity of online interactions and  comes with some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In our model, users are passive entities, and influencers are stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Moreover, homophily is the only driver of individuals’ interaction, as no other relationship structure was  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Nonetheless, despite the simplifying assumptions, the emergent behavior of the model proved  29  rich enough to reveal the effects of content personalization and shed light on influencer popularity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Our work points to several research directions, such as viewing users as active agents capable of publishing  their own posts and forwarding (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', sharing) posts from influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This can pose significant challenges  in terms of analytic tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Another promising direction could be to look at influencers as “strategic”  players aiming at maximizing their popularity on the platform by exploiting the internal mechanisms of the  platform itself (such as algorithmic content filtering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Declaration of Competing Interest  The authors declare that they have no known competing financial interests or personal relationships that  could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Description of the dataset  We collected data from real online social networks to support the hypotheses of our model and compare  emergent behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We focus on two popular social networks: Facebook (FB) and Instagram (IG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Facebook  has long been the most popular social media application, while Instagram has undergone a surge in popularity  in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In Facebook and Instagram, a profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', a social network user, can be followed by other profiles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=',  its followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' A profile with a large number of followers is also called an influencer - we consider profiles  with more than ten thousand followers as influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Influencers post content (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', posts) consisting of a  photo, a video, plain text, or a combination of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The profile’s followers and anyone registered on the  platform can see, like and comment on the influencer’s posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Note that when we use the term influencer,  we do not only mean individuals but also groups, soccer teams, newspapers, or companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In this work, we are interested in the plain-text messages of influencers, their temporal sequence,  and metadata describing the features of the influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  To get the list of such popular profiles, we  exploited the online analytics platform hypeauditor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='com for IG, and www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='socialbakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='com and www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  pubblicodelirio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='it for FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We restricted the analysis to influencers with at least 10, 000 followers on  June 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The lists of 649 influencers we used are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  For each monitored profile,  we downloaded the corresponding metadata, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', the profile information and all generated posts, using the  CrowdTangle tool and its API6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' CrowdTangle is a content discovery and social analytics tool owned by Meta  and available to researchers and analysts worldwide to support research, subject to a partnership agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  For each influencer, we downloaded all the data related to the posts published between January 1, 2016,  and June 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Finally, we stored the data, which takes around 110 GB of disk space, on a Hadoop-based  cluster, and we used PySpark for scalable processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Details on post classification  One of the novelties introduced in this work is the concept of  reference direction, which states that  influencers have a preferred topic of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' To confirm this hypothesis, we developed a classifier that  can categorize posts according to a particular set of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' First, we arbitrarily identified a subset of topics  that sufficiently characterize the discussions on the monitored profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Specifically, these topics are sports,  politics, food and cooking, music, and pandemics, which are intentionally loose and relatively uncorrelated  to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We developed a keyword classifier to classify the posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For each topic, we manually defined  a list of representative keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For example, if we consider pandemic, we search for words like COVID,  pandemic, and coronavirus in Italian (and commonly used terms in other languages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We search for the  topic-specific terms in the text corpus of the post, and if we find a match, we mark the post as belonging  to the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Notice that since keywords of various topics may be present in the same corpus, we can flag a  5https://mplanestore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='it:5001/sharing/P4WnRClQn  6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='com/CrowdTangle/API  30  message as discussing multiple topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this work, we discard posts marked as multiple and only consider  posts associated with a single topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We are not interested in classifying all posts by an influencer, first because our list of topics does not  cover all possible ones, and second because we only need a large enough subsample of posts to make some  statistical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Conversely, it is of utmost importance that the accuracy of the classifier is high  since misclassified posts could lead to wrong conclusions about the distribution among the available topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Therefore, we manually validate the accuracy of our methodology for topic detection, as described in the  following paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Classifier Precision Evaluation  We empirically evaluated the accuracy of the classifier by taking a random subsample of the labeled  posts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=', 100 posts for each topic for a total of 500 messages, and manually classifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' To this  end, we defined a lower and upper bound for accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Indeed, even for a human being, it is challenging  to univocally classify posts based on their content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore, we defined three possible states for each  classification decision: “t” correct classification, “f” incorrect classification, and “ncc” standing for not  completely correct (indicating that the assigned topic is related to the post but may not be the main topic  of the post or the classification of the post is difficult).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Given this states subdivision, the precision bounds  are as follows:  PL =  Nt  Nt + Nf + Nncc  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1)  PU =  Nt + Nncc  Nt + Nf + Nncc  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2)  We refined our term selection for each topic to improve precision based on this analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7 The classifier’s  precision is subject-dependent but was consistently above 80% considering the upper bound defined in  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The classification is particularly effective in the case of politics and pandemic, where the precision  goes above 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1 summarises the bounds on precision achieved by the procedure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  These results are sufficient to use the classification to support our modelling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1: Per-topic Precision  Topic  Precision l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Precision u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Sports  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='9  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2  Politics  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='0  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  Music  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  Food  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='5  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='4  Pandemic  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='6  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1  The average percentage of messages classified is 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='8% for all influencers in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Considering  the final classifier and the analysed dataset, we automatically flagged about one million posts8 with at least  one topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Of these, only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='7% of the posts were flagged with multiple labels, indicating the message dealt  with more than one topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' We decided to consider in the rest of the work only influencers for whom it was  possible to classify more than a thousand posts in the observed period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' At the end of this filtering process,  we could keep 237 influencers for whom the average posts’ classification percentage is 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  The dataset used contains a subset of Italian politicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  To check the correctness of the labelling  procedure, we checked whether the derived reference topic for all politicians was politics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' It turned out that  7We make the final list of terms available at https://mplanestore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='it:5001/sharing/0wD5oU6xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  81167963 posts were tagged with at least one label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  31  two politicians did not have politics as reference: Vincenzo De Luca had pandemic, and Renata Briano had  food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' However, this is entirely understandable as the latter runs a food blog and the former was known for  his firm and frequent statements on the pandemic situation during the COVID -19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Proofs of Theorems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2)  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Proof of Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1)  Let us start assuming ki(0) > 0 ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  In such a case we denote with i0 = arg maxi ki(1) and with  K := ki0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' First we show that the problem:  ki(yi) − cyi = 0,  with yi ∈ [0, 1] ∀i  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1)  admits a solution for any c ≥ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Indeed by choosing c ≥ K we have that necessarily ki(1) ≤ ki0(1) ≤ c·1 ∀i  while ki(0) > c · 0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' therefore a zero zi(c) must exist for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' This zero is unique as a consequence  of the concavity of ki(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' The set of zeros zi(c)i provides a solution of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Now to get a solution of the  original problem (12) we need to show that there exist a c such that {zi(c)}i are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Observe that  for c = K by construction zi0(K) = 1 while 0 < zi(K) ≤ 1 for i ̸= i0, therefore �  i zi(K) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Now, due  to the monotonicity and concavity of ki(·), zi(c) is by construction decreasing with respect to c, moreover  zi(c) → 0 as c → ∞ ∀i, therefore since �  i zi(·) is a continuous function of its argument, there will necessarily  be a c0 in correspondence of which �  i zi(c0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In the case in which ki(0) = 0, observe that 0 is a solution  of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1) for any c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' zi(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Moreover for any c ≥ K a second zero may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' For example, by  construction, zi0(K) = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Therefore for c = K, as before, we can always choose as set of zeros {zi(K)}i,  such that zi(K) = 0 if ki(0) = 0, and i ̸= i0, zi0(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' By construction �  i zi(K) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In particular  �  i zi(K) > 1 is there exists a i such that ki(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' In this latter case, by increasing c all the non null zeros  decrease, therefore, as before, there will necessarily be a c0 in correspondence of which �  i zi(c0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  □  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Proof of Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2)  We first show that ||¯π(1) − ¯π(2)||L∞ = maxi |¯π(1)  i  − ¯π(2)  i  | = ||G(F1(x, z)) − G(F2(x, z))||L∞ ≤ M||F1(x) −  F2(x)||L∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' then we show that we can always enforce: ||F1(x, z) − F2(x, z)||L∞ = ||H(¯π(1)) − H(¯π(2))|| ≤  1/(2M)||¯π(1) − ¯π(2)||L∞ by properly choosing ω(·, ·) and θ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Therefore, we can conclude that ||H ◦  G(F1(x, z)) − H ◦ G(F2(x, z))|| ≤ 1/(2M)||(G(F1(x)) − (G(F2(x))|| ≤ M/(2M)||F1(x, z) − F2(x, z)|| =  1/2||F1(x, z) − F2(x, z)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  First note that ||F1(x, z) − F2(x, z)||L∞ = supx |F1(x, z) − F2(x, z)| coincides with the Kolmogorov  distance between the two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Let us denote with  ki(y, F1(x, z)) = λf (i)  � �  θ(|x − xi|)ρ(¯πi, |x − xi|)dF1(x, z),  and similarly for ki(y, F2(x, z)) we assume that:  sup  y∈[0,1],i  |ki(y, F1(x, z)) − ki(y, F2(x, z))| := ∆K(F1, F2) ≤ a||F1(x, z) − F2(x, z)||L∞  a ∈ R+  and  dki(y, F1(x, z))  dy  |y=0< max  i  ki(1, F1(x, z))  dki(y, F2(x, z))  dy  |y=0< max  i  ki(1, F2(x, z))  ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Without lack of generality we assume maxi ki(1, F1(x)) ≥ maxi ki(1, F2(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Let the pair (¯π(1) = {¯π(1)  i  }i, c1)  be the solution of  ki(yi, F1(x, z)) − cyi = 0  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='t  �  i  yi = 1, yi ≥ 0, ∀i  32  now let ({�p(2)  i }i) the non necessarily normalized solution of  ki(yi, F2(x, z)) − c1yi = 0  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='t yi ≥ 0, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  by means of elementary geometric considerations we can bound:  |¯π(1)  i  − �p(2)  i | ≤ ∆K(F1, F2)  c1 − h1  where h1 =  dki(y,F1(x))  dy  |y=min(¯π(2)  i  ,�p(2)  i  )≤  dki(y,F1(x))  dy  |y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  We recall that by construction (see proof of  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1) we have c1 > maxi ki(1, F1(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Denoting with |�p(2)| = �  i �p(2)  i  , we have  1 −  �  i  |�p(2)  i  − ¯π(1)  i  | ≤ |�p(2)| ≤ 1 +  �  i  |�p(2)  i  − ¯π(1)  i  |  Now denoted with ({¯π(2)  i  }i, c2) the solution of  ki(yi, F2(x, z)) − cyi = 0  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='t  �  i  yi = 1, yi ≥ 0, ∀i  we have, by construction, that:  1  max(1, |�p(2)|) < c1  c2  <  1  min(1, |�p(2)|)  and therefore, exploiting again elementary geometrical arguments, we can bound:  |�p(2)  i  − ¯π(2)  i  | ≤  ����  �c1 − h2  c2 − h2  − 1  �  �p(2)  i  ����  where h2 = dki(y,F2(x,z))  dy  |y=min(�p(2)  i  ,¯π(2)  i  )= dki(y,F2(x,z))  dy  |y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Putting everything together, we have proved  that:  max  i  |||¯π(1)  i  − ¯π(2)  i  || = ||G(F1(x, z)) − G(F2(x, z))||L∞ ≤ M||F1(x, z) − F2(x, z)||L∞  To conclude the proof, first note that by properly choosing ρ(·, ·) and θ(·) we can assume vx(x, z) and σ2  x(x, z)  to depend sufficiently smoothly on ¯π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' ∀ε > 0 we can assume:  sup  x  ���  ���v(1)  x (x, z) − v(2)  x (x, z)  ���  ���  L∞  ≤ ε||¯π(1) − ¯π(2)||L∞  ∀z,  sup  x  ���  ���σ2,(1)  x  (x, z) − σ2,(2)  x  (x, z)  ���  ���  L∞ ≤ ε||¯π(1) − ¯π(2)||L∞  ∀z,  and  sup  x  �����  �����  ∂σ2,(1)  x  (x, z)  ∂x  − ∂σ2,(2)  x  (x, z)  ∂x  �����  �����  L∞  ≤ ε||¯π(1) − ¯π(2)||L∞  ∀z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  Then observe that the solution of the Fokker-Planck equation given in (9) on a compact interval (and so  also its primitive) depends smoothly on function vx(x, z), function σ2  x(x, z) and its first derivative, as long as  infx,z σ2  x(x, z) is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' As final remark note that the set of weakly-increasing functions  F(x), such that F(a) = 0 and F(b) = 1 equipped with the L∞-norm forms a closed set in a complete metric  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' □  33  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Cinelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Morales, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Galeazzi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Quattrociocchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Starnini, The echo chamber effect on social media,  Proceedings of the National Academy of Sciences 118 (9) (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2023301118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='2023301118  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Fortunato, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Castellano, Scaling and universality in proportional elections (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='PHYSICS/  0612140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content=' Risken, Fokker-planck equation, in: The Fokker-Planck Equation, Springer, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' 63–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content=' Cohen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+page_content=' Zhang, Analyzing opinion dynamics in online social networks, Big Data &  Information Analytics 1 (4) (2016) 279–298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
+page_content='  35' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'}
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+Detecting Pump&Dump Stock Market Manipulation from Online
+Forums
+D. Nam
+D.B. Skillicorn
+School of Computing
+Queen’s University
+Kingston. Canada
+skill@queensu.ca
+Abstract
+The intersection of social media, low-cost trading platforms, and naive investors has created an
+ideal situation for information-based market manipulations, especially pump&dumps. Manipulators
+accumulate small-cap stocks, disseminate false information on social media to inflate their price,
+and sell at the peak.
+We collect a dataset of stocks whose price and volume profiles have the
+characteristic shape of a pump&dump, and social media posts for those same stocks that match the
+timing of the initial price rises. From these we build predictive models for pump&dump events based
+on the language used in the social media posts.
+There are multiple difficulties: not every post will cause the intended market reaction, some
+pump&dump events may be triggered by posts in other forums, and there may be accidental con-
+fluences of post timing and market movements. Nevertheless, our best model achieves a prediction
+accuracy of 85% and an F1-score of 62%. Such a tool can provide early warning to investors and
+regulators that a pump&dump may be underway.
+1
+Introduction
+New financial products and technologies have allowed naive investors to easily enter financial mar-
+kets.
+This has increased the risk of manipulation, and detecting and investigating fraudulent
+activities has become much more difficult. Many go undetected [8].
+Social media has created new methods for manipulating markets. A scheme known as Pump
+and Dump (P&D) is one popular mechanism. Fraudsters buy quantities of a stock, disseminate
+false information about it to artificially raise its price, and then sell their purchased shares at the
+higher price. Social media provides a channel for rapid dissemination and a pool of investors with
+little knowledge or experience who may not detect that the information is false.
+Conventional approaches to detecting manipulation look for known patterns, and for anomalous
+activity such as exceeded thresholds for prices and trading volumes. Suspicious activities can be
+detected using sets of rules and triggers that cause notifications of potential manipulation. However,
+those methods struggle in the presence of behaviours that deviate from historical patterns [16].
+Previous work has also focused on detecting manipulations so that regulators can penalise those
+who carry them out. This does little to help investors, either to prevent their being deceived or
+recovering their investments.
+1
+arXiv:2301.11403v1  [cs.SI]  26 Jan 2023
+
+Data-analytic techniques have the potential to detect false information as it being disseminated
+[11, 25]. Natural language analytics can detect the posts in social media that are intended to pump
+particular stocks, providing a real-time warning to potential investors. We investigate how well
+P&D schemes can be detected in posts on social media, by matching the language patterns in the
+posts to the pattern of stock price corresponding to a P&D manipulation.
+A penny stock is a stock that is traded by a small public company for less than $5 per share
+[24].
+Many of these companies are known for their volatility due to their limited coverage by
+analysts and interest from institutional buyers. Because of their low price, retail investors can buy
+large quantities of these stocks without having to invest much money. This, however, makes their
+prices volatile and so creates the potential for large returns on investments; but also leaves them
+vulnerable to manipulation by malicious actors. One study found that 50% of manipulated stocks
+are those with a small market capitalization [1].
+It might be supposed that the connection between a social media post and a P&D event is too
+tenuous to be detected – after all, not every post will have the desired effect, and a P&D might be
+triggered by some less visible social media activity. We show that, at least for penny stocks, the
+connection is reasonably detectable, and we achieve prediction accuracies (that a post is intended
+to cause a P&D event) of 85%, with an F1 score of 67% (± 12 percentage points) from posts alone,
+and 62% (± 3 percentage points) from posts and comments.
+2
+Tools
+Stance detection is a technique to determine the attitude or viewpoint of a text towards a target.
+It aims to detect whether the author of the text is in support of or against a given entity [21].
+Some applications of stance detection have been in political debates, fake news, and social media
+[15, 26, 30].
+Empath is a tool that was developed by Fast et al. [13] for researchers to generate and validate
+new lexical categories on demand. It uses deep learning to establish connections between words
+and phrases used in modern fiction. Given a small set of seed words that represents a category,
+Empath can provide new related terms using its neural embeddings. It also employs the use of
+crowd-sourcing to validate the terms that it considers are related. Along with the ability to create
+new categories, Empath comes with 200 built-in, pre-validated categories for common topics (e.g.,
+neglect, government, social media).
+SHAP (SHapley Additive exPlanation) is a tool that was developed by Lundberg and Lee [22]
+to determine the impact of each attribute on the output of a predictive model. It is based on
+Shapley values, a concept from game theory that determines a fair way to distribute the payoff for
+players that have worked in coalition towards an outcome [33].
+Extreme Gradient Boosting is a decision-tree based ensemble algorithm that has become known
+for its speed and performance [5]. Decision trees are built sequentially so that each one reduces
+the errors of the previous one [35]. Random Forests is a decision-tree based ensemble algorithm
+with each tree built from a subset of the rows and columns of the dataset [34]. This allows for
+variation among the trees and results in lower correlation among their predictions [37]. Support
+Vector Machines are a supervised learning algorithm that finds a hyperplane that best separates
+the data points from two classes [14].
+Artificial Neural Networks are computational networks that are inspired by the biological ner-
+vous system [10]. ANNs excel at prediction for data where the amount of information in each
+2
+
+Figure 1: Stages of Pump and Dump
+attribute is small and there are non-linear interactions among them. Deep learning models are a
+class of extensions to ANNs that have solved long standing prediction problems in image recogni-
+tion and natural language [20]. Convolutional Neural Networks (CNNs) are a class of deep learning
+networks that were designed initially to work with images but work surprisingly well with sequence
+data such as texts as well. Long Short-Term Memory (LSTM) deep learning networks are a type of
+recurrent neural network designed to handle the long-term dependencies present in sequence pre-
+diction problems [4]. Understanding text often requires looking ahead (think of verbs in German)
+and so processing text in both directions, using a biLSTM, provides better results for language [6].
+3
+Experiments
+Within a typical online forum, there are two different categories of texts. The first is a post, which
+initiates a discussion.
+The second is a set of comments responding to the post.
+For example,
+an individual may post saying that, in their opinion, a stock’s price is about to rise, with others
+respond by sharing their opinions in the same thread. Responders may agree with the original post,
+or disagree.
+P&D is an information-based manipulation, artificially raising the price of a stock through the
+dissemination of false information. As shown in Figure 1, this manipulation strategy involves three
+different stages [19]. The operators of the scheme first purchase the stock that they are planning
+to manipulate (Accumulation). Once they have acquired enough shares, they will disseminate false
+information to make it appear more desirable, driving up the price (Pump). Once the price has
+risen to the desired level of profit, the operators sell off their shares before anyone uncovers that
+the information has no basis or the hype dies down (Dump).
+To identify P&Ds within the market, patterns associated with the scheme must be established.
+3
+
+Price
+Accumulation
+Pump
+Dump
+TimeWhile the method of conducting a P&D may vary, two indicators that can identify them are sharp
+changes in price and volume [19]. A P&D will cause a significant price increase within a short
+amount of time, larger than the fluctuations that the stock typically experiences; followed by a
+decrease once the dump phase has begun. The volume also increases as the stock gains interest
+among investors during and after the dissemination phase. However, the volume will typically not
+immediately experience as sharp a decline as the price when the operators begin to dump their
+shares because of the reluctance of investors to believe that the price is illusory.
+If the profile of a P&D manipulation can be detected in the market, then the post that putatively
+caused it can be straightforwardly labelled and its language patterns investigated. (Of course, it is
+possible that some of the apparent connections are spurious, but it is relatively unlikely that a post
+touting a particular stock will be disseminated exactly when the stock’s price and volume begin a
+sharp rise).
+Labelling comments is more complex, since the comments may agree with the original post, or
+disagree. Only the language of those that agree can contribute to predicting a P&D event.
+3.1
+Data Sources
+Two different data sources were utilized. The first is the popular online website Reddit, where users
+discuss the stock market. The second is Yahoo Finance, a financial market website that provides
+historical data about companies.
+Reddit contains forums referred to as subreddits, each dedicated to the discussion of a specific
+topic. Popular forums for the discussions of stocks are r/pennystocks, r/wallstreetbets, r/stocks,
+r/RobinHoodPennyStocks, r/TheWallStreet. We use r/pennystocks and r/RobinHoodPennyStocks,
+Yahoo Finance is a website provided by Yahoo for investors to access financial news, market
+data, and basic financial tools. Given a stock symbol or company name, it provides the relevant
+market data.
+Classification techniques such as Extreme Gradient Boosting (XGBoost), Random Forests, Sup-
+port Vector Machine (SVM), and Artificial Neural Networks (ANNs) were used to learn predictive
+models, and then to identify which attributes (i.e. words) are most predictive. Figure 2 shows the
+experimental workflow.
+Figure 2: Experiment workflow
+Data from Reddit and Yahoo Finance were collected daily for the period October 1, 2019, to
+June 28, 2020. A breakdown of the data is shown in Table 1. The majority of the data is retrieved
+4
+
+Redldit
+Yahoo!
+Finance
+Anomaly
+Detection
+Text
+Labelling
+Model
+Model
+Data
+Preprocessing
+Training
+Testing
+Historical Data
+Agreement
+Model
+Data Retriever
+Data Preparation
+Dataset
+Modelling
+Model
+Comparison/
+EvaluationSubreddit
+Number of Posts
+Number of Comments
+Total
+r/pennystocks
+12,049
+234,149
+246,198
+r/RobinHoodPennyStocks
+6,506
+78,429
+84,935
+Total
+18,555
+312,578
+331,133
+Table 1: Breakdown of records collected from subreddits
+Figure 3: Data Collection Volumes
+from r/pennystocks, with about a third from r/RobinHoodPennyStocks. The number of comments
+is much larger than the number of posts, with posts making up only about 5% of the texts.
+As shown in Figure 3, there was a sharp increase in the number of submissions over the period
+of data collection:
+• r/pennystocks - 139,000 Members ⇒ 257,000 Members
+• r/RobinHoodPennyStocks - 52,000 Members ⇒ 133,0000 Members
+This seems to reflect an increase in amateur stock market investing because of the covid-19 pan-
+demic, and a corresponding increase in manipulation. i.e, as manipulators look to take advantage of
+new, naive investors during the pandemic. Alerts and press releases by the SEC and the Canadian
+Securities Administrators warned new investors to be vigilant about the increasing number of P&D
+schemes that have occurred around that time [9, 28, 29].
+The median number of words per post or comment was 22, and the total number of distinct
+words was 4,862.
+Replacing stock symbols by the market sector to which each business belongs allows us to see
+which sectors are discussed the most, and which are the targets of P&D. Figure 4 shows that
+healthcare stocks are the most mentioned, followed by technology stocks. The pandemic clearly
+had an effect on both attention to markets and manipulations. Temporal trends in the healthcare
+5
+
+10000
+8000
+Number of Records
+6000
+40.00
+2000
+DatesFigure 4: Histogram of market sectors discussed within subreddits
+sector, Figure 5 , show an increase in online activity at the beginning of the pandemic, and then a
+further increase in the middle of 2020. Figure 6 shows that P&D manipulations also increased in
+2020.
+Table 2 shows the information collected for each post and comment.
+Data from Yahoo Finance was scraped using the yfinance tool [2]. Stock symbols were extracted
+from Reddit posts. This step is non-trivial and required regular expression extraction, and look ups
+against the publicly traded exchanges. Posts which mentioned more than one stock were discarded,
+partly because of the complexity of deciding which stock may be being touted, and partly because
+P&D posts typically focus on one particular stock they are pumping. If a stock symbol was found,
+yfinance was used to collect the financial information described in Table 3.
+As shown in Figure 7, the daily Open, High, Low, Close, and Volume (OHLCV) data was
+collected over nine business days surrounding an event. Data was collected over five days before each
+post event to establish a baseline for price and volume. Penny stocks almost always shows minor
+variation in price and volume so this baseline is typically quite flat. The remaining four days contain
+the pump event (sharp increase) followed by a decrease in price and a slower decrease in volume.
+6
+
+00008
+70000
+60000
+of Records
+50000
+40000
+30000
+20000
+10000
+SectorConglomerates
+ SectorServices
+SectorUtilities
+SectorConsumerDefensive
+SectorBasicMaterials
+ SectorRealEstate
+ SectorFinancialServices
+SectorEnergy
+ Sectorlndustrials
+SectorCommunicationServices
+SectorUnknown
+SectorTechnology
+ SectorHealthcare
+SectorsFigure 5: Trend of posts and comments that discussed healthcare stocks
+Feature
+Description
+Post Title
+Title of the post.
+Post ID
+Unique identification code for post.
+Post Author
+Author of the post.
+Post Created
+Unix Timestamp of when post was submitted.
+Post Body
+Text of the post.
+Comment ID
+Unique identification code for comment.
+Comment Author
+Author of the comment.
+Comment Created
+Unix Timestamp of when comment was submit-
+ted.
+Comment Body
+Text of the comment.
+Table 2: Features of collected Reddit data
+Sabherwal et al. [27] studied the effects of online message boards on market manipulation and
+found that dumps typically occur within four days and this is plausible because the manipulators
+want to sell as soon as the price reaches a peak.
+Texts from subreddits were preprocessed using the following steps: remove URLs, expand con-
+tractions, remove HTML Tags, remove punctuation, remove extra whitespaces, remove numbers,
+lemmatization, and remove stopwords.
+Stock symbols within the text were replaced by dummy stock names representing the market
+sector associated with each business. This is required because the name of the particular stock
+being pumped and dumped in one case has nothing to do with the name of the stock being used
+in another case – but there might be correspondences within sectors. Here is an example:
+7
+
+4000
+3500 
+3000
+ of Records
+25:00
+Yumber
+2000
+15:00
+1000 
+500 
+DatesFigure 6: Trend of posts that have been labelled as P&D
+Feature
+Description
+Open
+Opening price of the stock for the given period.
+High
+Highest price for the stock within the given pe-
+riod.
+Low
+Lowest price for the stock within the given pe-
+riod.
+Close
+Closing price of the stock for the given period.
+Volume
+Total number of shares traded within the given
+period.
+Market Sector
+Associated industry that the company is in.
+Market Capitalization
+Total market value of the company’s outstand-
+ing shares.
+Table 3: Features of Yahoo! Finance data
+• “AYTU perfect time to buy” ⇒ “SectorHealthcare perfect time to buy”
+3.2
+Data Labelling
+To label each post, stock data surrounding the day in which the post was submitted to Reddit
+were analyzed. If the market data exhibited that pattern associated with P&D (a notable rise from
+the time of the post, followed by a sharp drop) then the post was labelled accordingly. A rise was
+detected by calculating the average price and volume in the five-day window before the post. The
+8
+
+120
+100
+Posts
+Number of i
+60
+40
+20
+DatesFigure 7: Time window used to collect market data.
+daily average price (DAP) of the values was first calculated for each of the five days.
+DAP(Xt) = 1
+4(Xtopen + Xthigh + Xtlow + Xtclose)
+(1)
+and then the baseline average price (BAP) was calculated by
+BAP(Xest) = 1
+5 ·
+T1
+�
+t=T0
+DAP(Xt)
+(2)
+The baseline average volume (BAV) was calculated by taking the average of the volume values
+over the estimation window.
+BAV (Xest) = 1
+5 ·
+T1
+�
+t=T0
+Xtvolume
+(3)
+A threshold was set at two standard deviations above the average price within the five-day
+estimation window. Price increases above this threshold were considered to be pump events. A
+similar threshold was used to define a volume anomaly. Events were considered to be the result of
+P&D if they exceeded the threshold for both price and volume. Figure 8 shows a comparison of
+the stock behaviours labelled using this approach.
+A sudden price rise or volume increase might coincide with a post, but is not necessarily caused
+by it. The rising region of each stock trend of a potential P&D event was min-max normalised,
+9
+
+Reddit Post Date
+Price
+Event
+To
+T2
+Volume
+27
+29
+May
+5
+11
+Estimation
+Event
+Window (5 Days)
+Window (4 Days)Figure 8: Comparison of stock behaviours that have been labelled using anomaly detection
+and its slope calculated. Steep price increases are more likely to arise from genuine information
+and less likely to have resulted from a single manipulation post, so the median slope across the
+entire dataset was calculated, and only slopes below the median were considered as potential P&D
+events. Figure 9 shows the distribution of stock price trend slopes from the entire the dataset. The
+median value is 0.18.
+3.3
+Agreement Model
+The comments associated with the P&D post cannot all be labelled as examples of P&D language,
+since not all of them will be supportive of the post they are responding to. Manipulators, of course,
+will post comments in support of the post, either from the same identity or from others.
+We developed an agreement model, using ideas from stance detection. This was done using
+Empath to generate a lexicon of agreement, seeding it with the words: bought, agree, positive,
+increasing, good, and now. Empath returned the words listed in Table 4. Posts touting stocks
+also use a specialised vocabulary, shown in these examples.
+• “probably go to shoot up tomorrow”
+10
+
+TRNX2020-04-16
+CCO2020-03-31
+USWS2020-06-07
+GNUS2020-06-09
+Stockbehaviours
+Stockbehaviours
+labelledasP&D
+labelledasnotP&DFigure 9: Distribution of stock price trend slopes
+only
+done
+better
+true
+knew
+besides
+like
+maybe
+wanted
+liked
+also
+important
+buying
+understand
+good
+understood
+needed
+work
+because
+successful
+knowing
+grateful
+plus
+much
+reasonable
+should
+give
+happy
+course
+glad
+well
+considering
+anyway
+agree
+meaning
+great
+probably
+sure
+thought
+guaranteed
+more
+honestly
+positive
+thankful
+actually
+agreed
+special
+doubt
+guess
+though
+bet
+buy
+surpass
+worth
+suppose
+although
+especially
+definitely
+certain
+figured
+given
+means
+Table 4: List of generated agreement words from Empath
+• “this bad boy just rocket”
+• “i will see you on the moon”
+An extended lexicon was determined manually by inspecting posts associated with manipulation.
+Table 5 contains the list of words that were chosen using this approach.
+Comments were labelled as associated with pumping if they contained two or more of the
+11
+
+1200
+1000
+800
+Number of Posts
+600
+400
+200moon
+fast
+massive
+rich
+surprise
+rocket
+profit
+top
+easy
+move
+pump
+rally
+peak
+early
+load
+soar
+climb
+worth
+shoot
+quick
+jump
+rise
+sale
+money
+burst
+pop
+high
+gain
+breakout
+drive
+hype
+spike
+run
+cash
+nice
+fly
+go
+up
+hit
+bank
+awesome
+confident
+surpass
+more
+zoom
+big
+great
+potential
+advantage
+Table 5: List of custom words used in the Agreement Model
+agreement words, or if they were (visibly) authored by the original poster. The following are some
+examples of comments that were labelled as not P&D related based on the agreement model:
+• “it be the american dream to fall for snake oil salesman and then lose everything it be a story
+as old as humanity”
+• “clearly a pump and dump scheme”
+• “do not touch it if the chart look like a hockey stick”
+This labelling of comments is limited by the completeness of the agreement lexicon, and also does
+not account for negations.
+P&D posts and comments are relatively rare and so the dataset is naturally imbalanced. Tech-
+niques such as SMOTE [3] and ADASYN [17] were tried but proved ineffective. Instead, where
+predictors allowed it, class weight parameters were set to penalise mistakes in the minority class.
+3.4
+Modelling
+The following predictors were used:
+• Extreme Gradient Boosting (XGBoost)
+• Random Forest (RF)
+• Support Vector Machine (SVM)
+• Artificial Neural Networks
+– Multilayer Perceptron (MLP)
+– Convolutional Neural Network (CNN)
+– Bidirectional Long Short Term Memory (BiLSTM)
+In each case the standard performance measures (accuracy, precision, recall, F1-Score, confusion
+matrix) were calculated, as well as the Shapley values which rank words by their importance to the
+predictions.
+12
+
+Record Type
+P&D
+Not P&D
+Total
+Posts
+3,006
+15,549
+18,555
+Comment
+26,727
+285,851
+312,578
+Total
+29,733
+312,142
+331,133
+Table 6: Dataset class distribution
+Model
+TP
+FP
+TN
+FN
+Accuracy
+Precision
+Recall
+F1-Score
+XGBoost Posts
+1728
+6615
+8934
+1278
+57.46 (±3.73)
+20.71 (±0.48)
+57.49 (±0.68)
+30.45 (±2.25)
+XGBoost Posts and Comments
+2007
+7646
+7903
+999
+53.41 (±1.42)
+20.79 (±0.85)
+66.77 (±1.58)
+31.71 (±0.96)
+RF Posts
+271
+646
+14903
+2735
+81.78 (±0.51)
+29.55 (±1.40)
+9.01 (±0.52)
+13.81 (±0.78)
+RF Posts and Comments
+414
+211
+15338
+2592
+84.89 (±0.69)
+66.24 (±1.69)
+13.77 (±0.47)
+22.80 (±0.75)
+SVM Posts
+1752
+5263
+10286
+1254
+64.88 (±1.14)
+24.98 (±0.76)
+58.28 (±1.05)
+34.97 (±1.16)
+SVM Posts and Comments
+2125
+4559
+10990
+881
+70.6 (±0.49)
+31.79 (±0.43)
+70.69 (±0.56)
+43.86 (±0.57)
+MLP Posts
+2382
+1718
+13831
+624
+87.38 (±6.66)
+58.10 (±11.65)
+79.24 (±12.76)
+67.04 (±12.12)
+MLP Posts and Comments
+2103
+2602
+12947
+903
+81.11 (±3.71)
+44.70 (±4.28)
+69.96 (±3.80)
+54.55 (±4.36)
+CNN Posts
+2373
+1709
+13840
+633
+87.38 (±7.04)
+58.13 (±12.02)
+78.94 (±12.76)
+66.96 (±12.37)
+CNN Posts and Comments
+2304
+2068
+13481
+702
+85.07 (±1.25)
+52.70 (±2.33)
+76.65 (±3.45)
+62.46 (±2.64)
+biLSTM Posts
+2297
+2495
+13054
+709
+82.73 (±8.11)
+47.93 (±9.92)
+76.41 (±10.94)
+58.91 (±10.82)
+biLSTM Posts and Comments
+2288
+2370
+13179
+718
+83.36 (±2.27)
+49.12 (±3.25)
+76.11 (±3.86)
+59.71 (±3.54)
+Table 7: Summary of model performance
+4
+Results
+Table 6 shows the class distribution for the dataset. Less than 9% of the records are labelled as
+being P&D. This is typical of datasets where fraud is present; indeed it is striking that the rate of
+fraud is this high.
+The results of each of the predictive model are reported in Table 7 using 5-fold cross validation
+and upweighting the fraud class when the model permits it.
+The neural network models perform well as expected.
+Models such as XGBoost, Random
+Forests, and SVM had disappointing performance, and a heterogeneous stacked classifier combining
+their predictions did not improve on the performance of the individual predictors, suggesting that
+they make their errors on the same records.
+At first glance, the ANN models using posts perform better than those using posts and com-
+ments. However, the standard deviations of the performance numbers show that the inclusion of
+comments provides stability for correctly identifying P&D posts. The best performing model over-
+all is CNN, especially with comments included. Its precision is relatively low; of all the records that
+the model predicts to be P&D, only 52.7% are actually correct. If we look at the rate at which each
+class is predicted to be positive, a better outlook of the model is provided. Given a positive P&D
+text, the model has a 76.65% chance of classifying it correctly, whereas, if it is given a negative
+text, it has a 13.3% chance of classifying it incorrectly as positive. It is perhaps a little surprising
+that biLSTM did not perform best since they are typically strong predictors for natural language
+problems.
+The SHAP Explainers produce diagrams that rank the attributes by their impact on outcomes.
+Figure 10 shows the diagram for the CNN predictor for posts and comments and the 30 most
+impactful words. Although the influence of any single word is inevitably weak, there are visible
+red dots to the right for many of these words, indicating that higher frequencies of these words are
+associated with P&D events. The names of the popular sectors are indicator of P&Ds, as are words
+13
+
+Predicted Label
+Actual Label
+Misclassified Post
+P&D
+Not P&D
+sectorunknown about to soar
+P&D
+Not P&D
+sectorunknown fitness equipment maker owner
+of bow flex completely sell out of most retail
+store how be this look just buy in share
+P&D
+Not P&D
+quick all in sectorcommunicationservices pump
+my first time actually do something right the
+lambos go to be green for gain
+P&D
+Not P&D
+blast off look like gold and oil will be big player
+this i also suggest look at sectortechnology
+P&D
+Not P&D
+sectorenergy drop time to buy it be drop below
+which be its day low be it a good time to buy
+Not P&D
+P&D
+sectortechnology release patent news on thermal
+tech could be a mark sympathy play bust out
+over
+Not P&D
+P&D
+sectorhealthcare do anyone understand why sec-
+torhealthcare shoot up soo much i be not able
+to find any real catalyst
+Not P&D
+P&D
+sectorhealthcare on the move this have potential
+reach today
+Not P&D
+P&D
+sectorhealthcare to the moon
+Not P&D
+P&D
+any thought on when to sell sectorenergy bought
+in late i be up after hour should i wait til tomor-
+row or sell as soon as possible in the am
+Table 8: Examples of misclassified posts from CNN model
+from the agreement model such as “buy” and “go”. Across the best performing models, the same
+set of words emerge as the most impactful features (not shown).
+Misclassifications by the model have different impacts depending on how and where it is used.
+For an ordinary investor, a false positive (a post predicted to be a P&D when it isn’t) means a
+missed opportunity for profit, but a false negative means a financial loss. For a regulatory body, a
+false positive is problematic, but a false negative less so. Table 8 shows some of the examples of
+misclassifications by the CNN model.
+Some false positives, predicted to be P&D from the text, but without a corresponding market
+movement may be instances where the post failed to attract enough attention to cause a measurable
+market movement, or was so blatant that it was not credible to typical investors.
+Some false
+negatives may be because the posts were too short to contain the required two words, because the
+pumping took place on another platform or because a market movement happened to match the
+timing of the post.
+14
+
+5
+Related work
+The application of data analytics for detecting market manipulation is a relatively new in the
+field of finance. Most research has focused on detecting trade-based manipulation because it is
+most common [32]. Huang and Chang found that of the manipulation cases prosecuted in Taiwan
+from 1991 to 2010, 96.61% were trade-based, and only 3.39% were information-based [18]. Some
+examples detecting trade-based manipulation are: Ogut et al. [38] in the emerging Istanbul Stock
+Exchange, Wang et al. [32] for prosecuted manipulation cases reported by the China Securities
+Regulatory Commission, Cao et al. [7] using real trading data from four popular NASDAQ stocks
+with synthetic cases of manipulation (spoofing and quote stuffing), Cao et al. [36] using seven
+popular NASDAQ and LSE stocks data injecting ten simulated stock price manipulations, Diaz et
+al. [12] using manipulation cases pursued by the U.S. Securities and Exchange Commission (SEC)
+in 2003, and Golomohammadi et al. [16] trying to detect three groups of manipulation schemes:
+marking the close, wash trades, and cornering the market.
+For information-based manipulation, Victor and Hagemann [31] looked at 149 confirmed P&D
+schemes coordinated through Telegram chats and pumped via Twitter. Using XGBoost, they built
+a model that achieved a sensitivity of 85% and specificity of 99%. They concluded that P&Ds were
+frequent among cryptocurrencies that had a market capitalization of $50 million or below and often
+involved trading volumes of several hundred thousand dollars within a short time-frame.
+Mirtaheri et al. [23] looked specifically at forecasting P&Ds by combining the information from
+Twitter and Telegram. They manually labelled known P&D operation messages on Telegram, and
+then used SVMs with a stochastic gradient descent optimizer to label the remaining messages as
+P&D or not. They used Random Forests to detect whether a manipulation event was going to take
+place within the market. Their results showed that they were able to detect, with reasonable accu-
+racy, whether there is an unfolding manipulation scheme occurring on Telegram. Their proposed
+model was able to achieve an accuracy of 87% and an F1-Score of 90%.
+Some partially automated tools have also been developed.
+These flag suspicious activities
+that can then by investigated by regulators.
+Delort et al.
+[11] used Naive Bayes classifiers to
+examine collected messages from HotCopper, an Australian stock message board. They successfully
+identified messages of concern, but the number of false positives was too high to use the model
+in an automated way. Owda et al. [25] compared messages to lexicon templates of known illegal
+financial activities (e.g. Pump and Dump, Insider Information). They found that, of the 3000
+comments that were collected on a daily basis, 0.2% were deemed suspicious.
+6
+Conclusion
+The intersection of social media with low-cost trading platforms and naive investors has made
+market manipulation an attractive strategy.
+Pump&dump is particularly simple to implement
+since it requires only the dissemination of fictional information about the future prospects for a
+stock. This is particular easy for penny stocks where validating information is difficult for ordinary
+investors, and where relatively small purchase volumes can cause large price movements.
+We investigate protecting investors, and assisting regulators, by building predictive models that
+label social media posts (and the responses they elicit) as potential drivers of P&D events. We
+do this by collecting posts and comments, developing a model for a P&D event based on patterns
+of price and volume changes, using the match between posts and P&D events to label posts, and
+15
+
+extending this labelling to comments using an agreement model. Natural language predictors then
+learn the language patterns associated with P&D manipulations, so that new manipulations can
+be detected before they affect the market.
+Data is imbalanced, since manipulations are rare, but our best predictive model achieves an
+F1-score of 62% and an accuracy of 85%. Improvements in performance are limited by potential
+coincidences between a post and a price and volume change that mimics a P&D, posts that fail to
+reach a sufficient audience to cause the desired buying behaviour, and natural language issues that
+arise from informal and short texts, and a specialised vocabulary used in stock discussion forums.
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+emerging market: The case of Turkey. Expert Systems with Applications, 36(9):11944–11949,
+November 2009.
+19
+
+Figure 10: CNN SHAP Summary Plot for posts and comments
+20
+
+High
+buy
+sectorhealthcare
+stock
+get
+one
+sectorindustrials
+go
+look
+poob
+share
+sell
+sectortechnology
+price
+sectorconsumercyclical
+Feature value
+day
+megathread
+would
+company
+like
+make
+sectorcommunicationservices
+news
+see
+hold
+today
+post
+week
+market
+sectorunknown
+think
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+LOW
+SHAP value (impact on model output)
\ No newline at end of file
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+Optimization of Hybrid Power Plants:
+When Is a Detailed Electrolyzer Model Necessary?
+Manuel Tobias Baumhof, Enrica Raheli, Andrea Gloppen Johnsen, and Jalal Kazempour
+Department of Wind and Energy Systems, Technical University of Denmark, Kgs. Lyngby, Denmark
+{mtba, enrah, anglopj, jalal}@dtu.dk
+Abstract—Hybrid power plants comprising renewable power
+sources and electrolyzers are envisioned to play a key role in
+accelerating the transition towards decarbonization. It is common
+in the current literature to use simplified operational models for
+electrolyzers. It is still an open question whether this is a good
+practice, and if not, when a more detailed operational model is
+necessary. This paper answers it by assessing the impact of adding
+different levels of electrolyzer details, i.e., physics and operational
+constraints, to the optimal dispatch problem of a hybrid power
+plant in the day-ahead time stage. Our focus lies on the number
+of operating states (on, off, standby) as well as the number
+of linearization segments used for approximating the non-linear
+hydrogen production curve. For that, we develop several mixed-
+integer linear models, each representing a different level of
+operational details. We conduct a thorough comparative ex-post
+performance analysis under different price conditions, wind farm
+capacities, and minimum hydrogen demand requirements, and
+discuss under which operational circumstances a detailed model
+is necessary. In particular, we provide a case under which a
+simplified model, compared to a detailed one, results in a decrease
+in profit of 1.8% and hydrogen production of 13.5% over a year.
+The key lesson learned is that a detailed model potentially earns
+a higher profit in circumstances under which the electrolyzer
+operates with partial loading. This could be the case for a certain
+range of electricity and hydrogen prices, or limited wind power
+availability. The detailed model also provides a better estimation
+of true hydrogen production, facilitating the logistics required.
+Index Terms—hybrid power plants, electrolyzer, hydrogen,
+mixed-integer linear programming
+I. INTRODUCTION
+A. Background
+In order to limit global warming to a maximum of 1.5 °C,
+greenhouse gas emissions must be reduced to net zero by 2050,
+as called for in the European Green Deal 2019 [1]. Renewable
+hydrogen produced through electrolysis could aid in two
+major challenges on the path towards the net zero goal. First,
+electrolyzers can act as flexible loads and therefore potential
+frequency restoration ancillary service providers, contributing
+to maintaining the power balance in power systems with
+increased penetration of renewable energy sources. Second,
+renewable hydrogen can be further synthesized into other
+green fuels, eventually enabling decarbonization in the hard-
+to-abate sectors, such as heavy transport and industry.
+Hybrid power plants comprising of renewable power sources
+(wind and/or solar) and electrolyzers are the key components
+to accelerate the current energy transition through hydrogen
+[2]. Nonetheless, uncertainties in terms of the cost-benefit of
+electrolyzers in the long run have challenged the widespread
+investment in said technologies and thereby large-scale pro-
+duction of renewable-based green hydrogen [3]. In Denmark,
+there is currently a special focus on green hydrogen at the
+governmental level and also, among the regulator, system
+operator, and many industry stakeholders, envisioning a large
+deployment of electrolyzers and other power-to-X facilities in
+the coming years. In 2021 the Danish government published
+a strategy for the national power-to-X development, aiming to
+build 4 to 6 GW of electrolysis capacity by 2030, doubling the
+current Danish peak demand [4]. This emerging trend is not
+limited to Denmark, and many other countries both in Europe
+and globally see hydrogen as a key solution for the realization
+of green societies of the future [2], [5].
+B. Aim and Literature Review
+It is a common practice in the current literature to use a
+simplified operational model for electrolyzers e.g., by using
+a constant power-to-hydrogen conversion ratio irrespective of
+whether the electrolyzer operates in full capacity or not [6]–
+[9]. In addition, some papers do not consider operational states
+of the electrolyzer [6], [9]. This paper challenges these simpli-
+fication practices. While a simplified model works satisfacto-
+rily under certain operational circumstances, there are several
+other circumstances under which a simplified one yields a
+sub-optimal operation of electrolyzers, underestimating their
+value. This paper answers when a detailed operational model
+should be applied, and to what extent the profit and hydrogen
+production can be increased by using a detailed model. We will
+also discuss to what extent a detailed model brings additional
+computational burden.
+In general, two main physical aspects of electrolyzers need
+to be modeled for operation in the day-ahead time stage:
+1) Electrolyzer efficiency: The power-to-hydrogen conver-
+sion efficiency is a function of the power consumption
+of the electrolyzer. To accurately model the hydrogen
+production of the electrolyzer, the varying efficiency
+should be captured, which introduces non-linearities to
+the model. The simple models usually use a constant
+efficiency, while more accurate modeling incorporates
+the non-linearities, which can be later linearized.
+2) Number of operating states: Proper operational modeling
+of electrolyzers may require introducing three states,
+namely on, off, and standby, to ensure no hydrogen pro-
+duction below a given minimum allowed partial loading,
+for which additional binary variables are needed. Many
+1
+arXiv:2301.05310v1  [math.OC]  12 Jan 2023
+
+papers in the literature do not even model states, thus
+assuming the electrolyzer is always on, or model two
+states only, i.e., on and off, similar to conventional power
+generators1.
+Various studies have incorporated different levels of opera-
+tional details of the electrolyzer into their optimization prob-
+lems. In [7] and [8], a constant efficiency is applied but two
+and three states are modeled, respectively, by adding binary
+variables. In [10], three states are modeled, while assuming a
+linear hydrogen production curve, despite showing that the
+production curve is not well approximated by a first-order
+interpolation. A hybrid power plant including an electrolyzer
+is modeled in [11], where the non-linear hydrogen production
+is linearized between two points, with a single binary variable
+representing the on/off state of the electrolyzer. In [12] a
+quadratic production curve is applied and the resulting non-
+linear program is eventually solved by a heuristic. In [13],
+three states are included, and differently from the other papers,
+the operating temperature is considered as a variable, provid-
+ing an extra degree of freedom in the electrolyzer operation.
+This model allows to take into account the temperature impact
+on the conversion efficiency and the quality of the generated
+heat. The non-linear hydrogen production is then linearized
+around a fixed reference operating point to formulate the
+problem as a mixed-integer linear program (MILP).
+C. Contributions and Paper Organization
+To the best of our knowledge, there is a lack of a com-
+prehensive analysis in the current literature, identifying the
+operational circumstances under which a simple model ends
+up in a sub-optimal operation of electrolyzers, resulting in a
+reduced profit and hydrogen production2. This paper bridges
+such a gap through the following contributions:
+• To embed constraints describing the physics of electrolyz-
+ers while keeping the final model as a MILP,
+• To thoroughly investigate ex-post the impact of the in-
+clusion of different operational details on the final profit
+of the hybrid power plant and the amount of hydrogen
+produced,
+• and finally, to provide a set of recommendations in
+terms of including operational details of electrolyzers,
+depending on the application, the range of electricity
+prices, and the hydrogen price.
+Without loss of generality, this paper focuses on alkaline
+electrolyzers, as they are currently the most mature tech-
+nology [14]. The proposed model can be extended to other
+low-temperature electrolyzers, such as polymer electrolyte
+membrane (PEM). More operational characteristics may be
+necessary for modeling solid-oxide electrolyzers (SOEC).
+1We will discuss later in Section IV that under some operational conditions,
+a two-state model including on and standby states works well too. In contrast,
+the two-state model on-off is not satisfactory neither in terms of dispatch
+decisions nor the computational performance.
+2Reference [13] provides a similar analysis, however, the Faraday efficiency
+is assumed to be one. The consequences of this assumption will be further
+discussed in Section II-B.
+The rest of the paper is organized as follows. Section II
+describes the electrolyzer physics, focusing on the operating
+states and the hydrogen production curve. Section III provides
+the proposed MILP, representing all three states of the elec-
+trolyzer. Section IV discusses the impact of the electrolyzer
+modeling choices by means of a test case and a thorough
+sensitivity analysis. Section V concludes the paper. Finally,
+Appendices A and B provide two MILPs (simpler than the
+one proposed in Section III), both representing two states of
+the electrolyzer only, where one is a model with on-off states,
+and the other one is a model with on-standby states.
+II. ELECTROLYZER PHYSICS
+The core of the renewable-hydrogen hybrid power plant is
+the electrolyzer, where water is decomposed into hydrogen
+and oxygen by means of electrical power. The physics and
+operating characteristics of alkaline electrolyzers are described
+in this section and will be formulated as a set of mixed-integer
+linear constraints in Section III.
+A. States
+To describe and model the real operation of an alkaline
+electrolyzer, it is necessary to distinguish three different states:
+1) On state: the electrolyzer operates within its feasible
+load range, consuming power and producing hydrogen with a
+conversion efficiency that depends on the partial load, which
+will be explained in Section II-B. The minimum operating
+power for alkaline electrolyzers is around 15-20% of the
+nominal power, below which the electrolyzer must go into
+standby or off.
+2) Standby state: the electrolyzer does not produce any
+hydrogen but consumes the power needed to maintain the
+system temperature and pressure so that it can rapidly resume
+production. The value of the standby power consumption is
+not usually disclosed by manufacturers, but values between
+1-5% of the electrolyzer full load capacity have been adopted
+in the literature [7], [8], [10]. The time needed to switch from
+standby to on, i.e., a warm start-up is of the order of 30 seconds
+[8].
+3) Off state: the electrolyzer is shut down completely and
+does not consume any power nor produce any hydrogen. How-
+ever, to switch back to on, a significant amount of electricity
+is needed, corresponding to a cold start-up cost. Moreover,
+at least 20 minutes are necessary before resuming hydrogen
+production [8]. Apart from the introduced cold start-up cost
+and start-up time, the frequent shut down of the electrolyzer
+may have a negative impact on the device degradation and
+lifetime [15].
+B. Efficiency and Production Curve
+The conversion efficiency of electricity into hydrogen is not
+constant but depends on the partial load, i.e., the ratio between
+power consumption at a specific time and the nominal power
+of the electrolyzer. The variation of the efficiency based on the
+operating set-point is mainly due to two phenomena: (i) the
+current-voltage relationship, also called the polarization curve,
+2
+
+10
+20
+30
+40
+50
+Power [MW]
+17.5
+18.0
+18.5
+19.0
+19.5
+Efficiency [kg/MWh]
+(a)
+10
+20
+30
+40
+50
+Power [MW]
+200
+400
+600
+800
+Hydrogen [kg/h]
+(b)
+Non-linear curve
+Approximated curve
+pe *
+h*
+hr
+} h
+Fig. 1. Plot (a): the efficiency curve, and plot (b): the hydrogen production
+curve of a 52.25-MW alkaline electrolyzer, as a function of the electric power
+consumption, working at 90 °C and 30 bar. The black curves represent the
+original non-linear curves. Approximated by two segments, the red curve in
+plot (b) is the piecewise linearized hydrogen production curve. The non-linear
+efficiency curve corresponding to this piecewise linearization is represented
+by the red curve in plot (a). In our formulation, we will only use the red
+piecewise linear production curve in plot (b). The inner plot of (b) shows
+the hydrogen production discrepancy ∆h between original and approximated
+curves, for a given power consumption level.
+and (ii) the Faraday efficiency. We explain both phenomena in
+the following.
+The current-voltage relationship describes the voltage in-
+crease (also called over-voltage or over-potential) with increas-
+ing current density, due to different losses, as explained in [16]
+and [13]. Ulleberg [17] introduced a widely adopted empirical
+formulation that describes the relationships between voltage,
+current density, and electrolyzer operating temperature. To fur-
+ther take into account the operating pressure, this formulation
+was modified by Sanchez et al. [18]. For a given temperature
+and pressure, this can be formulated as
+U cell(i) = U rev + K1i + K2log(K3i + 1),
+(1)
+where U cell(i) is the cell voltage as a function of the current
+density i. In addition, U rev is the open-circuit voltage (i.e.,
+voltage corresponding to current density equal to zero). The
+parameters K1, K2, K3 are constants obtained from experi-
+mental data and can be found in [18]. Voltage U rev can be
+calculated for a specific operating temperature according to
+an empirical equation that can be found in [18]. The power
+consumed by the electrolyzer pe(i) can be calculated as
+pe(i) = U cell(i)iA,
+(2)
+where A is the total area of the cells composing the elec-
+trolyzer. The Faraday law calculates the hydrogen production
+h(i) of the electrolyzer as
+h(i) = 3600 · ηF(i)M H2iA
+2F
+,
+(3)
+where h(i) is the hydrogen production rate in kg/h, M H2 is the
+molar mass of hydrogen in kg/mol, F is the Faraday constant,
+and ηF(i) is the Faraday efficiency as a function of current
+density. The latter is defined as the ratio between the actual and
+the theoretical maximum amount of hydrogen produced. The
+difference between actual and theoretical output is explained
+in [17], and it increases significantly when the electrolyzer
+is working at low-current densities. In [18], an empirical
+expression that captures the relationship between the Faraday
+efficiency and the current density at a given temperature is
+provided: ηF(i) is close to one for higher current densities, and
+it drops to zero when reducing the current. The electrolyzer
+efficiency is defined as
+η(i) = h(i)
+pe(i),
+(4)
+where generally η(i) is expressed in kg/MWh. For different
+values of i, the black curve in Figure 1(a) shows efficiency η(i)
+versus power consumption pe(i). In addition, the black curve
+in Figure 1(b) shows the hydorgen production h(i) versus
+power consumption pe(i). For notational clarity, we drop (i)
+in the rest of the paper. The black curves in Figure 1 show that
+the model is non-linear. The efficiency has a peak at around
+30% of the load. This characteristic peak in the efficiency
+curve is not captured when a constant conversion efficiency is
+used, as done in [6], [8], [10], or when the Faraday efficiency
+is assumed to be equal to one in the entire feasible operating
+range, as done in [13].
+To keep the final problem a MILP, but describe the hydrogen
+production with more details, we use a piecewise linearization
+of the hydrogen production curve as shown by the red curve in
+Figure 1(b), for two linearization segments. For each segment
+s ∈ S, the As (slope) and Bs (intercept) coefficients of the
+line can be calculated such that the approximated hydrogen
+production is Aspe + Bs. Later we will define a binary
+variable indicating which segment is active. The proposed
+approximation is exact only at the segment endpoints (i.e.,
+linearization points), otherwise, it is an underestimation of
+the original non-linear curve. For example, the optimal power
+set-point pe∗ in the inset of Figure 1(b) corresponds to the
+hydrogen production h∗ according to the proposed piecewise
+linear model with two segments3. However, the actual hydro-
+gen realization based on the electrolyzer physics is hr. The
+hydrogen production difference ∆h is reduced by increasing
+the number of segments, and the effect of the hydrogen surplus
+obtained when choosing only one segment, as done in [10], is
+discussed in Section IV.
+According to this piecewise linear formulation for the
+hydrogen production curve, the efficiency η for segment s
+can be calculated based on (4), resulting in η = As + Bs
+pe .
+This is depicted by the red dotted curve in Figure 1(a), given
+two linearization segments used. Note that it does not present
+a linear behavior. However, this non-linear efficiency curve
+does not appear in our optimization problem. The hydrogen
+production curve is used instead, which is linearized through
+segments, as illustrated by the red dotted curve in Figure 1(b).
+III. PROBLEM FORMULATION
+We consider a hybrid power plant, as depicted in Figure 2,
+consisting of a wind farm, an electrolyzer, a hydrogen com-
+pressor, and a hydrogen storage. The generated wind power
+can be either sold to the grid at the electricity market price,
+3Symbol ∗ refers to the optimal value.
+3
+
+Wind farm
+Grid
+Electrolyzer
+Compressor
+Hydrogen 
+storage
+Hydrogen 
+demand
+Electricity
+Hydrogen
+Fig. 2. Schematic representation of a hybrid power plant.
+or consumed by the electrolyzer to produce 100% renewable-
+based green hydrogen. The hydrogen produced can either be
+directly delivered to the demand or temporarily stored in an on-
+site hydrogen storage, with an associated cost for compressing
+the gas. The dashed blue line in Figure 2 represents the option
+to buy electricity from the grid only to supply the electrolyzer’s
+standby power when there is no wind power.
+The hydrogen price is assumed to be a single-value constant,
+and the hybrid power plant serves a minimum daily hydrogen
+demand. We assume the plant has perfect foresight of future
+wind power production and electricity price. Given the 1-hour
+time resolution in our model, we neglect the ramping limitation
+which are typically around ±20% of the nominal power per
+second [10], as well as the warm and cold start-up times of
+the electrolyzer.
+For the optimal operation of the hybrid power plant, we
+develop a complete MILP in Section III-A accounting for
+three states of the electrolyzer and then provide two simplified
+counterparts in Section III-B, each with two states of the
+electrolyzer.
+Notation: All parameters are upper-case or Greek letters,
+whereas all variables are lower-case letters. All binary vari-
+ables are noted by z.
+A. Three-state Model
+The most complete MILP includes the objective function
+(6) constrained by (7)-(29).
+1) Objective function: Over the set of hours t ∈ T , the
+objective function (6) maximizes the total profit of the hybrid
+power plant as
+max
+x
+�
+t∈T
+ptλDA
+t
++ dtλh − pin
+t λin
+t − zsu
+t λsu,
+(6)
+where the variable set x will be defined later. The first term
+corresponds to selling power pt to the grid at the day-ahead
+electricity market price λDA
+t
+. The second term pertains to
+delivered hydrogen dt at a fixed price λh. The third term
+represents the cost for purchasing standby power pin
+t to support
+the electrolyzer’s standby state in case the wind power is
+insufficient. The corresponding price is λin
+t
+= λDA
+t
++ λTSO,
+where λTSO is the grid tariff imposed by the Transmission
+System Operator (TSO). Finally, the fourth term corresponds
+to the cold start-up cost of the electrolyzer, where the binary
+variable zsu
+t
+indicates the start-up at hour t, associated with
+the cost per startup λsu.
+2) Power balance: In every hour t, the power pt sold in the
+day-ahead market is equal to the wind farm power production
+P w
+t plus power pin
+t bought from the grid to support the standby
+state of the electrolyzer, subtracted by the power consumption
+pe
+t of the electrolyzer and the power consumption pc
+t of the
+compressor, such that
+pt = P w
+t + pin
+t − pe
+t − pc
+t
+∀ t ∈ T .
+(7)
+3) Limit on pin
+t : The input power pin
+t
+is limited by the
+standby state consumption of the electrolyzer, implying that
+power cannot be bought from the grid to produce hydrogen:
+pin
+t ≤ P sbzsb
+t
+∀ t ∈ T ,
+(8)
+where the parameter P sb is the standby consumption, and the
+binary variable zsb
+t
+indicates whether the electrolyzer is in the
+standby mode in hour t.
+4) Electrolyzer operational states: Constraint (9) ensures
+that the electrolyzer can take only one out of three states at
+any hour t, namely online, standby, or off:
+zon
+t
++ zoff
+t
++ zsb
+t
+= 1
+∀ t ∈ T ,
+(9)
+where similar to zsb
+t , binary variables zon
+t
+and zoff
+t
+indicate
+whether in hour t the electrolyzer is on and off, respectively.
+The states are activated based on the electricity consumption of
+the electrolyzer. In the online state, the electricity consumption
+pe
+t of the electrolyzer can neither exceed the capacity Ce nor
+go below a minimum load limit P min. In the standby state, the
+electricity consumption must be equal to the standby power
+consumption P sb. These constraints are enforced by
+pe
+t ≤ Cezon
+t
++ P sbzsb
+t
+∀ t ∈ T
+(10)
+pe
+t ≥ P minzon
+t
++ P sbzsb
+t
+∀ t ∈ T .
+(11)
+To represent the cold start-up of the electrolyzer, the binary
+variable zsu
+t
+is defined, taking the value 1 in the case of
+a transition from off to on state in hour t, as enforces by
+constraints (12) and (13). Further, constraint (14) ensures
+that the transition from an off-state to a standby-state is not
+allowed, to avoid bypassing of the start-up cost.
+zsu
+t
+≥ zon
+t
+− zon
+t−1 − zsb
+t−1
+∀ t ∈ T \1,
+(12)
+zsu
+t=1 = 0,
+(13)
+zoff
+t−1 + zsb
+t
+≤ 1
+∀ t ∈ T \1.
+(14)
+5) Electrolyzer hydrogen production: The hydrogen pro-
+duction ht is a function of the electricity consumption of the
+electrolyzer. As explained in Section II-B, for each segment
+s ∈ S, a linear function of the segment power consumption
+ˆpe
+ts with slope As and intercept Bs is defined, such that
+ht =
+�
+s∈S
+(Asˆpe
+ts + Bszh
+ts)
+∀ t ∈ T ,
+(15)
+where the binary variable zh
+ts defines which segment s is active
+in hour t. Each segment is valid within a pre-defined interval
+of upper P s and lower P s power consumption levels, i.e.,
+P szh
+ts ≤ ˆpe
+ts ≤ P szh
+ts
+∀ t ∈ T , s ∈ S.
+(16)
+4
+
+VectorStock
+VectorStock.com/24756804shutterstock.com • 1658641081Constraint (17) ensures that hydrogen production happens
+in the online state only, while one segment only can be active
+at any hour t. In addition, (18) computes the total power
+consumption of the electrolyzer:
+zon
+t
+=
+�
+s∈S
+zh
+t,s
+∀ t ∈ T
+(17)
+pe
+t =
+�
+s∈S
+ˆpe
+ts + P sbzsb
+t
+∀ t ∈ T .
+(18)
+6) Hydrogen storage: Constraints (19)-(25) represent the
+storage operation:
+ht = hd
+t + sin
+t
+∀ t ∈ T ,
+(19)
+dt = hd
+t + sout
+t
+∀ t ∈ T ,
+(20)
+sout
+t
+≤ Sout
+∀ t ∈ T ,
+(21)
+pc
+t = Kcsin
+t
+∀ t ∈ T ,
+(22)
+st=1 = Sini + sin
+t=1 − sout
+t=1
+(23)
+st = st−1 + sin
+t − sout
+t
+∀ t ∈ T \1,
+(24)
+st ≤ Cs
+∀ t ∈ T .
+(25)
+The hydrogen produced ht can either go directly to the demand
+hd
+t or be injected into the hydrogen storage sin
+t , as enforced
+by (19). The total hydrogen dt delivered to the demand is
+equal to the sum of hydrogen directly from the electrolyzer
+and that from the storage sout
+t
+, as per (20). The storage
+output of every hour is limited by the output flow capacity
+Sout in (21). Further, the compressor consumes power pc to
+compress the hydrogen injected into the storage. Assuming
+adiabatic compression, the compression coefficient Kc can be
+calculated, as proposed by [13]. The power consumption for
+compression is then (22). The state of charge of the hydrogen
+storage in the initial and following hours is calculated by
+(23) and (24), where Sini is the hydrogen initially stored in
+the storage at the beginning of time horizon T . The storage
+hydrogen mass capacity Cs is enforced by (25). Note that we
+do not impose any constraint for the energy stored at the end
+of time horizon T . Therefore, pursuing profit maximization in
+this time horizon, the hybrid power plant will leave the storage
+empty in the last hour4.
+7) Hydrogen demand: Imagine within the underlying time
+horizon T , which could be, for example, a year, there are N
+number of time subsets, e.g., 365 days, indexed by n, such
+that there is a minimum hydrogen demand for each n:
+�
+t∈Hn
+dt ≥ Dmin
+n
+∀ n ∈ {1, ..., N},
+(26)
+where Hn is the set of hours within time subset n.
+8) Variable declaration: Constraint (27) declares the non-
+negativity conditions:
+dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts, st, sin
+t , sout
+t
+∈ R+.
+(27)
+4One can enforce a constraint on the minimum stored hydrogen at the end
+of the time horizon, or add a value for this stored energy to the objective
+function.
+Constraint (28) lists binary variables:
+zsu
+t , zh
+ts, zon
+t , zoff
+t , zsb
+t
+∈ {0, 1}.
+(28)
+Therefore, the total number of binary variables is |T |(4 +
+|S|) binaries, where |T | and |S|, respectively, are the number
+of hours and the number of segments used to linearize the
+hydrogen production curve. Finally, the variable set x is
+defined as
+x = {dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts,
+sin
+t , st, sout
+t
+, zsu
+t , zh
+ts, zon
+t , zoff
+t , zsb
+t }.
+(29)
+Accordingly, in addition to |T |(4+|S|) number of binary vari-
+ables, we have |T |(9 + |S|) number of continuous variables.
+B. Two-state Models
+The optimal operation problem (6)-(29) of the hybrid power
+plant accounting for three states of the electrolyzer can be
+simplified if two states only are considered, either on-off states
+or on-standby states. Both result in MILPs.
+In the latter, i.e., the MILP with on-off states, one binary
+variable (instead of three) per hour t is sufficient, such that it
+indicates whether the electrolyzer in the given hour is on or
+off. The resulting MILP is provided in Appendix A. The total
+number of binary variables in this MILP is |T |(2 + |S|).
+Similarly, a single binary variable per hour t is enough
+in the MILP with on-standby states, indicating whether the
+electrolyzer is online or in standby mode. Also, the start-up
+binary variable is not needed. The corresponding MILP is
+given in Appendix B, where among three MILPs, we need
+the lowest number of binary variables, i.e., |T |(1 + |S|).
+IV. NUMERICAL STUDY
+We apply the proposed MILPs of Section III to a case study
+and investigate how the optimal operation of the hybrid power
+and the resulting profit change by adding more operational
+details of the electrolyzer. All source codes and input data are
+publicly shared5. We consider several options for the number
+of linearization segments, i.e., |S|, used to approximate the
+hydrogen production curve of the electrolyzer, including 1, 2,
+4, 8, and 12 segments. Also, we consider three options for
+the number of electrolyzer states: three states on-off-standby
+(OOS), two states on-standby (OS), and two states on-off
+(OO). In the rest of this section, we will refer to various
+models as, for example, OOS-12, implying we consider three
+states (OOS) with 12 segments. Finally, we conduct a sensitiv-
+ity analysis to explore the impact of various input parameters,
+such as wind farm capacity, hydrogen demand, and hydrogen
+price, on the operation of the hybrid power plant.
+A. Case Study
+We consider a hybrid power plant whose structure equals
+the one in Figure 2, and its input data is provided in Table I.
+The capacity of the wind farm is 104.5 MW, corresponding to
+11 V164-9.5 MW™ Vestas turbines, located in Køge Bay,
+5GitHub: https://github.com/mtba-dtu/detailed-electrolyzer-model
+5
+
+TABLE I
+INPUT DATA FOR THE CASE STUDY
+Wind farm
+Capacity
+Cw
+104.5
+MW
+Electrolyzer
+Capacity
+Ce
+50%
+of Cw
+Standby load
+P sb
+1%
+of Ce
+Minimum load
+P min
+15%
+of Ce
+Pressure
+30
+bar
+Temperature
+90
+°C
+Max. current density
+5,000
+A/m2
+Start-up cost
+λsu
+2,612.50
+C [10]
+TSO tariff
+λTSO
+15.06
+C/MWh
+Storage
+Capacity
+Cs
+22,000
+kg
+Maximum output
+Sout
+912.13
+kg/h
+Compressor
+Inlet temperature
+40
+°C
+Inlet pressure
+30
+bar
+Outlet pressure
+200
+bar
+Mechanical efficiency
+75%
+Hydrogen
+Price
+λh
+2.10
+C/kg
+Minimum demand
+Dmin
+n
+3,667
+kg/day
+Denmark. The electrolyzer capacity is set to 50% of the
+wind farm capacity, amounting to 52.25 MW. The modeling
+horizon spans one year with an hourly temporal resolution.
+We apply hourly electricity price data for 2019, as price data
+for the following years might be distorted by macroeconomic
+impacts, such as COVID-19. Day-ahead electricity prices for
+the East Denmark area (DK2) are obtained from ENTSO-e
+Transparency platform [19] and hourly historical wind capac-
+ity factors at the given location for 2019 are retrieved from
+the Renewable.ninja web platform [20]. The average yearly
+capacity factor for the selected location is 43.7%. The hybrid
+power plant is only allowed to buy power from the grid to
+keep the electrolyzer in standby mode, in case the wind power
+is insufficient. In that case, the electricity is bought at the
+hourly day-ahead market price plus the grid tariff of the TSO.
+Since the wind farm is located in DK2, the consumption tariff
+imposed by the Danish TSO, Energinet, is applied [21]. The
+minimum daily demand can be met by the full-load operation
+of the electrolyzer for around four hours. The hydrogen storage
+is scaled to store all hydrogen produced if the electrolyzer
+operates at full capacity for 24 consecutive hours.
+B. Impacts of the Number of Segments
+Let us consider the OOS case with three states, for which
+we solve the proposed MILP (6)-(29). We start with OOS-1,
+where |S| = 1. This means the original non-linear hydrogen
+production curve, depicted in Figure 1(b), is approximated by
+a single linear curve. Here, the minimum power consumption
+P min and the capacity Ce of the electrolyzer are taken as
+two endpoints. By moving to OOS-2, where the number of
+segments |S| is 2, we consider an additional point P η,max,
+which refers to the power consumption level corresponding to
+the peak in the efficiency curve in Figure 1(a). By increasing
+|S| to 4, and then to 8, the mean load value between existing
+points is added, splitting one segment into two. The same
+procedure but only on the right side of P η,max is applied
+when we move from OOS-8 to OOS-12, as this side covers
+1
+6
+12
+18
+24
+0
+20
+40
+60
+Electrolyzer power [MW]
+(a)
+1
+6
+12
+18
+24
+Time [h]
+(b)
+1
+6
+12
+18
+24
+(c)
+Ce
+P , max
+Pmin
+Psb
+pe
+t
+DA
+t
+25
+32
+38
+45
+Day-ahead price [ /MWh]
+Fig. 3.
+The power consumption schedule of the electrolyzer (pe
+t) in an
+example high-wind day when its hydrogen production curve is linearized
+by (a) 1, (b) 4, and (c) 12 segments. These three plots, from left to right,
+correspond to cases OOS-1, OOS-4, and OOS-12, respectively.
+over around 70% of the feasible operating range. With the
+adoption of this procedure, all cases from OOS-2 to OOS-12
+include the point P η,max. In addition, points are not removed
+when refining the discretization. By adding more segments, the
+hydrogen production curve and thus the electrolyzer efficiency
+with partial loading is more accurately represented.
+The increase in the number of segments |S| enables the
+electrolyzer to consume power more flexibly, as depicted in
+Figure 3, where the optimal power consumption schedule of
+the electrolyzer for one example day of the year is shown
+for three different numbers of segments (1, 4, and 12). It is
+observed that when the optimal power consumption of the
+electrolyzer is not constrained by wind production shortage,
+as on the chosen day, the optimal consumption level is always
+one of the piecewise linearization points. There are instances,
+e.g., hour 5 in Figure 3, where OOS-1 goes into the standby
+state as the day-ahead price is too high for profitable hydrogen
+production. In contrast, OOS-4 and OOS-12 continue the
+operation in the on state, but at the power consumption level
+corresponding to the maximum efficiency, where hydrogen
+production is still profitable.
+The number of segments |S| plays an important role in the
+optimal dispatch decision when the day-ahead price lies within
+a specific price range. The upper bound of this price range
+corresponds to the highest price for which the production of
+hydrogen is still profitable. The lower bound is the price below
+which the optimal dispatch decision is always the maximum
+electrolyzer consumption. Figure 4 shows the distribution of
+the day-ahead price λDA
+t
+over 8,760 hours of year 2019 in
+DK2 with the bounds of the price range of interest are marked
+by the red and green dotted lines. The upper bound is found
+as the day-ahead price for which the hydrogen production is
+only feasible at the maximum efficiency, denoted by α in the
+inner plot of Figure 4. The lower bound corresponds to the
+efficiency at the full load, denoted by β. If the day-ahead price
+of a given hour lies outside of this range, the dispatch decision
+for any number of segments would be the same; produce at
+the maximum possible load or cease the production, and there
+would be no added value of a detailed production curve 6.
+This will be further investigated in Section IV-F.
+6These two price thresholds are calculated by multiplying the hydrogen
+price and the efficiency at points β and α, respectively.
+6
+
+25
+30
+35
+40
+45
+50
+55
+Day-ahead price [ /MWh]
+0
+50
+100
+150
+200
+250
+300
+350
+Frequency
+Price
+Price mean
+Power [%]
+Efficiency 
+[kg/MWh]
+Fig. 4. Histogram of the day-ahead electricity price over 8,760 hours of year
+2019 in DK2. Prices λα and λβ correspond to electricity prices for which
+the electrolyzer operates at points α and β, indicated in the inner plot.
+C. Impacts of the States
+We consider three cases OOS, OO, and OS, each for both
+1 and 12 segments. Recall that their corresponding MILPs
+are different7. Comparing the results of MILPs with the same
+number of segments, we observe OS and OOS perform almost
+equally, as observed in Figure 5. The reason for this is the low
+frequency of consecutive hours of too high day-ahead prices,
+where a complete shut-off would be preferred over the standby
+state. Over 8,760 hours, OOS-1 starts up only 2 times, with a
+total of 286 hours offline. The difference in results obtained for
+OS and OOS increases if a higher standby power consumption
+or lower cold start-up cost for the electrolyzer is assumed,
+which would lead to more frequent shut-offs. On the contrary,
+OO earns the lowest profit, mainly due to the high start-up
+cost, which decreases the operational flexibility as even a short
+pause in production incurs a high cost.
+D. Ex-post Performance Analysis
+Recall that three MILPs solve the problem based on the
+linearized hydrogen curve. Through the following ex-post
+performance analysis, it is seen that this leads to both sub-
+optimal dispatch decisions and an underestimation of the true
+amount of hydrogen produced. We have already observed in
+Figure 1(b) that the linearized red curve is below the original
+black non-linear hydrogen production curve, implying that the
+hydrogen production might be underestimated. This means that
+we can expect to produce more hydrogen than what MILPs
+calculate. Such a difference is expected to be reduced by
+using more segments |S| to approximate the original non-
+linear hydrogen production curve.
+Pursuing a fair comparison among models, we conduct an
+ex-post performance analysis. Once the MILPs are solved and
+the optimal power consumption pe∗
+t
+of the electrolyzer ob-
+tained, we re-calculate the true amount of hydrogen produced
+based on the original non-linear hydrogen production curve.
+Note that we do not re-optimize the problem8. We refer to
+the amount of extra hydrogen and its corresponding profit as
+7While we solve the proposed MILP (6)-(29) for OOS, the MILPs
+presented in Appendixes A and B are solved for OO and OS, respectively.
+8To avoid re-optimization, we assume the extra hydrogen is directly sold to
+the demand and is not stored in the hydrogen storage. Otherwise, one needs to
+re-optimize a posteriori to optimize the operation of storage and compressor.
+12
+8
+4
+2
+1
+Number of segments
+97
+98
+99
+100
+Profit [%]
+OOS
+OS
+OO
+States
+15.8
+15.9
+16.1
+16.2
+Profit [million ]
+OOS
+Realized surplus
+1 seg.
+12 seg.
+Fig. 5.
+Estimated and realized surplus profit. The first five bars from the
+left correspond to OOS-1 to OOS-12. The next six bars show the results for
+OO-1, OO-12, OS-1, OS-12, OOS-1, and OOS-12, respectively. The right
+vertical axis is the profit in million C, whereas the left vertical axis is the
+relative profit in % in comparison to the highest profit achieved by OOS-12.
+12
+8
+4
+2
+1
+Number of segments
+80
+85
+90
+95
+100
+H2 production [%]
+OOS
+OS
+OO
+States
+2.3
+2.5
+2.6
+2.8
+2.9
+H2 production [thousand tons]
+OOS
+Realized surplus
+1 seg.
+12 seg.
+Fig. 6. Estimated and realized surplus hydrogen produced. The first five bars
+from the left correspond to OOS-1 to OOS-12. The next six bars show the
+results for OO-1, OO-12, OS-1, OS-12, OOS-1, and OOS-12, respectively.
+“realized surplus”. We assume that all extra hydrogen is sold
+at the same constant price, i.e., C2.10/kg.
+Figure 5 provides the estimated and realized surplus profit
+among different cases. The estimated profit (gray area) is
+the optimal value obtained for the objective function of the
+corresponding MILP, while the realized profit (dark area),
+calculated ex-post, takes into account the profit of selling extra
+hydrogen. Similarly, Figure 6 shows the total estimated and
+realized surplus hydrogen produced. Note that the compressor
+would need to consume more power (around 1 MWh/ton) due
+to extra hydrogen. We draw two conclusions from Figures 5
+and 6:
+(1) Realized surplus: This surplus for profit and hydrogen
+production is reduced by increasing the number of segments,
+due to the improved approximation of the original non-linear
+curve. The realized surplus profit decreases from C71,199
+(0.44%) for OOS-1 to C602 (below 0.01%) for OOS-12.
+Similarly, the hydrogen production surplus is significantly
+decreased, yielding a realized surplus of ∼ 34 tons (1.27%) for
+OOS-1 and only 0.3 tons (0.01%) for OOS-12. By choosing
+a low number of segments, the hydrogen production is under-
+estimated which may lead to logistic issues and inefficiencies
+in the real-life operation of the hybrid power plant.
+(2) Ex-post profit and hydrogen production: Adding more
+electrolyzer details (segments or/and states) always leads to
+an increase in the ex-post profit. To compare various models,
+7
+
+TABLE II
+COMPUTATIONAL ASPECTS
+Case
+Computational time [s]
+No. of binary variables
+OS-1
+1.4
+2×8760
+OS-12
+12.7
+13×8760
+OOS-1
+137.8
+5×8760
+OOS-2
+135.8
+6×8760
+OOS-4
+236.3
+8×8760
+OOS-8
+350.3
+12×8760
+OOS-12
+473.7
+16×8760
+OO-1
+767.1
+3×8760
+OO-12
+1,763.1
+14×8760
+OOS-12 is taken as a benchmark, as it leads to the highest
+profit. First, the impact of the number of segments is examined,
+while keeping the number of states fixed and equal to 3. The
+ex-post profit reduction applying 1 instead of 12 segments is
+0.72%, corresponding to around 117.6 kC for the entire hybrid
+power plant. The ex-post hydrogen production is increased by
+8.32%, corresponding to around 241 tons. This percentage
+deviation is notably higher in part because the increase in
+hydrogen profit is dampened by the reduction in electric-
+ity profit (3.86% electricity profit increase for 1 segment
+compared to 12 segments). For OOS-1, the profit share of
+selling hydrogen is much lower than the profit share of selling
+electricity (around 34%). By introducing more segments, the
+contribution of hydrogen sales is increased to 38% at the
+expense of electricity sales. More profit and different business
+models are therefore unlocked by including more electrolyzer
+details in the MILP formulation. Figures 5 and 6 show that the
+errors are considerably reduced by implementing 4 segments
+instead of 1. Second, we assess the impact of the states on the
+ex-post profit and hydrogen production. While OS performs
+just as well as OOS as described in Section IV-C, OO with 12
+segments results in a 1.22% lower ex-post profit, and in a 4%
+lower hydrogen production. For OO-1, a profit reduction of
+around 1.8% and a reduced hydrogen production of 13.5% are
+observed, compared to the benchmark. Finally, we observe that
+neglecting the standby state in the model formulation leads to
+the worst outcome in terms of profit and hydrogen production
+potential.
+E. Computational Analysis
+All MILPs have been solved using the Gurobi solver in Julia
+on a MacBook Pro M1 2020 with 16 GB RAM. The optimality
+gap is fixed to 0.01% when we solve every MILP. The
+increase in the number of linearization segments |S| leads to an
+increase in computational time due to introducing more binary
+variables. For OOS, the computational time is increased from
+138 seconds for 1 segment to 474 seconds for 12 segments,
+as reported in Table II. Removing the off state significantly
+reduces the computational time, with OS-1 being by far the
+fastest MILP to be solved (1.4 seconds). The OO models
+require the highest computational time, although they embody
+fewer binary variables than their OS and OOS counterparts.
+We hypothesize the reason is that the start-up cost constraints
+with inter-temporal nature are more often active when the
+option of standby state is not present. Therefore, we do
+not recommend using OO as its corresponding profit is the
+lowest among all cases (Figure 5), and it is being solved
+comparatively slower. Further, if computational efficiency is
+crucial, it may be beneficial to neglect the off state and run
+the OS model for improved computational performances. In
+general, the computational time increases with the number
+of segments but is deemed reasonable for the OS and OOS
+models, considering that our optimization problem is run over
+8,760 hours. As operational problems are typically solved
+for a shorter time horizon, e.g., 24 hours for day-ahead
+scheduling, the computational cost of adding more details to
+the electrolyzer would be minimal.
+F. Sensitivity Analysis with Respect to Input Data
+In the previous sections, we have shown that adopting a
+simplified electrolyzer model can lead to an underestimation
+of the profit and hydrogen production for the hybrid power
+plant. We have also shown that the benefit of added details is
+case-specific, and depends on the input parameters. We now
+aim at assessing the impact of input parameters and system
+configuration on these results, through a sensitivity analysis.
+In particular, we will focus on wind over electrolyzer capacity
+ratio, hydrogen demand over electrolyzer capacity ratio, and
+the hydrogen price. The sensitivity analysis is performed on
+the OOS-1 and OOS-12 models.
+1) Wind size: Recall from Table I that the wind farm
+capacity is 2 times that of the electrolyzer. To assess the impact
+of the wind-to-electrolyzer capacity ratio, two additional cases
+are considered, under which such a ratio is 1, 2 (reference),
+and 8. When this ratio is reduced from 2 to 1, the number
+of hours where the power input to the electrolyzer is limited
+by the wind availability is increased from 5,326 to all hours.
+Conversely, when the ratio is increased from 2 to 8, the number
+of power-limited hours is reduced to 1,236. We observe that
+the realized surplus for hydrogen production increases with
+the number of hours with limited wind power. The reason for
+this is that the piecewise approximation is exact only on the
+linearization points, and the limited wind availability forces
+the electrolyzer to operate out of those points. Conversely,
+when the number of wind power-limited hours is reduced,
+the electrolyzer operates more often on the linearization
+points, where the approximation is exact. It follows that the
+underestimation of hydrogen production is greater the more
+the electrolyzer is limited from operating at the linearization
+points. With a wind-to-electrolyzer ratio of 1, the difference
+in ex-post hydrogen production between 1 and 12 segments
+is 13%, which is reduced to 3% when the ratio increases to
+8. Therefore, incorporating electrolyzer details is crucial for
+hybrid power plants where the wind-to-electrolyzer capacity
+ratio is small.
+2) Hydrogen demand size: To investigate the sensitivity of
+optimization outcomes with respect to the hydrogen demand,
+the minimum daily demand is doubled, corresponding to
+around 8 full-load hours of hydrogen production. We observe
+that the impact of adding more segments to the electrolyzer
+8
+
+production curve diminishes when the demand constraint is
+tighter, i.e., with a higher minimum daily demand. For the case
+with the reference demand, the difference between the ex-post
+profit for OOS-12 and OOS-1 is 8%. This difference, when the
+hydrogen demand is doubled, is reduced to 2%. The increase
+in demand forces the electrolyzer to operate more frequently
+at its maximum load, where both OOS-1 and OOS-12 share
+the same linearization point and efficiency.
+3) Hydrogen price: To explore the impact of the hydrogen
+price, we increase it from C2.10/kg to C5.00/kg. As already
+discussed in Section IV-B, adding more segments impact
+the optimal solution and profit as long as the electricity
+price in the given hour is in the range [λβ, λα], shown in
+Figure 4. Since λα and λβ are proportional to the hydrogen
+price, by increasing the hydrogen price, the range [λβ, λα] is
+widened and moved towards higher electricity prices, where
+the frequency of occurrence is reduced. When the MILP is
+solved with the hydrogen price of C5/kg, it is more frequently
+optimal to operate the electrolyzer at full load (39% of the
+time, compared to 11% for the case with the hydrogen price of
+C2.1/kg) and the linearization segments are utilized less. This
+also results in a significantly decreased computational time
+(below 20 seconds for OOS-12). The profit contribution from
+the hydrogen sale is increased significantly to 92%. The ex-
+post profit and hydrogen production difference between OOS-
+1 and OOS-12 are reduced to 0.01% and 0.03%, respectively
+(they are 0.72% and 8.32% for the C2.1/kg case).
+The modeling of segments is relevant if higher hydrogen
+prices are coupled with also higher electricity prices. In
+this way, the electricity price range [λβ, λα] would still be
+overlapping with the majority of day-ahead price occurrences.
+For example, we test an artificial case where the day-ahead
+electricity price time series was multiplied by a constant factor
+to increase the mean price to around C90/MWh (similar to
+the mean value for 2021 in DK2). In this case, with the
+hydrogen price of C5/kg, similar results to the 2019 test case
+with the hydrogen price of C2.1/kg were obtained in terms of
+the impact of the number of segments. For a given hydrogen
+price and efficiency curve, checking if the price range [λβ, λα]
+overlaps with the expected electricity price is therefore crucial
+to assess a priori the impact of choosing a simplified model
+for the production curve (e.g., 1 linearization segment only)
+and support the modeling choices.
+V. DISCUSSION AND CONCLUSION
+Several studies have focused on the optimal dispatch of
+hybrid renewable-hydrogen power plants assuming simplified
+models for the electrolyzer component. This paper investigates
+the impact of choosing different levels of operational details
+for the electrolyzer model on the dispatch decisions, profit, the
+amount of hydrogen produced, and computational time. The
+impact of two modeling choices is considered: the operating
+states (on, off, standby), and the number of segments used
+to linearize the hydrogen production curve. The problems are
+formulated as MILPs, where the number of binary variables
+depends on the number of states and segments.
+For fixed states, adding more linearization segments for
+approximating the hydrogen production curve results in a
+higher profit, and a reduced surplus in the ex-post profit
+calculation, meaning that the model is able to estimate the
+actual cost and revenue streams more accurately. Moreover,
+a better estimation of the produced hydrogen is achieved.
+In fact, the linearization results in an underestimation of the
+produced hydrogen, but the underestimation is reduced by
+increasing the number of segments. Apart from introducing
+errors in the actual realized profit, thus potentially impacting
+the investment decisions in these types of technologies, the
+systematic underestimation of the hydrogen produced by the
+electrolyzer might introduce logistical inefficiencies, e.g., truck
+scheduling, and storage discharging/filling.
+The impact of adding more piecewise segments to the
+hydrogen production curve depends on the distribution of day-
+ahead electricity prices in the given time horizon. The model
+formulations with 1 and 12 segments take significantly dif-
+ferent dispatch decisions when the day-ahead electricity price
+is within a certain range, which depends on the electrolyzer
+efficiency (minimum and maximum) and the hydrogen price.
+Out of this day-ahead electricity price range, the model with 1
+and 12 segments takes the same dispatch decisions. Therefore,
+the value of adding more details to the hydrogen production
+curve could differ by varying input data and case studies. It
+is observed that this value decreases when the electrolyzer
+operates less at partial loading, e.g. when the input power is
+less limited by available wind power or with high-demand
+constraints. In this paper, revenues from other than the day-
+ahead market are not considered but this may also impact the
+dispatch strategy and therefore benefit from more segments.
+Choosing to represent only on and off states leads to the
+highest profit underestimation and worst ex-post performance
+while modeling only on and standby states lead to similar
+profit and dispatch decisions to the three-state model. This
+result is, however, significantly affected by the assumption
+made on the standby power consumption of the electrolyzer
+and its start-up cost. These parameters are highly uncertain
+due to the lack of data on large-scale electrolyzers.
+In conclusion, adopting more simplified models for the
+electrolyzer always leads to a reduced profit and sub-optimal
+scheduling. However, the impact of adding more details may
+vary depending on the case study considered and especially
+the range of day-ahead electricity prices, hydrogen price, wind
+power production compared to the electrolyzer installed capac-
+ity, standby power consumption, and start-up cost. Among all
+considered models, the most complete one (three states with
+12 segments) was solved for a 1-year horizon in less than
+10 minutes. The increase in computational time by adding
+more details would be marginal if a day-ahead scheduling
+problem is considered instead. Moreover, reducing the three-
+state model to two states only is not always faster, as it was
+observed that the two-state on-off model with 12 segments
+was the longest to solve among all the cases considered. A
+more detailed representation of the electrolyzers should be
+preferred for operational problems. For investment problems,
+9
+
+we hypothesize that it may be adequate to adopt a more
+simplified model of the electrolyzer, but this should be further
+assessed and it was out of the scope of the current paper.
+Further research should be conducted to assess the impact of
+modeling choices when additional revenue streams are consid-
+ered, such as flexibility provisions in ancillary service markets,
+which may impact the dispatch decisions of the hybrid power
+plant. Additionally, as there is a high uncertainty related to the
+start-up and standby costs, the sensitivity of these parameters
+on the impact of added details should be assessed further.
+Moreover, the level of detail needed for investment problems
+should be further investigated. The modeling of electrolyzer
+cell degradation over time should be investigated and included
+in the model with additional constraints. Finally, uncertainties
+in wind power supply and electricity prices should be included.
+ACKNOWLEDGEMENT
+This research was supported by the Energy Cluster Den-
+mark through the “Sustainable P2X Business Model” project,
+and by the Danish Energy Development Programme (EUDP)
+through the HOMEY project (64021-7010). We would like to
+thank Jens Jakob Sørensen (Ørsted), Alexander Holm Kiilerich
+(Ørsted), Roar Hestbek Nicolaisen (Hybrid Greentech), Yan-
+nick Werner (DTU), and Matˇej Novotn´y for collaborations,
+thoughtful discussions, and constructive feedback.
+APPENDIX
+A. The simplified MILP with On-Off States
+This appendix provides the MILP (A.30a), where the on
+and off states of the electrolyzer are only modeled. This is a
+simplified model compared to the one proposed in Section III
+with three states of the electrolyzer.
+max
+Ω
+�
+t∈T
+ptλDA
+t
++ dtλh − zsu
+t λsu
+(A.30a)
+s.t.
+pt = P w
+t − pe
+t − pc
+t
+∀ t ∈ T ,
+(A.30b)
+pe
+t ≤ Cezoo
+t
+∀ t ∈ T ,
+(A.30c)
+pe
+t ≥ P minzoo
+t
+∀ t ∈ T ,
+(A.30d)
+zsu
+t
+≥ zoo
+t
+− zoo
+t−1
+∀ t ∈ T \1,
+(A.30e)
+zoo
+t
+=
+�
+s∈S
+zh
+t,s
+∀ t ∈ T ,
+(A.30f)
+pe
+t =
+�
+s∈S
+ˆpe
+ts
+∀ t ∈ T ,
+(A.30g)
+(13), (15) − (16), (19) − (26),
+(A.30h)
+dt, ht, hd
+t , pt, pc
+t, ˆpe
+ts, st, sin
+t , sout
+t
+∈ R+,
+(A.30i)
+zsu
+t , zh
+ts, zoo
+t
+∈ {0, 1},
+(A.30j)
+Ω = {dt, ht, hd
+t , pt, pc
+t, ˆpe
+ts, sin
+t , sout
+t
+, zsu
+t , zh
+ts, zoo
+t }.
+(A.30k)
+B. The simplified MILP with On-Standby States
+This appendix presents the simplified MILP (A.31), taking
+into account on and standby states of the electrolyzer.
+max
+Γ
+�
+t∈T
+ptλDA
+t
++ dtλh − pin
+t λin
+t
+(A.31a)
+s.t.
+pin
+t ≤ P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31b)
+pe
+t ≤ Cezos
+t + P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31c)
+pe
+t ≥ P minzos
+t + P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31d)
+zos
+t =
+�
+s∈S
+zh
+t,s
+∀ t ∈ T ,
+(A.31e)
+pe
+t =
+�
+s∈S
+ˆpe
+ts + P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31f)
+(7), (15) − (16), (19) − (26),
+(A.31g)
+dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts, st, sin
+t , sout
+t
+∈ R+,
+(A.31h)
+zh
+ts, zos
+t ∈ {0, 1},
+(A.31i)
+Γ = {dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts, sin
+t , st, sout
+t
+, zsu
+t , zos
+t }.
+(A.31j)
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+
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+page_content='  Manuel Tobias Baumhof, Enrica Raheli, Andrea Gloppen Johnsen, and Jalal Kazempour  Department of Wind and Energy Systems, Technical University of Denmark, Kgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Lyngby, Denmark  {mtba, enrah, anglopj, jalal}@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='dk  Abstract—Hybrid power plants comprising renewable power  sources and electrolyzers are envisioned to play a key role in  accelerating the transition towards decarbonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' It is common  in the current literature to use simplified operational models for  electrolyzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' It is still an open question whether this is a good  practice, and if not, when a more detailed operational model is  necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This paper answers it by assessing the impact of adding  different levels of electrolyzer details, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', physics and operational  constraints, to the optimal dispatch problem of a hybrid power  plant in the day-ahead time stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Our focus lies on the number  of operating states (on, off, standby) as well as the number  of linearization segments used for approximating the non-linear  hydrogen production curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For that, we develop several mixed-  integer linear models, each representing a different level of  operational details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We conduct a thorough comparative ex-post  performance analysis under different price conditions, wind farm  capacities, and minimum hydrogen demand requirements, and  discuss under which operational circumstances a detailed model  is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In particular, we provide a case under which a  simplified model, compared to a detailed one, results in a decrease  in profit of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='8% and hydrogen production of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5% over a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The key lesson learned is that a detailed model potentially earns  a higher profit in circumstances under which the electrolyzer  operates with partial loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This could be the case for a certain  range of electricity and hydrogen prices, or limited wind power  availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The detailed model also provides a better estimation  of true hydrogen production, facilitating the logistics required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Index Terms—hybrid power plants, electrolyzer, hydrogen,  mixed-integer linear programming  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' INTRODUCTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Background  In order to limit global warming to a maximum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5 °C,  greenhouse gas emissions must be reduced to net zero by 2050,  as called for in the European Green Deal 2019 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Renewable  hydrogen produced through electrolysis could aid in two  major challenges on the path towards the net zero goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' First,  electrolyzers can act as flexible loads and therefore potential  frequency restoration ancillary service providers, contributing  to maintaining the power balance in power systems with  increased penetration of renewable energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Second,  renewable hydrogen can be further synthesized into other  green fuels, eventually enabling decarbonization in the hard-  to-abate sectors, such as heavy transport and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Hybrid power plants comprising of renewable power sources  (wind and/or solar) and electrolyzers are the key components  to accelerate the current energy transition through hydrogen  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Nonetheless, uncertainties in terms of the cost-benefit of  electrolyzers in the long run have challenged the widespread  investment in said technologies and thereby large-scale pro-  duction of renewable-based green hydrogen [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In Denmark,  there is currently a special focus on green hydrogen at the  governmental level and also, among the regulator, system  operator, and many industry stakeholders, envisioning a large  deployment of electrolyzers and other power-to-X facilities in  the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In 2021 the Danish government published  a strategy for the national power-to-X development, aiming to  build 4 to 6 GW of electrolysis capacity by 2030, doubling the  current Danish peak demand [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This emerging trend is not  limited to Denmark, and many other countries both in Europe  and globally see hydrogen as a key solution for the realization  of green societies of the future [2], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Aim and Literature Review  It is a common practice in the current literature to use a  simplified operational model for electrolyzers e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', by using  a constant power-to-hydrogen conversion ratio irrespective of  whether the electrolyzer operates in full capacity or not [6]–  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In addition, some papers do not consider operational states  of the electrolyzer [6], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This paper challenges these simpli-  fication practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' While a simplified model works satisfacto-  rily under certain operational circumstances, there are several  other circumstances under which a simplified one yields a  sub-optimal operation of electrolyzers, underestimating their  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This paper answers when a detailed operational model  should be applied, and to what extent the profit and hydrogen  production can be increased by using a detailed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We will  also discuss to what extent a detailed model brings additional  computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  In general, two main physical aspects of electrolyzers need  to be modeled for operation in the day-ahead time stage:  1) Electrolyzer efficiency: The power-to-hydrogen conver-  sion efficiency is a function of the power consumption  of the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' To accurately model the hydrogen  production of the electrolyzer, the varying efficiency  should be captured, which introduces non-linearities to  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The simple models usually use a constant  efficiency, while more accurate modeling incorporates  the non-linearities, which can be later linearized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  2) Number of operating states: Proper operational modeling  of electrolyzers may require introducing three states,  namely on, off, and standby, to ensure no hydrogen pro-  duction below a given minimum allowed partial loading,  for which additional binary variables are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Many  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='05310v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='OC]  12 Jan 2023  papers in the literature do not even model states, thus  assuming the electrolyzer is always on, or model two  states only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', on and off, similar to conventional power  generators1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Various studies have incorporated different levels of opera-  tional details of the electrolyzer into their optimization prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In [7] and [8], a constant efficiency is applied but two  and three states are modeled, respectively, by adding binary  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In [10], three states are modeled, while assuming a  linear hydrogen production curve, despite showing that the  production curve is not well approximated by a first-order  interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' A hybrid power plant including an electrolyzer  is modeled in [11], where the non-linear hydrogen production  is linearized between two points, with a single binary variable  representing the on/off state of the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In [12] a  quadratic production curve is applied and the resulting non-  linear program is eventually solved by a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In [13],  three states are included, and differently from the other papers,  the operating temperature is considered as a variable, provid-  ing an extra degree of freedom in the electrolyzer operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  This model allows to take into account the temperature impact  on the conversion efficiency and the quality of the generated  heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The non-linear hydrogen production is then linearized  around a fixed reference operating point to formulate the  problem as a mixed-integer linear program (MILP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Contributions and Paper Organization  To the best of our knowledge, there is a lack of a com-  prehensive analysis in the current literature, identifying the  operational circumstances under which a simple model ends  up in a sub-optimal operation of electrolyzers, resulting in a  reduced profit and hydrogen production2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This paper bridges  such a gap through the following contributions:  To embed constraints describing the physics of electrolyz-  ers while keeping the final model as a MILP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  To thoroughly investigate ex-post the impact of the in-  clusion of different operational details on the final profit  of the hybrid power plant and the amount of hydrogen  produced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  and finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' to provide a set of recommendations in  terms of including operational details of electrolyzers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  depending on the application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' the range of electricity  prices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' and the hydrogen price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Without loss of generality, this paper focuses on alkaline  electrolyzers, as they are currently the most mature tech-  nology [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The proposed model can be extended to other  low-temperature electrolyzers, such as polymer electrolyte  membrane (PEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' More operational characteristics may be  necessary for modeling solid-oxide electrolyzers (SOEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  1We will discuss later in Section IV that under some operational conditions,  a two-state model including on and standby states works well too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In contrast,  the two-state model on-off is not satisfactory neither in terms of dispatch  decisions nor the computational performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  2Reference [13] provides a similar analysis, however, the Faraday efficiency  is assumed to be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The consequences of this assumption will be further  discussed in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Section II  describes the electrolyzer physics, focusing on the operating  states and the hydrogen production curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Section III provides  the proposed MILP, representing all three states of the elec-  trolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Section IV discusses the impact of the electrolyzer  modeling choices by means of a test case and a thorough  sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Section V concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Finally,  Appendices A and B provide two MILPs (simpler than the  one proposed in Section III), both representing two states of  the electrolyzer only, where one is a model with on-off states,  and the other one is a model with on-standby states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' ELECTROLYZER PHYSICS  The core of the renewable-hydrogen hybrid power plant is  the electrolyzer, where water is decomposed into hydrogen  and oxygen by means of electrical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The physics and  operating characteristics of alkaline electrolyzers are described  in this section and will be formulated as a set of mixed-integer  linear constraints in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' States  To describe and model the real operation of an alkaline  electrolyzer, it is necessary to distinguish three different states:  1) On state: the electrolyzer operates within its feasible  load range, consuming power and producing hydrogen with a  conversion efficiency that depends on the partial load, which  will be explained in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The minimum operating  power for alkaline electrolyzers is around 15-20% of the  nominal power, below which the electrolyzer must go into  standby or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  2) Standby state: the electrolyzer does not produce any  hydrogen but consumes the power needed to maintain the  system temperature and pressure so that it can rapidly resume  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The value of the standby power consumption is  not usually disclosed by manufacturers, but values between  1-5% of the electrolyzer full load capacity have been adopted  in the literature [7], [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The time needed to switch from  standby to on, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', a warm start-up is of the order of 30 seconds  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  3) Off state: the electrolyzer is shut down completely and  does not consume any power nor produce any hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' How-  ever, to switch back to on, a significant amount of electricity  is needed, corresponding to a cold start-up cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Moreover,  at least 20 minutes are necessary before resuming hydrogen  production [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Apart from the introduced cold start-up cost  and start-up time, the frequent shut down of the electrolyzer  may have a negative impact on the device degradation and  lifetime [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Efficiency and Production Curve  The conversion efficiency of electricity into hydrogen is not  constant but depends on the partial load, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', the ratio between  power consumption at a specific time and the nominal power  of the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The variation of the efficiency based on the  operating set-point is mainly due to two phenomena: (i) the  current-voltage relationship, also called the polarization curve,  2  10  20  30  40  50  Power [MW]  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='0  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5  Efficiency [kg/MWh]  (a)  10  20  30  40  50  Power [MW]  200  400  600  800  Hydrogen [kg/h]  (b)  Non-linear curve  Approximated curve  pe *  h*  hr  } h  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Plot (a): the efficiency curve, and plot (b): the hydrogen production  curve of a 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='25-MW alkaline electrolyzer, as a function of the electric power  consumption, working at 90 °C and 30 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The black curves represent the  original non-linear curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Approximated by two segments, the red curve in  plot (b) is the piecewise linearized hydrogen production curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The non-linear  efficiency curve corresponding to this piecewise linearization is represented  by the red curve in plot (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In our formulation, we will only use the red  piecewise linear production curve in plot (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The inner plot of (b) shows  the hydrogen production discrepancy ∆h between original and approximated  curves, for a given power consumption level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  and (ii) the Faraday efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We explain both phenomena in  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The current-voltage relationship describes the voltage in-  crease (also called over-voltage or over-potential) with increas-  ing current density, due to different losses, as explained in [16]  and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Ulleberg [17] introduced a widely adopted empirical  formulation that describes the relationships between voltage,  current density, and electrolyzer operating temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' To fur-  ther take into account the operating pressure, this formulation  was modified by Sanchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For a given temperature  and pressure, this can be formulated as  U cell(i) = U rev + K1i + K2log(K3i + 1),  (1)  where U cell(i) is the cell voltage as a function of the current  density i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In addition, U rev is the open-circuit voltage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=',  voltage corresponding to current density equal to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The  parameters K1, K2, K3 are constants obtained from experi-  mental data and can be found in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Voltage U rev can be  calculated for a specific operating temperature according to  an empirical equation that can be found in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The power  consumed by the electrolyzer pe(i) can be calculated as  pe(i) = U cell(i)iA,  (2)  where A is the total area of the cells composing the elec-  trolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The Faraday law calculates the hydrogen production  h(i) of the electrolyzer as  h(i) = 3600 · ηF(i)M H2iA  2F  ,  (3)  where h(i) is the hydrogen production rate in kg/h, M H2 is the  molar mass of hydrogen in kg/mol, F is the Faraday constant,  and ηF(i) is the Faraday efficiency as a function of current  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The latter is defined as the ratio between the actual and  the theoretical maximum amount of hydrogen produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The  difference between actual and theoretical output is explained  in [17], and it increases significantly when the electrolyzer  is working at low-current densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In [18], an empirical  expression that captures the relationship between the Faraday  efficiency and the current density at a given temperature is  provided: ηF(i) is close to one for higher current densities, and  it drops to zero when reducing the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The electrolyzer  efficiency is defined as  η(i) = h(i)  pe(i),  (4)  where generally η(i) is expressed in kg/MWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For different  values of i, the black curve in Figure 1(a) shows efficiency η(i)  versus power consumption pe(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In addition, the black curve  in Figure 1(b) shows the hydorgen production h(i) versus  power consumption pe(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For notational clarity, we drop (i)  in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The black curves in Figure 1 show that  the model is non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The efficiency has a peak at around  30% of the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This characteristic peak in the efficiency  curve is not captured when a constant conversion efficiency is  used, as done in [6], [8], [10], or when the Faraday efficiency  is assumed to be equal to one in the entire feasible operating  range, as done in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  To keep the final problem a MILP, but describe the hydrogen  production with more details, we use a piecewise linearization  of the hydrogen production curve as shown by the red curve in  Figure 1(b), for two linearization segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For each segment  s ∈ S, the As (slope) and Bs (intercept) coefficients of the  line can be calculated such that the approximated hydrogen  production is Aspe + Bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Later we will define a binary  variable indicating which segment is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The proposed  approximation is exact only at the segment endpoints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=',  linearization points), otherwise, it is an underestimation of  the original non-linear curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For example, the optimal power  set-point pe∗ in the inset of Figure 1(b) corresponds to the  hydrogen production h∗ according to the proposed piecewise  linear model with two segments3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' However, the actual hydro-  gen realization based on the electrolyzer physics is hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The  hydrogen production difference ∆h is reduced by increasing  the number of segments, and the effect of the hydrogen surplus  obtained when choosing only one segment, as done in [10], is  discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  According to this piecewise linear formulation for the  hydrogen production curve, the efficiency η for segment s  can be calculated based on (4), resulting in η = As + Bs  pe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  This is depicted by the red dotted curve in Figure 1(a), given  two linearization segments used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Note that it does not present  a linear behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' However, this non-linear efficiency curve  does not appear in our optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The hydrogen  production curve is used instead, which is linearized through  segments, as illustrated by the red dotted curve in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' PROBLEM FORMULATION  We consider a hybrid power plant, as depicted in Figure 2,  consisting of a wind farm, an electrolyzer, a hydrogen com-  pressor, and a hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The generated wind power  can be either sold to the grid at the electricity market price,  3Symbol ∗ refers to the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  3  Wind farm  Grid  Electrolyzer  Compressor  Hydrogen  storage  Hydrogen  demand  Electricity  Hydrogen  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Schematic representation of a hybrid power plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  or consumed by the electrolyzer to produce 100% renewable-  based green hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The hydrogen produced can either be  directly delivered to the demand or temporarily stored in an on-  site hydrogen storage, with an associated cost for compressing  the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The dashed blue line in Figure 2 represents the option  to buy electricity from the grid only to supply the electrolyzer’s  standby power when there is no wind power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The hydrogen price is assumed to be a single-value constant,  and the hybrid power plant serves a minimum daily hydrogen  demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We assume the plant has perfect foresight of future  wind power production and electricity price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Given the 1-hour  time resolution in our model, we neglect the ramping limitation  which are typically around ±20% of the nominal power per  second [10], as well as the warm and cold start-up times of  the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  For the optimal operation of the hybrid power plant, we  develop a complete MILP in Section III-A accounting for  three states of the electrolyzer and then provide two simplified  counterparts in Section III-B, each with two states of the  electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Notation: All parameters are upper-case or Greek letters,  whereas all variables are lower-case letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' All binary vari-  ables are noted by z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Three-state Model  The most complete MILP includes the objective function  (6) constrained by (7)-(29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  1) Objective function: Over the set of hours t ∈ T , the  objective function (6) maximizes the total profit of the hybrid  power plant as  max  x  �  t∈T  ptλDA  t  + dtλh − pin  t λin  t − zsu  t λsu,  (6)  where the variable set x will be defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The first term  corresponds to selling power pt to the grid at the day-ahead  electricity market price λDA  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The second term pertains to  delivered hydrogen dt at a fixed price λh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The third term  represents the cost for purchasing standby power pin  t to support  the electrolyzer’s standby state in case the wind power is  insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The corresponding price is λin  t  = λDA  t  + λTSO,  where λTSO is the grid tariff imposed by the Transmission  System Operator (TSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Finally, the fourth term corresponds  to the cold start-up cost of the electrolyzer, where the binary  variable zsu  t  indicates the start-up at hour t, associated with  the cost per startup λsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  2) Power balance: In every hour t, the power pt sold in the  day-ahead market is equal to the wind farm power production  P w  t plus power pin  t bought from the grid to support the standby  state of the electrolyzer, subtracted by the power consumption  pe  t of the electrolyzer and the power consumption pc  t of the  compressor, such that  pt = P w  t + pin  t − pe  t − pc  t  ∀ t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (7)  3) Limit on pin  t : The input power pin  t  is limited by the  standby state consumption of the electrolyzer, implying that  power cannot be bought from the grid to produce hydrogen:  pin  t ≤ P sbzsb  t  ∀ t ∈ T ,  (8)  where the parameter P sb is the standby consumption, and the  binary variable zsb  t  indicates whether the electrolyzer is in the  standby mode in hour t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  4) Electrolyzer operational states: Constraint (9) ensures  that the electrolyzer can take only one out of three states at  any hour t, namely online, standby, or off:  zon  t  + zoff  t  + zsb  t  = 1  ∀ t ∈ T ,  (9)  where similar to zsb  t , binary variables zon  t  and zoff  t  indicate  whether in hour t the electrolyzer is on and off, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The states are activated based on the electricity consumption of  the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In the online state, the electricity consumption  pe  t of the electrolyzer can neither exceed the capacity Ce nor  go below a minimum load limit P min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In the standby state, the  electricity consumption must be equal to the standby power  consumption P sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' These constraints are enforced by  pe  t ≤ Cezon  t  + P sbzsb  t  ∀ t ∈ T  (10)  pe  t ≥ P minzon  t  + P sbzsb  t  ∀ t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (11)  To represent the cold start-up of the electrolyzer, the binary  variable zsu  t  is defined, taking the value 1 in the case of  a transition from off to on state in hour t, as enforces by  constraints (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Further, constraint (14) ensures  that the transition from an off-state to a standby-state is not  allowed, to avoid bypassing of the start-up cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  zsu  t  ≥ zon  t  − zon  t−1 − zsb  t−1  ∀ t ∈ T \\1,  (12)  zsu  t=1 = 0,  (13)  zoff  t−1 + zsb  t  ≤ 1  ∀ t ∈ T \\1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (14)  5) Electrolyzer hydrogen production: The hydrogen pro-  duction ht is a function of the electricity consumption of the  electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' As explained in Section II-B, for each segment  s ∈ S, a linear function of the segment power consumption  ˆpe  ts with slope As and intercept Bs is defined, such that  ht =  �  s∈S  (Asˆpe  ts + Bszh  ts)  ∀ t ∈ T ,  (15)  where the binary variable zh  ts defines which segment s is active  in hour t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Each segment is valid within a pre-defined interval  of upper P s and lower P s power consumption levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=',  P szh  ts ≤ ˆpe  ts ≤ P szh  ts  ∀ t ∈ T , s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (16)  4  VectorStock  VectorStock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='com/24756804shutterstock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='com • 1658641081Constraint (17) ensures that hydrogen production happens  in the online state only, while one segment only can be active  at any hour t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In addition, (18) computes the total power  consumption of the electrolyzer:  zon  t  =  �  s∈S  zh  t,s  ∀ t ∈ T  (17)  pe  t =  �  s∈S  ˆpe  ts + P sbzsb  t  ∀ t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (18)  6) Hydrogen storage: Constraints (19)-(25) represent the  storage operation:  ht = hd  t + sin  t  ∀ t ∈ T ,  (19)  dt = hd  t + sout  t  ∀ t ∈ T ,  (20)  sout  t  ≤ Sout  ∀ t ∈ T ,  (21)  pc  t = Kcsin  t  ∀ t ∈ T ,  (22)  st=1 = Sini + sin  t=1 − sout  t=1  (23)  st = st−1 + sin  t − sout  t  ∀ t ∈ T \\1,  (24)  st ≤ Cs  ∀ t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (25)  The hydrogen produced ht can either go directly to the demand  hd  t or be injected into the hydrogen storage sin  t , as enforced  by (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The total hydrogen dt delivered to the demand is  equal to the sum of hydrogen directly from the electrolyzer  and that from the storage sout  t  , as per (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The storage  output of every hour is limited by the output flow capacity  Sout in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Further, the compressor consumes power pc to  compress the hydrogen injected into the storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Assuming  adiabatic compression, the compression coefficient Kc can be  calculated, as proposed by [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The power consumption for  compression is then (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The state of charge of the hydrogen  storage in the initial and following hours is calculated by  (23) and (24), where Sini is the hydrogen initially stored in  the storage at the beginning of time horizon T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The storage  hydrogen mass capacity Cs is enforced by (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Note that we  do not impose any constraint for the energy stored at the end  of time horizon T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Therefore, pursuing profit maximization in  this time horizon, the hybrid power plant will leave the storage  empty in the last hour4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  7) Hydrogen demand: Imagine within the underlying time  horizon T , which could be, for example, a year, there are N  number of time subsets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', 365 days, indexed by n, such  that there is a minimum hydrogen demand for each n:  �  t∈Hn  dt ≥ Dmin  n  ∀ n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', N},  (26)  where Hn is the set of hours within time subset n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  8) Variable declaration: Constraint (27) declares the non-  negativity conditions:  dt, ht, hd  t , pt, pc  t, pin  t , ˆpe  ts, st, sin  t , sout  t  ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (27)  4One can enforce a constraint on the minimum stored hydrogen at the end  of the time horizon, or add a value for this stored energy to the objective  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Constraint (28) lists binary variables:  zsu  t , zh  ts, zon  t , zoff  t , zsb  t  ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (28)  Therefore, the total number of binary variables is |T |(4 +  |S|) binaries, where |T | and |S|, respectively, are the number  of hours and the number of segments used to linearize the  hydrogen production curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Finally, the variable set x is  defined as  x = {dt, ht, hd  t , pt, pc  t, pin  t , ˆpe  ts,  sin  t , st, sout  t  , zsu  t , zh  ts, zon  t , zoff  t , zsb  t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (29)  Accordingly, in addition to |T |(4+|S|) number of binary vari-  ables, we have |T |(9 + |S|) number of continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Two-state Models  The optimal operation problem (6)-(29) of the hybrid power  plant accounting for three states of the electrolyzer can be  simplified if two states only are considered, either on-off states  or on-standby states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Both result in MILPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  In the latter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', the MILP with on-off states, one binary  variable (instead of three) per hour t is sufficient, such that it  indicates whether the electrolyzer in the given hour is on or  off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The resulting MILP is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The total  number of binary variables in this MILP is |T |(2 + |S|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Similarly, a single binary variable per hour t is enough  in the MILP with on-standby states, indicating whether the  electrolyzer is online or in standby mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Also, the start-up  binary variable is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The corresponding MILP is  given in Appendix B, where among three MILPs, we need  the lowest number of binary variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', |T |(1 + |S|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' NUMERICAL STUDY  We apply the proposed MILPs of Section III to a case study  and investigate how the optimal operation of the hybrid power  and the resulting profit change by adding more operational  details of the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' All source codes and input data are  publicly shared5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We consider several options for the number  of linearization segments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', |S|, used to approximate the  hydrogen production curve of the electrolyzer, including 1, 2,  4, 8, and 12 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Also, we consider three options for  the number of electrolyzer states: three states on-off-standby  (OOS), two states on-standby (OS), and two states on-off  (OO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In the rest of this section, we will refer to various  models as, for example, OOS-12, implying we consider three  states (OOS) with 12 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Finally, we conduct a sensitiv-  ity analysis to explore the impact of various input parameters,  such as wind farm capacity, hydrogen demand, and hydrogen  price, on the operation of the hybrid power plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Case Study  We consider a hybrid power plant whose structure equals  the one in Figure 2, and its input data is provided in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The capacity of the wind farm is 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5 MW, corresponding to  11 V164-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5 MW™ Vestas turbines, located in Køge Bay,  5GitHub: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='com/mtba-dtu/detailed-electrolyzer-model  5  TABLE I  INPUT DATA FOR THE CASE STUDY  Wind farm  Capacity  Cw  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5  MW  Electrolyzer  Capacity  Ce  50%  of Cw  Standby load  P sb  1%  of Ce  Minimum load  P min  15%  of Ce  Pressure  30  bar  Temperature  90  °C  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' current density  5,000  A/m2  Start-up cost  λsu  2,612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='50  C [10]  TSO tariff  λTSO  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='06  C/MWh  Storage  Capacity  Cs  22,000  kg  Maximum output  Sout  912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='13  kg/h  Compressor  Inlet temperature  40  °C  Inlet pressure  30  bar  Outlet pressure  200  bar  Mechanical efficiency  75%  Hydrogen  Price  λh  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='10  C/kg  Minimum demand  Dmin  n  3,667  kg/day  Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The electrolyzer capacity is set to 50% of the  wind farm capacity, amounting to 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='25 MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The modeling  horizon spans one year with an hourly temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  We apply hourly electricity price data for 2019, as price data  for the following years might be distorted by macroeconomic  impacts, such as COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Day-ahead electricity prices for  the East Denmark area (DK2) are obtained from ENTSO-e  Transparency platform [19] and hourly historical wind capac-  ity factors at the given location for 2019 are retrieved from  the Renewable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='ninja web platform [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The average yearly  capacity factor for the selected location is 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The hybrid  power plant is only allowed to buy power from the grid to  keep the electrolyzer in standby mode, in case the wind power  is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In that case, the electricity is bought at the  hourly day-ahead market price plus the grid tariff of the TSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Since the wind farm is located in DK2, the consumption tariff  imposed by the Danish TSO, Energinet, is applied [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The  minimum daily demand can be met by the full-load operation  of the electrolyzer for around four hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The hydrogen storage  is scaled to store all hydrogen produced if the electrolyzer  operates at full capacity for 24 consecutive hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Impacts of the Number of Segments  Let us consider the OOS case with three states, for which  we solve the proposed MILP (6)-(29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We start with OOS-1,  where |S| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This means the original non-linear hydrogen  production curve, depicted in Figure 1(b), is approximated by  a single linear curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Here, the minimum power consumption  P min and the capacity Ce of the electrolyzer are taken as  two endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' By moving to OOS-2, where the number of  segments |S| is 2, we consider an additional point P η,max,  which refers to the power consumption level corresponding to  the peak in the efficiency curve in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' By increasing  |S| to 4, and then to 8, the mean load value between existing  points is added, splitting one segment into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The same  procedure but only on the right side of P η,max is applied  when we move from OOS-8 to OOS-12, as this side covers  1  6  12  18  24  0  20  40  60  Electrolyzer power [MW]  (a)  1  6  12  18  24  Time [h]  (b)  1  6  12  18  24  (c)  Ce  P , max  Pmin  Psb  pe  t  DA  t  25  32  38  45  Day-ahead price [ /MWh]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The power consumption schedule of the electrolyzer (pe  t) in an  example high-wind day when its hydrogen production curve is linearized  by (a) 1, (b) 4, and (c) 12 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' These three plots, from left to right,  correspond to cases OOS-1, OOS-4, and OOS-12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  over around 70% of the feasible operating range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' With the  adoption of this procedure, all cases from OOS-2 to OOS-12  include the point P η,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In addition, points are not removed  when refining the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' By adding more segments, the  hydrogen production curve and thus the electrolyzer efficiency  with partial loading is more accurately represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The increase in the number of segments |S| enables the  electrolyzer to consume power more flexibly, as depicted in  Figure 3, where the optimal power consumption schedule of  the electrolyzer for one example day of the year is shown  for three different numbers of segments (1, 4, and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' It is  observed that when the optimal power consumption of the  electrolyzer is not constrained by wind production shortage,  as on the chosen day, the optimal consumption level is always  one of the piecewise linearization points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' There are instances,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', hour 5 in Figure 3, where OOS-1 goes into the standby  state as the day-ahead price is too high for profitable hydrogen  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In contrast, OOS-4 and OOS-12 continue the  operation in the on state, but at the power consumption level  corresponding to the maximum efficiency, where hydrogen  production is still profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The number of segments |S| plays an important role in the  optimal dispatch decision when the day-ahead price lies within  a specific price range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The upper bound of this price range  corresponds to the highest price for which the production of  hydrogen is still profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The lower bound is the price below  which the optimal dispatch decision is always the maximum  electrolyzer consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Figure 4 shows the distribution of  the day-ahead price λDA  t  over 8,760 hours of year 2019 in  DK2 with the bounds of the price range of interest are marked  by the red and green dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The upper bound is found  as the day-ahead price for which the hydrogen production is  only feasible at the maximum efficiency, denoted by α in the  inner plot of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The lower bound corresponds to the  efficiency at the full load, denoted by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' If the day-ahead price  of a given hour lies outside of this range, the dispatch decision  for any number of segments would be the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' produce at  the maximum possible load or cease the production, and there  would be no added value of a detailed production curve 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  This will be further investigated in Section IV-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  6These two price thresholds are calculated by multiplying the hydrogen  price and the efficiency at points β and α, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  6  25  30  35  40  45  50  55  Day-ahead price [ /MWh]  0  50  100  150  200  250  300  350  Frequency  Price  Price mean  Power [%]  Efficiency  [kg/MWh]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Histogram of the day-ahead electricity price over 8,760 hours of year  2019 in DK2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Prices λα and λβ correspond to electricity prices for which  the electrolyzer operates at points α and β, indicated in the inner plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Impacts of the States  We consider three cases OOS, OO, and OS, each for both  1 and 12 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Recall that their corresponding MILPs  are different7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Comparing the results of MILPs with the same  number of segments, we observe OS and OOS perform almost  equally, as observed in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The reason for this is the low  frequency of consecutive hours of too high day-ahead prices,  where a complete shut-off would be preferred over the standby  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Over 8,760 hours, OOS-1 starts up only 2 times, with a  total of 286 hours offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The difference in results obtained for  OS and OOS increases if a higher standby power consumption  or lower cold start-up cost for the electrolyzer is assumed,  which would lead to more frequent shut-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' On the contrary,  OO earns the lowest profit, mainly due to the high start-up  cost, which decreases the operational flexibility as even a short  pause in production incurs a high cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Ex-post Performance Analysis  Recall that three MILPs solve the problem based on the  linearized hydrogen curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Through the following ex-post  performance analysis, it is seen that this leads to both sub-  optimal dispatch decisions and an underestimation of the true  amount of hydrogen produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We have already observed in  Figure 1(b) that the linearized red curve is below the original  black non-linear hydrogen production curve, implying that the  hydrogen production might be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This means that  we can expect to produce more hydrogen than what MILPs  calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Such a difference is expected to be reduced by  using more segments |S| to approximate the original non-  linear hydrogen production curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Pursuing a fair comparison among models, we conduct an  ex-post performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Once the MILPs are solved and  the optimal power consumption pe∗  t  of the electrolyzer ob-  tained, we re-calculate the true amount of hydrogen produced  based on the original non-linear hydrogen production curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Note that we do not re-optimize the problem8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We refer to  the amount of extra hydrogen and its corresponding profit as  7While we solve the proposed MILP (6)-(29) for OOS, the MILPs  presented in Appendixes A and B are solved for OO and OS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  8To avoid re-optimization, we assume the extra hydrogen is directly sold to  the demand and is not stored in the hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Otherwise, one needs to  re-optimize a posteriori to optimize the operation of storage and compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  12  8  4  2  1  Number of segments  97  98  99  100  Profit [%]  OOS  OS  OO  States  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='8  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='9  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='1  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='2  Profit [million ]  OOS  Realized surplus  1 seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  12 seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Estimated and realized surplus profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The first five bars from the  left correspond to OOS-1 to OOS-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The next six bars show the results for  OO-1, OO-12, OS-1, OS-12, OOS-1, and OOS-12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The right  vertical axis is the profit in million C, whereas the left vertical axis is the  relative profit in % in comparison to the highest profit achieved by OOS-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  12  8  4  2  1  Number of segments  80  85  90  95  100  H2 production [%]  OOS  OS  OO  States  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='9  H2 production [thousand tons]  OOS  Realized surplus  1 seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  12 seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Estimated and realized surplus hydrogen produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The first five bars  from the left correspond to OOS-1 to OOS-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The next six bars show the  results for OO-1, OO-12, OS-1, OS-12, OOS-1, and OOS-12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  “realized surplus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We assume that all extra hydrogen is sold  at the same constant price, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='10/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Figure 5 provides the estimated and realized surplus profit  among different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The estimated profit (gray area) is  the optimal value obtained for the objective function of the  corresponding MILP, while the realized profit (dark area),  calculated ex-post, takes into account the profit of selling extra  hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Similarly, Figure 6 shows the total estimated and  realized surplus hydrogen produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Note that the compressor  would need to consume more power (around 1 MWh/ton) due  to extra hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We draw two conclusions from Figures 5  and 6:  (1) Realized surplus: This surplus for profit and hydrogen  production is reduced by increasing the number of segments,  due to the improved approximation of the original non-linear  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The realized surplus profit decreases from C71,199  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='44%) for OOS-1 to C602 (below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='01%) for OOS-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Similarly, the hydrogen production surplus is significantly  decreased, yielding a realized surplus of ∼ 34 tons (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='27%) for  OOS-1 and only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='3 tons (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='01%) for OOS-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' By choosing  a low number of segments, the hydrogen production is under-  estimated which may lead to logistic issues and inefficiencies  in the real-life operation of the hybrid power plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (2) Ex-post profit and hydrogen production: Adding more  electrolyzer details (segments or/and states) always leads to  an increase in the ex-post profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' To compare various models,  7  TABLE II  COMPUTATIONAL ASPECTS  Case  Computational time [s]  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' of binary variables  OS-1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='4  2×8760  OS-12  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='7  13×8760  OOS-1  137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='8  5×8760  OOS-2  135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='8  6×8760  OOS-4  236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='3  8×8760  OOS-8  350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='3  12×8760  OOS-12  473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='7  16×8760  OO-1  767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='1  3×8760  OO-12  1,763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='1  14×8760  OOS-12 is taken as a benchmark, as it leads to the highest  profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' First, the impact of the number of segments is examined,  while keeping the number of states fixed and equal to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The  ex-post profit reduction applying 1 instead of 12 segments is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='72%, corresponding to around 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='6 kC for the entire hybrid  power plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The ex-post hydrogen production is increased by  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='32%, corresponding to around 241 tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This percentage  deviation is notably higher in part because the increase in  hydrogen profit is dampened by the reduction in electric-  ity profit (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='86% electricity profit increase for 1 segment  compared to 12 segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For OOS-1, the profit share of  selling hydrogen is much lower than the profit share of selling  electricity (around 34%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' By introducing more segments, the  contribution of hydrogen sales is increased to 38% at the  expense of electricity sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' More profit and different business  models are therefore unlocked by including more electrolyzer  details in the MILP formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Figures 5 and 6 show that the  errors are considerably reduced by implementing 4 segments  instead of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Second, we assess the impact of the states on the  ex-post profit and hydrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' While OS performs  just as well as OOS as described in Section IV-C, OO with 12  segments results in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='22% lower ex-post profit, and in a 4%  lower hydrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For OO-1, a profit reduction of  around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='8% and a reduced hydrogen production of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='5% are  observed, compared to the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Finally, we observe that  neglecting the standby state in the model formulation leads to  the worst outcome in terms of profit and hydrogen production  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Computational Analysis  All MILPs have been solved using the Gurobi solver in Julia  on a MacBook Pro M1 2020 with 16 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The optimality  gap is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='01% when we solve every MILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The  increase in the number of linearization segments |S| leads to an  increase in computational time due to introducing more binary  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For OOS, the computational time is increased from  138 seconds for 1 segment to 474 seconds for 12 segments,  as reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Removing the off state significantly  reduces the computational time, with OS-1 being by far the  fastest MILP to be solved (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='4 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The OO models  require the highest computational time, although they embody  fewer binary variables than their OS and OOS counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  We hypothesize the reason is that the start-up cost constraints  with inter-temporal nature are more often active when the  option of standby state is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Therefore, we do  not recommend using OO as its corresponding profit is the  lowest among all cases (Figure 5), and it is being solved  comparatively slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Further, if computational efficiency is  crucial, it may be beneficial to neglect the off state and run  the OS model for improved computational performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In  general, the computational time increases with the number  of segments but is deemed reasonable for the OS and OOS  models, considering that our optimization problem is run over  8,760 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' As operational problems are typically solved  for a shorter time horizon, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', 24 hours for day-ahead  scheduling, the computational cost of adding more details to  the electrolyzer would be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Sensitivity Analysis with Respect to Input Data  In the previous sections, we have shown that adopting a  simplified electrolyzer model can lead to an underestimation  of the profit and hydrogen production for the hybrid power  plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We have also shown that the benefit of added details is  case-specific, and depends on the input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We now  aim at assessing the impact of input parameters and system  configuration on these results, through a sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  In particular, we will focus on wind over electrolyzer capacity  ratio, hydrogen demand over electrolyzer capacity ratio, and  the hydrogen price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The sensitivity analysis is performed on  the OOS-1 and OOS-12 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  1) Wind size: Recall from Table I that the wind farm  capacity is 2 times that of the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' To assess the impact  of the wind-to-electrolyzer capacity ratio, two additional cases  are considered, under which such a ratio is 1, 2 (reference),  and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' When this ratio is reduced from 2 to 1, the number  of hours where the power input to the electrolyzer is limited  by the wind availability is increased from 5,326 to all hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Conversely, when the ratio is increased from 2 to 8, the number  of power-limited hours is reduced to 1,236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We observe that  the realized surplus for hydrogen production increases with  the number of hours with limited wind power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The reason for  this is that the piecewise approximation is exact only on the  linearization points, and the limited wind availability forces  the electrolyzer to operate out of those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Conversely,  when the number of wind power-limited hours is reduced,  the electrolyzer operates more often on the linearization  points, where the approximation is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' It follows that the  underestimation of hydrogen production is greater the more  the electrolyzer is limited from operating at the linearization  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' With a wind-to-electrolyzer ratio of 1, the difference  in ex-post hydrogen production between 1 and 12 segments  is 13%, which is reduced to 3% when the ratio increases to  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Therefore, incorporating electrolyzer details is crucial for  hybrid power plants where the wind-to-electrolyzer capacity  ratio is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  2) Hydrogen demand size: To investigate the sensitivity of  optimization outcomes with respect to the hydrogen demand,  the minimum daily demand is doubled, corresponding to  around 8 full-load hours of hydrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We observe  that the impact of adding more segments to the electrolyzer  8  production curve diminishes when the demand constraint is  tighter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', with a higher minimum daily demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For the case  with the reference demand, the difference between the ex-post  profit for OOS-12 and OOS-1 is 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This difference, when the  hydrogen demand is doubled, is reduced to 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The increase  in demand forces the electrolyzer to operate more frequently  at its maximum load, where both OOS-1 and OOS-12 share  the same linearization point and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  3) Hydrogen price: To explore the impact of the hydrogen  price, we increase it from C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='10/kg to C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='00/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' As already  discussed in Section IV-B, adding more segments impact  the optimal solution and profit as long as the electricity  price in the given hour is in the range [λβ, λα], shown in  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Since λα and λβ are proportional to the hydrogen  price, by increasing the hydrogen price, the range [λβ, λα] is  widened and moved towards higher electricity prices, where  the frequency of occurrence is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' When the MILP is  solved with the hydrogen price of C5/kg, it is more frequently  optimal to operate the electrolyzer at full load (39% of the  time, compared to 11% for the case with the hydrogen price of  C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='1/kg) and the linearization segments are utilized less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This  also results in a significantly decreased computational time  (below 20 seconds for OOS-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The profit contribution from  the hydrogen sale is increased significantly to 92%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The ex-  post profit and hydrogen production difference between OOS-  1 and OOS-12 are reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='01% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='03%, respectively  (they are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='72% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='32% for the C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='1/kg case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The modeling of segments is relevant if higher hydrogen  prices are coupled with also higher electricity prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In  this way, the electricity price range [λβ, λα] would still be  overlapping with the majority of day-ahead price occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  For example, we test an artificial case where the day-ahead  electricity price time series was multiplied by a constant factor  to increase the mean price to around C90/MWh (similar to  the mean value for 2021 in DK2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In this case, with the  hydrogen price of C5/kg, similar results to the 2019 test case  with the hydrogen price of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='1/kg were obtained in terms of  the impact of the number of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For a given hydrogen  price and efficiency curve, checking if the price range [λβ, λα]  overlaps with the expected electricity price is therefore crucial  to assess a priori the impact of choosing a simplified model  for the production curve (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', 1 linearization segment only)  and support the modeling choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSION  Several studies have focused on the optimal dispatch of  hybrid renewable-hydrogen power plants assuming simplified  models for the electrolyzer component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This paper investigates  the impact of choosing different levels of operational details  for the electrolyzer model on the dispatch decisions, profit, the  amount of hydrogen produced, and computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The  impact of two modeling choices is considered: the operating  states (on, off, standby), and the number of segments used  to linearize the hydrogen production curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The problems are  formulated as MILPs, where the number of binary variables  depends on the number of states and segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  For fixed states, adding more linearization segments for  approximating the hydrogen production curve results in a  higher profit, and a reduced surplus in the ex-post profit  calculation, meaning that the model is able to estimate the  actual cost and revenue streams more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Moreover,  a better estimation of the produced hydrogen is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  In fact, the linearization results in an underestimation of the  produced hydrogen, but the underestimation is reduced by  increasing the number of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Apart from introducing  errors in the actual realized profit, thus potentially impacting  the investment decisions in these types of technologies, the  systematic underestimation of the hydrogen produced by the  electrolyzer might introduce logistical inefficiencies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=', truck  scheduling, and storage discharging/filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  The impact of adding more piecewise segments to the  hydrogen production curve depends on the distribution of day-  ahead electricity prices in the given time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The model  formulations with 1 and 12 segments take significantly dif-  ferent dispatch decisions when the day-ahead electricity price  is within a certain range, which depends on the electrolyzer  efficiency (minimum and maximum) and the hydrogen price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Out of this day-ahead electricity price range, the model with 1  and 12 segments takes the same dispatch decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Therefore,  the value of adding more details to the hydrogen production  curve could differ by varying input data and case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' It  is observed that this value decreases when the electrolyzer  operates less at partial loading, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' when the input power is  less limited by available wind power or with high-demand  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' In this paper, revenues from other than the day-  ahead market are not considered but this may also impact the  dispatch strategy and therefore benefit from more segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Choosing to represent only on and off states leads to the  highest profit underestimation and worst ex-post performance  while modeling only on and standby states lead to similar  profit and dispatch decisions to the three-state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This  result is, however, significantly affected by the assumption  made on the standby power consumption of the electrolyzer  and its start-up cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' These parameters are highly uncertain  due to the lack of data on large-scale electrolyzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  In conclusion, adopting more simplified models for the  electrolyzer always leads to a reduced profit and sub-optimal  scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' However, the impact of adding more details may  vary depending on the case study considered and especially  the range of day-ahead electricity prices, hydrogen price, wind  power production compared to the electrolyzer installed capac-  ity, standby power consumption, and start-up cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Among all  considered models, the most complete one (three states with  12 segments) was solved for a 1-year horizon in less than  10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The increase in computational time by adding  more details would be marginal if a day-ahead scheduling  problem is considered instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Moreover, reducing the three-  state model to two states only is not always faster, as it was  observed that the two-state on-off model with 12 segments  was the longest to solve among all the cases considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' A  more detailed representation of the electrolyzers should be  preferred for operational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' For investment problems,  9  we hypothesize that it may be adequate to adopt a more  simplified model of the electrolyzer, but this should be further  assessed and it was out of the scope of the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Further research should be conducted to assess the impact of  modeling choices when additional revenue streams are consid-  ered, such as flexibility provisions in ancillary service markets,  which may impact the dispatch decisions of the hybrid power  plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Additionally, as there is a high uncertainty related to the  start-up and standby costs, the sensitivity of these parameters  on the impact of added details should be assessed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Moreover, the level of detail needed for investment problems  should be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The modeling of electrolyzer  cell degradation over time should be investigated and included  in the model with additional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' Finally, uncertainties  in wind power supply and electricity prices should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  This research was supported by the Energy Cluster Den-  mark through the “Sustainable P2X Business Model” project,  and by the Danish Energy Development Programme (EUDP)  through the HOMEY project (64021-7010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' We would like to  thank Jens Jakob Sørensen (Ørsted), Alexander Holm Kiilerich  (Ørsted), Roar Hestbek Nicolaisen (Hybrid Greentech), Yan-  nick Werner (DTU), and Matˇej Novotn´y for collaborations,  thoughtful discussions, and constructive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The simplified MILP with On-Off States  This appendix provides the MILP (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30a), where the on  and off states of the electrolyzer are only modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' This is a  simplified model compared to the one proposed in Section III  with three states of the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  max  Ω  �  t∈T  ptλDA  t  + dtλh − zsu  t λsu  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  pt = P w  t − pe  t − pc  t  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30b)  pe  t ≤ Cezoo  t  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30c)  pe  t ≥ P minzoo  t  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30d)  zsu  t  ≥ zoo  t  − zoo  t−1  ∀ t ∈ T \\1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30e)  zoo  t  =  �  s∈S  zh  t,s  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30f)  pe  t =  �  s∈S  ˆpe  ts  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30g)  (13), (15) − (16), (19) − (26),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30h)  dt, ht, hd  t , pt, pc  t, ˆpe  ts, st, sin  t , sout  t  ∈ R+,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30i)  zsu  t , zh  ts, zoo  t  ∈ {0, 1},  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30j)  Ω = {dt, ht, hd  t , pt, pc  t, ˆpe  ts, sin  t , sout  t  , zsu  t , zh  ts, zoo  t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='30k)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' The simplified MILP with On-Standby States  This appendix presents the simplified MILP (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31), taking  into account on and standby states of the electrolyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  max  Γ  �  t∈T  ptλDA  t  + dtλh − pin  t λin  t  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  pin  t ≤ P sb(1 − zos  t )  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31b)  pe  t ≤ Cezos  t + P sb(1 − zos  t )  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31c)  pe  t ≥ P minzos  t + P sb(1 − zos  t )  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31d)  zos  t =  �  s∈S  zh  t,s  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31e)  pe  t =  �  s∈S  ˆpe  ts + P sb(1 − zos  t )  ∀ t ∈ T ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31f)  (7), (15) − (16), (19) − (26),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31g)  dt, ht, hd  t , pt, pc  t, pin  t , ˆpe  ts, st, sin  t , sout  t  ∈ R+,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31h)  zh  ts, zos  t ∈ {0, 1},  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31i)  Γ = {dt, ht, hd  t , pt, pc  t, pin  t , ˆpe  ts, sin  t , st, sout  t  , zsu  t , zos  t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='31j)  REFERENCES  [1] European  Commission,  “A  European  green  deal,”  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  https://commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='eu/strategy-and-policy/priorities-2019-  2024/european-green-deal en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
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+page_content='  https://energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
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+page_content='eu/system/files/2020-07/  hydrogen strategy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  [3] International  Energy  Agency,  “Global  hydrogen  review,”  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
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+page_content='blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
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+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
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+page_content='  [5] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Raj,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Lakhina,  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  Stranger,  “Harnessing  green  hydrogen opportunities for deep decarbonisation in India,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
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+page_content=' 114, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  [21] Energinet, “Aktuelle tariffer,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content=' https://energinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='dk/El/Elmarkedet/  Tariffer/Aktuelle-tariffer/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE4T4oBgHgl3EQf4A1b/content/2301.05310v1.pdf'}
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+arXiv:2301.04848v1  [math.FA]  12 Jan 2023
+τ-QUANTIZATION AND τ-COHEN CLASSES DISTRIBUTIONS
+OF FEICHTINGER OPERATORS
+FEDERICO BASTIANONI AND FRANZ LUEF
+Abstract. We investigate the τ-quantizations and Cohen’s class distributions
+of a suitable class of trace-class operators, called Feichtinger’s operators, and
+show that it is a convenient substitute for the class of Schwartz operators. Many
+well-known concepts and results for functions in time-frequency analysis have an
+operator-analog in our setting, e.g. that Cohen’s classes are convolutions of Wigner
+functions with distributions or characterization of the class of Schwartz operators
+as an intersection of weighted variants of the class of Feichtinger operators.
+Contents
+1.
+Introduction
+1
+2.
+Preliminaries
+3
+2.1.
+A family of time-frequency representations
+3
+2.2.
+Basics of QHA and novel tools
+5
+2.3.
+τ-quantization of functions
+6
+3.
+Feichtinger operators
+7
+3.1.
+τ-quantization of operators
+9
+3.2.
+A convenient environment for QHA
+13
+3.3.
+τ-Cohen’s class of operators
+23
+4.
+A characterization of Schwartz operators
+28
+Acknowledgments
+30
+References
+30
+1. Introduction
+There is a vast literature on the boundedness of pseudodifferential operators
+for certain classes of symbols in various quantization schemes along the lines of
+H¨ormander classes or alternatively using Sj¨ostrand’s class or Shubin’s classes, e.g.
+2010 Mathematics Subject Classification. 42B35;46E35;47G30;47B10.
+Key
+words
+and
+phrases. Cohen’s
+class,
+τ-quantization,
+Feichtinger’s
+algebra,
+Wigner
+distribution.
+1
+
+2
+FEDERICO BASTIANONI AND FRANZ LUEF
+[1, 3, 4, 11, 22]. In the present work, we put our focus on Shubin’s τ-quantization
+and the associated time-frequency representations, the τ-Cohen classes.
+Our approach to this circle of ideas is based on the framework of quantum har-
+monic analysis with the goal to lift the well-known results concerning functions to
+an appropriate class of functions, which we call Feichtinger operators, S0, and which
+is the operator analog of the well-known Feichtinger algebra S0.
+We also discuss the relation between Feichtinger operators S0 and the class of
+Schwartz operators introduced by Keyl, Kiukas and Werner in [14]. There the idea
+is put forward that one should look for analogs of function spaces in the setting of
+classes of operators, which has been realized in the case of Sobolev spaces in [15]
+and for modulation spaces in [6].
+For τ ∈ [0, 1] the τ-quantization of a symbol a ∈ S′(R2d), the space of tempered
+distributions, is given by
+(1)
+Opτ(a)f(t) :=
+�
+R2d e2πi(t−y)ξa((1 − τ)t + τy, ξ)f(y) dydξ f ∈ S(Rd),
+where the operator Opτ(a) is understood to be defined in the weak sense. A well-
+known fact is that one can relate ⟨Opτ(a)f, g⟩ to a time-frequency representation,
+Wτ(f, g), the cross-τ-Wigner distribution of f and g:
+⟨Opτ(a)f, g⟩ = ⟨a, Wτ(g, f)⟩,
+for all
+f, g ∈ S(Rd).
+Given an operator S, we denote by aS
+τ its τ-symbol, i.e. the tempered distribution
+such that Opτ
+�
+aS
+τ
+�
+= S and Opτ is called the τ-Shubin quantization. For f, g ∈
+L2(Rd) we denote the rank-one operator by f ⊗ g and note that af⊗g
+τ
+= Wτ(g, f),
+i.e. there is an intrinsic relation between quantization schemes and time-frequency
+representations.
+We show that for well-behaved operators, e.g. trace class operators or Feichtinger
+operators, this relation might be extended to operators.
+Recall that Wigner in
+his ground-breaking work on quasi-probability distributions introduced the cross-
+Wigner distribution for certain classes of operators [24], which was later extended
+to more general classes of operators by Moyal in [19].
+Let S be a continuous operator between the Feichtinger algebra S0 and its contin-
+uous dual space S′
+0. We denote by KS the kernel of S, which exists by Feichtinger’s
+kernel theorem and is a mild distribution on R2d.
+We define Feichtinger operators, S0, to be the following class of continuous and
+linear operators S0 := S : S′
+0(Rd) → S0(Rd) that map norm bounded w-∗ convergent
+sequences in S′
+0 into norm convergent sequences in S0. In [10] it was shown that
+these are precisely the linear continuous operators from S′
+0 to S0 that have a kernel
+in Feichtinger’s algebra, the so-called inner kernel theorem.
+One of our main tools is that Feichtinger operators have a nice spectral decomposi-
+tion. If S is in S0, then there exist two (non-unique) sequences {fn}n, {gn}n ⊆ S0(Rd)
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+3
+such that
+S =
+∞
+�
+n=1
+fn ⊗ gn,
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < ∞,
+KS =
+∞
+�
+n=1
+Kfn⊗gn.
+Hence, Feichtinger operators are trace class operators and we can compute their trace
+as follows tr(S) =
+�
+Rd KS(x, x) dx. In [7] operators having such a decomposition
+have been studied and called Feichtinger states in case tr(S) = 1, but there the link
+between these operators and the work [10] was not established, which is one of our
+main observations.
+Then the τ-Wigner distribution of S is defined in the following way
+(2)
+WτS(x, ω) :=
+�
+Rd e−2πitωKS(x + τt, x − (1 − τ)t) dt.
+Our key observation is the following identity:
+⟨a,WτS⟩ = tr(Opτ(a)S∗) =: ⟨Opτ(a),S⟩,
+for S in S0 or J 1, and WτS is the τ-Wigner distribution of S. Consequently, we
+interpret WτS as the τ-quantization of an operator in S0 or J 1.
+Note, that if S is the rank-one operator f ⊗ g this becomes the aforementioned
+relation between the τ-Wigner distribution and the Shubin τ-transform.
+Based on this framework we deduce operator analogs of well-known results on
+τ-Wigner distributions and τ-Shubin quantization, which indicates that this is a
+very convenient setting for this type of investigation. In addition, we extend the
+Cohen class of an operator, introduced in [17], to the τ-setting and show that it
+can be written as the convolution of the Wigner distribution of an operator with a
+distribution as in the function setting.
+We close our discussion with the introduction of weighted versions of S0 and prove
+that the intersection of all these is the class of Schwartz operators in [14]. As in the
+case of functions, we hope that this global description of the Schwartz operators will
+also turn out to be useful in subsequent studies and it also hints at operator analogs
+of Gelfand-Shilov classes or other classes of test functions and the corresponding
+class of ultradistributions.
+2. Preliminaries
+In this paper, the parameter τ always belongs to [0, 1], even when not specified.
+2.1. A family of time-frequency representations. For x, ω ∈ Rd we define the
+translation and modulation operator by
+Txf(t) := f(t − x),
+Mωf(t) := e2πiωtf(t),
+∀t ∈ Rd,
+respectively. Their composition is denoted by π(x, ω) := MωTx.
+
+4
+FEDERICO BASTIANONI AND FRANZ LUEF
+Given τ ∈ [0, 1], the τ-time-frequency shift (τ-TFS) at (x, ω) ∈ R2d is defined to
+be
+(3)
+πτ(x, ω) := e−2πiτxωMωTx = M(1−τ)ωTxMτω.
+For τ = 0 we recover the usual time-frequency shifts π0 = π. The following relations
+are consequences of elementary computations, which are left to the reader:
+πτ(x, ω)πτ(x′, ω′) = e−2πi[(1−τ)xω′−τx′ω]πτ(x + x′, ω + ω′),
+πτ(x, ω)πτ(x′, ω′) = e−2πi[xω′−x′ω]πτ(x′, ω′)πτ(x, ω),
+πτ(x, ω)∗ = π1−τ(−x, −ω) = e−2πi(1−τ)xωπ(−x, −ω).
+In the present paper the symbol ⟨·,·⟩ either denotes the inner product in L2(Rd)
+or a duality pairing between a Banach space X and its dual space X′, which is
+compatible with the latter, i.e. ⟨·,·⟩ is assumed to be linear in the first argument
+and conjugate-linear in the second one. In particular, the dual pairs considered in
+this work are (L2, L2), (S′
+0, S0), (S′
+0, S0), respectively.
+Above, S0 is the Feichtinger algebra (24), for the definitions of S0 and S′
+0 see the
+equations (25),(26) and (28). We introduce for f, g ∈ L2(Rd), or for any suitable
+dual pair, the τ-short-time Fourier transform (τ-STFT) of f w.r.t g:
+(4)
+V τ
+g f(x, ω) := ⟨f, πτ(x, ω)g⟩,
+∀x, ω ∈ Rd.
+As can be easily verified, the mapping
+πτ : R2d → U(L2(Rd)),
+where U(L2(Rd)) denotes the unitary operators on L2(Rd), is a projective represen-
+tation of R2d for any τ. Consequently, V τ is the wavelet transform associated to πτ,
+thus V τ
+g f is a continuous function.
+Remark 2.1. For τ = 0 we obtain the usual STFT V 0
+g f = Vgf and we have
+(5)
+V τ
+g f(x, ω) = e2πiτxωVgf(x, ω).
+By the preceding identity, we have that V
+1
+2
+g f is the cross-ambiguity function of f and
+g:
+(6)
+V
+1
+2
+g f(x, ω) = A(f, g)(x, ω).
+We recall another frequently used time-frequency representation, the so-called
+cross-τ-Wigner distribution of f and g in L2(Rd) defined by
+(7)
+Wτ(f, g)(x, ω) :=
+�
+Rd e−2πitωf(x + τt)g(x − (1 − τ)t) dt.
+We aim to extend the definition of Wτ from functions to operators, see (15).
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+5
+2.2. Basics of QHA and novel tools. In this subsection we introduce the ba-
+sic definitions of quantum harmonic analysis (QHA) following the seminal work of
+Werner [23].
+For z ∈ R2d and A ∈ B(L2(Rd)) the translation of the operator A by z is
+(8)
+αz(A) := π(z)Aπ(z)∗,
+which satisfies αzαz′ = αz+z′. By the parity operator, we mean
+(9)
+Pf(t) := ˇf(t) := f(−t),
+for any f ∈ L2(Rd), which induces an involution of A ∈ B(L2(Rd)):
+(10)
+ˇA := PAP.
+We denote by J 1 the space of all trace class operators on L2(Rd). Given a ∈ L1(R2d)
+and S ∈ J 1. The convolution between a and S is the operator
+(11)
+a ⋆ S := S ⋆ a :=
+�
+R2d a(z)αz(S) dz,
+were the integral may be interpreted in the weak sense. For operators S, T ∈ J 1,
+their convolution is the function defined for every z ∈ R2d as
+(12)
+S ⋆ T(z) := tr
+�
+Sαz( ˇT)
+�
+.
+In this paper, we reserve the symbol ⊗ for rank-one operators.
+Namely, given
+f, g ∈ L2(Rd):
+(13)
+(f ⊗ g)ψ := ⟨ψ, g⟩f,
+∀ψ ∈ L2(Rd).
+The kernel of an operator S will always be denoted by KS. Evidently, the kernel of
+the operator f ⊗ g is the tensor product of functions f(x)g(y):
+(f ⊗ g)ψ(t) = ⟨ψ, g⟩f(t) =
+�
+Rd f(t)g(x)ψ(x) dx.
+In the sequel we denote the tensor product of two functions by f(x)g(y), we shall
+adopt the notation
+(14)
+Kf⊗g(x, y) = f(x)g(y).
+We now interpret (7) as the cross-τ-Wigner distribution of the rank-one operator
+f ⊗ g.
+Hence, it is natural to define the τ-Wigner distribution of an operator S with kernel
+KS in the following way:
+(15)
+WτS(x, ω) :=
+�
+Rd e−2πitωKS(x + τt, x − (1 − τ)t) dt.
+For S ∈ J 1 and τ ∈ [0, 1], we define the Fourier-τ-Wigner transform of S to be:
+(16)
+FWτS(z) := tr (πτ(z)∗S) ,
+∀z ∈ R2d.
+
+6
+FEDERICO BASTIANONI AND FRANZ LUEF
+For τ = 1/2 we recover the usual Fourier-Wigner transform [23].
+The τ-spreading representation of S ∈ B(L2) is the decomposition
+(17)
+S =
+�
+R2d h(z)πτ(z) dz,
+where the integral is understood in the weak sense. The function h is called the
+τ-spreading function of S.
+In the following, we shall consider the τ-spreading representation as a quantization
+scheme that assigns to a function an operator. Namely, h ∈ L1(R2d) gets associated
+to the operator
+(18)
+SRτ(h) :=
+�
+R2d h(z)πτ(z) dz.
+Let Fσ denote the symplectic Fourier transform. In the following lemma we collect
+a number of important relations between these notions. The proofs are elementary
+computations and based on the spectral decomposition of the trace class operators
+S and T ([21]), which we leave to the interested reader.
+Lemma 2.2. Let f, g, ∈ L2(Rd), S, T ∈ J 1, a ∈ L1(R2d) and τ ∈ [0, 1]. Then:
+(i) Fσ(Wτ(f ⊗ g)) = V τ
+g f;
+(ii) FWτ(f ⊗ g) = V τ
+g f;
+(iii) WτS = FσFWτS;
+(iv) FWτS(x, ω) = e−2πi(1/2−τ)xωFW1/2S(x, ω);
+(v) Fσ(S ⋆ T) = FWτS · FW1−τT = FW1−τS · FWτT;
+(vi) FWτ(a ⋆ S) = Fσa · FWτS;
+(vii) FWτS is the τ-spreading function of S, i.e. S =
+�
+R2d FWτS(z)πτ(z) dz.
+We notice that if we consider the rank-one operator S = f ⊗g, then the assertions
+(iii) and (ii) of the previous lemma imply
+(19)
+Wτ(f, g) = Wτ(f ⊗ g) = FσV τ
+g f.
+2.3. τ-quantization of functions. The τ-quantization of a symbol a ∈ S′(R2d),
+the space of tempered distributions, is formally given by
+(20)
+Opτ(a)f(t) :=
+�
+R2d e2πi(t−y)ξa((1 − τ)t + τy, ξ)f(y) dydξ,
+where f ∈ S(Rd). Opτ(a) may be described rigorously in the weak sense:
+⟨Opτ(a)f, g⟩ = ⟨a, Wτ(g, f)⟩,
+∀f, g, ∈ S(Rd).
+Given an operator S, we denote by aS
+τ its τ-symbol, i.e. the tempered distribution
+such that
+Opτ
+�
+aS
+τ
+�
+= S.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+7
+Remark 2.3. Under suitable assumptions, for example a ∈ L1(R2d), straightforward
+calculations give
+Opτ(a) =
+�
+R2d Fσa(z)πτ(z) dz,
+and since also FWτ Opτ(a) is the τ-spreading function of Opτ(a), we have
+(21)
+a = FσFWτ Opτ(a).
+Hence, for S ∈ J 1
+(22)
+aS
+τ = FσFWτS = WτS.
+Given a ∈ S′
+0(R2d) and f, g ∈ S0(Rd), we recall the definition of cross-τ-Cohen’s
+class representation of f and g, with kernel a:
+(23)
+Qτ
+a(f, g) := a ∗ Wτ(f, g).
+3. Feichtinger operators
+In this section we summarize some important results concerning a class of op-
+erators studied in [10]. For such operators, introduced below, we adopt the name
+“Feichtinger operators” for reasons which will become evident later.
+We recall that the Feichtinger algebra over Rd [9] is the Banach space
+(24)
+S0(Rd) := {f ∈ L2(Rd) | Vgf ∈ L1(R2d)},
+for some g ∈ L2(Rd) ∖ {0}, endowed with the norm
+∥f∥S0 := ∥Vgf∥L1 =
+�
+R2d |Vgf(x, ω)| dxdω.
+We refer the reader to [13] for a detailed survey on S0(Rd). In this work, S′
+0(Rd)
+denotes the conjugate-dual of S0(Rd).
+Definition 3.1. The set of Feichtinger operators is defined to be
+S0 :={S : S′
+0(Rd) → S0(Rd) | S is linear, continuous and
+maps norm bounded w-∗ convergent sequences in S′
+0
+(25)
+into norm convergent sequences in S0}.
+We adopt the following notation:
+(26)
+S′
+0 := B(S0(Rd), S′
+0(Rd))
+and state the so called Outer Kernel Theorem [10, Theorem 1.1]:
+Theorem 3.2. The Banach space S′
+0 is isomorphic to S′
+0(R2d) via the map T �→ KT,
+where the relation between T and its kernel KT is given by
+⟨Tf,g⟩ = ⟨KT,Kg⊗f⟩,
+∀ f, g, ∈ S0(Rd).
+
+8
+FEDERICO BASTIANONI AND FRANZ LUEF
+The following statement goes under the name of Inner Kernel Theorem.
+We
+present it in our setting. To this end, we introduce the following notation: given
+σ, ν ∈ S′
+0(Rd), we denote by ν �⊗σ the unique element of S′
+0(R2d) such that
+⟨ν �⊗σ,Kψ⊗ϕ⟩ = ⟨ν,ψ⟩⟨σ,ϕ⟩,
+∀ ψ, ϕ ∈ S0(Rd).
+We refer the reader to [10, Theorem 1.3], Lemma 3.1 and Corollary 3.10, too.
+Theorem 3.3. The space of Feichtinger operators S0 is a Banach space if endowed
+with the norm of B(S′
+0, S0) and it is naturally isomorphic as Banach space to S0(R2d)
+through the map T �→ KT, where the relation between T and its kernel KT is given
+by
+⟨ν,Tσ⟩ = ⟨ν �⊗σ,KT⟩,
+∀ σ, ν, ∈ S′
+0(Rd).
+Moreover, S0 is Banach algebra under composition. If S, T ∈ S0, then
+(27)
+KS◦T(y, u) =
+�
+Rd KT(y, t)KS(t, u) dt.
+By the above theorems 3.2 and 3.3, S′
+0 is the (conjugate) topological dual of S0
+and the duality is given by
+(28)
+⟨T,S⟩ = ⟨KT,KS⟩.
+Lemma 3.4. Suppose S ∈ S0. Then there exist two non-unique sequences {fn}n, {gn}n ⊆
+S0(Rd) such that
+S =
+∞
+�
+n=1
+fn ⊗ gn,
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < +∞,
+KS =
+∞
+�
+n=1
+Kfn⊗gn.
+Moreover,
+S0 ֒→ J 1
+with
+tr(S) =
+�
+Rd KS(x, x) dx.
+Proof. We just have to prove the continuous inclusion of Feichtinger operators into
+J 1, all the remaining statements can be found in [10], see in particular Corollary
+3.15 and Remark 9. The claim follows from an elementary computation:
+∥S∥J 1 = |tr(A)| ≤
+�
+Rd
+∞
+�
+n=1
+|fn(x)gn(x)| dx =
+∞
+�
+n=1
+�
+Rd |fn(x)gn(x)| dx
+≤
+∞
+�
+n=1
+∥fn∥L2 ∥gn∥L2 ≲
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < ∞.
+Since S0(R2d) = S0(Rd)ˆ⊗S0(Rd), see e.g. [10, Lemma 2.1], we get
+∥S∥J 1 ≲ ∥KS∥S0 ≍ ∥S∥S0 ,
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+9
+which gives the desired assertion.
+□
+The preceding result and the observations in [10, p. 4] yield
+(29)
+S0 ֒→ J 1 ֒→ J 2 ֒→ B(L2(Rd)) ֒→ S′
+0.
+The fact that all Feichtinger operators are trace class implies the validity of Lemma
+2.2.
+3.1. τ-quantization of operators. The following remark is the key insight for the
+subsequent results concerning Opτ and Wτ.
+Remark 3.5. Let us consider f, g ∈ L2(Rd) such that f ̸= 0, a ∈ L2(R2d) and {fj}j
+o.n.b. for L2 with f1 = f. Then we compute as follows:
+⟨Opτ(a)f, g⟩ = ⟨Opτ(a)f,
+∞
+�
+j=1
+⟨g, fj⟩fj⟩ =
+∞
+�
+j=1
+⟨Opτ(a) (⟨fj, g⟩f) , fj⟩
+=
+∞
+�
+j=1
+⟨Opτ(a)(f ⊗ g)fj, fj⟩ = tr (Opτ(a)(f ⊗ g)) .
+Taking into account the weak definition of Opτ(a) and (15) we can write
+(30)
+⟨Opτ(a)f, g⟩ = ⟨a, Wτ((f ⊗ g)∗)⟩ = tr (Opτ(a)(f ⊗ g)) = ⟨Opτ(a), (f ⊗ g)∗⟩(J 1,J ∞).
+By computations similar to the ones above for S ∈ J 1 with the spectral decomposition
+�∞
+k=1 λkfk ⊗ gk after extending {fk}k to an orthonormal basis of L2(Rd) implies
+(31)
+⟨a, WτS⟩ = tr (Opτ(a)S∗) = ⟨Opτ(a), S⟩(J 1,J ∞).
+Theorem 3.6. For every τ ∈ [0, 1] the following mappings are linear and continu-
+ous:
+Opτ : L2(R2d) → J ∞,
+Wτ : J 1 → L2(R2d).
+Moreover, Opτ is the Banach space adjoint of Wτ: Opτ = W ∗
+τ .
+Proof. The boundedness of Opτ is evident; the proof of the continuity of Wτ follows
+by a similar reasoning as the proof of the subsequent Theorem 3.7. The last claim
+is just (31).
+□
+Theorem 3.7. For every τ ∈ [0, 1] the following mappings are linear and continu-
+ous:
+Opτ : S′
+0(R2d) → S′
+0,
+Wτ : S0 → S0(R2d).
+Moreover, Opτ is the Banach space adjoint of Wτ: Opτ = W ∗
+τ , i.e.
+for every
+a ∈ S′
+0(R2d) and S ∈ S0
+(32)
+⟨a,WτS⟩ = ⟨Opτ(a),S⟩.
+
+10
+FEDERICO BASTIANONI AND FRANZ LUEF
+Proof. The boundedness and linearity of Opτ follow from the definitions. By using
+the formal representation of Opτ(a) we can derive an expression for its kernel:
+(33)
+KOpτ (a)(t, x) =
+�
+Rd e2πi(t−x)ωa((1 − τ)t + τx, ω) dω.
+Let us consider first f, g ∈ S0. Then a standard argument, see e.g. [5, Proposition
+1.3.25], gives that
+Wτ(f ⊗ g) = Wτ(f, g) ∈ S0(R2d)
+with
+∥Wτ(f ⊗ g)∥S0 ≲ ∥f∥S0 ∥g∥S0 .
+Since Lemma 2.2 holds for S0, we write Wτ = FσFWτ and use the spectral decom-
+position for S of the form �∞
+n=1 fn ⊗ gn as shown in Lemma 3.4. Now, we compute:
+FWτS(z) = tr(πτ(z)∗S) = tr(
+∞
+�
+n=1
+πτ(z)∗(fn ⊗ gn))
+=
+∞
+�
+n=1
+⟨πτ(z)∗fn, gn⟩ =
+∞
+�
+n=1
+V τ
+gnfn(z).
+(34)
+Taking a suitable window for the norm on S0(R2d) [13, Theorem 5.3] we have
+∥FWτS∥S0 ≤
+∞
+�
+n=1
+��V τ
+gnfn
+��
+S0 =
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < +∞.
+Consequently,
+∥FWτS∥S0 ≤ inf{
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 , S =
+∞
+�
+n=1
+fn ⊗ gn}
+≤ inf{
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 , KS =
+∞
+�
+n=1
+Kfn⊗gn}
+= ∥KS∥S0 ≍ ∥S∥S0 .
+We proved the boundedness of FWτ : S0 → S0(R2d), the continuity of the symplectic
+Fourier transform Fσ : S0(R2d) → S0(R2d) is well-known, and thus the continuity of
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+11
+Wτ : S0 → S0(R2d) follows. Concerning the last claim, we proceed as follows:
+⟨Opτ(a),S⟩ = ⟨KOpτ (a),KS⟩ = ⟨KOpτ (a),
+∞
+�
+n=1
+Kf⊗gn⟩
+=
+∞
+�
+n=1
+⟨KOpτ (a),Kf⊗gn⟩ =
+∞
+�
+n=1
+⟨Opτ(a)gn,fn⟩
+=
+∞
+�
+n=1
+⟨a,Wτ(fn ⊗ gn)⟩ = ⟨a,
+∞
+�
+n=1
+Wτ(fn ⊗ gn)⟩
+= ⟨a,WτS⟩,
+which concludes the proof.
+□
+On account of Theorem 3.6 and 3.7, it seems reasonable to interpret WτS as the
+τ-quantization of an operator in S0 or J 1.
+Corollary 3.8.
+(i) For every τ ∈ [0, 1] the mapping Wτ : S0 → S0(R2d) is a
+topological isomorphism with inverse given by Opτ : S0(R2d) → S0;
+(ii) A linear and continuous operator S : S0(Rd) → S′
+0(Rd) belongs to S0 if and
+only if WτS ∈ S0(R2d) for some (and hence any) τ ∈ [0, 1].
+Proof. (i) We observed in (22) that WτS is just the τ-symbol aS
+τ of a trace class
+operator S, in particular this holds for S ∈ S0. Therefore,
+Opτ ◦WτS = Opτ(aS
+τ ) = S.
+We now show that if we start with a ∈ S0(R2d), then Opτ(a) belongs to S0. From
+(33), we have that the kernel of Opτ(a) can be written as
+KOpτ (a)(t, x) =
+�
+Rd e2πi(t−x)ωa((1 − τ)t + τx, ω) dω = ΨτF −1
+2 a(t, x),
+where F −1
+2
+is the inverse of the partial Fourier transform with respect to the second
+variable; Ψτ is the change of variables induced by the matrix
+(35)
+�
+1 − τ
+τ
+1
+−1
+�
+,
+ΨτF(t, x) := F((1 − τ)t + τx, t − x).
+From the assumption a in the Feichtinger algebra S0(R2d) we have F −1
+2 a ∈ S0(R2d),
+thus ΨτF −1
+2 a is in S0(R2d), i.e. Opτ(a) is an element of S0. The fact that Opτ is
+continuous from S0(R2d) into S0 is evident from the applications of F −1
+2
+and Ψτ.
+Hence we have shown that
+Wτ ◦ Opτ(a) = aOpτ (a)
+τ
+= a.
+(ii) The claim is a straightforward consequence of (i).
+□
+
+12
+FEDERICO BASTIANONI AND FRANZ LUEF
+Corollary 3.9.
+(i) For every τ ∈ [0, 1] FWτ : S0 → S0(R2d) is a topological
+isomorphisms with inverse given by the τ-spreading representation
+(36)
+SRτ : S0(R2d) → S0 , a �→
+�
+R2d a(z)πτ(z) dz;
+(ii) Let us define
+(37)
+SRτ : S′
+0(R2d) → S′
+0 a �→
+�
+R2d a(z)πτ(z) dz,
+where the integral has to be understood weakly as follows:
+⟨SRτ(a)f,g⟩ := ⟨a,V τ
+f g⟩,
+a ∈ S′
+0(R2d), f, g ∈ S0(Rd).
+Then SRτ as in (37) is well-defined, linear, continuous, extends (36) and it
+is the Banach space adjoint of FWτ in (i):
+(38)
+SRτ = F ∗
+Wτ,
+in the sense that for every a ∈ S′
+0(R2d) and S ∈ S0
+⟨a,FWτS⟩ = ⟨SRτ(a),S⟩ = ⟨KSRτ (a),KS⟩;
+(iii) Every function F ∈ S0(R2d) admits an expansion of the following type:
+F =
+∞
+�
+n=1
+V τ
+gnfn,
+for some sequences {fn}n, {gn}n ⊆ S0(Rd) such that �∞
+n=1 ∥fn∥S0 ∥gn∥S0 <
+∞.
+Proof. (i) First we notice that if we start with a ∈ S0(R2d), then SRτ(a) is the
+Feichtinger operator with kernel
+KSRτ (a)(y, u) =
+�
+Rd a(y − u, ω)e2πiyω dω = F −1
+2 [a(y − u, ·)](y).
+Clearly SRτ is continuous from S0(R2d) into S0.
+Since we have Wτ = FσFWτ and Fσ is an automorphism of S0(R2d), we can write
+FWτ = FσWτ and which is an isomorphism due to Corollary 3.8. To prove that SRτ
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+13
+is the inverse of FWτ we use (34), take S = �∞
+n=1 fn ⊗ gn ∈ S0 and ψ, ϕ ∈ S0(Rd):
+⟨(SRτ ◦ FWτS)ψ,ϕ⟩ =
+�
+R2d FWτS(z)⟨πτ(z)ψ,ϕ⟩ dz
+=
+∞
+�
+n=1
+�
+R2d V τ
+gnfn(z)V τ
+ψ ϕ(z) dz
+=
+∞
+�
+n=1
+⟨fn,ϕ⟩⟨gn,ψ⟩
+= ⟨
+∞
+�
+n=1
+⟨ψ,gn⟩fn,ϕ⟩
+= ⟨
+∞
+�
+n=1
+(fn ⊗ gn)ψ,ϕ⟩
+= ⟨Sψ,ϕ⟩,
+in the third equality we used Moyal’s identity. For the composition FWτ ◦SRτ, notice
+that this is the identity on S0(R2d) due lo Lemma 2.2 (vii).
+(ii) Well-posedness, linearity and continuity of SRτ from S′
+0(R2d) into S′
+0 are stan-
+dard. Trivially (37) extends (36). To see that SRτ is the Banach space adjoint of
+FWτ from S0 into S0(R2d), take a ∈ S′
+0(R2d) and S ∈ S0. In the following calcula-
+tions we use: the prior stated (34), the representation for Feichtinger operators and
+their kernel given in Lemma 3.4, the Outer and Inner Kernel Theorems:
+⟨a,FWτS⟩ =
+∞
+�
+n=1
+⟨a,V τ
+gnfn⟩ =
+∞
+�
+n=1
+⟨SRτ(a)gn,fn⟩
+=
+∞
+�
+n=1
+⟨KSRτ (a),Kfn⊗gn⟩ = ⟨KSRτ (a),KS⟩
+= ⟨SRτ(a),S⟩.
+(iii) The last claim is a direct consequence of the computations in (34) and the
+surjectivity of FWτ.
+□
+3.2. A convenient environment for QHA. In Section 2 we introduced convo-
+lutions between a function and an operator and two operators. Keyl, Kiukas and
+Werner [14] showed that such convolutions make sense for wider classes of (gen-
+eralized) functions and operators. We summarize here the main results; in what
+follows S denotes the set of pseudo-differential operators with Weyl symbol in the
+Schwartz class S(R2d) and S′ those pseudo-differential operators with Weyl symbol
+in S′(R2d). On account of the Schwartz Kernel Theorem we can identify S′ with
+the continuous and linear operators from S(Rd) into S′(Rd).
+
+14
+FEDERICO BASTIANONI AND FRANZ LUEF
+Proposition 3.10.
+(i) Suppose S, T ∈ S, A ∈ S′, b ∈ S(R2d) and a ∈ S′(R2d).
+Then the following convolutions are well-defined and they extend the ones
+defined in Subsection 2.2:
+S ⋆ T ∈ S(R2d),
+S ⋆ A ∈ S′(R2d),
+b ⋆ S ∈ S,
+a ⋆ S, b ⋆ A ∈ S′;
+(ii) The Fourier-Wigner transform can be extended to a topological isomorphism
+FW1/2 : S′ → S′(R2d);
+(iii) We have Fσ(S ⋆ T) = FW1/2S · FW1/2T and FW1/2(b ⋆ S) = Fσb · FW1/2S
+whenever S, T and b are such that the convolutions are defined as in part (i);
+(iv) The Weyl symbol of A ∈ S′ is given by FσFW1/2A.
+The authors of [14] proved that the class of so-called Schwartz operators S has
+the structure of a Fr´echet space. We propose that the Banach space of Feichtinger
+operators S0 is an alternative to S that is a much bigger class of “nice” operators.
+We start with some preliminaries on S0 and S0.
+Lemma 3.11. Given f ∈ S′
+0(Rd), there exists a sequence {fn}n ⊆ S0(Rd) which
+w-∗ converges to f and it is bounded by ∥f∥S′
+0, i.e.
+lim
+n→+∞⟨fn, g⟩ = ⟨f,g⟩
+∀ g ∈ S0(Rd),
+sup
+n ∥fn∥S0 ≤ ∥f∥S′
+0 .
+Proof. Let us fix f ∈ S′
+0(Rd) ∖ {0} and set R := ∥f∥S′
+0. By [13, Proposition 6.15],
+there exists a net {fα}α∈A ⊆ S0(Rd) which converges w-∗ to f in S′
+0 and such that
+∥fα∥S′
+0 ≤ R for every α ∈ A. Set
+BR :=
+�
+f ∈ S′
+0(Rd) | ∥f∥S′
+0 ≤ R
+�
+and
+ER := S0(Rd) ∩ BR,
+where S0 is identified with its natural embedding in S′
+0, i.e.
+ER ⊆ BR ⊆ ER
+w−∗.
+ER
+w−∗ is bounded in S′
+0(Rd).
+In fact, if f0 ∈ ER
+w−∗, then there exists a net {fα}α∈A ⊆ ER that it converges w-∗
+to f0. Hence, we obtain
+∥f0∥S′
+0 ≤ lim inf
+α∈A
+∥fα∥S′
+0 = lim
+α∈A inf{∥fβ∥S′
+0 | α ⪯ β} ≤ lim
+α∈A R = R.
+In particular, this shows that ER
+w−∗ ⊆ BR and we get
+ER
+w−∗ = BR.
+Since S0 separable, and the relative w-∗ topology on BR is induced by a metric by
+[18, Therem 2.6.23]. Hence the topological w-∗ closure of ER equals its sequential
+w-∗ closure. Consequently, there exists a sequence {fn}n ⊆ ER which converges w-∗
+to f in S′
+0(Rd).
+□
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+15
+Remark 3.12. The above lemma holds also for any LCA second countable group G
+replacing Rd, see [8, Theorem 2] for the separability of S0(G).
+Lemma 3.13. For any S ∈ S′
+0, there exists a sequence {Sn}n ⊆ S0 such that
+(i) ∥Sn∥S′
+0 ≲ ∥S∥S′
+0;
+(ii) limn→+∞ |⟨(S − Sn)f,g⟩| = 0 for all f, g ∈ S0(Rd).
+Proof. This is a straightforward application of the Kernel Theorems 3.2 and 3.2 and
+of Lemma 3.11.
+□
+Convergence as in item (ii) of the above lemma will be also denoted by
+Sn
+w−∗
+−→
+n
+S
+in
+S′
+0
+or
+S = w- ∗ -limn Sn
+in
+S′
+0.
+Lemma 3.14. Let S : S0 → S′
+0 be in S0. Then the Banach space adjoint S∗: S′
+0 →
+S0 is in S0 with kernel
+(39)
+KS∗(y, u) = KS(u, y).
+Proof. We take f, g ∈ S0(Rd), then
+⟨Sf,g⟩ =
+�
+R2d KT(y, u)g(y)f(u) dydu
+=
+�
+Rd f(u)
+�
+Rd KS(y, u)g(y) dy du
+= ⟨f,S∗g⟩.
+Hence, S∗g(y) =
+�
+Rd KS(u, y)g(u) du, i.e. KS∗(y, u) = KS(u, y) which is an element
+of S0(R2d).
+□
+Corollary 3.15. S0 is a Banach ∗-algebra.
+We notice that (S∗)ˇ= ( ˇS)∗, so that from now on we shall simply write ˇS∗ when
+necessary.
+Lemma 3.16.
+(i) The following applications are surjective isometries:
+(i − a) αz : S0 → S0, for any z = (x, ω) ∈ R2d, and
+(40)
+KαzS(y, u) = e2πi(y−u)ωKS(y − x, u − x);
+(i − b) ˇ·: S0 → S0 and
+(41)
+K ˇS(y, u) = KS(−y − u);
+(i − c) αz : S′
+0 → S′
+0, for any z ∈ R2d;
+(i − d) ˇ·: S′
+0 → S′
+0;
+(ii) Let S, T ∈ S0 and b ∈ S0(R2d). Then
+S ⋆ T ∈ S0(R2d),
+b ⋆ S ∈ S0;
+
+16
+FEDERICO BASTIANONI AND FRANZ LUEF
+(iii) The kernel of the mixed-state localization operator b ⋆ S is given by
+(42)
+Kb⋆S(y, u) =
+�
+Rd b(x, ω)e2πi(y−u)ωKS(y − x, u − x) dxdω;
+for very z = (x, ω) ∈ R2d the kernel of Sαz ˇT is
+(43)
+KSαz ˇT(y, u) =
+�
+Rd e2πi(y−t)ωKT(x − y, x − t)KS(t, u) dt.
+Proof. (i) We leave the elementary computations to the interest reader, and note that
+in order to prove αzS, ˇS ∈ S0 the result [10, Corollary 3.3] is useful. A continuous
+and linear operator S : S0 → S′
+0 is a Feichtinger operator if and only if
+�
+R2d
+�
+R2d |⟨Sπ(z)g1,π(w)g2⟩| dzdw
+is finite for any g1, g2 ∈ S0(Rd).
+(ii) We first address the convolution between two Feichtinger operators. By item
+(i) and the fact that S0 is a Banach algebra under composition, we have that Sαz ˇT
+is in S0 for any z = (x, ω) ∈ R2d. We have by [10, Corollary 3.15]:
+S ⋆ T(z) = tr(Sαz ˇT) =
+�
+Rd KSαz ˇT(y, y) dy =
+�
+R2d Kαz ˇT(y, t)KS(t, y) dtdy
+=
+�
+R2d e2πi(y−t)ωKT(x − y, x − t)KS(t, y) dtdy
+=
+�
+Rd
+��
+Rd KT(x − y, x − t)KS(t, y)e−2πitω dt
+�
+e2πiyω dy
+= F −1
+2 F1
+�
+ΦT(x,x)KT · KS
+�
+(ω, ω),
+where ΦF(t, y) := F(−y, −t), F1 and F2 are the partial Fourier transforms with re-
+spect to the first and second variable, respectively. Consider now f, g, h, l ∈ S0(Rd),
+it is useful to compute the following where P is the parity operator:
+F −1
+2 F1
+�
+ΦT(x,x)Kh⊗l · Kf⊗g
+�
+(ω, ω) =
+�
+Rd
+��
+Rd h(x − y)l(x − t)f(t)g(y)e−2πitω dt
+�
+e2πiyω dy
+=
+�
+Rd f(t)e−2πitωl(x − t) dt ·
+�
+Rd g(y)e2πiyωh(x − y) dy
+= VP lf(−x, ω) · VP hg(−x, ω).
+Hence F −1
+2 F1
+�
+ΦT(x,x)Kh⊗l · Kf⊗g
+�
+(ω, ω) is in S0(R2d) as a function of (x, ω). We
+consider now two representations S = �∞
+n=1 fn ⊗ gn and T = �∞
+n=1 hn ⊗ ln, see
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+17
+Lemma 3.4, so that
+KS =
+∞
+�
+n=1
+Kfn⊗gn,
+KT =
+∞
+�
+n=1
+Khn⊗ln.
+It follows that we can write
+S ⋆ T(z) = F −1
+2 F1
+�
+ΦT(x,x)
+∞
+�
+M
+Khm⊗lm ·
+∞
+�
+n=1
+Kfn⊗gn
+�
+(ω, ω)
+=
+∞
+�
+m=1
+∞
+�
+n=1
+F −1
+2 F1
+�
+ΦT(x,x)Khm⊗lm · Kfn⊗gn
+�
+(ω, ω)
+=
+∞
+�
+m=1
+∞
+�
+n=1
+VP lmfn(−x, ω) · VP hmgn(−x, ω) ∈ S0(R2d),
+the convergence is guaranteed by Lemma 3.4.
+Concerning b⋆S, the following estimate for any f, g ∈ S0(Rd) proves that b⋆S ∈ S′
+0:
+|⟨(b ⋆ S)f,g⟩| ≤
+�
+R2d |b(z)| |⟨Sπ(z)∗f,π(z)∗g⟩| dz ≲ ∥b∥L1 ∥S∥S′
+0 ∥f∥S0 ∥g∥S0 .
+We exploit [10, Theorem 3.2 (ii)] to show that b ⋆ S is in S0. For g1, g2 ∈ S0(Rd) we
+have
+�
+R2d
+�
+R2d |⟨(b ⋆ S)π(w)g1,π(u)g2⟩| dwdu ≤
+�
+R2d
+�
+R2d
+�
+R2d |b(z)|
+× |⟨Sπ(w − z)g1,π(u − z)g2⟩| dzdwdu
+=
+�
+R2d
+�
+R2d |⟨Sπ(w′)g1,π(u′)g2⟩| dw′du′ ·
+�
+R2d |b(z)| dz < +∞.
+(iii) We compute explicitly the kernel of the operator given by the convolution b⋆S:
+⟨(b ⋆ S)f,g⟩ =
+�
+R2d b(x, ω)
+�
+R2d KS(y, u)π(−z)g(y)π(−z)f(u) dydu dxdω
+=
+�
+R2d
+�
+R2d b(x, ω)e2πi(y−u)ωKS(y, u)g(y + x)f(u + x) dxdω dydu,
+for z = (x, ω) ∈ R2d. The change of variables y′ = y + u, u′ = u + x gives the desired
+result.
+The last claim is just a direct application of (40), (41) and the Banach
+algebra property for S0 [10, Lemma 3.10].
+□
+Corollary 3.17. Let S, T ∈ S0 with spectral decompositions S = �∞
+n=1 fn ⊗ gn and
+T = �∞
+n=1 hn⊗ln, where {fn}n, {gn}n, {hn}n, {ln}n ⊆ S0(Rd) with �∞
+n=1 ∥fn∥S0 ∥gn∥S0 <
+
+18
+FEDERICO BASTIANONI AND FRANZ LUEF
++∞, �∞
+n=1 ∥hn∥S0 ∥ln∥S0 < +∞. Then, with the notations introduced in the proof
+of Lemma 3.16, for every z = (x, ω) ∈ R2d:
+S ⋆ T(z) = F −1
+2 F1
+�
+ΦT(x,x)KT · KS
+�
+(ω, ω)
+=
+∞
+�
+m=1
+∞
+�
+n=1
+VP lmfn(−x, ω) · VP hmgn(−x, ω).
+(44)
+Definition 3.18. Let A ∈ S′
+0, a ∈ S′
+0(R2d), S ∈ S0 and b ∈ S0(R2d). Consider any
+sequences {An}n ⊆ S0 and {an}n ⊆ S0(R2d) such that
+An
+w−∗
+−→
+n
+A
+in
+S′
+0
+and
+an
+w−∗
+−→
+n
+a
+in
+S′
+0(R2d).
+Then we define:
+S ⋆ A := w- ∗ -limn S ⋆ An
+in
+S′
+0(R2d);
+(45)
+a ⋆ S := S ⋆ a := w- ∗ -limn an ⋆ S
+in
+S′
+0;
+(46)
+b ⋆ A := A ⋆ b := w- ∗ -limn b ⋆ An
+in
+S′
+0.
+(47)
+Remark 3.19. The reader may find it useful to keep in mind the following simple
+identities, which will be used in the proof of the subsequent proposition. Consider
+S ∈ S0, ψ, ϕ, f, g ∈ S0(Rd) and z ∈ R2d:
+αz(ψ ⊗ ϕ) = π(z)ψ ⊗ π(z)ϕ;
+(ψ ⊗ ϕ)(Kf⊗g) = ⟨f, ϕ⟩(ψ ⊗ g);
+(ψ ⊗ ϕ) ⋆ ˇS(z) = ⟨π(z)Sπ(z)∗ψ,ϕ⟩.
+Proposition 3.20. The convolutions introduced in Definition 3.18:
+(i) They do not depend on the sequences chosen; moreover, taking A, a, S, b as
+in Definition 3.18:
+⟨S ⋆ A,b⟩ = ⟨KA,Kb⋆ ˇS∗⟩;
+(48)
+⟨(a ⋆ S)f,g⟩ = ⟨a,(g ⊗ f) ⋆ ˇS∗⟩;
+(49)
+⟨(b ⋆ A)f,g⟩ = ⟨KA,Kb∗⋆(g⊗f)⟩,
+(50)
+where b∗(z) := b(−z);
+(ii) These extend the definitions given in Subsection 2.2;
+(iii) They are commutative;
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+19
+(iv) Moreover, they are associative. In particular, if z ∈ R2d, T, Q ∈ S0, σ ∈
+S0(R2d) and A, a, S, b as in Definition 3.18 then:
+(S ⋆ (T ⋆ b))(z) = ((S ⋆ T) ∗ b)(z);
+(51)
+S ⋆ (T ⋆ Q) = (S ⋆ T) ⋆ Q;
+(52)
+(S ⋆ b) ⋆ σ = S ⋆ (b ∗ σ);
+(53)
+S ⋆ (T ⋆ a) = (S ⋆ T) ∗ a;
+(54)
+A ⋆ (T ⋆ b) = (A ⋆ T) ⋆ b;
+(55)
+S ⋆ (T ⋆ A) = (S ⋆ T) ⋆ A;
+(56)
+in the above identities ∗ denotes the usual convolution between two functions
+or a function and a distribution.
+Proof. (i) It suffices to show (48), (49) and (50), since the other assertions in (i) are
+evident.
+We start with(48).
+Let b ∈ S0(R2d) and z = (x, ω) ∈ R2d, in the subsequent
+computations we use Lemma 3.14 and 3.16:
+⟨S ⋆ A,b⟩ = lim
+n→+∞⟨S ⋆ An,b⟩ = lim
+n→+∞
+�
+R2d tr(Sαz ˇAn)b(z) dz
+= lim
+n→+∞
+�
+R2d
+�
+Rd KSαz ˇ
+An(y, y) dyb(z)dz
+= lim
+n→+∞
+�
+R2d
+�
+Rd
+�
+Rd e2πi(y−t)ωKAn(x − y, x − t)KS(t, y) dtdy b(z) dz
+= lim
+n→+∞
+�
+R2d
+�
+Rd
+�
+Rd e2πi(t′−y′)ωKAn(y′, t′)KS(x − t′, x − y′) dt′dy′ b(z) dz
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t′)
+��
+R2d KS(x − t′, x − y′)e2πi(y′−t′)ωb(z) dz
+�
+dy′dt′
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t′)
+��
+R2d K ˇS(t′ − x, y′ − x)e2πi(y′−t′)ωb(z) dz
+�
+dy′dt′
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t′)
+��
+R2d K ˇS∗(y′ − x, t′ − x)e2πi(y′−t′)ωb(z) dz
+�
+dy′dt′
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t′)Kb⋆ ˇS∗(y′, t′) dy′dt′.
+
+20
+FEDERICO BASTIANONI AND FRANZ LUEF
+About (49), we take f, g ∈ S0(Rd) and compute directly keeping in mind Remark
+3.19:
+⟨(a ⋆ S)f,g⟩ =
+lim
+n→+∞
+�
+R2d an(z)⟨π(z)Sπ(z)∗f,g⟩ dz
+=
+lim
+n→+∞
+�
+R2d an(z)⟨π(z)S∗π(z)∗g,f⟩ dz
+=
+lim
+n→+∞
+�
+R2d an(z)(g ⊗ f) ⋆ ˇS∗(z) dz.
+Let us address (50):
+⟨(b ⋆ A)f,g⟩ =
+lim
+n→+∞⟨Kb⋆An,Kg⊗f⟩
+=
+lim
+n→+∞
+�
+R2d
+� �
+R2d b(x, ω)e2πi(y−u)ωKAn(y − x, u − x) dxdω
+�
+× g(y)f(u) dydu
+=
+lim
+n→+∞
+�
+R2d KAn(y′, u′)
+� �
+R2d b(x, ω)e−2πi(y′−u′)ω
+× g(y′ + x)f(u′ + x) dxdω
+�
+dy′du′
+=
+lim
+n→+∞
+�
+R2d KAn(y′, u′)
+� �
+R2d b∗(x′, ω′)e2πi(y′−u′)ω′
+× g(y′ − x′)f(u′ + x′) dx′dω′
+�
+dy′du′
+=
+lim
+n→+∞
+�
+R2d KAn(y′, u′)Kb∗⋆(g⊗f)(y′, u′) dy′du′,
+where for sake of brevity we set b∗(z) := b(−z).
+(ii) and (iii) are trivial.
+(iv) We prove just (51), (52) and (53). The remaining identities can be derived in
+a similar manner.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+21
+In order to show (51) we compute for z ∈ R2d:
+(S ⋆ (T ⋆ b))(z) = tr
+�
+S ◦ αz
+���
+R2d b(z)αwT dw
+�
+ˇ
+��
+= tr
+�
+S ◦
+��
+R2d b(w)αz ((αwT)ˇ) dw
+��
+= tr
+�
+S ◦
+�
+R2d b(w)αzα−w ˇT dw
+�
+= tr
+�
+S ◦
+�
+R2d b(−w′)αw′αz ˇT dw′
+�
+=
+�
+R2d b(−w′) tr
+�
+Sαw′+z ˇT
+�
+dw′,
+where the last equality is due, e.g., to [20, Proposition 2.9]. we can the rephrase the
+last right-side term as
+�
+R2d b(z − w′′) tr
+�
+Sαw′′ ˇT
+�
+dw′′ =
+�
+R2d b(z − w′′)(S ⋆ T)(w′′) dw′′
+= ((S ⋆ T) ∗ b)(z).
+For the proof of (52), the following property of the trace is useful:
+�
+R2d tr(SαwT) dw = tr(S) tr(T),
+where S, T ∈ J 1. Take now f, g ∈ S0(Rd):
+⟨(S ⋆ (T ⋆ Q))f,g⟩ =
+�
+R2d tr(Tαz ˇQ)⟨αzSf,g⟩ dz
+=
+�
+R2d tr(Qαz ˇT) tr((αzS)(f ⊗ g)) dz
+=
+�
+R2d
+�
+R2d tr(Q(αz ˇT)αw((αzS)(f ⊗ g))) dwdz
+=
+�
+R2d
+�
+R2d tr((f ⊗ g)(αwQ)αz((αw ˇT)S)) dzdw
+=
+�
+R2d tr(Sαw ˇT) tr((αwQ)(f ⊗ g)) dw
+= ⟨((S ⋆ T) ⋆ Q)f,g⟩.
+
+22
+FEDERICO BASTIANONI AND FRANZ LUEF
+Also the last identity (53) may be deduced by a direct computation. For f, g ∈
+S0(Rd) we have
+⟨((S ⋆ b) ⋆ σ)f,g⟩ =
+�
+R2d σ(z)⟨αz(S ⋆ b)f,g⟩ dz
+=
+�
+R2d σ(z)
+�
+R2d b(w)⟨(αwS)π(z)∗f,π(z)∗g⟩ dwdz
+=
+�
+R2d
+�
+R2d σ(z)b(w)⟨(αw+zS)f,g⟩ dwdz
+=
+�
+R2d
+�
+R2d σ(z)b(w) tr((αw+zS)(f ⊗ g)) dwdz
+=
+�
+R2d b(w)
+�
+R2d σ(z′ − w) tr((αz′S)(f ⊗ g)) dz′dw
+=
+�
+R2d(
+�
+R2d b(w)σ(z′ − w) dz′) tr((αz′S)(f ⊗ g)) dw
+=
+�
+R2d b ∗ σ(z′)⟨(αz′S)f,g⟩ dz′
+= ⟨(S ⋆ (b ∗ σ))f,g⟩.
+This concludes the proof.
+□
+Corollary 3.21. The mappings FWτ and Wτ defined on S0 can be extended to
+topological isomorphisms
+FWτ : S′
+0 → S′
+0(R2d)
+and
+Wτ : S′
+0 → S′
+0(R2d)
+by duality:
+(57)
+⟨FWτS,a⟩ := ⟨S,SRτa⟩,
+⟨WτS,a⟩ := ⟨S, Opτ a⟩,
+where S ∈ S′
+0 and a ∈ S0(R2d). The inverses are given by
+SRτ : S′
+0(R2d) → S′
+0
+and
+Opτ : S′
+0(R2d) → S′
+0,
+respectively.
+Proof. The definitions in (57) rely on the fact that Opτ = W ∗
+τ and SRτ = F ∗
+Wτ, see
+Theorem 3.7 and Corollary 3.9. It is straightforward to see that if S ∈ S′
+0, then
+FWτS and WτS defined as in (57) are in S′
+0(R2d). Also linearity and boundedness
+of FWτ : S′
+0 → S′
+0(R2d) and Wτ : S′
+0 → S′
+0(R2d) are easy to verify as well as the fact
+that they extend FWτ : S0 → S0(R2d) and Wτ : S0 → S0(R2d).
+We show that Wτ is an isomorphisms with inverse Opτ, then FWτ is treated in
+the same way.
+Wτ is injective because Opτ : S0(R2d) → S0 is an isomorphism.
+Fix now a ∈ S′
+0(R2d), there exists a sequence {an}n ⊆ S0(R2d) such that an
+w−∗
+−→
+n
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+23
+a
+in
+S′
+0(R2d). Since Wτ is an isomorphism between S0 and S0(R2d), there exists
+{An}n ⊆ S0 such that an = WτAn. We see that there is A ∈ S′
+0 such that An
+w−∗
+−→
+n
+A
+in
+S′
+0, in fact taking b ∈ S0(R2d)
+⟨a,b⟩ =
+lim
+n→+∞⟨WτAn,b⟩ = lim
+n→+∞⟨An, Opτ b⟩.
+Hence a = WτA, which proves that Wτ is onto. We show now that Wτ ◦ Opτ is the
+identity on S′
+0(R2d), take a ∈ S′
+0(R2d) and b ∈ S0(R2d):
+⟨Wτ ◦ Opτ a,b⟩ = ⟨Opτ a, Opτ b⟩ = ⟨a,Wτ ◦ Opτ b⟩ = ⟨a,b⟩.
+The first identity is just (57), the second one is (32) and the last one is (i) of
+Corollary 3.8. For the other direction, take S ∈ S′
+0 and T ∈ S0:
+⟨Opτ ◦WτS,T⟩ = ⟨WτS,WτT⟩ = ⟨S, Opτ ◦WτT⟩ = ⟨S,T⟩.
+The first identity is (32), the second one is (57) and the last one is (i) of Corollary
+3.8.
+□
+3.3. τ-Cohen’s class of operators. In the present subsection we define Qτ
+a(S)
+and recall the definition of Qτ
+S(f) from [17]. We shall see that Qτ
+a(S) relates to
+well-known objects and observe that it coincides with the τ-symbol of the mixed-
+state localization operator a ⋆ S. We continue with some statements concerning the
+interplay between the Gabor matrix of an operator Gϕ
+T, the τ-Cohen’s class, the
+trace and the τ-Wigner distribution.
+Definition 3.22. For a ∈ S′
+0(R2d) we define the τ-Cohen’s class distribution, with
+kernel a, of an operator S ∈ S0 as
+(58)
+Qτ
+a(S) := a ∗ WτS.
+Of course, the rank-one case f ⊗ g reduces to the definition given in (23). We
+recall also the definition given in [17] of Cohen’s class distribution of a function
+f ∈ S0(Rd) w.r.t. the operator S ∈ S′
+0 by
+(59)
+QSf := (f ⊗ f) ⋆ ˇS.
+It can be easily seen that for every z ∈ R2d
+QSf(z) = (f ⊗ f) ⋆ ˇS(z) = ⟨(αzS)f, f, ⟩.
+Remark 3.23. If a ∈ S′
+0(R2d) and S ∈ S0, then we see that the τ-Cohen’s class rep-
+resentation of S w.r.t. a is just the τ-symbol of the mixed-state localization operator
+a ⋆ S:
+aa⋆S
+τ
+= Wτ(a ⋆ S) = a ∗ WτS = Qτ
+a(S).
+
+24
+FEDERICO BASTIANONI AND FRANZ LUEF
+Lemma 3.24. Let S ∈ S0 have the spectral decomposition �∞
+n=1 fn⊗gn, for f, ϕ, ψ ∈
+S0(Rd) and {hn}n ⊆ S0(Rd) with
+∞
+�
+n=1
+∥hn∥2
+S0 < +∞.
+. Then for every z ∈ R2d:
+Qτ
+W1−τ ( ˇψ, ˇϕ)(S)(z) =
+∞
+�
+n=1
+Vϕfn(z)Vψgn(z);
+(60)
+Qτ
+W1−τ ( ˇϕ, ˇϕ)(
+∞
+�
+n=1
+hn ⊗ hn)(z) =
+∞
+�
+n=1
+|Vϕhn(z)|2 .
+(61)
+Proof. Clearly, it suffices to prove the first identity. We show first that for f, g ∈
+S0(Rd)
+(62)
+Qτ
+a(f, g) = (f ⊗ g) ⋆ Op1-τ(a).
+In fact, applying Fσ to the right-hand side first we get
+Fσ((f ⊗ g) ⋆ Op1-τ(a)) = FWτ(f ⊗ g) · FW1−τ Op1-τ(a) = V τ
+g f · Fσa.
+We apply Fσ a second time:
+(f ⊗ g) ⋆ Op1-τ(a) = FσV τ
+g f ∗ FσFσa = Wτ(f, g) ∗ a.
+We can now proceed as follows:
+Qτ
+W1−τ ( ˇψ, ˇϕ)(S) = W1−τ( ˇψ, ˇϕ) ∗ Wτ(
+∞
+�
+n=1
+fn ⊗ gn) =
+∞
+�
+n=1
+W1−τ( ˇψ, ˇϕ) ∗ Wτ(fn, gn)
+=
+∞
+�
+n=1
+(fn ⊗ gn) ⋆ Op1-τ(W1−τ( ˇψ, ˇϕ)) =
+∞
+�
+n=1
+(fn ⊗ gn) ⋆ ( ˇψ ⊗ ˇϕ)
+=
+∞
+�
+n=1
+Vϕfn(z)Vψgn(z),
+where the last equality is due to [17].
+□
+We call a bounded operator T on L2(Rd) positive, denoted by T ≥ 0, if
+⟨Tf, f⟩ ≥ 0,
+∀ f ∈ L2(Rd).
+An operator T ∈ J 1 and T ≥ 0 is also called a state in quantum mechanics.
+Let us take T ∈ S′
+0 and ϕ ∈ S, then the Gabor matrix of T (w.r.t. ϕ) is defined as
+(63)
+Gϕ
+T(z, w) := ⟨Tπ(w)ϕ, π(z)ϕ⟩,
+z = (x, ω), w = (u, v) ∈ R2d.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+25
+We notice that the Gabor matrix of an operator does not depend on τ, in the sense
+that
+Gϕ
+T(z, w) = ⟨Tπ(w)ϕ, π(z)ϕ⟩ = ⟨Tπτ(w)ϕ, πτ(z)ϕ⟩,
+∀ τ ∈ [0, 1].
+Remark 3.25. We point out that the diagonal of the Gabor matrix of T, w.r.t. ϕ,
+is the Cohen’s class representation of ϕ w.r.t. T up to a reflection:
+(64)
+Gϕ
+T(−z, −z) = QTϕ(z).
+In fact
+Gϕ
+T(−z, −z) = ⟨Tπ(−z)ϕ, π(−z)ϕ⟩ = ⟨Tπ(z)∗ϕ, π(z)∗ϕ⟩
+= ⟨(αzT)ϕ, ϕ, ⟩ = QT ϕ(z).
+Let F and H be functions of (z, w) ∈ R4d and let Θ be a real 4d × 4d matrix.
+Then the twisted convolution induced by Θ is defined as
+(65)
+F ♮Θ H(z, w) :=
+�
+R2d
+�
+R2d F(z′, w′)H(z − z′, w − w′)e2πi(z,w)Θ(z′,w′) dz′dw′.
+Lemma 3.26. Let T, S ∈ J 1, T, S ≥ 0. Then for every τ ∈ [0, 1] we have
+(66)
+tr(TS) =
+�
+R2d WτT(z)WτS(z) dz.
+Proof. Since T and S are trace-class and positive, they can be described as
+T =
+∞
+�
+n=1
+λnfn ⊗ fn,
+S =
+∞
+�
+n=1
+µngn ⊗ gn
+for some orthonormal sets {fn}n and {gn}n in L2 and λn, µn ≥ 0. Let {en}n be an
+o.n.b. for L2(Rd):
+tr(TS) =
+∞
+�
+n=1
+⟨TSen, en⟩ =
+∞
+�
+i,j
+λjµi |⟨fj, gi⟩|2 .
+On the other hand,
+�
+R2d WτT(z)WτS(z) dz =
+∞
+�
+i,j
+λjµi
+�
+R2d Wτfj(z)Wτgi(z) dz =
+∞
+�
+i,j
+λjµi |⟨fj, gi⟩|2 ,
+where the last equality is due to Moyal’s identity. This concludes the proof.
+□
+Remark 3.27. Since we assume S ≥ 0, S is self-adjoint and for τ = 1/2 we have
+that W1/2S is real-valued. In fact, using the representation given in the proof of
+Lemma 3.26:
+W1/2S =
+∞
+�
+n=1
+µnW1/2gn
+
+26
+FEDERICO BASTIANONI AND FRANZ LUEF
+with every W1/2gn real-valued and µn ≥ 0. Hence, for τ = 1/2 we recover [12,
+Lemma 2.7].
+Lemma 3.28. Let T ∈ J 1 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1. Then
+(67)
+tr T =
+�
+R2d⟨(αzT)ϕ, ϕ⟩ dz =
+�
+R2d QT ϕ(z) dz =
+�
+R2d Gϕ
+T(z, z) dz.
+Proof. The proof follows from a direct computation using the representations pre-
+sented in the proof of Lemma 3.26 and Moyal’s identity involving the function ϕ:
+⟨fj, gi⟩ = ⟨Vϕfj, Vϕgi⟩,
+we leave details to the interested reader.
+□
+Lemma 3.29. Let T ∈ J 1, T ≥ 0 and let ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1. Then
+for every z ∈ R2d:
+(68)
+QTϕ(z) =
+�
+R2d WτT(w)Wτϕ(z + w) dw = WτT ∗ (Wτϕ)∗(z),
+where (Wτϕ)∗(w) = Wτϕ(−w).
+Proof. We compute directly
+QT ϕ(z) = ⟨π(z)Tπ(z)∗ϕ, ϕ⟩ = tr(T(π(z)∗ϕ ⊗ π(z)∗ϕ))
+=
+�
+R2d WτT(w)Wτ(π(z)∗ϕ ⊗ π(z)∗ϕ)(w) dw,
+the last equation holds because of Lemma 3.26. An elementary calculation gives
+Wτ(π(z)∗ϕ ⊗ π(z)∗ϕ)(w) = Wτϕ(z + w),
+which is also known as covariance property and this concludes the proof.
+□
+Lemma 3.30. Let T ∈ J 1, T ≥ 0 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.
+Then for every z, w ∈ R2d:
+|Gϕ
+T(z, w)|2 ≤ QTϕ(−z)QT ϕ(−w).
+Proof. The claim follows from the Cauchy-Schwarz inequality for the inner product
+induced by the positive operator T and Remark 3.25.
+□
+Lemma 3.31. Let 0d and Id denote the zero and identity d×d matrices, respectively.
+Let us define
+Θ :=
+
+
+0d
+0d
+0d
+0d
+Id
+0d
+0d
+0d
+0d
+0d
+0d
+0d
+0d
+0d
+−Id
+0d
+
+ .
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+27
+Let T ∈ J 1 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1. For z = (x, ω), w =
+(u, v) ∈ R2d we have
+Gϕ
+T(z, w) = Gϕ
+T ♮Θ(Gϕ
+ϕ⊗ϕ)∗(z, w)
+(69)
+=
+�
+R2d
+�
+R2d Gϕ
+T(z′, w′)(Gϕ
+ϕ⊗ϕ)∗(z − z′, w − w′)e2πi(ωx′−u′v) dz′dw′,
+where z′ = (x′, ω′), w′ = (u′, v′) ∈ R2d.
+Proof. We apply twice Moyal’s identity:
+Gϕ
+T(z, w) =
+�
+R2d Vϕ[Tπ(w)ϕ](z′)Vϕ[π(z)ϕ](z′) dz′
+=
+�
+R2d
+�
+R2d Vϕ[π(w)ϕ](w′)Vϕ[T ∗π(z′)ϕ](w′)⟨π(z′)ϕ, π(z)ϕ⟩ dz′dw′
+=
+�
+R2d
+�
+R2d Gϕ
+T(z′, w′)⟨π(w)ϕ, π(w′)ϕ⟩⟨π(z′)ϕ, π(z)ϕ⟩ dz′dw′.
+It is then a direct, although tedious, calculation to show that
+⟨π(z)ϕ, π(z′)ϕ⟩⟨π(w′)ϕ, π(w)ϕ⟩ = (Gϕ
+ϕ⊗ϕ)∗(z − z′, w − w′)e2πi(ωx′−u′v).
+This concludes the proof.
+□
+Lemma 3.32. Let T ∈ J 1, T ≥ 0 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.
+Then for any τ ∈ [0, 1]:
+(70)
+WτT(z) =
+�
+R2d
+�
+R2d e−2πi[(ωx′−ω′x)+( 1
+2− 3
+4 τ)x′ω′+x′v]Gϕ
+T
+�z′
+2 − w, −z′
+2 − w
+�
+dwdz′,
+where z = (x, ω), z′ = (x′, ω′), w = (u, v) ∈ R2d.
+Proof. We start rephrasing the τ-Wigner distribution of T:
+WτT(z) = FσFWτT(z) =
+�
+R2d e−2πi(ωx′−ω′x) tr(πτ(z′)∗T) dz′.
+Recalling the properties for πτ, see Section 2, we see that
+πτ(z′/2 + z′/2) = e2πi[(1−τ) x′ω′
+4
+−τ x′ω′
+4
+]πτ(z′/2)πτ(z′/2)
+= e
+π
+2 i(1−2τ)x′ω′πτ(z′/2)πτ(z′/2).
+
+28
+FEDERICO BASTIANONI AND FRANZ LUEF
+Taking the adjoint we get πτ(z′)∗ = e− π
+2 i(1−2τ)x′ω′πτ(z′/2)∗πτ(z′/2)∗ and we write
+using Lemma 3.28:
+tr(πτ(z′)∗T) = e− π
+2 i(1−2τ)x′ω′ tr(πτ(z′/2)∗Tπτ(z′/2)∗)
+= e− π
+2 i(1−2τ)x′ω′ �
+R2d⟨Tπτ(z′/2)∗πτ(w)∗ϕ, πτ(z′/2)πτ(w)∗ϕ⟩ dw
+= e− π
+2 i(1−2τ)x′ω′e− π
+2 i(1−τ)x′ω′
+×
+�
+R2d⟨Tπτ(−z′/2)πτ(−w)ϕ, πτ(z′/2)πτ(−w)ϕ⟩ dw
+= e− π
+2 i(2−3τ)x′ω′ �
+R2d⟨Tπ(−z′/2)π(−w)ϕ, π(z′/2)π(−w)ϕ⟩ dw
+= e− π
+2 i(2−3τ)x′ω′ �
+R2d e−2πix′v⟨Tπ(−z′/2 − w)ϕ, π(z′/2 − w)ϕ⟩ dw.
+This concludes the argument.
+□
+4. A characterization of Schwartz operators
+In this section we introduce weighted versions of S0 and give an alternative de-
+scription of the class S. We use the polynomial weight
+(71)
+vs(z) := (1 + |z|2)
+s
+2,
+z ∈ R2d,
+where s ≥ 0. In order to avoid an extremely cumbersome notation, just for the
+weight functions vs we shall use the following:
+vs ⊗ vs(z, w) := Kvs⊗vs = vs(z)vs(w),
+∀z, w ∈ R2d.
+Definition 4.1. For s ≥ 0 we define the weighted class of Feichtinger operators as
+(72)
+M1
+s := {S : S′
+0(Rd) → S0(Rd) | S is linear, continuous with kernel KS ∈ M1
+vs⊗vs(R2d)}.
+For S in M1
+s we define the mapping
+(73)
+∥S∥M1s := ∥KS∥M1
+vs⊗vs .
+Remark 4.2.
+(i) For s = 0 we recover the Feichtinger operators S0;
+(ii) The mapping defined in (73) is a norm on M1
+s and it is easy to see that
+(M1
+s, ∥·∥M1s) is a Banach space and the following continuous inclusion holds
+true for every s ≥ 0:
+(74)
+M1
+s ֒→ S0.
+Lemma 4.3. For any S ∈ M1
+s there exist {fn}n, {gn}n ⊆ M1
+vs⊗vs(R2d) such that
+S =
+∞
+�
+n=1
+fn ⊗ gn,
+∞
+�
+n=1
+∥fn∥M1vs ∥gn∥M1vs ≤ +∞,
+KS =
+∞
+�
+n=1
+Kfn⊗gn.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+29
+Proof. The proof follows from the fact that
+M1
+vs⊗vs(R2d) = M1
+vs(Rd)ˆ⊗M1
+vs(Rd).
+See also the proof of Lemma 3.4.
+□
+Theorem 4.4. For every τ ∈ [0, 1] the mapping Wτ : M1
+s → M1
+vs⊗vs(R2d) is a topo-
+logical isomorphism with inverse given by Opτ : M1
+vs⊗vs(R2d) → M1
+s.
+Proof. The proof follows the same pattern as the ones of Theorem 3.7 and Corollary
+3.8.
+□
+Corollary 4.5. An operator S belongs to M1
+s if and only if for some (hence every)
+τ ∈ [0, 1] WτS ∈ M1
+vs⊗vs(R2d).
+Theorem 4.6. The following is true:
+(75)
+S =
+�
+s≥0
+M1
+s.
+Proof. By Corollary 4.5, S belongs to the set on the right-hand side if and only if
+WτS ∈
+�
+s≥0
+M1
+vs⊗vs(R2d) = S(R2d).
+The claim follows since W1/2S is the Weyl symbol of S, i.e. aS
+1/2 = W1/2S.
+□
+We recall that a function F on R2d is called rapidly decaying if for every multiindex
+α, β ∈ Nd
+0 we have
+sup
+x,ω∈Rd
+��xαωβF(x, ω)
+�� < +∞,
+where, if x = (x1, . . . , xd) and α = (α1, . . . , αd), xα stands for xα1
+1 · . . . · xαd
+d .
+In [12, Theorem 1.1] a sufficient condition is given for a positive trace-class op-
+erator to be in S.
+Namely, if T ∈ B(L2), T ≥ 0, is such that WτT exists for
+some τ ∈ [0, 1] and it is rapidly decreasing, then T ∈ S and WτT exists for every
+τ ∈ [0, 1]. In this spirit, we provide the following sufficient condition for a generic
+S ∈ B(L2). Observe that we do not not require S to be positive.
+Corollary 4.7. Let S ∈ B(L2) and assume that for some τ ∈ [0, 1] WτS exists.
+Suppose also that, w.r.t. some non-zero window in L2(R2d), the STFT of WτS is
+rapidly decaying. Then WτS exists for every τ ∈ [0, 1] and S is in S.
+Proof. Let us pick G ∈ L2(R2d) ∖ {0}. If VGWτS is rapidly decaying then S ∈ M1
+s
+for every s ≥ 0. The claim follows from Theorem 4.6.
+□
+
+30
+FEDERICO BASTIANONI AND FRANZ LUEF
+Acknowledgments
+The first author would like to thank Eduard Ortega for the financial support to
+visit Trondheim which led to this work.
+References
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+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+31
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+Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli
+Abruzzi 24, 10129 Torino, Italy
+Email address: federico.bastianoni@polito.it
+Department of Mathematics, NTNU Norwegian University of Science and Tech-
+nology, NO-7491 Trondheim, Norway
+Email address: franz.luef@ntnu.no
+
diff --git a/4tE4T4oBgHgl3EQfBAt_/content/tmp_files/load_file.txt b/4tE4T4oBgHgl3EQfBAt_/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..48c939228bc51243412fd66aed0e2acf5c87dd84
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@@ -0,0 +1,820 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf,len=819
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='04848v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='FA]  12 Jan 2023  τ-QUANTIZATION AND τ-COHEN CLASSES DISTRIBUTIONS  OF FEICHTINGER OPERATORS  FEDERICO BASTIANONI AND FRANZ LUEF  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We investigate the τ-quantizations and Cohen’s class distributions  of a suitable class of trace-class operators, called Feichtinger’s operators, and  show that it is a convenient substitute for the class of Schwartz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Many  well-known concepts and results for functions in time-frequency analysis have an  operator-analog in our setting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' that Cohen’s classes are convolutions of Wigner  functions with distributions or characterization of the class of Schwartz operators  as an intersection of weighted variants of the class of Feichtinger operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Preliminaries  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  A family of time-frequency representations  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Basics of QHA and novel tools  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-quantization of functions  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Feichtinger operators  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-quantization of operators  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  A convenient environment for QHA  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-Cohen’s class of operators  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  A characterization of Schwartz operators  28  Acknowledgments  30  References  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Introduction  There is a vast literature on the boundedness of pseudodifferential operators  for certain classes of symbols in various quantization schemes along the lines of  H¨ormander classes or alternatively using Sj¨ostrand’s class or Shubin’s classes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' 42B35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='46E35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='47G30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='47B10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Key  words  and  phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Cohen’s  class,  τ-quantization,  Feichtinger’s  algebra,  Wigner  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  1  2  FEDERICO BASTIANONI AND FRANZ LUEF  [1, 3, 4, 11, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In the present work, we put our focus on Shubin’s τ-quantization  and the associated time-frequency representations, the τ-Cohen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Our approach to this circle of ideas is based on the framework of quantum har-  monic analysis with the goal to lift the well-known results concerning functions to  an appropriate class of functions, which we call Feichtinger operators, S0, and which  is the operator analog of the well-known Feichtinger algebra S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We also discuss the relation between Feichtinger operators S0 and the class of  Schwartz operators introduced by Keyl, Kiukas and Werner in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' There the idea  is put forward that one should look for analogs of function spaces in the setting of  classes of operators, which has been realized in the case of Sobolev spaces in [15]  and for modulation spaces in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  For τ ∈ [0, 1] the τ-quantization of a symbol a ∈ S′(R2d), the space of tempered  distributions, is given by  (1)  Opτ(a)f(t) :=  �  R2d e2πi(t−y)ξa((1 − τ)t + τy, ξ)f(y) dydξ f ∈ S(Rd),  where the operator Opτ(a) is understood to be defined in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' A well-  known fact is that one can relate ⟨Opτ(a)f, g⟩ to a time-frequency representation,  Wτ(f, g), the cross-τ-Wigner distribution of f and g:  ⟨Opτ(a)f, g⟩ = ⟨a, Wτ(g, f)⟩,  for all  f, g ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Given an operator S, we denote by aS  τ its τ-symbol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' the tempered distribution  such that Opτ  �  aS  τ  �  = S and Opτ is called the τ-Shubin quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For f, g ∈  L2(Rd) we denote the rank-one operator by f ⊗ g and note that af⊗g  τ  = Wτ(g, f),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' there is an intrinsic relation between quantization schemes and time-frequency  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We show that for well-behaved operators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' trace class operators or Feichtinger  operators, this relation might be extended to operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Recall that Wigner in  his ground-breaking work on quasi-probability distributions introduced the cross-  Wigner distribution for certain classes of operators [24], which was later extended  to more general classes of operators by Moyal in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let S be a continuous operator between the Feichtinger algebra S0 and its contin-  uous dual space S′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We denote by KS the kernel of S, which exists by Feichtinger’s  kernel theorem and is a mild distribution on R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We define Feichtinger operators, S0, to be the following class of continuous and  linear operators S0 := S : S′  0(Rd) → S0(Rd) that map norm bounded w-∗ convergent  sequences in S′  0 into norm convergent sequences in S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In [10] it was shown that  these are precisely the linear continuous operators from S′  0 to S0 that have a kernel  in Feichtinger’s algebra, the so-called inner kernel theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  One of our main tools is that Feichtinger operators have a nice spectral decomposi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' If S is in S0, then there exist two (non-unique) sequences {fn}n, {gn}n ⊆ S0(Rd)  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  3  such that  S =  ∞  �  n=1  fn ⊗ gn,  ∞  �  n=1  ∥fn∥S0 ∥gn∥S0 < ∞,  KS =  ∞  �  n=1  Kfn⊗gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Hence, Feichtinger operators are trace class operators and we can compute their trace  as follows tr(S) =  �  Rd KS(x, x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In [7] operators having such a decomposition  have been studied and called Feichtinger states in case tr(S) = 1, but there the link  between these operators and the work [10] was not established, which is one of our  main observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Then the τ-Wigner distribution of S is defined in the following way  (2)  WτS(x, ω) :=  �  Rd e−2πitωKS(x + τt, x − (1 − τ)t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Our key observation is the following identity:  ⟨a,WτS⟩ = tr(Opτ(a)S∗) =: ⟨Opτ(a),S⟩,  for S in S0 or J 1, and WτS is the τ-Wigner distribution of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Consequently, we  interpret WτS as the τ-quantization of an operator in S0 or J 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Note, that if S is the rank-one operator f ⊗ g this becomes the aforementioned  relation between the τ-Wigner distribution and the Shubin τ-transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Based on this framework we deduce operator analogs of well-known results on  τ-Wigner distributions and τ-Shubin quantization, which indicates that this is a  very convenient setting for this type of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In addition, we extend the  Cohen class of an operator, introduced in [17], to the τ-setting and show that it  can be written as the convolution of the Wigner distribution of an operator with a  distribution as in the function setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We close our discussion with the introduction of weighted versions of S0 and prove  that the intersection of all these is the class of Schwartz operators in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' As in the  case of functions, we hope that this global description of the Schwartz operators will  also turn out to be useful in subsequent studies and it also hints at operator analogs  of Gelfand-Shilov classes or other classes of test functions and the corresponding  class of ultradistributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Preliminaries  In this paper, the parameter τ always belongs to [0, 1], even when not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' A family of time-frequency representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For x, ω ∈ Rd we define the  translation and modulation operator by  Txf(t) := f(t − x),  Mωf(t) := e2πiωtf(t),  ∀t ∈ Rd,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Their composition is denoted by π(x, ω) := MωTx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  4  FEDERICO BASTIANONI AND FRANZ LUEF  Given τ ∈ [0, 1], the τ-time-frequency shift (τ-TFS) at (x, ω) ∈ R2d is defined to  be  (3)  πτ(x, ω) := e−2πiτxωMωTx = M(1−τ)ωTxMτω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  For τ = 0 we recover the usual time-frequency shifts π0 = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The following relations  are consequences of elementary computations, which are left to the reader:  πτ(x, ω)πτ(x′, ω′) = e−2πi[(1−τ)xω′−τx′ω]πτ(x + x′, ω + ω′),  πτ(x, ω)πτ(x′, ω′) = e−2πi[xω′−x′ω]πτ(x′, ω′)πτ(x, ω),  πτ(x, ω)∗ = π1−τ(−x, −ω) = e−2πi(1−τ)xωπ(−x, −ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In the present paper the symbol ⟨·,·⟩ either denotes the inner product in L2(Rd)  or a duality pairing between a Banach space X and its dual space X′, which is  compatible with the latter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' ⟨·,·⟩ is assumed to be linear in the first argument  and conjugate-linear in the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In particular, the dual pairs considered in  this work are (L2, L2), (S′  0, S0), (S′  0, S0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Above, S0 is the Feichtinger algebra (24), for the definitions of S0 and S′  0 see the  equations (25),(26) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We introduce for f, g ∈ L2(Rd), or for any suitable  dual pair, the τ-short-time Fourier transform (τ-STFT) of f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='t g:  (4)  V τ  g f(x, ω) := ⟨f, πτ(x, ω)g⟩,  ∀x, ω ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  As can be easily verified, the mapping  πτ : R2d → U(L2(Rd)),  where U(L2(Rd)) denotes the unitary operators on L2(Rd), is a projective represen-  tation of R2d for any τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Consequently, V τ is the wavelet transform associated to πτ,  thus V τ  g f is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For τ = 0 we obtain the usual STFT V 0  g f = Vgf and we have  (5)  V τ  g f(x, ω) = e2πiτxωVgf(x, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  By the preceding identity, we have that V  1  2  g f is the cross-ambiguity function of f and  g:  (6)  V  1  2  g f(x, ω) = A(f, g)(x, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We recall another frequently used time-frequency representation, the so-called  cross-τ-Wigner distribution of f and g in L2(Rd) defined by  (7)  Wτ(f, g)(x, ω) :=  �  Rd e−2πitωf(x + τt)g(x − (1 − τ)t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We aim to extend the definition of Wτ from functions to operators, see (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Basics of QHA and novel tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In this subsection we introduce the ba-  sic definitions of quantum harmonic analysis (QHA) following the seminal work of  Werner [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  For z ∈ R2d and A ∈ B(L2(Rd)) the translation of the operator A by z is  (8)  αz(A) := π(z)Aπ(z)∗,  which satisfies αzαz′ = αz+z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' By the parity operator, we mean  (9)  Pf(t) := ˇf(t) := f(−t),  for any f ∈ L2(Rd), which induces an involution of A ∈ B(L2(Rd)):  (10)  ˇA := PAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We denote by J 1 the space of all trace class operators on L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Given a ∈ L1(R2d)  and S ∈ J 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The convolution between a and S is the operator  (11)  a ⋆ S := S ⋆ a :=  �  R2d a(z)αz(S) dz,  were the integral may be interpreted in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For operators S, T ∈ J 1,  their convolution is the function defined for every z ∈ R2d as  (12)  S ⋆ T(z) := tr  �  Sαz( ˇT)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In this paper, we reserve the symbol ⊗ for rank-one operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Namely, given  f, g ∈ L2(Rd):  (13)  (f ⊗ g)ψ := ⟨ψ, g⟩f,  ∀ψ ∈ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The kernel of an operator S will always be denoted by KS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Evidently, the kernel of  the operator f ⊗ g is the tensor product of functions f(x)g(y):  (f ⊗ g)ψ(t) = ⟨ψ, g⟩f(t) =  �  Rd f(t)g(x)ψ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In the sequel we denote the tensor product of two functions by f(x)g(y), we shall  adopt the notation  (14)  Kf⊗g(x, y) = f(x)g(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We now interpret (7) as the cross-τ-Wigner distribution of the rank-one operator  f ⊗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Hence, it is natural to define the τ-Wigner distribution of an operator S with kernel  KS in the following way:  (15)  WτS(x, ω) :=  �  Rd e−2πitωKS(x + τt, x − (1 − τ)t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  For S ∈ J 1 and τ ∈ [0, 1], we define the Fourier-τ-Wigner transform of S to be:  (16)  FWτS(z) := tr (πτ(z)∗S) ,  ∀z ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  6  FEDERICO BASTIANONI AND FRANZ LUEF  For τ = 1/2 we recover the usual Fourier-Wigner transform [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The τ-spreading representation of S ∈ B(L2) is the decomposition  (17)  S =  �  R2d h(z)πτ(z) dz,  where the integral is understood in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The function h is called the  τ-spreading function of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In the following, we shall consider the τ-spreading representation as a quantization  scheme that assigns to a function an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Namely, h ∈ L1(R2d) gets associated  to the operator  (18)  SRτ(h) :=  �  R2d h(z)πτ(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let Fσ denote the symplectic Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In the following lemma we collect  a number of important relations between these notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The proofs are elementary  computations and based on the spectral decomposition of the trace class operators  S and T ([21]), which we leave to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let f, g, ∈ L2(Rd), S, T ∈ J 1, a ∈ L1(R2d) and τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then:  (i) Fσ(Wτ(f ⊗ g)) = V τ  g f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) FWτ(f ⊗ g) = V τ  g f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iii) WτS = FσFWτS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iv) FWτS(x, ω) = e−2πi(1/2−τ)xωFW1/2S(x, ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (v) Fσ(S ⋆ T) = FWτS · FW1−τT = FW1−τS · FWτT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (vi) FWτ(a ⋆ S) = Fσa · FWτS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (vii) FWτS is the τ-spreading function of S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' S =  �  R2d FWτS(z)πτ(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We notice that if we consider the rank-one operator S = f ⊗g, then the assertions  (iii) and (ii) of the previous lemma imply  (19)  Wτ(f, g) = Wτ(f ⊗ g) = FσV τ  g f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' τ-quantization of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The τ-quantization of a symbol a ∈ S′(R2d),  the space of tempered distributions, is formally given by  (20)  Opτ(a)f(t) :=  �  R2d e2πi(t−y)ξa((1 − τ)t + τy, ξ)f(y) dydξ,  where f ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Opτ(a) may be described rigorously in the weak sense:  ⟨Opτ(a)f, g⟩ = ⟨a, Wτ(g, f)⟩,  ∀f, g, ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Given an operator S, we denote by aS  τ its τ-symbol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' the tempered distribution  such that  Opτ  �  aS  τ  �  = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  7  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Under suitable assumptions, for example a ∈ L1(R2d), straightforward  calculations give  Opτ(a) =  �  R2d Fσa(z)πτ(z) dz,  and since also FWτ Opτ(a) is the τ-spreading function of Opτ(a), we have  (21)  a = FσFWτ Opτ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Hence, for S ∈ J 1  (22)  aS  τ = FσFWτS = WτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Given a ∈ S′  0(R2d) and f, g ∈ S0(Rd), we recall the definition of cross-τ-Cohen’s  class representation of f and g, with kernel a:  (23)  Qτ  a(f, g) := a ∗ Wτ(f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Feichtinger operators  In this section we summarize some important results concerning a class of op-  erators studied in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For such operators, introduced below, we adopt the name  “Feichtinger operators” for reasons which will become evident later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We recall that the Feichtinger algebra over Rd [9] is the Banach space  (24)  S0(Rd) := {f ∈ L2(Rd) | Vgf ∈ L1(R2d)},  for some g ∈ L2(Rd) ∖ {0}, endowed with the norm  ∥f∥S0 := ∥Vgf∥L1 =  �  R2d |Vgf(x, ω)| dxdω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We refer the reader to [13] for a detailed survey on S0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In this work, S′  0(Rd)  denotes the conjugate-dual of S0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The set of Feichtinger operators is defined to be  S0 :={S : S′  0(Rd) → S0(Rd) | S is linear, continuous and  maps norm bounded w-∗ convergent sequences in S′  0  (25)  into norm convergent sequences in S0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We adopt the following notation:  (26)  S′  0 := B(S0(Rd), S′  0(Rd))  and state the so called Outer Kernel Theorem [10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1]:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The Banach space S′  0 is isomorphic to S′  0(R2d) via the map T �→ KT,  where the relation between T and its kernel KT is given by  ⟨Tf,g⟩ = ⟨KT,Kg⊗f⟩,  ∀ f, g, ∈ S0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  8  FEDERICO BASTIANONI AND FRANZ LUEF  The following statement goes under the name of Inner Kernel Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We  present it in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' To this end, we introduce the following notation: given  σ, ν ∈ S′  0(Rd), we denote by ν �⊗σ the unique element of S′  0(R2d) such that  ⟨ν �⊗σ,Kψ⊗ϕ⟩ = ⟨ν,ψ⟩⟨σ,ϕ⟩,  ∀ ψ, ϕ ∈ S0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We refer the reader to [10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='10, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The space of Feichtinger operators S0 is a Banach space if endowed  with the norm of B(S′  0, S0) and it is naturally isomorphic as Banach space to S0(R2d)  through the map T �→ KT, where the relation between T and its kernel KT is given  by  ⟨ν,Tσ⟩ = ⟨ν �⊗σ,KT⟩,  ∀ σ, ν, ∈ S′  0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Moreover, S0 is Banach algebra under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' If S, T ∈ S0, then  (27)  KS◦T(y, u) =  �  Rd KT(y, t)KS(t, u) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  By the above theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3, S′  0 is the (conjugate) topological dual of S0  and the duality is given by  (28)  ⟨T,S⟩ = ⟨KT,KS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Suppose S ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then there exist two non-unique sequences {fn}n, {gn}n ⊆  S0(Rd) such that  S =  ∞  �  n=1  fn ⊗ gn,  ∞  �  n=1  ∥fn∥S0 ∥gn∥S0 < +∞,  KS =  ∞  �  n=1  Kfn⊗gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Moreover,  S0 ֒→ J 1  with  tr(S) =  �  Rd KS(x, x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We just have to prove the continuous inclusion of Feichtinger operators into  J 1, all the remaining statements can be found in [10], see in particular Corollary  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='15 and Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The claim follows from an elementary computation:  ∥S∥J 1 = |tr(A)| ≤  �  Rd  ∞  �  n=1  |fn(x)gn(x)| dx =  ∞  �  n=1  �  Rd |fn(x)gn(x)| dx  ≤  ∞  �  n=1  ∥fn∥L2 ∥gn∥L2 ≲  ∞  �  n=1  ∥fn∥S0 ∥gn∥S0 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Since S0(R2d) = S0(Rd)ˆ⊗S0(Rd), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1], we get  ∥S∥J 1 ≲ ∥KS∥S0 ≍ ∥S∥S0 ,  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  9  which gives the desired assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  The preceding result and the observations in [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' 4] yield  (29)  S0 ֒→ J 1 ֒→ J 2 ֒→ B(L2(Rd)) ֒→ S′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The fact that all Feichtinger operators are trace class implies the validity of Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' τ-quantization of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The following remark is the key insight for the  subsequent results concerning Opτ and Wτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let us consider f, g ∈ L2(Rd) such that f ̸= 0, a ∈ L2(R2d) and {fj}j  o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' for L2 with f1 = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then we compute as follows:  ⟨Opτ(a)f, g⟩ = ⟨Opτ(a)f,  ∞  �  j=1  ⟨g, fj⟩fj⟩ =  ∞  �  j=1  ⟨Opτ(a) (⟨fj, g⟩f) , fj⟩  =  ∞  �  j=1  ⟨Opτ(a)(f ⊗ g)fj, fj⟩ = tr (Opτ(a)(f ⊗ g)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Taking into account the weak definition of Opτ(a) and (15) we can write  (30)  ⟨Opτ(a)f, g⟩ = ⟨a, Wτ((f ⊗ g)∗)⟩ = tr (Opτ(a)(f ⊗ g)) = ⟨Opτ(a), (f ⊗ g)∗⟩(J 1,J ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  By computations similar to the ones above for S ∈ J 1 with the spectral decomposition  �∞  k=1 λkfk ⊗ gk after extending {fk}k to an orthonormal basis of L2(Rd) implies  (31)  ⟨a, WτS⟩ = tr (Opτ(a)S∗) = ⟨Opτ(a), S⟩(J 1,J ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For every τ ∈ [0, 1] the following mappings are linear and continu-  ous:  Opτ : L2(R2d) → J ∞,  Wτ : J 1 → L2(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Moreover, Opτ is the Banach space adjoint of Wτ: Opτ = W ∗  τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The boundedness of Opτ is evident;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' the proof of the continuity of Wτ follows  by a similar reasoning as the proof of the subsequent Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The last claim  is just (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For every τ ∈ [0, 1] the following mappings are linear and continu-  ous:  Opτ : S′  0(R2d) → S′  0,  Wτ : S0 → S0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Moreover, Opτ is the Banach space adjoint of Wτ: Opτ = W ∗  τ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  for every  a ∈ S′  0(R2d) and S ∈ S0  (32)  ⟨a,WτS⟩ = ⟨Opτ(a),S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  10  FEDERICO BASTIANONI AND FRANZ LUEF  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The boundedness and linearity of Opτ follow from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' By using  the formal representation of Opτ(a) we can derive an expression for its kernel:  (33)  KOpτ (a)(t, x) =  �  Rd e2πi(t−x)ωa((1 − τ)t + τx, ω) dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let us consider first f, g ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then a standard argument, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' [5, Proposition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='25], gives that  Wτ(f ⊗ g) = Wτ(f, g) ∈ S0(R2d)  with  ∥Wτ(f ⊗ g)∥S0 ≲ ∥f∥S0 ∥g∥S0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Since Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2 holds for S0, we write Wτ = FσFWτ and use the spectral decom-  position for S of the form �∞  n=1 fn ⊗ gn as shown in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Now, we compute:  FWτS(z) = tr(πτ(z)∗S) = tr(  ∞  �  n=1  πτ(z)∗(fn ⊗ gn))  =  ∞  �  n=1  ⟨πτ(z)∗fn, gn⟩ =  ∞  �  n=1  V τ  gnfn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (34)  Taking a suitable window for the norm on S0(R2d) [13, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3] we have  ∥FWτS∥S0 ≤  ∞  �  n=1  ��V τ  gnfn  ��  S0 =  ∞  �  n=1  ∥fn∥S0 ∥gn∥S0 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Consequently,  ∥FWτS∥S0 ≤ inf{  ∞  �  n=1  ∥fn∥S0 ∥gn∥S0 , S =  ∞  �  n=1  fn ⊗ gn}  ≤ inf{  ∞  �  n=1  ∥fn∥S0 ∥gn∥S0 , KS =  ∞  �  n=1  Kfn⊗gn}  = ∥KS∥S0 ≍ ∥S∥S0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We proved the boundedness of FWτ : S0 → S0(R2d), the continuity of the symplectic  Fourier transform Fσ : S0(R2d) → S0(R2d) is well-known, and thus the continuity of  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  11  Wτ : S0 → S0(R2d) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Concerning the last claim, we proceed as follows:  ⟨Opτ(a),S⟩ = ⟨KOpτ (a),KS⟩ = ⟨KOpτ (a),  ∞  �  n=1  Kf⊗gn⟩  =  ∞  �  n=1  ⟨KOpτ (a),Kf⊗gn⟩ =  ∞  �  n=1  ⟨Opτ(a)gn,fn⟩  =  ∞  �  n=1  ⟨a,Wτ(fn ⊗ gn)⟩ = ⟨a,  ∞  �  n=1  Wτ(fn ⊗ gn)⟩  = ⟨a,WτS⟩,  which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  On account of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='7, it seems reasonable to interpret WτS as the  τ-quantization of an operator in S0 or J 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i) For every τ ∈ [0, 1] the mapping Wτ : S0 → S0(R2d) is a  topological isomorphism with inverse given by Opτ : S0(R2d) → S0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) A linear and continuous operator S : S0(Rd) → S′  0(Rd) belongs to S0 if and  only if WτS ∈ S0(R2d) for some (and hence any) τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' (i) We observed in (22) that WτS is just the τ-symbol aS  τ of a trace class  operator S, in particular this holds for S ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Therefore,  Opτ ◦WτS = Opτ(aS  τ ) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We now show that if we start with a ∈ S0(R2d), then Opτ(a) belongs to S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' From  (33), we have that the kernel of Opτ(a) can be written as  KOpτ (a)(t, x) =  �  Rd e2πi(t−x)ωa((1 − τ)t + τx, ω) dω = ΨτF −1  2 a(t, x),  where F −1  2  is the inverse of the partial Fourier transform with respect to the second  variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Ψτ is the change of variables induced by the matrix  (35)  �  1 − τ  τ  1  −1  �  ,  ΨτF(t, x) := F((1 − τ)t + τx, t − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  From the assumption a in the Feichtinger algebra S0(R2d) we have F −1  2 a ∈ S0(R2d),  thus ΨτF −1  2 a is in S0(R2d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Opτ(a) is an element of S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The fact that Opτ is  continuous from S0(R2d) into S0 is evident from the applications of F −1  2  and Ψτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Hence we have shown that  Wτ ◦ Opτ(a) = aOpτ (a)  τ  = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) The claim is a straightforward consequence of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  12  FEDERICO BASTIANONI AND FRANZ LUEF  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i) For every τ ∈ [0, 1] FWτ : S0 → S0(R2d) is a topological  isomorphisms with inverse given by the τ-spreading representation  (36)  SRτ : S0(R2d) → S0 , a �→  �  R2d a(z)πτ(z) dz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) Let us define  (37)  SRτ : S′  0(R2d) → S′  0 a �→  �  R2d a(z)πτ(z) dz,  where the integral has to be understood weakly as follows:  ⟨SRτ(a)f,g⟩ := ⟨a,V τ  f g⟩,  a ∈ S′  0(R2d), f, g ∈ S0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Then SRτ as in (37) is well-defined, linear, continuous, extends (36) and it  is the Banach space adjoint of FWτ in (i):  (38)  SRτ = F ∗  Wτ,  in the sense that for every a ∈ S′  0(R2d) and S ∈ S0  ⟨a,FWτS⟩ = ⟨SRτ(a),S⟩ = ⟨KSRτ (a),KS⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iii) Every function F ∈ S0(R2d) admits an expansion of the following type:  F =  ∞  �  n=1  V τ  gnfn,  for some sequences {fn}n, {gn}n ⊆ S0(Rd) such that �∞  n=1 ∥fn∥S0 ∥gn∥S0 <  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' (i) First we notice that if we start with a ∈ S0(R2d), then SRτ(a) is the  Feichtinger operator with kernel  KSRτ (a)(y, u) =  �  Rd a(y − u, ω)e2πiyω dω = F −1  2 [a(y − u, ·)](y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Clearly SRτ is continuous from S0(R2d) into S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Since we have Wτ = FσFWτ and Fσ is an automorphism of S0(R2d), we can write  FWτ = FσWτ and which is an isomorphism due to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' To prove that SRτ  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  13  is the inverse of FWτ we use (34), take S = �∞  n=1 fn ⊗ gn ∈ S0 and ψ, ϕ ∈ S0(Rd):  ⟨(SRτ ◦ FWτS)ψ,ϕ⟩ =  �  R2d FWτS(z)⟨πτ(z)ψ,ϕ⟩ dz  =  ∞  �  n=1  �  R2d V τ  gnfn(z)V τ  ψ ϕ(z) dz  =  ∞  �  n=1  ⟨fn,ϕ⟩⟨gn,ψ⟩  = ⟨  ∞  �  n=1  ⟨ψ,gn⟩fn,ϕ⟩  = ⟨  ∞  �  n=1  (fn ⊗ gn)ψ,ϕ⟩  = ⟨Sψ,ϕ⟩,  in the third equality we used Moyal’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For the composition FWτ ◦SRτ, notice  that this is the identity on S0(R2d) due lo Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2 (vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) Well-posedness, linearity and continuity of SRτ from S′  0(R2d) into S′  0 are stan-  dard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Trivially (37) extends (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' To see that SRτ is the Banach space adjoint of  FWτ from S0 into S0(R2d), take a ∈ S′  0(R2d) and S ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In the following calcula-  tions we use: the prior stated (34), the representation for Feichtinger operators and  their kernel given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='4, the Outer and Inner Kernel Theorems:  ⟨a,FWτS⟩ =  ∞  �  n=1  ⟨a,V τ  gnfn⟩ =  ∞  �  n=1  ⟨SRτ(a)gn,fn⟩  =  ∞  �  n=1  ⟨KSRτ (a),Kfn⊗gn⟩ = ⟨KSRτ (a),KS⟩  = ⟨SRτ(a),S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iii) The last claim is a direct consequence of the computations in (34) and the  surjectivity of FWτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' A convenient environment for QHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In Section 2 we introduced convo-  lutions between a function and an operator and two operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Keyl, Kiukas and  Werner [14] showed that such convolutions make sense for wider classes of (gen-  eralized) functions and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We summarize here the main results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' in what  follows S denotes the set of pseudo-differential operators with Weyl symbol in the  Schwartz class S(R2d) and S′ those pseudo-differential operators with Weyl symbol  in S′(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' On account of the Schwartz Kernel Theorem we can identify S′ with  the continuous and linear operators from S(Rd) into S′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  14  FEDERICO BASTIANONI AND FRANZ LUEF  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i) Suppose S, T ∈ S, A ∈ S′, b ∈ S(R2d) and a ∈ S′(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Then the following convolutions are well-defined and they extend the ones  defined in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2:  S ⋆ T ∈ S(R2d),  S ⋆ A ∈ S′(R2d),  b ⋆ S ∈ S,  a ⋆ S, b ⋆ A ∈ S′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) The Fourier-Wigner transform can be extended to a topological isomorphism  FW1/2 : S′ → S′(R2d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iii) We have Fσ(S ⋆ T) = FW1/2S · FW1/2T and FW1/2(b ⋆ S) = Fσb · FW1/2S  whenever S, T and b are such that the convolutions are defined as in part (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iv) The Weyl symbol of A ∈ S′ is given by FσFW1/2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The authors of [14] proved that the class of so-called Schwartz operators S has  the structure of a Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We propose that the Banach space of Feichtinger  operators S0 is an alternative to S that is a much bigger class of “nice” operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We start with some preliminaries on S0 and S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Given f ∈ S′  0(Rd), there exists a sequence {fn}n ⊆ S0(Rd) which  w-∗ converges to f and it is bounded by ∥f∥S′  0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  lim  n→+∞⟨fn, g⟩ = ⟨f,g⟩  ∀ g ∈ S0(Rd),  sup  n ∥fn∥S0 ≤ ∥f∥S′  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let us fix f ∈ S′  0(Rd) ∖ {0} and set R := ∥f∥S′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' By [13, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='15],  there exists a net {fα}α∈A ⊆ S0(Rd) which converges w-∗ to f in S′  0 and such that  ∥fα∥S′  0 ≤ R for every α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Set  BR :=  �  f ∈ S′  0(Rd) | ∥f∥S′  0 ≤ R  �  and  ER := S0(Rd) ∩ BR,  where S0 is identified with its natural embedding in S′  0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  ER ⊆ BR ⊆ ER  w−∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  ER  w−∗ is bounded in S′  0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In fact, if f0 ∈ ER  w−∗, then there exists a net {fα}α∈A ⊆ ER that it converges w-∗  to f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Hence, we obtain  ∥f0∥S′  0 ≤ lim inf  α∈A  ∥fα∥S′  0 = lim  α∈A inf{∥fβ∥S′  0 | α ⪯ β} ≤ lim  α∈A R = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In particular, this shows that ER  w−∗ ⊆ BR and we get  ER  w−∗ = BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Since S0 separable, and the relative w-∗ topology on BR is induced by a metric by  [18, Therem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Hence the topological w-∗ closure of ER equals its sequential  w-∗ closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Consequently, there exists a sequence {fn}n ⊆ ER which converges w-∗  to f in S′  0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  15  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The above lemma holds also for any LCA second countable group G  replacing Rd, see [8, Theorem 2] for the separability of S0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For any S ∈ S′  0, there exists a sequence {Sn}n ⊆ S0 such that  (i) ∥Sn∥S′  0 ≲ ∥S∥S′  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) limn→+∞ |⟨(S − Sn)f,g⟩| = 0 for all f, g ∈ S0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' This is a straightforward application of the Kernel Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2 and  of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Convergence as in item (ii) of the above lemma will be also denoted by  Sn  w−∗  −→  n  S  in  S′  0  or  S = w- ∗ -limn Sn  in  S′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let S : S0 → S′  0 be in S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then the Banach space adjoint S∗: S′  0 →  S0 is in S0 with kernel  (39)  KS∗(y, u) = KS(u, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We take f, g ∈ S0(Rd), then  ⟨Sf,g⟩ =  �  R2d KT(y, u)g(y)f(u) dydu  =  �  Rd f(u)  �  Rd KS(y, u)g(y) dy du  = ⟨f,S∗g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Hence, S∗g(y) =  �  Rd KS(u, y)g(u) du, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' KS∗(y, u) = KS(u, y) which is an element  of S0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' S0 is a Banach ∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We notice that (S∗)ˇ= ( ˇS)∗, so that from now on we shall simply write ˇS∗ when  necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i) The following applications are surjective isometries:  (i − a) αz : S0 → S0, for any z = (x, ω) ∈ R2d, and  (40)  KαzS(y, u) = e2πi(y−u)ωKS(y − x, u − x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i − b) ˇ·: S0 → S0 and  (41)  K ˇS(y, u) = KS(−y − u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i − c) αz : S′  0 → S′  0, for any z ∈ R2d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i − d) ˇ·: S′  0 → S′  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) Let S, T ∈ S0 and b ∈ S0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then  S ⋆ T ∈ S0(R2d),  b ⋆ S ∈ S0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  16  FEDERICO BASTIANONI AND FRANZ LUEF  (iii) The kernel of the mixed-state localization operator b ⋆ S is given by  (42)  Kb⋆S(y, u) =  �  Rd b(x, ω)e2πi(y−u)ωKS(y − x, u − x) dxdω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  for very z = (x, ω) ∈ R2d the kernel of Sαz ˇT is  (43)  KSαz ˇT(y, u) =  �  Rd e2πi(y−t)ωKT(x − y, x − t)KS(t, u) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' (i) We leave the elementary computations to the interest reader, and note that  in order to prove αzS, ˇS ∈ S0 the result [10, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3] is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' A continuous  and linear operator S : S0 → S′  0 is a Feichtinger operator if and only if  �  R2d  �  R2d |⟨Sπ(z)g1,π(w)g2⟩| dzdw  is finite for any g1, g2 ∈ S0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) We first address the convolution between two Feichtinger operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' By item  (i) and the fact that S0 is a Banach algebra under composition, we have that Sαz ˇT  is in S0 for any z = (x, ω) ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We have by [10, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='15]:  S ⋆ T(z) = tr(Sαz ˇT) =  �  Rd KSαz ˇT(y, y) dy =  �  R2d Kαz ˇT(y, t)KS(t, y) dtdy  =  �  R2d e2πi(y−t)ωKT(x − y, x − t)KS(t, y) dtdy  =  �  Rd  ��  Rd KT(x − y, x − t)KS(t, y)e−2πitω dt  �  e2πiyω dy  = F −1  2 F1  �  ΦT(x,x)KT · KS  �  (ω, ω),  where ΦF(t, y) := F(−y, −t), F1 and F2 are the partial Fourier transforms with re-  spect to the first and second variable, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Consider now f, g, h, l ∈ S0(Rd),  it is useful to compute the following where P is the parity operator:  F −1  2 F1  �  ΦT(x,x)Kh⊗l · Kf⊗g  �  (ω, ω) =  �  Rd  ��  Rd h(x − y)l(x − t)f(t)g(y)e−2πitω dt  �  e2πiyω dy  =  �  Rd f(t)e−2πitωl(x − t) dt ·  �  Rd g(y)e2πiyωh(x − y) dy  = VP lf(−x, ω) · VP hg(−x, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Hence F −1  2 F1  �  ΦT(x,x)Kh⊗l · Kf⊗g  �  (ω, ω) is in S0(R2d) as a function of (x, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We  consider now two representations S = �∞  n=1 fn ⊗ gn and T = �∞  n=1 hn ⊗ ln, see  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  17  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='4, so that  KS =  ∞  �  n=1  Kfn⊗gn,  KT =  ∞  �  n=1  Khn⊗ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  It follows that we can write  S ⋆ T(z) = F −1  2 F1  �  ΦT(x,x)  ∞  �  M  Khm⊗lm ·  ∞  �  n=1  Kfn⊗gn  �  (ω, ω)  =  ∞  �  m=1  ∞  �  n=1  F −1  2 F1  �  ΦT(x,x)Khm⊗lm · Kfn⊗gn  �  (ω, ω)  =  ∞  �  m=1  ∞  �  n=1  VP lmfn(−x, ω) · VP hmgn(−x, ω) ∈ S0(R2d),  the convergence is guaranteed by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Concerning b⋆S, the following estimate for any f, g ∈ S0(Rd) proves that b⋆S ∈ S′  0:  |⟨(b ⋆ S)f,g⟩| ≤  �  R2d |b(z)| |⟨Sπ(z)∗f,π(z)∗g⟩| dz ≲ ∥b∥L1 ∥S∥S′  0 ∥f∥S0 ∥g∥S0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We exploit [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2 (ii)] to show that b ⋆ S is in S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For g1, g2 ∈ S0(Rd) we  have  �  R2d  �  R2d |⟨(b ⋆ S)π(w)g1,π(u)g2⟩| dwdu ≤  �  R2d  �  R2d  �  R2d |b(z)|  × |⟨Sπ(w − z)g1,π(u − z)g2⟩| dzdwdu  =  �  R2d  �  R2d |⟨Sπ(w′)g1,π(u′)g2⟩| dw′du′ ·  �  R2d |b(z)| dz < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iii) We compute explicitly the kernel of the operator given by the convolution b⋆S:  ⟨(b ⋆ S)f,g⟩ =  �  R2d b(x, ω)  �  R2d KS(y, u)π(−z)g(y)π(−z)f(u) dydu dxdω  =  �  R2d  �  R2d b(x, ω)e2πi(y−u)ωKS(y, u)g(y + x)f(u + x) dxdω dydu,  for z = (x, ω) ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The change of variables y′ = y + u, u′ = u + x gives the desired  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The last claim is just a direct application of (40), (41) and the Banach  algebra property for S0 [10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let S, T ∈ S0 with spectral decompositions S = �∞  n=1 fn ⊗ gn and  T = �∞  n=1 hn⊗ln, where {fn}n, {gn}n, {hn}n, {ln}n ⊆ S0(Rd) with �∞  n=1 ∥fn∥S0 ∥gn∥S0 <  18  FEDERICO BASTIANONI AND FRANZ LUEF  +∞, �∞  n=1 ∥hn∥S0 ∥ln∥S0 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then, with the notations introduced in the proof  of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='16, for every z = (x, ω) ∈ R2d:  S ⋆ T(z) = F −1  2 F1  �  ΦT(x,x)KT · KS  �  (ω, ω)  =  ∞  �  m=1  ∞  �  n=1  VP lmfn(−x, ω) · VP hmgn(−x, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (44)  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let A ∈ S′  0, a ∈ S′  0(R2d), S ∈ S0 and b ∈ S0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Consider any  sequences {An}n ⊆ S0 and {an}n ⊆ S0(R2d) such that  An  w−∗  −→  n  A  in  S′  0  and  an  w−∗  −→  n  a  in  S′  0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Then we define:  S ⋆ A := w- ∗ -limn S ⋆ An  in  S′  0(R2d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (45)  a ⋆ S := S ⋆ a := w- ∗ -limn an ⋆ S  in  S′  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (46)  b ⋆ A := A ⋆ b := w- ∗ -limn b ⋆ An  in  S′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (47)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The reader may find it useful to keep in mind the following simple  identities, which will be used in the proof of the subsequent proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Consider  S ∈ S0, ψ, ϕ, f, g ∈ S0(Rd) and z ∈ R2d:  αz(ψ ⊗ ϕ) = π(z)ψ ⊗ π(z)ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ψ ⊗ ϕ)(Kf⊗g) = ⟨f, ϕ⟩(ψ ⊗ g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ψ ⊗ ϕ) ⋆ ˇS(z) = ⟨π(z)Sπ(z)∗ψ,ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The convolutions introduced in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='18:  (i) They do not depend on the sequences chosen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' moreover, taking A, a, S, b as  in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='18:  ⟨S ⋆ A,b⟩ = ⟨KA,Kb⋆ ˇS∗⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (48)  ⟨(a ⋆ S)f,g⟩ = ⟨a,(g ⊗ f) ⋆ ˇS∗⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (49)  ⟨(b ⋆ A)f,g⟩ = ⟨KA,Kb∗⋆(g⊗f)⟩,  (50)  where b∗(z) := b(−z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) These extend the definitions given in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iii) They are commutative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  19  (iv) Moreover, they are associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In particular, if z ∈ R2d, T, Q ∈ S0, σ ∈  S0(R2d) and A, a, S, b as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='18 then:  (S ⋆ (T ⋆ b))(z) = ((S ⋆ T) ∗ b)(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (51)  S ⋆ (T ⋆ Q) = (S ⋆ T) ⋆ Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (52)  (S ⋆ b) ⋆ σ = S ⋆ (b ∗ σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (53)  S ⋆ (T ⋆ a) = (S ⋆ T) ∗ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (54)  A ⋆ (T ⋆ b) = (A ⋆ T) ⋆ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (55)  S ⋆ (T ⋆ A) = (S ⋆ T) ⋆ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (56)  in the above identities ∗ denotes the usual convolution between two functions  or a function and a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' (i) It suffices to show (48), (49) and (50), since the other assertions in (i) are  evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We start with(48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let b ∈ S0(R2d) and z = (x, ω) ∈ R2d, in the subsequent  computations we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='16:  ⟨S ⋆ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='b⟩ = lim  n→+∞⟨S ⋆ An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='b⟩ = lim  n→+∞  �  R2d tr(Sαz ˇAn)b(z) dz  = lim  n→+∞  �  R2d  �  Rd KSαz ˇ  An(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' y) dyb(z)dz  = lim  n→+∞  �  R2d  �  Rd  �  Rd e2πi(y−t)ωKAn(x − y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' x − t)KS(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' y) dtdy b(z) dz  = lim  n→+∞  �  R2d  �  Rd  �  Rd e2πi(t′−y′)ωKAn(y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' t′)KS(x − t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' x − y′) dt′dy′ b(z) dz  = lim  n→+∞  �  Rd  �  Rd KAn(y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' t′)  ��  R2d KS(x − t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' x − y′)e2πi(y′−t′)ωb(z) dz  �  dy′dt′  = lim  n→+∞  �  Rd  �  Rd KAn(y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' t′)  ��  R2d K ˇS(t′ − x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' y′ − x)e2πi(y′−t′)ωb(z) dz  �  dy′dt′  = lim  n→+∞  �  Rd  �  Rd KAn(y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' t′)  ��  R2d K ˇS∗(y′ − x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' t′ − x)e2πi(y′−t′)ωb(z) dz  �  dy′dt′  = lim  n→+∞  �  Rd  �  Rd KAn(y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' t′)Kb⋆ ˇS∗(y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' t′) dy′dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  20  FEDERICO BASTIANONI AND FRANZ LUEF  About (49), we take f, g ∈ S0(Rd) and compute directly keeping in mind Remark  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='19:  ⟨(a ⋆ S)f,g⟩ =  lim  n→+∞  �  R2d an(z)⟨π(z)Sπ(z)∗f,g⟩ dz  =  lim  n→+∞  �  R2d an(z)⟨π(z)S∗π(z)∗g,f⟩ dz  =  lim  n→+∞  �  R2d an(z)(g ⊗ f) ⋆ ˇS∗(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let us address (50):  ⟨(b ⋆ A)f,g⟩ =  lim  n→+∞⟨Kb⋆An,Kg⊗f⟩  =  lim  n→+∞  �  R2d  � �  R2d b(x, ω)e2πi(y−u)ωKAn(y − x, u − x) dxdω  �  × g(y)f(u) dydu  =  lim  n→+∞  �  R2d KAn(y′, u′)  � �  R2d b(x, ω)e−2πi(y′−u′)ω  × g(y′ + x)f(u′ + x) dxdω  �  dy′du′  =  lim  n→+∞  �  R2d KAn(y′, u′)  � �  R2d b∗(x′, ω′)e2πi(y′−u′)ω′  × g(y′ − x′)f(u′ + x′) dx′dω′  �  dy′du′  =  lim  n→+∞  �  R2d KAn(y′, u′)Kb∗⋆(g⊗f)(y′, u′) dy′du′,  where for sake of brevity we set b∗(z) := b(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) and (iii) are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (iv) We prove just (51), (52) and (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The remaining identities can be derived in  a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  21  In order to show (51) we compute for z ∈ R2d:  (S ⋆ (T ⋆ b))(z) = tr  �  S ◦ αz  ���  R2d b(z)αwT dw  �  ˇ  ��  = tr  �  S ◦  ��  R2d b(w)αz ((αwT)ˇ) dw  ��  = tr  �  S ◦  �  R2d b(w)αzα−w ˇT dw  �  = tr  �  S ◦  �  R2d b(−w′)αw′αz ˇT dw′  �  =  �  R2d b(−w′) tr  �  Sαw′+z ˇT  �  dw′,  where the last equality is due, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=', to [20, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' we can the rephrase the  last right-side term as  �  R2d b(z − w′′) tr  �  Sαw′′ ˇT  �  dw′′ =  �  R2d b(z − w′′)(S ⋆ T)(w′′) dw′′  = ((S ⋆ T) ∗ b)(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  For the proof of (52), the following property of the trace is useful:  �  R2d tr(SαwT) dw = tr(S) tr(T),  where S, T ∈ J 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Take now f, g ∈ S0(Rd):  ⟨(S ⋆ (T ⋆ Q))f,g⟩ =  �  R2d tr(Tαz ˇQ)⟨αzSf,g⟩ dz  =  �  R2d tr(Qαz ˇT) tr((αzS)(f ⊗ g)) dz  =  �  R2d  �  R2d tr(Q(αz ˇT)αw((αzS)(f ⊗ g))) dwdz  =  �  R2d  �  R2d tr((f ⊗ g)(αwQ)αz((αw ˇT)S)) dzdw  =  �  R2d tr(Sαw ˇT) tr((αwQ)(f ⊗ g)) dw  = ⟨((S ⋆ T) ⋆ Q)f,g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  22  FEDERICO BASTIANONI AND FRANZ LUEF  Also the last identity (53) may be deduced by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For f, g ∈  S0(Rd) we have  ⟨((S ⋆ b) ⋆ σ)f,g⟩ =  �  R2d σ(z)⟨αz(S ⋆ b)f,g⟩ dz  =  �  R2d σ(z)  �  R2d b(w)⟨(αwS)π(z)∗f,π(z)∗g⟩ dwdz  =  �  R2d  �  R2d σ(z)b(w)⟨(αw+zS)f,g⟩ dwdz  =  �  R2d  �  R2d σ(z)b(w) tr((αw+zS)(f ⊗ g)) dwdz  =  �  R2d b(w)  �  R2d σ(z′ − w) tr((αz′S)(f ⊗ g)) dz′dw  =  �  R2d(  �  R2d b(w)σ(z′ − w) dz′) tr((αz′S)(f ⊗ g)) dw  =  �  R2d b ∗ σ(z′)⟨(αz′S)f,g⟩ dz′  = ⟨(S ⋆ (b ∗ σ))f,g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The mappings FWτ and Wτ defined on S0 can be extended to  topological isomorphisms  FWτ : S′  0 → S′  0(R2d)  and  Wτ : S′  0 → S′  0(R2d)  by duality:  (57)  ⟨FWτS,a⟩ := ⟨S,SRτa⟩,  ⟨WτS,a⟩ := ⟨S, Opτ a⟩,  where S ∈ S′  0 and a ∈ S0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The inverses are given by  SRτ : S′  0(R2d) → S′  0  and  Opτ : S′  0(R2d) → S′  0,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The definitions in (57) rely on the fact that Opτ = W ∗  τ and SRτ = F ∗  Wτ, see  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='7 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' It is straightforward to see that if S ∈ S′  0, then  FWτS and WτS defined as in (57) are in S′  0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Also linearity and boundedness  of FWτ : S′  0 → S′  0(R2d) and Wτ : S′  0 → S′  0(R2d) are easy to verify as well as the fact  that they extend FWτ : S0 → S0(R2d) and Wτ : S0 → S0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We show that Wτ is an isomorphisms with inverse Opτ, then FWτ is treated in  the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Wτ is injective because Opτ : S0(R2d) → S0 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Fix now a ∈ S′  0(R2d), there exists a sequence {an}n ⊆ S0(R2d) such that an  w−∗  −→  n  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  23  a  in  S′  0(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Since Wτ is an isomorphism between S0 and S0(R2d), there exists  {An}n ⊆ S0 such that an = WτAn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We see that there is A ∈ S′  0 such that An  w−∗  −→  n  A  in  S′  0, in fact taking b ∈ S0(R2d)  ⟨a,b⟩ =  lim  n→+∞⟨WτAn,b⟩ = lim  n→+∞⟨An, Opτ b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Hence a = WτA, which proves that Wτ is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We show now that Wτ ◦ Opτ is the  identity on S′  0(R2d), take a ∈ S′  0(R2d) and b ∈ S0(R2d):  ⟨Wτ ◦ Opτ a,b⟩ = ⟨Opτ a, Opτ b⟩ = ⟨a,Wτ ◦ Opτ b⟩ = ⟨a,b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The first identity is just (57), the second one is (32) and the last one is (i) of  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For the other direction, take S ∈ S′  0 and T ∈ S0:  ⟨Opτ ◦WτS,T⟩ = ⟨WτS,WτT⟩ = ⟨S, Opτ ◦WτT⟩ = ⟨S,T⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The first identity is (32), the second one is (57) and the last one is (i) of Corollary  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' τ-Cohen’s class of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In the present subsection we define Qτ  a(S)  and recall the definition of Qτ  S(f) from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We shall see that Qτ  a(S) relates to  well-known objects and observe that it coincides with the τ-symbol of the mixed-  state localization operator a ⋆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We continue with some statements concerning the  interplay between the Gabor matrix of an operator Gϕ  T, the τ-Cohen’s class, the  trace and the τ-Wigner distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For a ∈ S′  0(R2d) we define the τ-Cohen’s class distribution, with  kernel a, of an operator S ∈ S0 as  (58)  Qτ  a(S) := a ∗ WτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Of course, the rank-one case f ⊗ g reduces to the definition given in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We  recall also the definition given in [17] of Cohen’s class distribution of a function  f ∈ S0(Rd) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' the operator S ∈ S′  0 by  (59)  QSf := (f ⊗ f) ⋆ ˇS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  It can be easily seen that for every z ∈ R2d  QSf(z) = (f ⊗ f) ⋆ ˇS(z) = ⟨(αzS)f, f, ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' If a ∈ S′  0(R2d) and S ∈ S0, then we see that the τ-Cohen’s class rep-  resentation of S w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' a is just the τ-symbol of the mixed-state localization operator  a ⋆ S:  aa⋆S  τ  = Wτ(a ⋆ S) = a ∗ WτS = Qτ  a(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  24  FEDERICO BASTIANONI AND FRANZ LUEF  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let S ∈ S0 have the spectral decomposition �∞  n=1 fn⊗gn, for f, ϕ, ψ ∈  S0(Rd) and {hn}n ⊆ S0(Rd) with  ∞  �  n=1  ∥hn∥2  S0 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then for every z ∈ R2d:  Qτ  W1−τ ( ˇψ, ˇϕ)(S)(z) =  ∞  �  n=1  Vϕfn(z)Vψgn(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (60)  Qτ  W1−τ ( ˇϕ, ˇϕ)(  ∞  �  n=1  hn ⊗ hn)(z) =  ∞  �  n=1  |Vϕhn(z)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (61)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Clearly, it suffices to prove the first identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We show first that for f, g ∈  S0(Rd)  (62)  Qτ  a(f, g) = (f ⊗ g) ⋆ Op1-τ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In fact, applying Fσ to the right-hand side first we get  Fσ((f ⊗ g) ⋆ Op1-τ(a)) = FWτ(f ⊗ g) · FW1−τ Op1-τ(a) = V τ  g f · Fσa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We apply Fσ a second time:  (f ⊗ g) ⋆ Op1-τ(a) = FσV τ  g f ∗ FσFσa = Wτ(f, g) ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  We can now proceed as follows:  Qτ  W1−τ ( ˇψ, ˇϕ)(S) = W1−τ( ˇψ, ˇϕ) ∗ Wτ(  ∞  �  n=1  fn ⊗ gn) =  ∞  �  n=1  W1−τ( ˇψ, ˇϕ) ∗ Wτ(fn, gn)  =  ∞  �  n=1  (fn ⊗ gn) ⋆ Op1-τ(W1−τ( ˇψ, ˇϕ)) =  ∞  �  n=1  (fn ⊗ gn) ⋆ ( ˇψ ⊗ ˇϕ)  =  ∞  �  n=1  Vϕfn(z)Vψgn(z),  where the last equality is due to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  We call a bounded operator T on L2(Rd) positive, denoted by T ≥ 0, if  ⟨Tf, f⟩ ≥ 0,  ∀ f ∈ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  An operator T ∈ J 1 and T ≥ 0 is also called a state in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let us take T ∈ S′  0 and ϕ ∈ S, then the Gabor matrix of T (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' ϕ) is defined as  (63)  Gϕ  T(z, w) := ⟨Tπ(w)ϕ, π(z)ϕ⟩,  z = (x, ω), w = (u, v) ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  25  We notice that the Gabor matrix of an operator does not depend on τ, in the sense  that  Gϕ  T(z, w) = ⟨Tπ(w)ϕ, π(z)ϕ⟩ = ⟨Tπτ(w)ϕ, πτ(z)ϕ⟩,  ∀ τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We point out that the diagonal of the Gabor matrix of T, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' ϕ,  is the Cohen’s class representation of ϕ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' T up to a reflection:  (64)  Gϕ  T(−z, −z) = QTϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In fact  Gϕ  T(−z, −z) = ⟨Tπ(−z)ϕ, π(−z)ϕ⟩ = ⟨Tπ(z)∗ϕ, π(z)∗ϕ⟩  = ⟨(αzT)ϕ, ϕ, ⟩ = QT ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let F and H be functions of (z, w) ∈ R4d and let Θ be a real 4d × 4d matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Then the twisted convolution induced by Θ is defined as  (65)  F ♮Θ H(z, w) :=  �  R2d  �  R2d F(z′, w′)H(z − z′, w − w′)e2πi(z,w)Θ(z′,w′) dz′dw′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let T, S ∈ J 1, T, S ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then for every τ ∈ [0, 1] we have  (66)  tr(TS) =  �  R2d WτT(z)WτS(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Since T and S are trace-class and positive, they can be described as  T =  ∞  �  n=1  λnfn ⊗ fn,  S =  ∞  �  n=1  µngn ⊗ gn  for some orthonormal sets {fn}n and {gn}n in L2 and λn, µn ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let {en}n be an  o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' for L2(Rd):  tr(TS) =  ∞  �  n=1  ⟨TSen, en⟩ =  ∞  �  i,j  λjµi |⟨fj, gi⟩|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  On the other hand,  �  R2d WτT(z)WτS(z) dz =  ∞  �  i,j  λjµi  �  R2d Wτfj(z)Wτgi(z) dz =  ∞  �  i,j  λjµi |⟨fj, gi⟩|2 ,  where the last equality is due to Moyal’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Since we assume S ≥ 0, S is self-adjoint and for τ = 1/2 we have  that W1/2S is real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In fact, using the representation given in the proof of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='26:  W1/2S =  ∞  �  n=1  µnW1/2gn  26  FEDERICO BASTIANONI AND FRANZ LUEF  with every W1/2gn real-valued and µn ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Hence, for τ = 1/2 we recover [12,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let T ∈ J 1 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then  (67)  tr T =  �  R2d⟨(αzT)ϕ, ϕ⟩ dz =  �  R2d QT ϕ(z) dz =  �  R2d Gϕ  T(z, z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The proof follows from a direct computation using the representations pre-  sented in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='26 and Moyal’s identity involving the function ϕ:  ⟨fj, gi⟩ = ⟨Vϕfj, Vϕgi⟩,  we leave details to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let T ∈ J 1, T ≥ 0 and let ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then  for every z ∈ R2d:  (68)  QTϕ(z) =  �  R2d WτT(w)Wτϕ(z + w) dw = WτT ∗ (Wτϕ)∗(z),  where (Wτϕ)∗(w) = Wτϕ(−w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We compute directly  QT ϕ(z) = ⟨π(z)Tπ(z)∗ϕ, ϕ⟩ = tr(T(π(z)∗ϕ ⊗ π(z)∗ϕ))  =  �  R2d WτT(w)Wτ(π(z)∗ϕ ⊗ π(z)∗ϕ)(w) dw,  the last equation holds because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' An elementary calculation gives  Wτ(π(z)∗ϕ ⊗ π(z)∗ϕ)(w) = Wτϕ(z + w),  which is also known as covariance property and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let T ∈ J 1, T ≥ 0 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Then for every z, w ∈ R2d:  |Gϕ  T(z, w)|2 ≤ QTϕ(−z)QT ϕ(−w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The claim follows from the Cauchy-Schwarz inequality for the inner product  induced by the positive operator T and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let 0d and Id denote the zero and identity d×d matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Let us define  Θ :=  \uf8ee  \uf8ef\uf8ef\uf8f0  0d  0d  0d  0d  Id  0d  0d  0d  0d  0d  0d  0d  0d  0d  −Id  0d  \uf8f9  \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  27  Let T ∈ J 1 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For z = (x, ω), w =  (u, v) ∈ R2d we have  Gϕ  T(z, w) = Gϕ  T ♮Θ(Gϕ  ϕ⊗ϕ)∗(z, w)  (69)  =  �  R2d  �  R2d Gϕ  T(z′, w′)(Gϕ  ϕ⊗ϕ)∗(z − z′, w − w′)e2πi(ωx′−u′v) dz′dw′,  where z′ = (x′, ω′), w′ = (u′, v′) ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We apply twice Moyal’s identity:  Gϕ  T(z, w) =  �  R2d Vϕ[Tπ(w)ϕ](z′)Vϕ[π(z)ϕ](z′) dz′  =  �  R2d  �  R2d Vϕ[π(w)ϕ](w′)Vϕ[T ∗π(z′)ϕ](w′)⟨π(z′)ϕ, π(z)ϕ⟩ dz′dw′  =  �  R2d  �  R2d Gϕ  T(z′, w′)⟨π(w)ϕ, π(w′)ϕ⟩⟨π(z′)ϕ, π(z)ϕ⟩ dz′dw′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  It is then a direct, although tedious, calculation to show that  ⟨π(z)ϕ, π(z′)ϕ⟩⟨π(w′)ϕ, π(w)ϕ⟩ = (Gϕ  ϕ⊗ϕ)∗(z − z′, w − w′)e2πi(ωx′−u′v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let T ∈ J 1, T ≥ 0 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Then for any τ ∈ [0, 1]:  (70)  WτT(z) =  �  R2d  �  R2d e−2πi[(ωx′−ω′x)+( 1  2− 3  4 τ)x′ω′+x′v]Gϕ  T  �z′  2 − w, −z′  2 − w  �  dwdz′,  where z = (x, ω), z′ = (x′, ω′), w = (u, v) ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We start rephrasing the τ-Wigner distribution of T:  WτT(z) = FσFWτT(z) =  �  R2d e−2πi(ωx′−ω′x) tr(πτ(z′)∗T) dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Recalling the properties for πτ, see Section 2, we see that  πτ(z′/2 + z′/2) = e2πi[(1−τ) x′ω′  4  −τ x′ω′  4  ]πτ(z′/2)πτ(z′/2)  = e  π  2 i(1−2τ)x′ω′πτ(z′/2)πτ(z′/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  28  FEDERICO BASTIANONI AND FRANZ LUEF  Taking the adjoint we get πτ(z′)∗ = e− π  2 i(1−2τ)x′ω′πτ(z′/2)∗πτ(z′/2)∗ and we write  using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='28:  tr(πτ(z′)∗T) = e− π  2 i(1−2τ)x′ω′ tr(πτ(z′/2)∗Tπτ(z′/2)∗)  = e− π  2 i(1−2τ)x′ω′ �  R2d⟨Tπτ(z′/2)∗πτ(w)∗ϕ, πτ(z′/2)πτ(w)∗ϕ⟩ dw  = e− π  2 i(1−2τ)x′ω′e− π  2 i(1−τ)x′ω′  ×  �  R2d⟨Tπτ(−z′/2)πτ(−w)ϕ, πτ(z′/2)πτ(−w)ϕ⟩ dw  = e− π  2 i(2−3τ)x′ω′ �  R2d⟨Tπ(−z′/2)π(−w)ϕ, π(z′/2)π(−w)ϕ⟩ dw  = e− π  2 i(2−3τ)x′ω′ �  R2d e−2πix′v⟨Tπ(−z′/2 − w)ϕ, π(z′/2 − w)ϕ⟩ dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  This concludes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' A characterization of Schwartz operators  In this section we introduce weighted versions of S0 and give an alternative de-  scription of the class S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' We use the polynomial weight  (71)  vs(z) := (1 + |z|2)  s  2,  z ∈ R2d,  where s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In order to avoid an extremely cumbersome notation, just for the  weight functions vs we shall use the following:  vs ⊗ vs(z, w) := Kvs⊗vs = vs(z)vs(w),  ∀z, w ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For s ≥ 0 we define the weighted class of Feichtinger operators as  (72)  M1  s := {S : S′  0(Rd) → S0(Rd) | S is linear, continuous with kernel KS ∈ M1  vs⊗vs(R2d)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  For S in M1  s we define the mapping  (73)  ∥S∥M1s := ∥KS∥M1  vs⊗vs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (i) For s = 0 we recover the Feichtinger operators S0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  (ii) The mapping defined in (73) is a norm on M1  s and it is easy to see that  (M1  s, ∥·∥M1s) is a Banach space and the following continuous inclusion holds  true for every s ≥ 0:  (74)  M1  s ֒→ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For any S ∈ M1  s there exist {fn}n, {gn}n ⊆ M1  vs⊗vs(R2d) such that  S =  ∞  �  n=1  fn ⊗ gn,  ∞  �  n=1  ∥fn∥M1vs ∥gn∥M1vs ≤ +∞,  KS =  ∞  �  n=1  Kfn⊗gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS  29  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The proof follows from the fact that  M1  vs⊗vs(R2d) = M1  vs(Rd)ˆ⊗M1  vs(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  See also the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' For every τ ∈ [0, 1] the mapping Wτ : M1  s → M1  vs⊗vs(R2d) is a topo-  logical isomorphism with inverse given by Opτ : M1  vs⊗vs(R2d) → M1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The proof follows the same pattern as the ones of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='7 and Corollary  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' An operator S belongs to M1  s if and only if for some (hence every)  τ ∈ [0, 1] WτS ∈ M1  vs⊗vs(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The following is true:  (75)  S =  �  s≥0  M1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='5, S belongs to the set on the right-hand side if and only if  WτS ∈  �  s≥0  M1  vs⊗vs(R2d) = S(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  The claim follows since W1/2S is the Weyl symbol of S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' aS  1/2 = W1/2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  We recall that a function F on R2d is called rapidly decaying if for every multiindex  α, β ∈ Nd  0 we have  sup  x,ω∈Rd  ��xαωβF(x, ω)  �� < +∞,  where, if x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' , xd) and α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' , αd), xα stands for xα1  1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' · xαd  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  In [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='1] a sufficient condition is given for a positive trace-class op-  erator to be in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Namely, if T ∈ B(L2), T ≥ 0, is such that WτT exists for  some τ ∈ [0, 1] and it is rapidly decreasing, then T ∈ S and WτT exists for every  τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' In this spirit, we provide the following sufficient condition for a generic  S ∈ B(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Observe that we do not not require S to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let S ∈ B(L2) and assume that for some τ ∈ [0, 1] WτS exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Suppose also that, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' some non-zero window in L2(R2d), the STFT of WτS is  rapidly decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Then WτS exists for every τ ∈ [0, 1] and S is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Let us pick G ∈ L2(R2d) ∖ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' If VGWτS is rapidly decaying then S ∈ M1  s  for every s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' The claim follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  □  30  FEDERICO BASTIANONI AND FRANZ LUEF  Acknowledgments  The first author would like to thank Eduard Ortega for the financial support to  visit Trondheim which led to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  References  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Bastianoni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Cordero and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Decay and smoothness for eigenfunctions of local-  ization operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' 492, 124480, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  [2] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Christensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' An introduction to frames and Riesz bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Applied and Numerical Harmonic  Analysis, Birkh¨auser Basel, Second Edition, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Cordero and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Time-frequency analysis of localization operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=', 205(1):107–131, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Cordero and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Sharp integral bounds for Wigner distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
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+page_content=' (Singap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' ), 15(3):353–389, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Werner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Quantum harmonic analysis on phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' 25(5), 1404–1411,  1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Wigner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' On the quantum correction for thermodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content=' 40,  749–759, 1932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='  Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli  Abruzzi 24, 10129 Torino, Italy  Email address: federico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='bastianoni@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='it  Department of Mathematics, NTNU Norwegian University of Science and Tech-  nology, NO-7491 Trondheim, Norway  Email address: franz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='luef@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
+page_content='no' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfBAt_/content/2301.04848v1.pdf'}
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+arXiv:2301.04640v1  [math.GM]  2 Jan 2023
+Properties of the multi-index special function W(¯α,¯ν)(z)
+R. Drogheia
+aLiceo Scientifico Francesco Severi, Viale Europa,36, 03100 Frosinone (FR), ITALY
+ABSTRACT
+In this paper, we investigate some properties related to a multi-index special func-
+tion W(¯α,¯ν) that arose from an eigenvalue problem for a multi-order fractional hyper-
+Bessel operator, involving Caputo fractional derivatives. We show that for particular
+values of the parameters involved in this special function W(¯α,¯ν), this leads to the
+hyper-Bessel function of Delerue. The Laplace transform of the W(¯α,¯ν) is discussed
+obtaining, in particular cases, the well-known functional relation between hyper-
+Bessel function and multi-index Mittag-Leffler function, or, quite simply, between
+classical Wright and Mittag-Leffler functions. Moreover, it is shown that the multi-
+index special function satisfies the recurrence relation involving fractional deriva-
+tives. In a particular case, we derive, to the best of our knowledge, a new differential
+recurrence relation for the Mittag-Leffler function. We also provide derivatives of the
+3-parameters function Wα,β,ν with respect to parameters, leading to infinite power
+series with coefficients being quotients of digamma and gamma functions.
+KEYWORDS
+Special Function of Fractional Calculus; hyper-Bessel type operators; Wright and
+Mittag-Leffler functions; Caputo derivatives; recurrence relations of special
+functions; hyper-Bessel functions
+1. Introduction
+Nowadays, the interest in fractional differential equations is increasing because these
+are becoming more adequate than those of integer order to investigate various problems
+in different fields of physics, engineering and economics [1], [2], [3]. They have indeed
+the fundamental characteristic to describe memory and heredity properties of many
+materials. Some of them have been introduced within the framework of partition theory
+in solving number theory problems. This is the case of the Wright function, introduced
+by E. M. Wright in his articles on the asymptotic partition formulae[4], [5], [6] and [7],
+[8], [9].
+Recently, many authors are dealing with multi-indices special functions (SF) of
+fractional calculus (FC) appearing in solution of differential equations and systems of
+fractional multi-order type (e.g. hyper-Bessel and quasi-Bessel operators) [10], [11].
+Among them, the most general functions we just want to refer to are the Fox H-
+function and the Wright generalized hypergeometric function [12]. Indeed, one gets the
+classical SF setting their parameters with integer values.
+In the previous paper [13] the author investigated a hyper-Bessel-type operator in-
+volving Caputo derivatives. Solving the eigenvalue problem associated with this frac-
+tional operator, the author introduced a function, written in series expansion, that in
+specific cases is possible to refer to the well-known special function of the fractional
+CONTACT R. Droghei. Email: riccardo.droghei@francescoseveri.org
+
+calculus. According to the information we have, this special function was not studied
+by now. But as seen, it is reduced in particular cases to some known special func-
+tions, which on their side are cases of the Bessel and hyper-Bessel functions and more
+generally, of the multi-index Mittag-Leffer functions.
+This multi-index special function, called in the previous paper m-p generalized
+Wright function, plays an important role in nonlinear fractional differential equations,
+and in their isochronous ω-modified version[13],[14]. It is also a natural generalization
+of the applications of the Laguerre derivatives and the Laguerre-type exponentials [15],
+[16], [17], [18]. In this survey article, firstly, we want to examine several properties as-
+sociated with the multi-index special function investigated in [13].
+The outline of this work is as follows. In Section 2, we recall the definition of the
+multi-index function W(¯α,¯ν) introduced in [13] and its connection with the Hyper -
+Bessel function. Moreover, the simpler function in the only 3-parameters case Wα,β,ν
+is described. In Section 3 we computed the Laplace Transform of the function W(¯α,¯ν)
+and, using it, we derived some new functional relations between this function and
+other known special functions. The main result of this work is described in Section 4.
+Here we showed the recurrence relations of the function Wα,β,ν obtaining, we suppose,
+new differential recurrence relation for the Mittag-Leffler function. In Section 5 we
+investigated the derivatives of Wα,β,ν with respect to the parameters.
+2. Multi-index special function W(¯α,¯ν)(z)
+The multi-index special function W(¯α,¯ν)(z) investigated in [13], is defined by series
+representation as a function of the complex variable z and parameters αj, j = 1, ..., n+
+1 and νj, j = 1, ..., n:
+W(¯α,¯ν)(z) =
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+Γ(αn+1i + aj)
+Γ(αn+1i + bj) ·
+zk
+Γ(αn+1k + bn+1)
+.
+(1)
+where
+aj = 1 +
+j
+�
+m=1
+(νm−1 − αm) ;
+bj = 1 +
+j
+�
+m=1
+(νm−1 − αm−1) .
+(2)
+and the relation aj = bj − αj with j = 1..n + 1.
+The W(¯α,¯ν)(z) is an entire function for αj > 0, j = 1..n + 1; νj ∈ C, j = 1..n and
+α0 = ν0 = 0.
+Theorem 2.1. The multi-index special function W(¯α,¯ν)(λxαn+1) with λ ∈ R, x ≥
+0, αj > 0, j = 1, ..., n + 1 and νj > 0, j = 1..n satisfy the following fractional differ-
+ential equation involving fractional hyper-Bessel-type operator.[see [13] for the proof]
+2
+
+ˆD(¯α,¯ν)
+nL
+W(¯α,¯ν)(λxαn+1) = λW(¯α,¯ν)(λxαn+1);
+(3)
+where
+ˆD(¯α,¯ν)
+nL
+= x
+�n
+s=1(αs−νs) dαn+1
+dxαn+1 xνn dαn
+dxαn xνn−1 dαn−1
+dxαn−1 · · · xν1 dα1
+dxα1 .
+(4)
+2.1. Hyper-Bessel function as a particular case
+The hyper-Bessel function of Delerue (or a multi-index analogue of Bessel function) of
+order d with indices µ1, ..., µd, introduced in 1953 by Delereu [19] as a generalization
+of the Bessel function of the first type (see also [20]) is defined by
+Jµd(z) = z−
+µ1+...+µd
+d+1
+Jµd((d + 1)
+d+1√z) =
+�
+k≥0
+(−1)kzk
+k! �d
+j=1 Γ(k + µj + 1)
+.
+(5)
+Setting αj = 1, j = 1, ..., n + 1 in the multi-index special function W(¯α,¯ν), we obtain
+the relation
+W(¯1,¯ν)(z) =
+n
+�
+j=1
+Γ(1 + aj)Jan(−z), .
+(6)
+with aj defined in (2). It is not surprising because the hyper-Bessel function satisfies the
+so-called hyper-Bessel differential operators of higher order, introduced by Dimovski
+and Kiryakova [21], [22], and obtained from (3) setting all parameters αj = 1 with
+j = 1..n + 1, i.e. derivatives of integer order.
+2.2. 3-parameters function Wα,β,ν
+In this section we analyse the simpler case of (1) with n = 1, α2 = β, α1 = α and
+ν1 = ν:
+Wα,β,ν(xβ) =
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+xβk
+Γ(βk + 1 − α + ν).
+(7)
+Proposition 2.2. Obviously, the above function (7) satisfies the following fractional
+differential equation
+ˆDα,β,νf(x) = xα−ν dβ
+dxβ
+�
+xν dα
+dxα f(x)
+�
+= f(x),
+(8)
+involving two fractional derivatives in the sense of Caputo of orders α, β ∈ (0, 1).
+Where
+f(x) = Wα,β,ν(xβ)
+3
+
+.
+Remark 1. The Weinstein and Bessel-Clifford operators Setting α = β = 1
+and ν = k, k ≥ 1 the operator ˆDα,β,ν becomes
+ˆD1,1,k = xBk = x
+� d2
+dx2 + k
+x
+d
+dx
+�
+= x−k+1 d
+dxxk d
+dx
+where Bk is the well known Weinstein operator (or Bessel operator) from the so-
+called Darboux-Weinstein relation [23], [24]. In [25] Hayek studied in details exactly
+the operator ˆD1,1,k+1 calling its solution as Bessel-Clifford function of second order
+Cν(x) = x− ν−1
+2 Iν−1(2√x) =
+1
+Γ(ν+1) 0F1(ν + 1; −x), where Iν(x) is the modified Bessel
+function of the first kind. Later, in [26] he introduced the two indices Bessel-Clifford
+functions of the third order modifying the hyper-Bessel function J(2)
+µ,ν(x):
+Cµ,ν(x) = x− µ+ν
+3 J(2)
+µ,ν(3
+3√x) =
+1
+Γ(µ + 1)Γ(ν + 1) 0
+F2(µ + 1, ν + 1; −x);
+(9)
+satisfying the third-order Bessel-Clifford differential equation related to the operator
+ˆBµ,ν = x−ν d
+dxxµ−ν+1 d
+dxxν+1 d
+dx.
+(10)
+As it is simple to see, the two-parameter operator ˆBµ,ν is equivalent to the operator
+(4), ˆD({α1,α2,α3},{ν1,ν2})
+2L
+with α1 = α2 = α3 = 1; ν1 = ν + 1 and ν2 = µ − ν + 1; and
+then the Bessel-Clifford of the third order function (9) is equal to
+Cµ,ν(x) =
+1
+Γ(ν + 1)W({1,1,1},{ν+1,µ−ν+1})(x).
+These differential operators appear very often in the PDEs of mathematical physics
+(especially in fluid mechanics, elasticity, and transonic flow), for instance in the gen-
+eralized Bessel heat equation and other equations of generalized axially symmetric
+potentials (GASP) theory [27].
+2.2.1. Particular cases of Wα,β,ν
+For α = 1, β = λ and ν = µ the function corresponds to the Classical Wright function
+W1,λ,µ(xλ) = Wλ,µ
+�xλ
+λ
+�
+=
+∞
+�
+k=0
+�
+xλ
+λ
+�k
+k!Γ(λk + µ).
+(11)
+For α = 0, β → α, ν → β − 1 the function corresponds to the generalized Mittag-
+Leffler function
+W0,α,β−1(z) = Eα,β(z) =
+∞
+�
+k=0
+zk
+Γ(αk + β).
+(12)
+4
+
+In case of α = β = ν holds the relation
+Wν,ν,ν(xν) = E1;ν,1(xν)
+where Eα;ν,γ(x) = �∞
+k=0
+xk
+Γα+1(νk+γ) is the α-Mittag-Leffler function.
+In Addition, we present some examples of the 3-parameters function Wα,β,ν in the
+following table, and in Figure 1 we represent the behavior of this function for different
+values of the parameters α, β, ν;
+Integer order derivatives
+Fractional order derivatives
+W0,1,0(x) = ex
+W 1
+2 , 1
+2, 1
+2 (√x) = +I0(2√x) + L0(2√x)
+W0,1,n(x) = ex
+xn − �n−1
+i=0
+xi−n
+i!
+with n ∈ N
+W 1
+2, 1
+2, 3
+2 (√x) = +I1(2√x) + L1(√x)
+W1,1,0(x) = √xI1(2√x)
+W 1
+2, 1
+2,1(√x) = sinh(2√x)+cosh(2√x)−1
+√πx
+W1,1,ν(x) = x− ν−1
+2 Iν−1(2√x)
+W 1
+2, 1
+2,2(√x) = (2√x−1)e2√x−2x+1
+2x√πx
+where Iα(x) = i−αJα(ix) = �∞
+m=0
+1
+m!Γ(m+α+1)(x
+2)2m+α is the modified Bessel func-
+tion of the first kind and Lα(x) =
+� x
+2
+�ν+1 �∞
+m=0
+( x
+2)
+2m
+Γ(m+ 3
+2)Γ(m+ν+ 3
+2 ) is the modified Struve
+function.
+(a)
+Plot
+of
+the
+function
+W0,1,ν(x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+(b)
+Plot
+of
+the
+function
+W1,1,ν(x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+(c)
+Plot
+of
+the
+function
+W 1
+2 , 1
+2 ,ν(√x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+(d)
+Plot
+of
+the
+function
+W 1
+2 ,1,ν(x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+Figure 1.
+3. Laplace Transform
+Let us compute the Laplace transform of the W(¯α,¯ν)(λx)
+5
+
+L
+�
+W(¯α,¯ν)(λxαn+1), s
+�
+=
+� ∞
+0
+e−sx
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+Γ(αn+1i + aj)
+Γ(αn+1i + bj)
+λkxαn+1k
+Γ(αn+1k + bn+1)dx
+=
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+λkΓ(αn+1i + aj)
+Γ(αn+1i + bj)Γ(αn+1k + bn+1)
+� ∞
+0
+e−sxxαn+1kdx
+=
+1
+s
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+Γ(αn+1i + aj)Γ(αn+1k + 1)
+Γ(αn+1i + bj)Γ(αn+1k + bn+1)
+�
+λ
+sαn+1
+�k
+. (13)
+th analytical properties of the W(¯α,¯ν) provides that the resulting Laplace transform
+turns out to be an analytic function, vanishing at infinity and exhibiting an essential
+singularity at s = 0.
+Remark 2. In case we set αj = 1, j = 1, ..., n + 1, the multi-index special functions
+W(¯α,¯ν) will be related to the hyper-Bessel functions as is showed in (6). After some
+calculations, we obtain the following functional relation between the Laplace transform
+of the Hyper-Bessel function and the multi-index Mittag-Leffler function. A more
+general relation between these two functions can be found in the article of Kiryakova
+and Luchko [28].
+L
+�
+W(¯1,¯ν)(λx), s
+�
+=
+n
+�
+j=1
+Γ(1 + aj)1
+s
+∞
+�
+k=0
+1
+�n
+j=1 Γ(k + aj+1 + 1)
+�λ
+s
+�k
+=
+n
+�
+j=1
+Γ(1 + aj)1
+sE(n)
+(1,1,...,1),(aj+1+1)
+�λ
+s
+�
+.
+(14)
+Remark 3. The Laplace transform of Wα,β,ν(x) can be obtained as a special case of
+the (13) as follows:
+L (Wα,β,ν(λxρ), s) = 1
+s
+∞
+�
+k=0
+k
+�
+i=1
+βΓ(βi + 1 − α)
+Γ(βi + 1)
+Γ(ρk + 1)
+Γ(βk + 1 − α + ν)
+� λ
+sρ
+�k
+;
+(15)
+and, in the case of the parameter α = 1, we obtain the well-known Laplace transform
+of the Wright function which can be expressed in terms of the two-parameter Mittag-
+Leffler function.
+L (W1,β,ν(λx), s) = L (Wβ,ν(λx), s) = 1
+sEβ,ν
+� λ
+β s
+�
+.
+(16)
+•
+6
+
+4. Recurrence relations of Wα,β,ν
+A recurrence relation is an equation that recursively defines a sequence of values; given
+one or more initial terms, each further term of the sequence is defined as a function of
+the previous terms. Differential recurrence relation of the generalized Wright function
+can be used in the study of fractional differential equations, and it is obtained directly
+from series representation.
+xα+β dβ
+dxβ
+�
+xν−α+βWα,β,ν+β(xβ)
+�
+− 2xν+βWα,β,ν(xβ) + xα+ν dα
+dxα Wα,β,ν−β(xβ) = 0.
+(17)
+Remark 4. In case α = β = 1; ν = n + 1 with n ∈ N0 and
+d
+dxW1,1,n(x) = W1,1,n+1(x) = Cn(x);
+we obtain the well known three-term recurrence relation for the Bessel-Clifford func-
+tion Cn(x)
+xCn+2(x) + (n + 1)Cn+1(x) = Cn(x).
+(18)
+Remark 5. Recurrence fractional derivatives relation for the Wright and
+Mittag-Leffler functions. From the relation (11) between the generalized Wright
+function and the classical Wright function the relation (17) becomes
+dλ
+dzλ
+�
+zλ+ν−1Wλ,λ+ν
+�zλ
+λ
+��
+= zν−1Wλ,ν
+�zλ
+λ
+�
+;
+(19)
+using the formula
+d
+dz Wλ,ν−λ
+�zλ
+λ
+�
+= zλ−1Wλ,ν
+�zλ
+λ
+�
+.
+(20)
+In case α = 0, and β → α, ν → β − 1 the generalized Wright function is related to
+the Mittag-Leffler function by the following relation:
+W0,α,β−1(z) = Eα,β(z).
+(21)
+In particular, from the recurrence relation (17), we obtain the new recurrence relation
+involving fractional derivatives for M-L functions.
+zα dα
+dzα
+�
+zα+β−1Eα,α+β(zα)
+�
+− 2zα+β−1Eα,β(zα) + zβ−1Eα,β−α(zα) = 0.
+(22)
+7
+
+5. Partial derivatives of Wα,β,ν with respect to the parameters
+In this section, taking inspiration from the works of Apelblat and Mainardi [29], [30] we
+analyse the derivatives of Wα,β,ν respect the three parameters included in the function.
+We can treat parameters as variables and hence the derivatives with respect to them
+can be obtained. These derivatives lead to infinite power series involving digamma (ψ)
+and gamma functions.
+∂
+∂ν Wα,β,ν(z) = −
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+ψ(βk + 1 − α + ν)
+Γ(βk + 1 − α + ν)zk;
+(23)
+∂
+∂β Wα,β,ν(z) =
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+zk
+Γ(βk + 1 − α + ν) ·
+·
+
+
+k
+�
+j=1
+j [ψ(βj + 1 − α) − ψ(βj + 1)] − kψ(βk + 1 − α + ν)
+
+ ;
+(24)
+∂
+∂αWα,β,ν(z) =
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+zk
+Γ(βk + 1 − α + ν)
+
+−
+k
+�
+j=1
+ψ(βj + 1 − α) + ψ(βk + 1 − α + ν)
+
+ ;
+(25)
+where ψ(z) = Γ′(z)
+Γ(z) denotes the digamma function.
+Remark 6. In the case α = 1 and considering the property of the digamma function
+ψ(z+1) = ψ(z)+ 1
+z; we obtain the formula (5) and (6) of the Apelblat-Mainardi article
+([30]) for the classical Wright function
+∂
+∂β W1,β,ν(z) =
+� ∂
+∂β Wβ,ν
+�
+(βz) = −
+∞
+�
+k=0
+� ψ(βk + ν)
+k!Γ(βk + ν)
+�
+kzk;
+∂
+∂ν W1,β,ν(z) =
+� ∂
+∂ν Wβ,ν
+�
+(βz) = −
+∞
+�
+k=0
+� ψ(βk + ν)
+k!Γ(βk + ν)
+�
+zk.
+By setting the parameters, α = 0, β → α and ν → β − 1, we obtain the formulas
+(95) and (96) of the Apelblat paper ([29])
+∂
+∂αW0,α,β−1(z) = ∂
+∂αEα,β(z) = −
+∞
+�
+k=0
+�kψ(αk + β)
+Γ(αk + β)
+�
+zk;
+∂
+∂β W0,α,β−1(z) = ∂
+∂β Eα,β(z) = −
+∞
+�
+k=0
+�ψ(αk + β)
+Γ(αk + β)
+�
+zk.
+8
+
+6. Conclusion
+The aim of this paper is to investigate several properties related to the multi-index
+special function W(¯α,¯ν) and its 3-parameters version. An important result was finding
+the connection between the W(¯α,¯ν) and the hyper-Bessel function of Delerue. Here we
+analyzed the Laplace transform, recurrence relation and derivatives of the function
+with respect to the parameters. In particular, we found new findings that, for special
+values of the parameters, retrieve some well-known relations. Indeed, a simple func-
+tional relation is obtained between the Laplace transform of the hyper-Bessel function
+and the multi-index Mittag-Leffler.
+Disclosure statement
+No potential conflict of interest was reported by the author.
+Acknowledgements
+The author is grateful to Dr Roberto Garra for providing essential information, help
+and advice.
+References
+[1] Podlubny I. Fractional Differential Equations. Academic Press, San Diego; 1999.
+[2] Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introduction to
+mathematical models. World Scientific; 2010.
+[3] Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV. Mittag-Leffler functions, related topics
+and applications (p. 540). New York, NY, USA: Springer; 2020.
+[4] Wright E. M. Asymptotic partition formulae: I. plane partitions. The Quarterly Journal of
+Mathematics, Volume os-2, Issue 1; 1931; p. 177–189. https://doi.org/10.1093/qmath/os-
+2.1.177
+[5] Wright E. M. Asymptotic partition formulae:(II) weighted partitions. Proceedings of the
+London Mathematical Society, 2(1); 1934; p. 117-141. https://doi.org/10.1112/plms/s2-
+36.1.117
+[6] Wright E. M. Asymptotic partition formulae. III. Partitions intok-th powers. Acta Math-
+ematica, 63(1); 1934; p.143-191. https://doi.org/10.1007/BF02547353
+[7] Wright EM. On the coefficients of power series having exponential singularities. Journal
+London Math. Soc. 8; 1933; p. 71–79.
+[8] Wright EM. The asymptotic expansion of the generalized Bessel function. Proc. London
+Math. Soc. (Ser. II) 38; 1935; p. 257–270.
+[9] Wright EM. The asymptotic expansion of the generalized hypergeometric function. Jour-
+nal London Math. Soc. 10; 1935; p. 287–293.
+[10] Garra R, Polito F. On some operators involving Hadamard derivatives. Integral Trans-
+forms and Special Functions; 2013. https://doi.org/10.1080/10652469.2012.756875.
+[11] Dubovski PB, Slepoi JA. Construction and analysis of series solutions for frac-
+tional
+quasi-Bessel
+equations.
+Fract
+Calc
+Appl
+Anal
+25;
+2022;
+p.1229–1249.
+https://doi.org/10.1007/s13540-022-00045-z
+[12] Kiryakova V. Fractional calculus of some ”new” but not new special function: K-, multi-
+index-, and S-analogues. AIP Conference Proceedings. 2172, 050008; 2019.
+9
+
+[13] Droghei R. On a Solution of a Fractional Hyper-Bessel Differential Equation by Means
+of a Multi-Index Special Function. Fract Calc Appl Anal 24; 2021; p. 1559–1570.
+https://doi.org/10.1515/fca-2021-0065
+[14] Droghei R, Garra R. Isochronous fractional PDEs. Lecture Notes of TICMI 21; 2020; p.
+43–51.
+[15] Dattoli G, Ricci PE. Laguerre-type exponentials, and the relevant-circular and-hyperbolic
+functions. Georgian Mathematical Journal, 10(3); 2003; p. 481-494.
+[16] Bretti G, Ricci PE. Laguerre-type special functions and population dynamics. Applied
+mathematics and computation, 187(1); 2007; p. 89-100.
+[17] Ricci PE. Laguerre-Type Exponentials, Laguerre Derivatives and Applications. A Survey.
+Mathematics 8, 2054; 2020.
+[18] Garra R, Tomovski Z. Exact results on some nonlinear Laguerre-type diffusion equations.
+Mathematical Modelling and Analysis, 26(1); 2021; p. 72-81.
+[19] Delerue P., Sur le calcul symbolique `a n variables et fonctions hyperbesseliennes (II). Ann.
+Soc. Sci. Brux. 3; 1953; p. 229–274.
+[20] Kiryakova V. Generalized Fractional Calculus and Applications. Longman – J. Wiley,
+Harlow, N. York; 1994.
+[21] Dimowski I, Kiryakova V. Generalized Poisson transmutations and corresponding repre-
+sentations of hyper-Bessel functions. C. R. Acad. Bulg. Sci. 39, N. 10; 1986; p. 20-32.
+[22] Dimowski I, Kiryakova V. Generalized Poisson representations of hyper-geometric func-
+tions pFq , p < q using fractional integrals. In: Proc.16th Spring Conf Union Bulg. Math.
+Sofia; 1987; p. 205-212.
+[23] Weinstein A. The generalized radiation problem and the Euler-Poisson-Darboux equation.
+Summa Brazil Math. 3; 1955; p. 125-147.
+[24] Kiryakova V, Hernandez-Suarez V. Bessel-Clifford third order differential operator and
+corresponding Laplace type integral transform. Dissertationes Mathematicae 340; 1995);
+p. 143-161.
+[25] Hayek N. Estudio de la ecuaci`on diferencial xy′′ + (ν + 1)y′ + y = 0 y de sus aplicaciones.
+Collect. Math. 18, No 1-2; 1967; p. 57-174.
+[26] Hayek N. Funciones de Bessel-Cliff`ord de tercer orden. Actas XII Jornadas Luso-Esp. de
+Mat. (Braga); 1987; p. 346-351.
+[27] Weinstein A. Generalized axially symmetric potential theory. Bull.AMS 59, 20; 1955.
+[28] Kiryakova V, Luchko Yu. The Multiindex MittagLeffler Functions and Their Applications
+for Solving Fractional Order Problems in Applied Analysis. AIP Conf. Proc. 1301, 597;
+2010; doi: 10.1063/1.3526661.
+[29] Apelblat A. Differentiation of the Mittag-Leffler functions with respect to parameters in
+the Laplace transform approach. Mathematics, 8(5), 657; 2020.
+[30] Apelblat A, Mainardi F. Differentiation of the Wright functions with respect to parameters
+and other results. arXiv e-prints, arXiv-2009; 2020.
+Appendix A. Fractional calculus
+In order to make the papar self-contained, we briefly recall main definitions and prop-
+erties of fractional calculus operators.
+Let γ ∈ R+. The Riemann-Liouville fractional integral is defined by
+Jγ
+x f(x) =
+1
+Γ(γ)
+� x
+0
+(x − x′)γ−1f(x′)dx′,
+(A1)
+10
+
+where
+Γ(γ) =
+� +∞
+0
+xγ−1e−xdx,
+is the Euler Gamma function.
+Note that, by definition, J0
+xf(x) = f(x).
+Moreover it satisfies the semigroup property, i.e. Jα
+x Jβ
+x f(x) = Jα+β
+x
+f(x).
+There are different definitions of fractional derivative (see e.g. [1]). In this paper we
+used the fractional derivatives in the sense of Caputo, that is
+Dγ
+xf(x) = Jm−γ
+x
+Dm
+x f(x) =
+1
+Γ(m − γ)
+� x
+0
+(x−x′)m−γ−1
+dm
+d(x′)m f(x′) dx′, γ ̸= m. (A2)
+It is simple to prove the following properties of fractional derivatives and integrals
+(see e.g. [1]) that will be used in the analysis:
+Dγ
+xJγ
+x f(x) = f(x),
+γ > 0,
+(A3)
+Jγ
+x Dγ
+xf(x) = f(x) −
+m−1
+�
+k=0
+f (k)(0)xk
+k! ,
+γ > 0, x > 0,
+(A4)
+Jγ
+x xδ =
+Γ(δ + 1)
+Γ(δ + γ + 1)xδ+γ
+γ > 0, δ > −1, t > 0,
+(A5)
+Dγ
+xxδ =
+Γ(δ + 1)
+Γ(δ − γ + 1)xδ−γ
+γ > 0, δ > −1, t > 0.
+(A6)
+11
+
diff --git a/5dE3T4oBgHgl3EQfpQo9/content/tmp_files/load_file.txt b/5dE3T4oBgHgl3EQfpQo9/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..44bec744e610553e7ef7e758716842b798bf6694
--- /dev/null
+++ b/5dE3T4oBgHgl3EQfpQo9/content/tmp_files/load_file.txt
@@ -0,0 +1,468 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf,len=467
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='04640v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='GM]  2 Jan 2023  Properties of the multi-index special function W(¯α,¯ν)(z)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Drogheia  aLiceo Scientifico Francesco Severi, Viale Europa,36, 03100 Frosinone (FR), ITALY  ABSTRACT  In this paper, we investigate some properties related to a multi-index special func-  tion W(¯α,¯ν) that arose from an eigenvalue problem for a multi-order fractional hyper-  Bessel operator, involving Caputo fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' We show that for particular  values of the parameters involved in this special function W(¯α,¯ν), this leads to the  hyper-Bessel function of Delerue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Laplace transform of the W(¯α,¯ν) is discussed  obtaining, in particular cases, the well-known functional relation between hyper-  Bessel function and multi-index Mittag-Leffler function, or, quite simply, between  classical Wright and Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Moreover, it is shown that the multi-  index special function satisfies the recurrence relation involving fractional deriva-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In a particular case, we derive, to the best of our knowledge, a new differential  recurrence relation for the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' We also provide derivatives of the  3-parameters function Wα,β,ν with respect to parameters, leading to infinite power  series with coefficients being quotients of digamma and gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  KEYWORDS  Special Function of Fractional Calculus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' hyper-Bessel type operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Wright and  Mittag-Leffler functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Caputo derivatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' recurrence relations of special  functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' hyper-Bessel functions  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Introduction  Nowadays, the interest in fractional differential equations is increasing because these  are becoming more adequate than those of integer order to investigate various problems  in different fields of physics, engineering and economics [1], [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' They have indeed  the fundamental characteristic to describe memory and heredity properties of many  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Some of them have been introduced within the framework of partition theory  in solving number theory problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' This is the case of the Wright function, introduced  by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Wright in his articles on the asymptotic partition formulae[4], [5], [6] and [7],  [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Recently, many authors are dealing with multi-indices special functions (SF) of  fractional calculus (FC) appearing in solution of differential equations and systems of  fractional multi-order type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' hyper-Bessel and quasi-Bessel operators) [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Among them, the most general functions we just want to refer to are the Fox H-  function and the Wright generalized hypergeometric function [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Indeed, one gets the  classical SF setting their parameters with integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  In the previous paper [13] the author investigated a hyper-Bessel-type operator in-  volving Caputo derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Solving the eigenvalue problem associated with this frac-  tional operator, the author introduced a function, written in series expansion, that in  specific cases is possible to refer to the well-known special function of the fractional  CONTACT R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Droghei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Email: riccardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='droghei@francescoseveri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org  calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' According to the information we have, this special function was not studied  by now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' But as seen, it is reduced in particular cases to some known special func-  tions, which on their side are cases of the Bessel and hyper-Bessel functions and more  generally, of the multi-index Mittag-Leffer functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  This multi-index special function, called in the previous paper m-p generalized  Wright function, plays an important role in nonlinear fractional differential equations,  and in their isochronous ω-modified version[13],[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' It is also a natural generalization  of the applications of the Laguerre derivatives and the Laguerre-type exponentials [15],  [16], [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In this survey article, firstly, we want to examine several properties as-  sociated with the multi-index special function investigated in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  The outline of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In Section 2, we recall the definition of the  multi-index function W(¯α,¯ν) introduced in [13] and its connection with the Hyper -  Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Moreover, the simpler function in the only 3-parameters case Wα,β,ν  is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In Section 3 we computed the Laplace Transform of the function W(¯α,¯ν)  and, using it, we derived some new functional relations between this function and  other known special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The main result of this work is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Here we showed the recurrence relations of the function Wα,β,ν obtaining, we suppose,  new differential recurrence relation for the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In Section 5 we  investigated the derivatives of Wα,β,ν with respect to the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Multi-index special function W(¯α,¯ν)(z)  The multi-index special function W(¯α,¯ν)(z) investigated in [13], is defined by series  representation as a function of the complex variable z and parameters αj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n+  1 and νj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n:  W(¯α,¯ν)(z) =  ∞  �  k=0  k  �  i=1  n  �  j=1  Γ(αn+1i + aj)  Γ(αn+1i + bj) ·  zk  Γ(αn+1k + bn+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (1)  where  aj = 1 +  j  �  m=1  (νm−1 − αm) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  bj = 1 +  j  �  m=1  (νm−1 − αm−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (2)  and the relation aj = bj − αj with j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  The W(¯α,¯ν)(z) is an entire function for αj > 0, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' νj ∈ C, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n and  α0 = ν0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The multi-index special function W(¯α,¯ν)(λxαn+1) with λ ∈ R, x ≥  0, αj > 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n + 1 and νj > 0, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n satisfy the following fractional differ-  ential equation involving fractional hyper-Bessel-type operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' [see [13] for the proof]  2  ˆD(¯α,¯ν)  nL  W(¯α,¯ν)(λxαn+1) = λW(¯α,¯ν)(λxαn+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (3)  where  ˆD(¯α,¯ν)  nL  = x  �n  s=1(αs−νs) dαn+1  dxαn+1 xνn dαn  dxαn xνn−1 dαn−1  dxαn−1 · · · xν1 dα1  dxα1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Hyper-Bessel function as a particular case  The hyper-Bessel function of Delerue (or a multi-index analogue of Bessel function) of  order d with indices µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', µd, introduced in 1953 by Delereu [19] as a generalization  of the Bessel function of the first type (see also [20]) is defined by  Jµd(z) = z−  µ1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='+µd  d+1  Jµd((d + 1)  d+1√z) =  �  k≥0  (−1)kzk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' �d  j=1 Γ(k + µj + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (5)  Setting αj = 1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n + 1 in the multi-index special function W(¯α,¯ν), we obtain  the relation  W(¯1,¯ν)(z) =  n  �  j=1  Γ(1 + aj)Jan(−z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (6)  with aj defined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' It is not surprising because the hyper-Bessel function satisfies the  so-called hyper-Bessel differential operators of higher order, introduced by Dimovski  and Kiryakova [21], [22], and obtained from (3) setting all parameters αj = 1 with  j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' derivatives of integer order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 3-parameters function Wα,β,ν  In this section we analyse the simpler case of (1) with n = 1, α2 = β, α1 = α and  ν1 = ν:  Wα,β,ν(xβ) =  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  xβk  Γ(βk + 1 − α + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (7)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Obviously, the above function (7) satisfies the following fractional  differential equation  ˆDα,β,νf(x) = xα−ν dβ  dxβ  �  xν dα  dxα f(x)  �  = f(x),  (8)  involving two fractional derivatives in the sense of Caputo of orders α, β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Where  f(x) = Wα,β,ν(xβ)  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Weinstein and Bessel-Clifford operators Setting α = β = 1  and ν = k, k ≥ 1 the operator ˆDα,β,ν becomes  ˆD1,1,k = xBk = x  � d2  dx2 + k  x  d  dx  �  = x−k+1 d  dxxk d  dx  where Bk is the well known Weinstein operator (or Bessel operator) from the so-  called Darboux-Weinstein relation [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In [25] Hayek studied in details exactly  the operator ˆD1,1,k+1 calling its solution as Bessel-Clifford function of second order  Cν(x) = x− ν−1  2 Iν−1(2√x) =  1  Γ(ν+1) 0F1(ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' −x), where Iν(x) is the modified Bessel  function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Later, in [26] he introduced the two indices Bessel-Clifford  functions of the third order modifying the hyper-Bessel function J(2)  µ,ν(x):  Cµ,ν(x) = x− µ+ν  3 J(2)  µ,ν(3  3√x) =  1  Γ(µ + 1)Γ(ν + 1) 0  F2(µ + 1, ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' −x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (9)  satisfying the third-order Bessel-Clifford differential equation related to the operator  ˆBµ,ν = x−ν d  dxxµ−ν+1 d  dxxν+1 d  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (10)  As it is simple to see, the two-parameter operator ˆBµ,ν is equivalent to the operator  (4), ˆD({α1,α2,α3},{ν1,ν2})  2L  with α1 = α2 = α3 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' ν1 = ν + 1 and ν2 = µ − ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' and  then the Bessel-Clifford of the third order function (9) is equal to  Cµ,ν(x) =  1  Γ(ν + 1)W({1,1,1},{ν+1,µ−ν+1})(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  These differential operators appear very often in the PDEs of mathematical physics  (especially in fluid mechanics, elasticity, and transonic flow), for instance in the gen-  eralized Bessel heat equation and other equations of generalized axially symmetric  potentials (GASP) theory [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Particular cases of Wα,β,ν  For α = 1, β = λ and ν = µ the function corresponds to the Classical Wright function  W1,λ,µ(xλ) = Wλ,µ  �xλ  λ  �  =  ∞  �  k=0  �  xλ  λ  �k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(λk + µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (11)  For α = 0, β → α, ν → β − 1 the function corresponds to the generalized Mittag-  Leffler function  W0,α,β−1(z) = Eα,β(z) =  ∞  �  k=0  zk  Γ(αk + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (12)  4  In case of α = β = ν holds the relation  Wν,ν,ν(xν) = E1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='ν,1(xν)  where Eα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='ν,γ(x) = �∞  k=0  xk  Γα+1(νk+γ) is the α-Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  In Addition, we present some examples of the 3-parameters function Wα,β,ν in the  following table, and in Figure 1 we represent the behavior of this function for different  values of the parameters α, β, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Integer order derivatives  Fractional order derivatives  W0,1,0(x) = ex  W 1  2 , 1  2, 1  2 (√x) = +I0(2√x) + L0(2√x)  W0,1,n(x) = ex  xn − �n−1  i=0  xi−n  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  with n ∈ N  W 1  2, 1  2, 3  2 (√x) = +I1(2√x) + L1(√x)  W1,1,0(x) = √xI1(2√x)  W 1  2, 1  2,1(√x) = sinh(2√x)+cosh(2√x)−1  √πx  W1,1,ν(x) = x− ν−1  2 Iν−1(2√x)  W 1  2, 1  2,2(√x) = (2√x−1)e2√x−2x+1  2x√πx  where Iα(x) = i−αJα(ix) = �∞  m=0  1  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(m+α+1)(x  2)2m+α is the modified Bessel func-  tion of the first kind and Lα(x) =  � x  2  �ν+1 �∞  m=0  ( x  2)  2m  Γ(m+ 3  2)Γ(m+ν+ 3  2 ) is the modified Struve  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (a)  Plot  of  the  function  W0,1,ν(x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (b)  Plot  of  the  function  W1,1,ν(x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (c)  Plot  of  the  function  W 1  2 , 1  2 ,ν(√x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (d)  Plot  of  the  function  W 1  2 ,1,ν(x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Laplace Transform  Let us compute the Laplace transform of the W(¯α,¯ν)(λx)  5  L  �  W(¯α,¯ν)(λxαn+1), s  �  =  � ∞  0  e−sx  ∞  �  k=0  k  �  i=1  n  �  j=1  Γ(αn+1i + aj)  Γ(αn+1i + bj)  λkxαn+1k  Γ(αn+1k + bn+1)dx  =  ∞  �  k=0  k  �  i=1  n  �  j=1  λkΓ(αn+1i + aj)  Γ(αn+1i + bj)Γ(αn+1k + bn+1)  � ∞  0  e−sxxαn+1kdx  =  1  s  ∞  �  k=0  k  �  i=1  n  �  j=1  Γ(αn+1i + aj)Γ(αn+1k + 1)  Γ(αn+1i + bj)Γ(αn+1k + bn+1)  �  λ  sαn+1  �k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' (13)  th analytical properties of the W(¯α,¯ν) provides that the resulting Laplace transform  turns out to be an analytic function, vanishing at infinity and exhibiting an essential  singularity at s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In case we set αj = 1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n + 1, the multi-index special functions  W(¯α,¯ν) will be related to the hyper-Bessel functions as is showed in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' After some  calculations, we obtain the following functional relation between the Laplace transform  of the Hyper-Bessel function and the multi-index Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' A more  general relation between these two functions can be found in the article of Kiryakova  and Luchko [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  L  �  W(¯1,¯ν)(λx), s  �  =  n  �  j=1  Γ(1 + aj)1  s  ∞  �  k=0  1  �n  j=1 Γ(k + aj+1 + 1)  �λ  s  �k  =  n  �  j=1  Γ(1 + aj)1  sE(n)  (1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=',1),(aj+1+1)  �λ  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (14)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Laplace transform of Wα,β,ν(x) can be obtained as a special case of  the (13) as follows:  L (Wα,β,ν(λxρ), s) = 1  s  ∞  �  k=0  k  �  i=1  βΓ(βi + 1 − α)  Γ(βi + 1)  Γ(ρk + 1)  Γ(βk + 1 − α + ν)  � λ  sρ  �k  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (15)  and, in the case of the parameter α = 1, we obtain the well-known Laplace transform  of the Wright function which can be expressed in terms of the two-parameter Mittag-  Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  L (W1,β,ν(λx), s) = L (Wβ,ν(λx), s) = 1  sEβ,ν  � λ  β s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (16) 6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Recurrence relations of Wα,β,ν  A recurrence relation is an equation that recursively defines a sequence of values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' given  one or more initial terms, each further term of the sequence is defined as a function of  the previous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Differential recurrence relation of the generalized Wright function  can be used in the study of fractional differential equations, and it is obtained directly  from series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  xα+β dβ  dxβ  �  xν−α+βWα,β,ν+β(xβ)  �  − 2xν+βWα,β,ν(xβ) + xα+ν dα  dxα Wα,β,ν−β(xβ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (17)  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In case α = β = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' ν = n + 1 with n ∈ N0 and  d  dxW1,1,n(x) = W1,1,n+1(x) = Cn(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  we obtain the well known three-term recurrence relation for the Bessel-Clifford func-  tion Cn(x)  xCn+2(x) + (n + 1)Cn+1(x) = Cn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (18)  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Recurrence fractional derivatives relation for the Wright and  Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' From the relation (11) between the generalized Wright  function and the classical Wright function the relation (17) becomes  dλ  dzλ  �  zλ+ν−1Wλ,λ+ν  �zλ  λ  ��  = zν−1Wλ,ν  �zλ  λ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (19)  using the formula  d  dz Wλ,ν−λ  �zλ  λ  �  = zλ−1Wλ,ν  �zλ  λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (20)  In case α = 0, and β → α, ν → β − 1 the generalized Wright function is related to  the Mittag-Leffler function by the following relation:  W0,α,β−1(z) = Eα,β(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (21)  In particular, from the recurrence relation (17), we obtain the new recurrence relation  involving fractional derivatives for M-L functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  zα dα  dzα  �  zα+β−1Eα,α+β(zα)  �  − 2zα+β−1Eα,β(zα) + zβ−1Eα,β−α(zα) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (22)  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Partial derivatives of Wα,β,ν with respect to the parameters  In this section, taking inspiration from the works of Apelblat and Mainardi [29], [30] we  analyse the derivatives of Wα,β,ν respect the three parameters included in the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  We can treat parameters as variables and hence the derivatives with respect to them  can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' These derivatives lead to infinite power series involving digamma (ψ)  and gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  ∂  ∂ν Wα,β,ν(z) = −  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  ψ(βk + 1 − α + ν)  Γ(βk + 1 − α + ν)zk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (23)  ∂  ∂β Wα,β,ν(z) =  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  zk  Γ(βk + 1 − α + ν) · \uf8ee  \uf8f0  k  �  j=1  j [ψ(βj + 1 − α) − ψ(βj + 1)] − kψ(βk + 1 − α + ν)  \uf8f9  \uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (24)  ∂  ∂αWα,β,ν(z) =  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  zk  Γ(βk + 1 − α + ν)  \uf8ee  \uf8f0−  k  �  j=1  ψ(βj + 1 − α) + ψ(βk + 1 − α + ν)  \uf8f9  \uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (25)  where ψ(z) = Γ′(z)  Γ(z) denotes the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In the case α = 1 and considering the property of the digamma function  ψ(z+1) = ψ(z)+ 1  z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' we obtain the formula (5) and (6) of the Apelblat-Mainardi article  ([30]) for the classical Wright function  ∂  ∂β W1,β,ν(z) =  � ∂  ∂β Wβ,ν  �  (βz) = −  ∞  �  k=0  � ψ(βk + ν)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(βk + ν)  �  kzk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  ∂  ∂ν W1,β,ν(z) =  � ∂  ∂ν Wβ,ν  �  (βz) = −  ∞  �  k=0  � ψ(βk + ν)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(βk + ν)  �  zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  By setting the parameters, α = 0, β → α and ν → β − 1, we obtain the formulas  (95) and (96) of the Apelblat paper ([29])  ∂  ∂αW0,α,β−1(z) = ∂  ∂αEα,β(z) = −  ∞  �  k=0  �kψ(αk + β)  Γ(αk + β)  �  zk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  ∂  ∂β W0,α,β−1(z) = ∂  ∂β Eα,β(z) = −  ∞  �  k=0  �ψ(αk + β)  Γ(αk + β)  �  zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Conclusion  The aim of this paper is to investigate several properties related to the multi-index  special function W(¯α,¯ν) and its 3-parameters version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' An important result was finding  the connection between the W(¯α,¯ν) and the hyper-Bessel function of Delerue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Here we  analyzed the Laplace transform, recurrence relation and derivatives of the function  with respect to the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In particular, we found new findings that, for special  values of the parameters, retrieve some well-known relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Indeed, a simple func-  tional relation is obtained between the Laplace transform of the hyper-Bessel function  and the multi-index Mittag-Leffler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Disclosure statement  No potential conflict of interest was reported by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Acknowledgements  The author is grateful to Dr Roberto Garra for providing essential information, help  and advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  References  [1] Podlubny I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Fractional Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Academic Press, San Diego;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [2] Mainardi F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Fractional calculus and waves in linear viscoelasticity: an introduction to  mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' World Scientific;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' Mittag-Leffler functions, related topics  and applications (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 540).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' New York, NY, USA: Springer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [4] Wright E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Asymptotic partition formulae: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' plane partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Quarterly Journal of  Mathematics, Volume os-2, Issue 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1931;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 177–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1093/qmath/os-  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='177  [5] Wright E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Asymptotic partition formulae:(II) weighted partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Proceedings of the  London Mathematical Society, 2(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1934;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1112/plms/s2-  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='117  [6] Wright E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Asymptotic partition formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Partitions intok-th powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Acta Math-  ematica, 63(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1934;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1007/BF02547353  [7] Wright EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' On the coefficients of power series having exponential singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Journal  London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' 1933;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 71–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [8] Wright EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' London  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' 1935;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content='  [9] Wright EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The asymptotic expansion of the generalized hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Jour-  nal London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' 1935;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' On some operators involving Hadamard derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Integral Trans-  forms and Special Functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content='  [11] Dubovski PB, Slepoi JA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Construction and analysis of series solutions for frac-  tional  quasi-Bessel  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Fract  Calc  Appl  Anal  25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content='1007/s13540-022-00045-z  [12] Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Fractional calculus of some ”new” but not new special function: K-, multi-  index-, and S-analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' AIP Conference Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' Fract Calc Appl Anal 24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 89-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [17] Ricci PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Laguerre-Type Exponentials, Laguerre Derivatives and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' A Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Mathematics 8, 2054;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [18] Garra R, Tomovski Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Exact results on some nonlinear Laguerre-type diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Mathematical Modelling and Analysis, 26(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 72-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [19] Delerue P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', Sur le calcul symbolique `a n variables et fonctions hyperbesseliennes (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Brux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 229–274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [20] Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized Fractional Calculus and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Longman – J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Wiley,  Harlow, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' York;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [21] Dimowski I, Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized Poisson transmutations and corresponding repre-  sentations of hyper-Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Bulg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 39, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 20-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [22] Dimowski I, Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized Poisson representations of hyper-geometric func-  tions pFq , p < q using fractional integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='16th Spring Conf Union Bulg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Sofia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 205-212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [23] Weinstein A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The generalized radiation problem and the Euler-Poisson-Darboux equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Summa Brazil Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1955;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 125-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [24] Kiryakova V, Hernandez-Suarez V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Bessel-Clifford third order differential operator and  corresponding Laplace type integral transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Dissertationes Mathematicae 340;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 143-161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [25] Hayek N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Estudio de la ecuaci`on diferencial xy′′ + (ν + 1)y′ + y = 0 y de sus aplicaciones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 18, No 1-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 57-174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [26] Hayek N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Funciones de Bessel-Cliff`ord de tercer orden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Actas XII Jornadas Luso-Esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' de  Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' (Braga);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 346-351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [27] Weinstein A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized axially symmetric potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='AMS 59, 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [28] Kiryakova V, Luchko Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Multiindex MittagLeffler Functions and Their Applications  for Solving Fractional Order Problems in Applied Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1301, 597;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='3526661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [29] Apelblat A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Differentiation of the Mittag-Leffler functions with respect to parameters in  the Laplace transform approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Mathematics, 8(5), 657;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [30] Apelblat A, Mainardi F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Differentiation of the Wright functions with respect to parameters  and other results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' arXiv e-prints, arXiv-2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Fractional calculus  In order to make the papar self-contained, we briefly recall main definitions and prop-  erties of fractional calculus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Let γ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Riemann-Liouville fractional integral is defined by  Jγ  x f(x) =  1  Γ(γ)  � x  0  (x − x′)γ−1f(x′)dx′,  (A1)  10  where  Γ(γ) =  � +∞  0  xγ−1e−xdx,  is the Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Note that, by definition, J0  xf(x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Moreover it satisfies the semigroup property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Jα  x Jβ  x f(x) = Jα+β  x  f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  There are different definitions of fractional derivative (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In this paper we  used the fractional derivatives in the sense of Caputo, that is  Dγ  xf(x) = Jm−γ  x  Dm  x f(x) =  1  Γ(m − γ)  � x  0  (x−x′)m−γ−1  dm  d(x′)m f(x′) dx′, γ ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' (A2)  It is simple to prove the following properties of fractional derivatives and integrals  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' [1]) that will be used in the analysis:  Dγ  xJγ  x f(x) = f(x),  γ > 0,  (A3)  Jγ  x Dγ  xf(x) = f(x) −  m−1  �  k=0  f (k)(0)xk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' ,  γ > 0, x > 0,  (A4)  Jγ  x xδ =  Γ(δ + 1)  Γ(δ + γ + 1)xδ+γ  γ > 0, δ > −1, t > 0,  (A5)  Dγ  xxδ =  Γ(δ + 1)  Γ(δ − γ + 1)xδ−γ  γ > 0, δ > −1, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+Beckman Defense
+A V Subramanyam
+IIITD
+subramanyam@iiitd.ac.in
+Abstract
+Optimal transport (OT) based distributional robust
+optimisation (DRO) has received some traction in
+the recent past. However, it is at a nascent stage but
+has a sound potential in robustifying the deep learn-
+ing models. Interestingly, OT barycenters demon-
+strate a good robustness against adversarial attacks.
+Owing to the computationally expensive nature of
+OT barycenters, they have not been investigated
+under DRO framework.
+In this work, we pro-
+pose a new barycenter, namely Beckman barycen-
+ter, which can be computed efficiently and used
+for training the network to defend against adver-
+sarial attacks in conjunction with adversarial train-
+ing. We propose a novel formulation of Beckman
+barycenter and analytically obtain the barycenter
+using the marginals of the input image. We show
+that the Beckman barycenter can be used to train
+adversarially trained networks to improve the ro-
+bustness. Our training is extremely efficient as it re-
+quires only a single epoch of training. Elaborate ex-
+periments on CIFAR-10, CIFAR-100 and Tiny Im-
+ageNet demonstrate that training an adversarially
+robust network with Beckman barycenter can sig-
+nificantly increase the performance. Under auto at-
+tack, we get a a maximum boost of 10% in CIFAR-
+10, 8.34% in CIFAR-100 and 11.51% in Tiny Ima-
+geNet. Our code is available at http://bitly.ws/yvgh.
+1
+Introduction
+Optimal mass transport (OT), originally proposed by Monge
+in his seminal work [Monge,
+1781],
+has gathered a
+widespread interest in the field of learning representations.
+The original deterministic OT problem was later relaxed by
+Kantorovich [Kantorovich, 1942] and considered a proba-
+bilistic transport problem. This formulation seeks solution
+for the optimal transport plan which can transport mass be-
+tween two measures by incurring the minimum cost and is
+solved using a linear program. The modern day OT is also at-
+tributed to the phenomenal work of Kantorovich. Following
+the OT theory, barycenters in Wasserstein space was proposed
+by Agueh and Carlier in their remarkable work [Agueh and
+Carlier, 2011]. Further, using entropic regularization [Cuturi,
+2013], a fast method of computing barycenters was proposed
+by Cuturi and Doucet [Cuturi and Doucet, 2014]. Recent
+works addresses the challenge of computational complexity
+of barycenters using neural networks [Lacombe et al., 2021].
+In this work, we investigate the barycenters towards robust
+learning of deep learning models.
+Deep learning systems have shown impressive perfor-
+mance in various applications. However, these systems are
+vulnerable to adversarial perturbations [Wong et al., 2020],
+[Croce and Hein, 2020], [Xie et al., 2019].
+In order to
+counter these attacks, several defense mechanisms have also
+been proposed.
+In one of the early works, Szegedy et
+al. [Szegedy et al., 2013] formulated the adversarial attack
+as an optimization problem and obtained the adversarial sam-
+ple using L-BFGS. Several adversarial attacks have been
+proposed since Szegedy’ work [Goodfellow et al., 2014;
+Kurakin et al., 2016]. On the other hand, strong defense mea-
+sures have been studied in [Madry et al., 2017], [Theagarajan
+et al., 2019], [Wong et al., 2020], [Rebuffi et al., 2021].
+Rotated  
+samples of
+Classical AT
+Barycentric
+Training 
+ 
+Barycenter of  
+adversarial sample
+Inference
+Figure 1: Illustration: Classical defense methods use Adversarial
+Training (AT) as a major defense technique. Our method obtains
+barycenter from rotated inputs and uses them for training the model
+using a cross-entropy loss. During inference time also we compute
+barycenter of the given sample. The dashed boundary of barycenter
+indicates that the barycenter is close to input samples in terms of
+appearance but there are some differences. In the computation of
+barycenter of adversarial sample, the barycenter shows the changes
+in same color as that of the background to imply that barycenter
+suppresses the adversarial noise.
+In the field of adversarial attacks and defense, lP space has
+arXiv:2301.01495v1  [cs.LG]  4 Jan 2023
+
+been extensively studied. However, only a few works investi-
+gate attacks under OT framework [Wong et al., 2019], [Li et
+al., 2021]. There are even fewer works which investigate ro-
+bustness using OT theory [Kwon et al., 2020], [Subramanyam
+and Raj, 2022]. Distinct from these works, we first intro-
+duce Beckman barycenter, a concept analogous to Wasser-
+stein barycenter. We use proximal operator methods to solve
+for the barycenter. The barycenters obtained from the clean
+samples are used to train a pretrained adversarially robust net-
+work. We note that in the absence of adversarial samples in
+the training, the model would give a better clean accuracy but
+will suffer in terms of adversarial accuracy. Therefore, we
+use a pre-trained adversarially robust network to overcome
+this challenge. An abstract illustration of our method is given
+in Figure 1.
+Beckman barycenter is obtained from input marginals via
+a non-linear interpolation. The input marginals are linearly
+transformed versions of the input and thus interfere with the
+adversarial noise. Using these marginals the barycenter gen-
+erates a sample which is similar in appearance to the input
+and is closer in terms of class label. Thus, the class label
+is preserved when the input is a clean sample, whereas, the
+adversarial noise gets suppressed when the input is an ad-
+versarial sample. Further, the network needs to be trained
+with barycenter of clean samples so as to correctly classify
+them. However, this training is cheap as a single epoch is
+sufficient. We prove our hypothesis using extensive qualita-
+tive and quantitative experiments.
+2
+Related Works
+Adversarial Attacks Given an adversarial sample x with la-
+bel y, a target network f parameterized by θ, the adversary
+tries to find xadv by adding an adversarial noise such that
+the prediction fθ(xadv) ̸= fθ(x) = y. Some of the robust at-
+tacks are iterative FGSM [Kurakin et al., 2016], PGD [Madry
+et al., 2017], Carlini and Wagner attacks [Carlini and Wagner,
+2017], Jacobian based attack [Papernot et al., 2016], physical
+attack Athalye [Athalye et al., 2018], and Autoattack [Croce
+and Hein, 2020]. These attacks are primarily focused in lp
+domain.
+Adversarial Defense In response to adversarial attacks, sev-
+eral defenses been proposed. One of the best defense ap-
+proach is adversarial training [Szegedy et al., 2013], [Good-
+fellow et al., 2014], [Moosavi-Dezfooli et al., 2016]. Madry
+et al. [Madry et al., 2017] formally studied adversarial train-
+ing and proposed that such training allows network to de-
+fend well against first order adversary. Adversarial logit pair-
+ing uses a pair of logits from clean and adversarial examples
+to defend against adversarial samples [Kannan et al., 2018].
+TRADES [Zhang et al., 2019] prove the bounds based on
+regularization term which minimizes the difference in pre-
+diction between clean and adversarial examples. In [Wong et
+al., 2020], authors proposed to effectively combine FGSM
+and random initialization to demonstrate better adversarial
+training. RST [Carmon et al., 2019] propose a self-training
+technique using unlabelled samples to improve the robust-
+ness. Observing the correlation between flatness of weight
+loss landscape and adversarial robustness, Wu et al. proposed
+adversarial weight perturbation (AWP) to regularize the flat-
+ness of weight loss [Wu et al., 2020]. On similar lines, [Yu et
+al., 2022] propose a criterion called Loss Stationary Condi-
+tion (LSC) for constrained perturbation, which regulates the
+weight perturbation to prevent overfitting. LBGAT [Cui et al.,
+2021] constrains the logits of a robust model, trained with ad-
+versarial examples, to be similar to the logits of a clean model
+trained on natural data.
+While adversarial training uses all the samples, many tech-
+niques propose that naively using adversarial samples in ad-
+versarial training is not efficient.
+This primarily involves
+training the model with a weak attack first, and then grad-
+ually increasing the strength of the adversary - CAT [Cai et
+al., 2018], DART [Wang et al., 2019a], MART [Wang et al.,
+2019b], FAT [Zhang et al., 2020]. Aforementioned methods
+rely on pre-determined attack parameters for adversarial sam-
+ple generation. However, this restricts the model’s robust-
+ness. To address this issue, LAS-AT [Jia et al., 2022] propose
+a framework for adversarial training that introduces the no-
+tion of learnable attack strategy. It is composed of two com-
+ponents: a target network that uses adversarial examples for
+training to improve robustness, and a strategy network that
+produces attack strategies to control adversarial sample gen-
+eration. In similar spirit, A2 [Xu et al., 2022] and [Cheng
+et al., 2022] have also been proposed. A classical review of
+defense methods can be obtained in [Bai et al., 2021].
+In a parallel line of defense works, input purification has
+also been explored. At the test time, these techniques try to
+remove the adversarial noise [Shi et al., 2021], TRADESSSL
+[Mao et al., 2021], HedgeRST [Wu et al., 2021]. Score based
+generative models such as [Yoon et al., 2021] and [Nie et al.,
+2022] have also been used to purify the images before sending
+them for classification.
+Our work is inspired from two different theories, namely,
+OT barycenters and distributional robust optimization. We
+discuss these theories in the following.
+Wasserstein Barycenter In the following we discuss Wasser-
+stein distance and barycenter. Given probability distributions,
+µ1, µ2 ∈ Ω, the Wasserstein distance is defined as,
+W(µ1, µ2) = inf
+Ω×Ω c(x, y)π(x, y)dxdy,
+(1)
+s.t.
+�
+Ω
+π(x, y)dx = µ1(x),
+�
+Ω
+π(x, y)dy = µ2(y),
+where the cost matrix c(x, y) = ∥x − y∥1 and π denotes the
+transport plan. This is also known as Earth Mover’ Distance
+(EMD). This form is also used to compute barycenter [Cuturi
+and Peyr´e, 2016] wherein the summation of Wasserstein dis-
+tance between the barycenter and each input marginal is con-
+sidered. However, barycenters are costly to compute and the
+best known complexity scales exponentially with the number
+of marginals [Fan et al., 2022].
+EMD can also be represented as dual of the dual of Eq 1
+in variational form popularly introduced by Beckman [Beck-
+
+mann, 1952], [Li et al., 2018], [Lee et al., 2020],
+W(µ1, µ2) = inf
+M
+�
+Ω
+∥M∥
+(2)
+s.t. div(M) + µ1 − µ2 = 0
+M.n = 0 ∀x ∈ ∂Ω; n is normal to ∂Ω
+Under appropriate discretisation, M = (Mx, My), M ∈
+Rn×2 is flux vector satisfying zero flux boundary conditions.
+µ1, µ2 ∈ Rn, and,
+div(M) = (Mx[i, j]−Mx[i−1, j])+(My[i, j]−My[i, j−1])
+and the zero-flux boundary conditions mean that Mx[i, j] =
+My[i, j] = 0 outside the boundary. Eq 2 is favorable com-
+pared to Eq 1 as it reduces the complexity from O(n2) to
+O(n) [Li et al., 2018]. Motivated by the recent developments
+of OT barycenters, we make use of Eq 2 to propose Beckman
+barycenter as they can be efficiently solved using well known
+techniques like [Goldstein and Osher, 2009], [Chambolle and
+Pock, 2011].
+DRO One of the influential works in DRO was proposed by
+Scarf [Scarf, 1957]. Following this work, significant research
+has been done in this field [Ben-Tal et al., 2009], [Duchi et
+al., 2021], [Staib and Jegelka, 2017]. DRO aims to address
+the problem of uncertainty or shift in the data distribution that
+can arise due to measurement errors and admits a solution
+for the worst case scenario. Let L(θ, x) be the loss function
+where θ are network parameters. Then, DRO solves for,
+inf
+θ sup
+Q∈Q
+EQL(θ, x)
+(3)
+Here, Q is the distribution against which DRO minimizes
+the loss. For instance, Q can be considered as a distribu-
+tion set which contains perturbations of input samples x.
+Here we note that adversarial training can be considered to
+be a specific instance of DRO wherein the distribution Q is
+drawn from adversarial samples. In our case, we consider the
+barycenters as the samples drawn from the distribution Q and
+thus provide robustness against perturbed samples.
+2.1
+Proposed Algorithm
+In this work, we propose a novel Beckman Barycenter for-
+mulation and derive the barycenter analytically. We use the
+barycenter to demonstrate that it can be applied for adversar-
+ial defense. We first obtain the barycenter using the marginals
+from the given input image and then train the network using
+barycenter.
+While OT barycenters are a good choice for the distri-
+bution Q in Eq 3, computing OT barycenter suffers from
+high complexity and exponentially increases with the number
+marginals [Fan et al., 2022]. To counter this high complexity
+challenge, we first discuss an analogous barycenter problem
+by building upon the formulation given in Eq 2.
+inf
+M1,M2
+r1,r2,µ
+∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1
+(4)
++∥r2∥1) + β∥µ∥1
+s.t. div(M1) + µ1 − µ = r1
+div(M2) + µ2 − µ = r2
+where, r1, r2, µ ∈ Rn. Our formulation is loosely inspired
+from the Beckman OT formulation that are given in [Li et al.,
+2018], [Lee et al., 2020]. There are notable changes in Eq 4
+from Eq 2. First we solve for Beckman barycenter µ in ad-
+dition to other variables. Similar to Wasserstein barycenter
+which acts as a representative of marginals using Wasserstein
+metric, the Beckman barycenter µ minimizes the flux with
+respect to input marginals µ1 and µ2. In our experiments,
+these marginals are obtained by rotating the input image with
+±4◦. Second, the variables r1 and r2 allow the mass to be
+created or destroyed [Lee et al., 2020] and the regularization
+over r1, r2 and µ ensure that these variable do not take ar-
+bitrarily large values. Third, Eq 4 can be easily converted to
+Lagrange formulation and solved in linear time using primal-
+dual method of Chambolle and Pock [Chambolle and Pock,
+2011].
+In order to make the objective strongly convex, we first
+apply proximal operators. The l2 regularizer makes the ob-
+jective strongly convex. Using the proximal operator,
+inf
+M1,M2,r1
+r2,µ′
+1,µ′
+2,µ
+∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1 (5)
++∥r2∥1) + 1
+2ρ(∥µ′
+1 − µ1∥2 + ∥µ′
+2 − µ2∥2) + β∥µ∥1
+s.t. div(M1) + µ′
+1 − µ = r1
+div(M2) + µ′
+2 − µ = r2
+The Lagrangian of Eq 5 is given as,
+inf
+M1,M2,r1
+r2,µ′
+1,µ′
+2,µ
+∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1 (6)
++∥r2∥1) + 1
+2ρ(∥µ′
+1 − µ1∥2 + ∥µ′
+2 − µ2∥2) + β∥µ∥1
++
+�
+i
+⟨λi, div(Mi) + µ′
+i − µ − ri⟩
+Eq 6 can be solved using first-order primal dual method of
+Chambolle and Pock [Chambolle and Pock, 2011]1.
+Mt+1
+i
+← arg min
+Mi
+∥Mi∥2,1 + ⟨λi, div(Mi) + µ′
+i−
+µ − ri⟩ + 1
+2τ1
+∥Mi − Mt
+i∥2
+∀i = {1, 2}
+µ′
+i
+t+1 ← arg min
+µ′
+i
+1
+2τ1
+(∥µ′
+i − µi∥2) + ⟨λi, µ′
+i⟩
++ 1
+2τ1
+∥µ′
+i − µ′
+i
+t∥2
+rt+1
+i
+← arg min
+ri
+α∥ri∥1 + ⟨λt
+i, ri⟩ + 1
+2τ1
+∥ri − rt
+i∥2
+µt+1 ← arg min
+µ
+∥µ∥1 + ⟨λt
+i, µ⟩ + 1
+2τ1
+∥µ − µt∥2
+λt+1
+i
+← arg max
+λi
+⟨λi, κt+1⟩ − 1
+2τ2
+∥λi − λt
+i∥2,
+1We use similar notations to that of [Li et al., 2018], [Chambolle
+and Pock, 2011] for consistency and simplicity.
+
+where, κt+1 = 2(div(Mi)t+1+µ′
+i
+t+1−rt+1
+i
+)−(div(Mi)t+
+µ′
+i
+t − rt
+i)
+We now discuss the solution of each individual optimiza-
+tion.
+Solving for Mi: The rows mij of Mi can be expressed
+and solved using l21 norm shrinkage operator,
+mt+1
+ij
+← shrinkl2
+τ1(mt
+ij − τ1div∗(λt
+i)j)
+(7)
+Here,
+div∗
+denotes the adjoint of div operator,
+and
+shrinkl2
+τ1η = max(∥η∥2 − τ1, 0) ⊙
+η
+(∥η∥2). “⊙” denotes the
+Hadamard product.
+Solving for µ′
+i:
+µ′
+i
+t+1 ← max{0,
+ρτ1
+1 + ρτ1
+µ′
+i +
+1
+1 + ρτ1
+(µ′
+i
+t − τ1λt
+i)}, (8)
+Solving for ri: We use an l1 shrinkage operator.
+rt+1
+i
+← shrinkl1
+ατ1(rt
+i + τ1λt
+i)
+(9)
+Here, shrinkl1
+ατ1(η) = sign(η) ⊙ max(∥η∥ − ατ1, 0).
+Solving for barycenter µ:
+µt+1 ← shrinkl1
+βτ1(µt + τ1(λt
+1 + λt
+2))
+(10)
+Solving for λ:
+λt+1
+i
+← λt
+i + τ2κt+1
+(11)
+2.2
+Toy example
+We demonstrate the barycenter computation using a Gaussian
+image in Figure 2. The barycenter of clean samples, sample
+with random noise and adversarial sample are shown. As we
+see, for the clean case the barycenter is very similar to that of
+the original image. In the second column where random noise
+is added, the barycenter reduces the noise.
+Similar effect
+is also seen for the case where adversarial noise is present.
+This indicates that non-linear interpolation of Barycenter sup-
+presses the adversarial noise.
+(a) Clean
+(b) Random
+(c) Adversarial
+Figure 2: Top: Clean image, noisy image, adversarial image. Bot-
+tom: Barycenter of clean image, noisy image, adversarial image.
+2.3
+Training
+Let a model be given by fθ, the barycenter of clean samples
+be denoted by x and its labels as y. We then optimize the
+following loss
+arg min
+θ
+1
+n
+n
+�
+i=1
+LCE(fθ(xi), yi)
+where LCE is the cross-entropy loss. We would like to em-
+phasize that we do not perform adversarial training. Instead
+we use an adversarially pretrained model. Thus, fθ is an ad-
+versarial robust model and our training further enhances the
+robustness. We also note that this optimization falls under
+DRO as the samples used are barycenters which belong to the
+distribution Q.
+2.4
+Theoretical analysis
+We first present a convergence analysis of Eq 6.
+Theorem 1.
+Let τ1τ2(λmax(∇2) + 3)
+<
+1, where
+λmax(∇2) denotes the largest eigenvalue of discrete Lapla-
+cian operator ∇2 = DD⊤, where D is the matrix repre-
+senting div operator. Then, the iterations Mt
+i, µ′
+i
+t
+i, µt, rt
+i, λt
+converge to the saddle point solution of the Lagrangian
+M∗
+i , µ∗
+i , µ∗, r∗
+i , λ∗.
+Proof: Let u = {M1, M2, µ2, µ2, µ, r}. Then, we write
+Eq 6 as
+L(u, λ) = G(u) + ⟨λ, ˜Kb⟩
+where λ
+=
+[λ1; λ2],
+K
+=
+[D, I, −I, −I],
+˜K
+=
+[K, 0; 0, K], b = [b1; b2], b1 = [vec(M1); µ′
+1; µ; r1]; b2 =
+[vec(M2); µ′
+2; µ; r2].
+The function G
+=
+∥M1∥2,1 +
+∥M2∥2,1 + α(∥r1∥1 + ∥r2∥1) +
+1
+2ρ(∥µ′
+1 − µ1∥2 + ∥µ′
+2 −
+µ2∥2) + β∥µ∥1 is convex and ˜K is a linear operator. These
+conditions satisfy Theorem 1 of [Chambolle and Pock, 2011].
+If λmax(∇2) is the max eigenvalue of DD⊤, then the max
+eigenvalue of [D, ±I][D, ±I]⊤ is λmax(∇2) + 1. Similarly,
+for KK⊤, it is λmax(∇2) + 3. Since ˜K is obtained from
+K by padding zeros only, ˜K has the same max eigenvalue
+as that of K. Further, since ∥ ˜K ˜K⊤∥2
+2 ≥ λmax( ˜K ˜K⊤) =
+λmax(∇2) + 3, we can also write the convergence criteria as
+τ1τ2∥ ˜K ˜K⊤∥2
+2 < 1.
+Since we solve for the Lagrangian dual function, we anal-
+yse the primal dual gap which is given as [Jacobs et al., 2019]
+G(u, λ) =
+sup
+∥λ′−λ0∥≤R1
+L(u, λ′) −
+inf
+∥u′−u0∥≤R2L(u′, λ)
+Theorem 2.
+Suppose the step sizes τ1 and τ2 satisfy
+τ1τ2∥ ˜K ˜K⊤∥2
+2 < 1. Let uN =
+1
+N
+�N
+n=1 un and λN =
+1
+N
+�N
+n=1 λn, where un and λn are sequences generated from
+Eqns 7 - 11. Then after N iterations, we have,
+G(u, λ) ≤ sup
+u,λ
+1
+2N
+�
+∥u − u0∥2
+τ1
++ ∥λ − λ0∥2
+τ2
+�
+This rate is similar to convergence rates in gradient descent
+and shows that the gap converges with rate O(1/N). For
+brevity, we omit the proof and it can be derived as an exten-
+sion of Theorem 1 [Chambolle and Pock, 2011].
+
+2.5
+Mutual Information
+In order to understand the underlying reason behind the per-
+formance of our method, we provide more insights using
+mutual information (MI). We first note that the MI between
+two random variables is given by I(X, y) = H(P(y)) −
+E
+P (x)[H(P(y|X))]. In our case, we take the random variables
+as model parameters θ and softmax output y. Then, given a
+sample x and dataset D,
+I(θ, y|D, x) = H(p(y|x, D)) −
+E
+p(θ|D)
+H(p(y|x, θ)) (12)
+Eq 12 measures the information shared between θ and y.
+A tractable way of computing I(θ, y|D, x) is given in [Smith
+and Gal, 2018], [Houlsby et al., 2011].
+I(θ, y|D, x) = 1
+C
+C
+�
+j=1
+1
+n
+n
+�
+i=1
+(pij − ˆp)2
+(13)
+where, ˆp ∈ [0, 1]C is computed as the mean of all softmax
+probabilities, C is the number of classes, pi ∈ [0, 1]C, pij ∈
+[0, 1] denotes the softmax probability for a particular class j.
+A higher I indicates that knowing θ (or y) gives a higher
+information about y (or θ). In other words, the model will
+perform better if the mutual information is high.
+In addition, we also compute MI between the predictions
+for the following two cases - (i) clean test set and adversarial
+test set, and (ii) barycenter of clean test set and barycenter of
+adversarial test set using [Ji et al., 2019]. Given a model f
+paramterised by θ, clean sample xi and its adversarial coun-
+terpart x′
+i, the joint probability distribution between natural
+and adversarial samples is given by the following C × C ma-
+trix,
+I(f(xi, θ), f(x′, θ)) =
+C
+�
+y=1
+C
+�
+y′=1
+Pyy′ ln Pyy′
+PyPy′
+(14)
+where, Pyy′ is given as,
+Pyy′ = 1
+n
+n
+�
+i=1
+f(xi, θ)f(x′
+i, θ)⊤
+(15)
+and the marginals Py, Py′ are obtained by row and column
+sum of Pyy′. A higher value of I(., .) indicates that know-
+ing about clean samples gives a higher amount of information
+about the adversarial samples.
+3
+Experiments
+We present elaborate experimental results on CIFAR-10,
+CIFAR-100 and Tiny ImageNet. We use strong baseline of
+LAS [Jia et al., 2022]. We also show improvements over
+other baselines - LBGAT [Cui et al., 2021], PGD-AT [Madry
+et al., 2017], TRADES [Zhang et al., 2019], RST [Carmon
+et al., 2019]. We compare against several popular adversarial
+training models, MART [Wang et al., 2019b], AWP-A2 [Xu
+et al., 2022], RST-RWT [Yu et al., 2022], TRADESAWP [Wu
+et al., 2020], AWP [Wu et al., 2020], LASAT, LASTRADES,
+LASAWP [Jia et al., 2022]. We also compare with adaptive test
+time defenses HedgeRST [Wu et al., 2021] and TRADESSSL
+[Mao et al., 2021]. In the Tables, we use “+B” to indicate the
+results obtained using our approach.
+3.1
+Implementation details
+In case of CIFAR10 and CIFAR100, WideResNet34-10 is
+used and for Tiny ImageNet PreActResnet18 is used. Ad-
+ditionally, we evaluate on CIFAR-10 with WideResNet28-
+10, WideResNet32-10, WideResNet70-16 and on CIFAR-
+100 with WideResNet34-20. We use these models for a fair
+comparison with existing works as these models are widely
+used for adversarial defense evaluation. We evaluate against
+different attacks namely FGSM, PGD-10, PGD-20, CW, and
+AA using l∞ attack with ϵ = 8/255. Our evaluation proto-
+cols are similar to the protocols given in [Zhang et al., 2019],
+[Jia et al., 2022]. We would like to emphasize that we use the
+checkpoints from the baseline models and perform a single
+epoch training using clean barycenters. Upon increasing the
+number of epochs, the clean accuracy improves, however, the
+adversarial accuracy becomes comparable to that of baseline
+and further increasing epochs leads to subsequent drop in ac-
+curacy against adversarial samples. We use SGD optimizer
+with a learning rate of 1e-4, momentum = 0.9 without any
+weight decay.
+In order to compute the barycenter, we set ρ = 5e−1, τ1 =
+1e − 1, τ2 = α = β = 1 and iterations is set to 200. While
+one can also attack the barycenter, we give experiments for
+the case where the clean image is attacked. This is because
+the barycenter itself lies at an ϵ which is greater than attacker’
+budget. Thus attacking barycenter has little incentive as in
+that case the attacked image will lie at an ϵ outside the given
+ϵ = 8/255 for the l∞ attack.
+3.2
+Comparison on CIFAR-10
+In Table 1, we observe that clean performance is better
+for the models trained with barycenters - TRADESAWP+B,
+LBGAT+B, LASTRADES+B and RST+B. Amongst WRN-
+28-10 models, RST has the best clean performance and our
+method enhances it by 1%. In PGD-10, there is a rise of
+2.57%. In case of AA, there is a boost of 6.49%.
+In case of WRN-34-10, LASAT+B shows a huge boost
+of 10% under AA. Further, LASAWP+B shows the best per-
+formance under PGD-10, PGD-20 and CW attack amongst
+WRN-34-10 models. Under AA it shows an improvement of
+7.71%.
+Comparison with Adversarial Purification models Our
+RST+B model outperforms Hedge∗
+RST under all the cases.
+Against AA, our approach gives 3.1% higher accuracy
+compared to Hedge∗
+RST.
+We also see that compared to
+TRADESSSL, TRADESAWP has a better performance.
+3.3
+Comparison on CIFAR-100
+In Table 2, we observe that our method gives a significant
+boost under all the cases. In case of strong baseline LASAWP,
+our method increases the performance by 0.85% under clean
+accuracy. For PGD-20, there is a rise of 0.91%. In case of
+CW, there is an increase of 18.35%. In other models such as
+LBGAT, we see a rise of 5.4% in clean accuracy.
+3.4
+Comparison with Curriculum based AT
+In Table 3, we compare against curriculum based AT meth-
+ods like CAT [Cai et al., 2018], FAT [Zhang et al., 2020] and
+
+Table 1: CIFAR-10.
+∗ indicates that the model uses WRN-28-10.
+Bold font is used to indicate the best performance amongst WRN-
+34-10 and Red color font is used to indicate the best performance
+amongst WRN-28-10.
+Method
+Clean
+PGD10
+PGD20
+CW
+AA
+Adversarial Training
+PGD-AT
+85.17
+56.07
+55.08
+53.91
+51.69
+TRADES
+85.72
+56.75
+56.10
+53.87
+53.40
+MART
+84.17
+58.98
+58.56
+54.58
+51.10
+AWP-A2
+87.54
+-
+59.50
+57.42
+54.86
+RST-RWT∗
+88.87
+-
+64.11
+62.03
+60.36
+Adversarial Purification
+TRADESSSL
+82.12
+-
+-
+-
+60.67
+Hedge∗
+RST
+88.64
+-
+-
+73.89
+63.10
+Adversarial and Barycentric Training
+TRADESAWP
+85.36
+59.58
+59.25
+57.07
+56.17
++B
+87.32
+62.60
+62.32
+75.85
+65.32
+LBGAT
+88.22
+56.25
+54.60
+54.29
+52.23
++B
+88.38
+59.28
+58.43
+74.61
+61.22
+LASAT
+86.23
+57.11
+56.41
+55.54
+53.58
++B
+86.21
+61.08
+60.64
+74.09
+63.59
+LASTRADES
+85.24
+57.66
+57.07
+55.45
+54.15
++B
+86.15
+60.32
+60.03
+73.75
+63.43
+LASAWP
+87.74
+61.09
+60.16
+58.22
+55.52
++B
+87.45
+63.66
+61.16
+74.81
+63.23
+RST∗
+89.69
+63.48
+62.51
+61.06
+59.71
++B∗
+90.68
+65.12
+64.38
+77.08
+66.20
+DART [Wang et al., 2019a]. Under FGSM, PGD-20 and CW,
+our model shows a huge improvement. In case of clean sam-
+ples, we see that the accuracy of FAT+B compared to FAT is
+less. This may be due to the fact that FAT employs curricu-
+lum learning in the training whereas our method does not use
+curriculum learning.
+3.5
+Comparison on Tiny ImageNet
+We present the results in Table 4. In comparison to baselines,
+our method shows significant improvement in all cases. For
+LASAWP, our method improves the performance under clean
+samples by 1.65%. In case of PGD-50, our method shows
+a rise of 1.28%, and in case of CW attack, our method al-
+most doubles the accuracy. Under AA, LASTRADES observes
+a maximum performance rise by 11.51%.
+3.6
+Analysis using Deeper and Wider Models
+We use WRN-70-16 and WRN-34-20 to analyse the effect
+when the models get deeper and wider. In particular, for clean
+samples, we can observe that the deeper and wider models
+give a better boost. In CIFAR-10, WRN-70-16 gives 88.87%
+for clean samples which is 3.21% better than LASAWP model’
+85.66%. In contrast, for WRN-34-10, our method gives accu-
+racy similar to that of LASAWP. In CIFAR-100, our method
+boosts the performance by 8.34% under AA. In other cases
+also we see that the barycenters improve the performance by
+a significant margin.
+Table 2: CIFAR-100 WRN-34-10.
+Method
+Clean
+PGD-10
+PGD-20
+CW
+PGD-AT
+60.89
+32.19
+31.69
+30.10
+TRADES
+58.61
+29.20
+28.66
+27.05
+TRADESAWP
+60.17
+33.81
+33.6
+57.07
++B
+63.67
+36.34
+36.15
+51.92
+LBGAT
+60.64
+35.13
+34.53
+30.65
++B
+66.04
+36.29
+36.01
+52.92
+LASAT
+61.8
+33.27
+32.83
+31.12
++B
+62.45
+36.60
+36.17
+49.60
+LASTRADES
+60.62
+32.82
+32.51
+29.51
++B
+62.58
+35.22
+34.96
+50.99
+LASAWP
+64.89
+37.11
+36.36
+33.92
++B
+65.50
+37.55
+37.27
+52.27
+Table 3: CIFAR-10 WRN-32-10.
+Method
+Clean
+FGSM
+PGD-20
+CW
+CAT
+77.43
+57.17
+46.06
+42.48
+DART
+85.03
+63.53
+48.70
+47.27
+FAT
+89.34
+65.52
+46.13
+46.82
++B
+84.59
+69.98
+57.02
+71.36
+3.7
+TSNE
+In Figure 3, we show the tsne plots for MNIST testset with
+classes 0 and 1. Here, we use a weak MNIST model which
+has only two dimensions before the classification layer. We
+deliberately chose a weak model so that we can easily show
+the effect in low dimensions. Though higher dmensions could
+be taken, the effect cannot be easily seen due to a highly non-
+linear transformation from high to low dimension of tsne. We
+can see that the two clusters yellow and purple are well sepa-
+rated for clean and barycenters of clean images. In case of ad-
+versarial samples, the points overlap on each other. However,
+when we take barycenter of adversarial samples, we again
+see that the clusters are well separated, similar to the case of
+clean images. Thus, it is evident that the barycenter nullifies
+the effect of adversarial noise.
+(a) Clean
+(b) Barycenter
+(c) Attacked
+(d) Adv.+Bary.
+Figure 3: Left to right: Plot of 2D features of Clean image, Barycen-
+ter of clean image, Attacked image, Barycenter of adversarial image.
+MNIST model obtains 51% accuracy and has only 2D feature vector
+before the classification layer.
+3.8
+Mutual Information
+In Table 6, we present the study of mutual information. We
+use LASAT and LASTRADES on CIFAR-10. The MI is com-
+puted using Eq 12 and Eq 14. Here we note that the MI for
+LASAT+B is more for training set compared to that of LASAT.
+
+Table 4: Tiny ImageNet PreActResNet18.
+Method
+Clean
+PGD-50
+CW
+AA
+LASAT
+44.86
+22.16
+18.54
+16.74
++B
+45.12
+24.54
+37.14
+27.78
+LASTRADES
+41.38
+18.36
+14.50
+14.08
++B
+43.07
+19.25
+35.13
+25.59
+LASAWP
+45.26
+23.42
+19.88
+18.42
++B
+46.91
+24.70
+37.93
+27.00
+Table 5: CIFAR-10 (C-10) WRN-70-16 and CIFAR-100 (C-100)
+WRN-34-20.
+Dataset
+Method
+Clean
+FGSM
+CW
+AA
+C-10
+LASAWP
+85.66
+70.25
+58.44
+57.61
++B
+88.87
+74.04
+75.40
+62.54
+C-100
+LBGAT
+62.55
+43.16
+31.72
+31.92
++B
+66.86
+50.92
+54.19
+40.26
+This indicates that the information available about the labels
+given the model parameters is high and in turn gives a better
+clean accuracy. In case of adversarial samples too, we see that
+the MI is higher for our case. This indicates that the model
+has better prediction for these samples. Further, the measure
+for test set is smaller compared to training set which is ex-
+pected as the model carries more information about train set
+compared to test set.
+Table 6: CIFAR-10 WRN-34-10.
+Method
+Train
+Test
+FGSM
+CW
+LASAT
+0.029
+0.026
+0.020
+0.019
++B
+0.034
+0.029
+0.023
+0.022
+LASTRADES
+0.048
+0.040
+0.033
+0.032
++B
+0.054
+0.045
+0.037
+0.036
+In Table 7, we present the results obtained using Eq 14. We
+observe that for the model trained with barycenter, the MI is
+higher between the barycenter of clean and adversarial sam-
+ples. Thus, the model does better on barycenter of adversarial
+samples compared to baseline LASAT and TRADESAWP. This
+is consistent across FGSM, PGD-10 and CW attacks.
+3.9
+Sensitivity to Barycenter Parameters
+In Figure 4, we demonstrate the sensitivity to different pa-
+rameters involved in the computation of barycenter. In the top
+row, we fix the number of iterations to 200 and τ1 = 1e − 1.
+Here we observe that increasing τ2 makes the barycenter
+brighter. In the second row, increasing τ1 makes the barycen-
+ter darker. Decreasing iterations has a similar effect in the last
+row. We see that unless there is a change of order of magni-
+tude, the appearance does not substantially change. Thus, our
+proposed Beckman barycenter is robust with respect to the
+parameter settings.
+4
+Conclusion
+In this work we introduce Beckman barycenter analogous to
+Wasserstein barycenter. We use the Beckman OT formula-
+Table 7: Mutial Information for CIFAR-10 WRN-34-10.
+Method
+FGSM
+PGD-10
+CW
+LASAT
+0.218
+0.198
+0.203
++B
+0.275
+0.241
+0.264
+LASTRADES
+0.576
+0.554
+0.563
++B
+0.629
+0.572
+0.618
+Figure 4: Top row: Blue boundary represent the given image.
+Barycenter for iterations = 200, τ1 = 1e − 1, τ2 = 1, 1e-1, 1e-2,
+1e-3. Second: barycenter for iterations = 200, τ1=1, 1e-1, 1e-2, 1e-
+3, τ2 = 1. Third: iterations = 200, 100, 50, 10, τ1=1e-1, τ2=1. The
+red boundary indicates the images obtained from default settings of
+the paramater which are used for all experiments.
+tion and analytically solve for the barycenter. Defining the
+baycenter using Beckman OT also has the advantage that the
+computational tools to obtain barycenter are well known and
+efficient. This overcomes the complexity in solving Wasser-
+stein barycenters. Further, we show that barycenter can be
+used for enhancing the performance of adversarially trained
+models. Our training is very efficient as we only need a sin-
+gle epoch. We theoretically show that our barycenters can
+help in defending against attacks. We perform rigorous qual-
+itative and quantitaive analysis to show the effectivenes of
+barycenter. Experimental analysis on CIFAR-10, CIFAR-100
+and Tiny ImageNet demonstrates state-of-art results against
+wide variety of attacks.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf,len=844
+page_content='Beckman Defense  A V Subramanyam  IIITD  subramanyam@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='in  Abstract  Optimal transport (OT) based distributional robust  optimisation (DRO) has received some traction in  the recent past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' However, it is at a nascent stage but  has a sound potential in robustifying the deep learn-  ing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Interestingly, OT barycenters demon-  strate a good robustness against adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Owing to the computationally expensive nature of  OT barycenters, they have not been investigated  under DRO framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In this work, we pro-  pose a new barycenter, namely Beckman barycen-  ter, which can be computed efficiently and used  for training the network to defend against adver-  sarial attacks in conjunction with adversarial train-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We propose a novel formulation of Beckman  barycenter and analytically obtain the barycenter  using the marginals of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We show  that the Beckman barycenter can be used to train  adversarially trained networks to improve the ro-  bustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Our training is extremely efficient as it re-  quires only a single epoch of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Elaborate ex-  periments on CIFAR-10, CIFAR-100 and Tiny Im-  ageNet demonstrate that training an adversarially  robust network with Beckman barycenter can sig-  nificantly increase the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Under auto at-  tack, we get a a maximum boost of 10% in CIFAR-  10, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='34% in CIFAR-100 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='51% in Tiny Ima-  geNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Our code is available at http://bitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='ws/yvgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  1  Introduction  Optimal mass transport (OT), originally proposed by Monge  in his seminal work [Monge,  1781],  has gathered a  widespread interest in the field of learning representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  The original deterministic OT problem was later relaxed by  Kantorovich [Kantorovich, 1942] and considered a proba-  bilistic transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This formulation seeks solution  for the optimal transport plan which can transport mass be-  tween two measures by incurring the minimum cost and is  solved using a linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The modern day OT is also at-  tributed to the phenomenal work of Kantorovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Following  the OT theory, barycenters in Wasserstein space was proposed  by Agueh and Carlier in their remarkable work [Agueh and  Carlier, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Further, using entropic regularization [Cuturi,  2013], a fast method of computing barycenters was proposed  by Cuturi and Doucet [Cuturi and Doucet, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Recent  works addresses the challenge of computational complexity  of barycenters using neural networks [Lacombe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In this work, we investigate the barycenters towards robust  learning of deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Deep learning systems have shown impressive perfor-  mance in various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' However, these systems are  vulnerable to adversarial perturbations [Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020],  [Croce and Hein, 2020], [Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In order to  counter these attacks, several defense mechanisms have also  been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In one of the early works, Szegedy et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' [Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2013] formulated the adversarial attack  as an optimization problem and obtained the adversarial sam-  ple using L-BFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Several adversarial attacks have been  proposed since Szegedy’ work [Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Kurakin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' On the other hand, strong defense mea-  sures have been studied in [Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2017], [Theagarajan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019], [Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020], [Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Rotated  samples of  Classical AT  Barycentric  Training  Barycenter of  adversarial sample  Inference  Figure 1: Illustration: Classical defense methods use Adversarial  Training (AT) as a major defense technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Our method obtains  barycenter from rotated inputs and uses them for training the model  using a cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' During inference time also we compute  barycenter of the given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The dashed boundary of barycenter  indicates that the barycenter is close to input samples in terms of  appearance but there are some differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In the computation of  barycenter of adversarial sample, the barycenter shows the changes  in same color as that of the background to imply that barycenter  suppresses the adversarial noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In the field of adversarial attacks and defense, lP space has  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='01495v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='LG]  4 Jan 2023  been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' However, only a few works investi-  gate attacks under OT framework [Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019], [Li et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' There are even fewer works which investigate ro-  bustness using OT theory [Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020], [Subramanyam  and Raj, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Distinct from these works, we first intro-  duce Beckman barycenter, a concept analogous to Wasser-  stein barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We use proximal operator methods to solve  for the barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The barycenters obtained from the clean  samples are used to train a pretrained adversarially robust net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We note that in the absence of adversarial samples in  the training, the model would give a better clean accuracy but  will suffer in terms of adversarial accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Therefore, we  use a pre-trained adversarially robust network to overcome  this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' An abstract illustration of our method is given  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Beckman barycenter is obtained from input marginals via  a non-linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The input marginals are linearly  transformed versions of the input and thus interfere with the  adversarial noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Using these marginals the barycenter gen-  erates a sample which is similar in appearance to the input  and is closer in terms of class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Thus, the class label  is preserved when the input is a clean sample, whereas, the  adversarial noise gets suppressed when the input is an ad-  versarial sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Further, the network needs to be trained  with barycenter of clean samples so as to correctly classify  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' However, this training is cheap as a single epoch is  sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We prove our hypothesis using extensive qualita-  tive and quantitative experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  2  Related Works  Adversarial Attacks Given an adversarial sample x with la-  bel y, a target network f parameterized by θ, the adversary  tries to find xadv by adding an adversarial noise such that  the prediction fθ(xadv) ̸= fθ(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Some of the robust at-  tacks are iterative FGSM [Kurakin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2016], PGD [Madry  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2017], Carlini and Wagner attacks [Carlini and Wagner,  2017], Jacobian based attack [Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2016], physical  attack Athalye [Athalye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2018], and Autoattack [Croce  and Hein, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' These attacks are primarily focused in lp  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Adversarial Defense In response to adversarial attacks, sev-  eral defenses been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' One of the best defense ap-  proach is adversarial training [Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2013], [Good-  fellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2014], [Moosavi-Dezfooli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Madry  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' [Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2017] formally studied adversarial train-  ing and proposed that such training allows network to de-  fend well against first order adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Adversarial logit pair-  ing uses a pair of logits from clean and adversarial examples  to defend against adversarial samples [Kannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  TRADES [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019] prove the bounds based on  regularization term which minimizes the difference in pre-  diction between clean and adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In [Wong et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020], authors proposed to effectively combine FGSM  and random initialization to demonstrate better adversarial  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' RST [Carmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019] propose a self-training  technique using unlabelled samples to improve the robust-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Observing the correlation between flatness of weight  loss landscape and adversarial robustness, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' proposed  adversarial weight perturbation (AWP) to regularize the flat-  ness of weight loss [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' On similar lines, [Yu et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022] propose a criterion called Loss Stationary Condi-  tion (LSC) for constrained perturbation, which regulates the  weight perturbation to prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' LBGAT [Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=',  2021] constrains the logits of a robust model, trained with ad-  versarial examples, to be similar to the logits of a clean model  trained on natural data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  While adversarial training uses all the samples, many tech-  niques propose that naively using adversarial samples in ad-  versarial training is not efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  This primarily involves  training the model with a weak attack first, and then grad-  ually increasing the strength of the adversary - CAT [Cai et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2018], DART [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019a], MART [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=',  2019b], FAT [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Aforementioned methods  rely on pre-determined attack parameters for adversarial sam-  ple generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' However, this restricts the model’s robust-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' To address this issue, LAS-AT [Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022] propose  a framework for adversarial training that introduces the no-  tion of learnable attack strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' It is composed of two com-  ponents: a target network that uses adversarial examples for  training to improve robustness, and a strategy network that  produces attack strategies to control adversarial sample gen-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In similar spirit, A2 [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022] and [Cheng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022] have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' A classical review of  defense methods can be obtained in [Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In a parallel line of defense works, input purification has  also been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' At the test time, these techniques try to  remove the adversarial noise [Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021], TRADESSSL  [Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021], HedgeRST [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Score based  generative models such as [Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021] and [Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=',  2022] have also been used to purify the images before sending  them for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Our work is inspired from two different theories, namely,  OT barycenters and distributional robust optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We  discuss these theories in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Wasserstein Barycenter In the following we discuss Wasser-  stein distance and barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Given probability distributions,  µ1, µ2 ∈ Ω, the Wasserstein distance is defined as,  W(µ1, µ2) = inf  Ω×Ω c(x, y)π(x, y)dxdy,  (1)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  �  Ω  π(x, y)dx = µ1(x),  �  Ω  π(x, y)dy = µ2(y),  where the cost matrix c(x, y) = ∥x − y∥1 and π denotes the  transport plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This is also known as Earth Mover’ Distance  (EMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This form is also used to compute barycenter [Cuturi  and Peyr´e, 2016] wherein the summation of Wasserstein dis-  tance between the barycenter and each input marginal is con-  sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' However, barycenters are costly to compute and the  best known complexity scales exponentially with the number  of marginals [Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  EMD can also be represented as dual of the dual of Eq 1  in variational form popularly introduced by Beckman [Beck-  mann, 1952], [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2018], [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020],  W(µ1, µ2) = inf  M  �  Ω  ∥M∥  (2)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' div(M) + µ1 − µ2 = 0  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='n = 0 ∀x ∈ ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' n is normal to ∂Ω  Under appropriate discretisation, M = (Mx, My), M ∈  Rn×2 is flux vector satisfying zero flux boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  µ1, µ2 ∈ Rn, and,  div(M) = (Mx[i, j]−Mx[i−1, j])+(My[i, j]−My[i, j−1])  and the zero-flux boundary conditions mean that Mx[i, j] =  My[i, j] = 0 outside the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Eq 2 is favorable com-  pared to Eq 1 as it reduces the complexity from O(n2) to  O(n) [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Motivated by the recent developments  of OT barycenters, we make use of Eq 2 to propose Beckman  barycenter as they can be efficiently solved using well known  techniques like [Goldstein and Osher, 2009], [Chambolle and  Pock, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  DRO One of the influential works in DRO was proposed by  Scarf [Scarf, 1957].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Following this work, significant research  has been done in this field [Ben-Tal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2009], [Duchi et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021], [Staib and Jegelka, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' DRO aims to address  the problem of uncertainty or shift in the data distribution that  can arise due to measurement errors and admits a solution  for the worst case scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Let L(θ, x) be the loss function  where θ are network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Then, DRO solves for,  inf  θ sup  Q∈Q  EQL(θ, x)  (3)  Here, Q is the distribution against which DRO minimizes  the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' For instance, Q can be considered as a distribu-  tion set which contains perturbations of input samples x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Here we note that adversarial training can be considered to  be a specific instance of DRO wherein the distribution Q is  drawn from adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In our case, we consider the  barycenters as the samples drawn from the distribution Q and  thus provide robustness against perturbed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='1  Proposed Algorithm  In this work, we propose a novel Beckman Barycenter for-  mulation and derive the barycenter analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We use the  barycenter to demonstrate that it can be applied for adversar-  ial defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We first obtain the barycenter using the marginals  from the given input image and then train the network using  barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  While OT barycenters are a good choice for the distri-  bution Q in Eq 3, computing OT barycenter suffers from  high complexity and exponentially increases with the number  marginals [Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' To counter this high complexity  challenge, we first discuss an analogous barycenter problem  by building upon the formulation given in Eq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  inf  M1,M2  r1,r2,µ  ∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1  (4)  +∥r2∥1) + β∥µ∥1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' div(M1) + µ1 − µ = r1  div(M2) + µ2 − µ = r2  where, r1, r2, µ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Our formulation is loosely inspired  from the Beckman OT formulation that are given in [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=',  2018], [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' There are notable changes in Eq 4  from Eq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' First we solve for Beckman barycenter µ in ad-  dition to other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Similar to Wasserstein barycenter  which acts as a representative of marginals using Wasserstein  metric, the Beckman barycenter µ minimizes the flux with  respect to input marginals µ1 and µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In our experiments,  these marginals are obtained by rotating the input image with  ±4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Second, the variables r1 and r2 allow the mass to be  created or destroyed [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020] and the regularization  over r1, r2 and µ ensure that these variable do not take ar-  bitrarily large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Third, Eq 4 can be easily converted to  Lagrange formulation and solved in linear time using primal-  dual method of Chambolle and Pock [Chambolle and Pock,  2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In order to make the objective strongly convex, we first  apply proximal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The l2 regularizer makes the ob-  jective strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Using the proximal operator,  inf  M1,M2,r1  r2,µ′  1,µ′  2,µ  ∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1 (5)  +∥r2∥1) + 1  2ρ(∥µ′  1 − µ1∥2 + ∥µ′  2 − µ2∥2) + β∥µ∥1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' div(M1) + µ′  1 − µ = r1  div(M2) + µ′  2 − µ = r2  The Lagrangian of Eq 5 is given as,  inf  M1,M2,r1  r2,µ′  1,µ′  2,µ  ∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1 (6)  +∥r2∥1) + 1  2ρ(∥µ′  1 − µ1∥2 + ∥µ′  2 − µ2∥2) + β∥µ∥1  +  �  i  ⟨λi, div(Mi) + µ′  i − µ − ri⟩  Eq 6 can be solved using first-order primal dual method of  Chambolle and Pock [Chambolle and Pock, 2011]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Mt+1  i  ← arg min  Mi  ∥Mi∥2,1 + ⟨λi, div(Mi) + µ′  i−  µ − ri⟩ + 1  2τ1  ∥Mi − Mt  i∥2  ∀i = {1, 2}  µ′  i  t+1 ← arg min  µ′  i  1  2τ1  (∥µ′  i − µi∥2) + ⟨λi, µ′  i⟩  + 1  2τ1  ∥µ′  i − µ′  i  t∥2  rt+1  i  ← arg min  ri  α∥ri∥1 + ⟨λt  i, ri⟩ + 1  2τ1  ∥ri − rt  i∥2  µt+1 ← arg min  µ  ∥µ∥1 + ⟨λt  i, µ⟩ + 1  2τ1  ∥µ − µt∥2  λt+1  i  ← arg max  λi  ⟨λi, κt+1⟩ − 1  2τ2  ∥λi − λt  i∥2,  1We use similar notations to that of [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2018], [Chambolle  and Pock, 2011] for consistency and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  where, κt+1 = 2(div(Mi)t+1+µ′  i  t+1−rt+1  i  )−(div(Mi)t+  µ′  i  t − rt  i)  We now discuss the solution of each individual optimiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Solving for Mi: The rows mij of Mi can be expressed  and solved using l21 norm shrinkage operator,  mt+1  ij  ← shrinkl2  τ1(mt  ij − τ1div∗(λt  i)j)  (7)  Here,  div∗  denotes the adjoint of div operator,  and  shrinkl2  τ1η = max(∥η∥2 − τ1, 0) ⊙  η  (∥η∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' “⊙” denotes the  Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Solving for µ′  i:  µ′  i  t+1 ← max{0,  ρτ1  1 + ρτ1  µ′  i +  1  1 + ρτ1  (µ′  i  t − τ1λt  i)}, (8)  Solving for ri: We use an l1 shrinkage operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  rt+1  i  ← shrinkl1  ατ1(rt  i + τ1λt  i)  (9)  Here, shrinkl1  ατ1(η) = sign(η) ⊙ max(∥η∥ − ατ1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Solving for barycenter µ:  µt+1 ← shrinkl1  βτ1(µt + τ1(λt  1 + λt  2))  (10)  Solving for λ:  λt+1  i  ← λt  i + τ2κt+1  (11)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='2  Toy example  We demonstrate the barycenter computation using a Gaussian  image in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The barycenter of clean samples, sample  with random noise and adversarial sample are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' As we  see, for the clean case the barycenter is very similar to that of  the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In the second column where random noise  is added, the barycenter reduces the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Similar effect  is also seen for the case where adversarial noise is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  This indicates that non-linear interpolation of Barycenter sup-  presses the adversarial noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  (a) Clean  (b) Random  (c) Adversarial  Figure 2: Top: Clean image, noisy image, adversarial image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Bot-  tom: Barycenter of clean image, noisy image, adversarial image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='3  Training  Let a model be given by fθ, the barycenter of clean samples  be denoted by x and its labels as y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We then optimize the  following loss  arg min  θ  1  n  n  �  i=1  LCE(fθ(xi), yi)  where LCE is the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We would like to em-  phasize that we do not perform adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Instead  we use an adversarially pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Thus, fθ is an ad-  versarial robust model and our training further enhances the  robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We also note that this optimization falls under  DRO as the samples used are barycenters which belong to the  distribution Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='4  Theoretical analysis  We first present a convergence analysis of Eq 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Let τ1τ2(λmax(∇2) + 3)  <  1, where  λmax(∇2) denotes the largest eigenvalue of discrete Lapla-  cian operator ∇2 = DD⊤, where D is the matrix repre-  senting div operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Then, the iterations Mt  i, µ′  i  t  i, µt, rt  i, λt  converge to the saddle point solution of the Lagrangian  M∗  i , µ∗  i , µ∗, r∗  i , λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Proof: Let u = {M1, M2, µ2, µ2, µ, r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Then, we write  Eq 6 as  L(u, λ) = G(u) + ⟨λ, ˜Kb⟩  where λ  =  [λ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' λ2],  K  =  [D, I, −I, −I],  ˜K  =  [K, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' 0, K], b = [b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' b2], b1 = [vec(M1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' µ′  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' r1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' b2 =  [vec(M2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' µ′  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' r2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  The function G  =  ∥M1∥2,1 +  ∥M2∥2,1 + α(∥r1∥1 + ∥r2∥1) +  1  2ρ(∥µ′  1 − µ1∥2 + ∥µ′  2 −  µ2∥2) + β∥µ∥1 is convex and ˜K is a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' These  conditions satisfy Theorem 1 of [Chambolle and Pock, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  If λmax(∇2) is the max eigenvalue of DD⊤, then the max  eigenvalue of [D, ±I][D, ±I]⊤ is λmax(∇2) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Similarly,  for KK⊤, it is λmax(∇2) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Since ˜K is obtained from  K by padding zeros only, ˜K has the same max eigenvalue  as that of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Further, since ∥ ˜K ˜K⊤∥2  2 ≥ λmax( ˜K ˜K⊤) =  λmax(∇2) + 3, we can also write the convergence criteria as  τ1τ2∥ ˜K ˜K⊤∥2  2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Since we solve for the Lagrangian dual function, we anal-  yse the primal dual gap which is given as [Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019]  G(u, λ) =  sup  ∥λ′−λ0∥≤R1  L(u, λ′) −  inf  ∥u′−u0∥≤R2L(u′, λ)  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Suppose the step sizes τ1 and τ2 satisfy  τ1τ2∥ ˜K ˜K⊤∥2  2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Let uN =  1  N  �N  n=1 un and λN =  1  N  �N  n=1 λn, where un and λn are sequences generated from  Eqns 7 - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Then after N iterations, we have,  G(u, λ) ≤ sup  u,λ  1  2N  �  ∥u − u0∥2  τ1  + ∥λ − λ0∥2  τ2  �  This rate is similar to convergence rates in gradient descent  and shows that the gap converges with rate O(1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' For  brevity, we omit the proof and it can be derived as an exten-  sion of Theorem 1 [Chambolle and Pock, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='5  Mutual Information  In order to understand the underlying reason behind the per-  formance of our method, we provide more insights using  mutual information (MI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We first note that the MI between  two random variables is given by I(X, y) = H(P(y)) −  E  P (x)[H(P(y|X))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In our case, we take the random variables  as model parameters θ and softmax output y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Then, given a  sample x and dataset D,  I(θ, y|D, x) = H(p(y|x, D)) −  E  p(θ|D)  H(p(y|x, θ)) (12)  Eq 12 measures the information shared between θ and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  A tractable way of computing I(θ, y|D, x) is given in [Smith  and Gal, 2018], [Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  I(θ, y|D, x) = 1  C  C  �  j=1  1  n  n  �  i=1  (pij − ˆp)2  (13)  where, ˆp ∈ [0, 1]C is computed as the mean of all softmax  probabilities, C is the number of classes, pi ∈ [0, 1]C, pij ∈  [0, 1] denotes the softmax probability for a particular class j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  A higher I indicates that knowing θ (or y) gives a higher  information about y (or θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In other words, the model will  perform better if the mutual information is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In addition, we also compute MI between the predictions  for the following two cases - (i) clean test set and adversarial  test set, and (ii) barycenter of clean test set and barycenter of  adversarial test set using [Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Given a model f  paramterised by θ, clean sample xi and its adversarial coun-  terpart x′  i, the joint probability distribution between natural  and adversarial samples is given by the following C × C ma-  trix,  I(f(xi, θ), f(x′, θ)) =  C  �  y=1  C  �  y′=1  Pyy′ ln Pyy′  PyPy′  (14)  where, Pyy′ is given as,  Pyy′ = 1  n  n  �  i=1  f(xi, θ)f(x′  i, θ)⊤  (15)  and the marginals Py, Py′ are obtained by row and column  sum of Pyy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' A higher value of I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=') indicates that know-  ing about clean samples gives a higher amount of information  about the adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3  Experiments  We present elaborate experimental results on CIFAR-10,  CIFAR-100 and Tiny ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We use strong baseline of  LAS [Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We also show improvements over  other baselines - LBGAT [Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021], PGD-AT [Madry  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2017], TRADES [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019], RST [Carmon  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We compare against several popular adversarial  training models, MART [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019b], AWP-A2 [Xu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022], RST-RWT [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022], TRADESAWP [Wu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020], AWP [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020], LASAT, LASTRADES,  LASAWP [Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We also compare with adaptive test  time defenses HedgeRST [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021] and TRADESSSL  [Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In the Tables, we use “+B” to indicate the  results obtained using our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='1  Implementation details  In case of CIFAR10 and CIFAR100, WideResNet34-10 is  used and for Tiny ImageNet PreActResnet18 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Ad-  ditionally, we evaluate on CIFAR-10 with WideResNet28-  10, WideResNet32-10, WideResNet70-16 and on CIFAR-  100 with WideResNet34-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We use these models for a fair  comparison with existing works as these models are widely  used for adversarial defense evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We evaluate against  different attacks namely FGSM, PGD-10, PGD-20, CW, and  AA using l∞ attack with ϵ = 8/255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Our evaluation proto-  cols are similar to the protocols given in [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019],  [Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We would like to emphasize that we use the  checkpoints from the baseline models and perform a single  epoch training using clean barycenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Upon increasing the  number of epochs, the clean accuracy improves, however, the  adversarial accuracy becomes comparable to that of baseline  and further increasing epochs leads to subsequent drop in ac-  curacy against adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We use SGD optimizer  with a learning rate of 1e-4, momentum = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='9 without any  weight decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In order to compute the barycenter, we set ρ = 5e−1, τ1 =  1e − 1, τ2 = α = β = 1 and iterations is set to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' While  one can also attack the barycenter, we give experiments for  the case where the clean image is attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This is because  the barycenter itself lies at an ϵ which is greater than attacker’  budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Thus attacking barycenter has little incentive as in  that case the attacked image will lie at an ϵ outside the given  ϵ = 8/255 for the l∞ attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='2  Comparison on CIFAR-10  In Table 1, we observe that clean performance is better  for the models trained with barycenters - TRADESAWP+B,  LBGAT+B, LASTRADES+B and RST+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Amongst WRN-  28-10 models, RST has the best clean performance and our  method enhances it by 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In PGD-10, there is a rise of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='57%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In case of AA, there is a boost of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  In case of WRN-34-10, LASAT+B shows a huge boost  of 10% under AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Further, LASAWP+B shows the best per-  formance under PGD-10, PGD-20 and CW attack amongst  WRN-34-10 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Under AA it shows an improvement of  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Comparison with Adversarial Purification models Our  RST+B model outperforms Hedge∗  RST under all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Against AA, our approach gives 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='1% higher accuracy  compared to Hedge∗  RST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  We also see that compared to  TRADESSSL, TRADESAWP has a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='3  Comparison on CIFAR-100  In Table 2, we observe that our method gives a significant  boost under all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In case of strong baseline LASAWP,  our method increases the performance by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='85% under clean  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' For PGD-20, there is a rise of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='91%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In case of  CW, there is an increase of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='35%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In other models such as  LBGAT, we see a rise of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='4% in clean accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='4  Comparison with Curriculum based AT  In Table 3, we compare against curriculum based AT meth-  ods like CAT [Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2018], FAT [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2020] and  Table 1: CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  ∗ indicates that the model uses WRN-28-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Bold font is used to indicate the best performance amongst WRN-  34-10 and Red color font is used to indicate the best performance  amongst WRN-28-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Method  Clean  PGD10  PGD20  CW  AA  Adversarial Training  PGD-AT  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='17  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='07  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='08  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='91  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='69  TRADES  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='72  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='75  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='10  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='87  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='40  MART  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='17  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='98  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='56  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='58  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='10  AWP-A2  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='54 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='50  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='42  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='86  RST-RWT∗  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='87 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='11  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='03  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='36  Adversarial Purification  TRADESSSL  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='12 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='67  Hedge∗  RST  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='64 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='89  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='10  Adversarial and Barycentric Training  TRADESAWP  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='36  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content='07  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='17  +B  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content='32  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='85  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='32  LBGAT  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='22  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='25  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='60  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='29  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='23  +B  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='38  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='28  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='43  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='61  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='22  LASAT  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='23  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='11  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='41  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='54  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='58  +B  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='21  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='08  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='64  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='09  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='59  LASTRADES  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='24  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='66  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='07  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='45  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='15  +B  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='15  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='32  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='03  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='75  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='43  LASAWP  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='74  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='09  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='16  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='22  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='52  +B  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='45  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='66  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='16  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='81  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='23  RST∗  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='69  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='48  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='51  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='06  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='71  +B∗  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='68  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='12  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='38  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='08  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='20  DART [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Under FGSM, PGD-20 and CW,  our model shows a huge improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In case of clean sam-  ples, we see that the accuracy of FAT+B compared to FAT is  less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This may be due to the fact that FAT employs curricu-  lum learning in the training whereas our method does not use  curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='5  Comparison on Tiny ImageNet  We present the results in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In comparison to baselines,  our method shows significant improvement in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' For  LASAWP, our method improves the performance under clean  samples by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In case of PGD-50, our method shows  a rise of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='28%, and in case of CW attack, our method al-  most doubles the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Under AA, LASTRADES observes  a maximum performance rise by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='51%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='6  Analysis using Deeper and Wider Models  We use WRN-70-16 and WRN-34-20 to analyse the effect  when the models get deeper and wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In particular, for clean  samples, we can observe that the deeper and wider models  give a better boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In CIFAR-10, WRN-70-16 gives 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='87%  for clean samples which is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='21% better than LASAWP model’  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='66%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In contrast, for WRN-34-10, our method gives accu-  racy similar to that of LASAWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In CIFAR-100, our method  boosts the performance by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='34% under AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In other cases  also we see that the barycenters improve the performance by  a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Table 2: CIFAR-100 WRN-34-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Method  Clean  PGD-10  PGD-20  CW  PGD-AT  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='89  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='19  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='69  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='10  TRADES  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='61  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='20  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='66  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='05  TRADESAWP  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='17  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='81  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='6  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='07  +B  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='67  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='34  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='15  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='92  LBGAT  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='64  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='13  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='53  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='65  +B  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='04  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='29  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='01  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='92  LASAT  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='8  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='27  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='83  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='12  +B  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='45  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='60  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='17  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='60  LASTRADES  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='62  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='82  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='51  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='51  +B  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='58  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='22  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='96  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='99  LASAWP  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='89  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='11  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='36  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='92  +B  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='50  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='55  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='27  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='27  Table 3: CIFAR-10 WRN-32-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Method  Clean  FGSM  PGD-20  CW  CAT  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='43  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='17  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='06  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='48  DART  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='03  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='53  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='70  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='27  FAT  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='34  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='52  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='13  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='82  +B  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='59  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='98  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='02  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='36  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='7  TSNE  In Figure 3, we show the tsne plots for MNIST testset with  classes 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Here, we use a weak MNIST model which  has only two dimensions before the classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We  deliberately chose a weak model so that we can easily show  the effect in low dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Though higher dmensions could  be taken, the effect cannot be easily seen due to a highly non-  linear transformation from high to low dimension of tsne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We  can see that the two clusters yellow and purple are well sepa-  rated for clean and barycenters of clean images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In case of ad-  versarial samples, the points overlap on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' However,  when we take barycenter of adversarial samples, we again  see that the clusters are well separated, similar to the case of  clean images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Thus, it is evident that the barycenter nullifies  the effect of adversarial noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  (a) Clean  (b) Barycenter  (c) Attacked  (d) Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='+Bary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Figure 3: Left to right: Plot of 2D features of Clean image, Barycen-  ter of clean image, Attacked image, Barycenter of adversarial image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  MNIST model obtains 51% accuracy and has only 2D feature vector  before the classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='8  Mutual Information  In Table 6, we present the study of mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We  use LASAT and LASTRADES on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The MI is com-  puted using Eq 12 and Eq 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Here we note that the MI for  LASAT+B is more for training set compared to that of LASAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Table 4: Tiny ImageNet PreActResNet18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Method  Clean  PGD-50  CW  AA  LASAT  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='86  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='16  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='54  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='74  +B  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='12  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='54  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='14  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='78  LASTRADES  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='38  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='36  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='50  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='08  +B  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='07  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='25  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='13  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='59  LASAWP  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='26  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='42  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='88  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='42  +B  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='91  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='70  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='93  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='00  Table 5: CIFAR-10 (C-10) WRN-70-16 and CIFAR-100 (C-100)  WRN-34-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Dataset  Method  Clean  FGSM  CW  AA  C-10  LASAWP  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='66  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='25  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='44  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='61  +B  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='87  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='04  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='40  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='54  C-100  LBGAT  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='55  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='16  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='72  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='92  +B  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='86  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='92  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='19  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='26  This indicates that the information available about the labels  given the model parameters is high and in turn gives a better  clean accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In case of adversarial samples too, we see that  the MI is higher for our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This indicates that the model  has better prediction for these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Further, the measure  for test set is smaller compared to training set which is ex-  pected as the model carries more information about train set  compared to test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Table 6: CIFAR-10 WRN-34-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Method  Train  Test  FGSM  CW  LASAT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='019  +B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='034  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='022  LASTRADES  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='048  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='033  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='032  +B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='037  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='036  In Table 7, we present the results obtained using Eq 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We  observe that for the model trained with barycenter, the MI is  higher between the barycenter of clean and adversarial sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Thus, the model does better on barycenter of adversarial  samples compared to baseline LASAT and TRADESAWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This  is consistent across FGSM, PGD-10 and CW attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='9  Sensitivity to Barycenter Parameters  In Figure 4, we demonstrate the sensitivity to different pa-  rameters involved in the computation of barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In the top  row, we fix the number of iterations to 200 and τ1 = 1e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Here we observe that increasing τ2 makes the barycenter  brighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In the second row, increasing τ1 makes the barycen-  ter darker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Decreasing iterations has a similar effect in the last  row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We see that unless there is a change of order of magni-  tude, the appearance does not substantially change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Thus, our  proposed Beckman barycenter is robust with respect to the  parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  4  Conclusion  In this work we introduce Beckman barycenter analogous to  Wasserstein barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We use the Beckman OT formula-  Table 7: Mutial Information for CIFAR-10 WRN-34-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Method  FGSM  PGD-10  CW  LASAT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='218  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='198  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='203  +B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='275  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='241  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='264  LASTRADES  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='576  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='554  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='563  +B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='629  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='572  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='618  Figure 4: Top row: Blue boundary represent the given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Barycenter for iterations = 200, τ1 = 1e − 1, τ2 = 1, 1e-1, 1e-2,  1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Second: barycenter for iterations = 200, τ1=1, 1e-1, 1e-2, 1e-  3, τ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Third: iterations = 200, 100, 50, 10, τ1=1e-1, τ2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' The  red boundary indicates the images obtained from default settings of  the paramater which are used for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  tion and analytically solve for the barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Defining the  baycenter using Beckman OT also has the advantage that the  computational tools to obtain barycenter are well known and  efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' This overcomes the complexity in solving Wasser-  stein barycenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Further, we show that barycenter can be  used for enhancing the performance of adversarially trained  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Our training is very efficient as we only need a sin-  gle epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We theoretically show that our barycenters can  help in defending against attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' We perform rigorous qual-  itative and quantitaive analysis to show the effectivenes of  barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' Experimental analysis on CIFAR-10, CIFAR-100  and Tiny ImageNet demonstrates state-of-art results against  wide variety of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content=' Barycenters in the wasserstein space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content=' Adversarial weight perturbation helps robust gen-  eralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' NeurIPS, 33:2958–2969, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content='  Attacking adversarial attacks as a defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  arXiv  preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content=' Im-  proving transferability of adversarial examples with input  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In CVPR, pages 2730–2739, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content='  arXiv preprint  arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='03543, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  [Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content=' Adversarial purification with score-based gen-  erative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In ICML, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content='  Ro-  bust weight perturbation for adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In IJCAI,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=', 2019] Hongyang Zhang, Yaodong Yu, Jiantao  Jiao, Eric Xing, Laurent El Ghaoui, and Michael Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  Theoretically principled trade-off between robustness and  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In ICML, pages 7472–7482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+page_content=' Attacks which do not kill training make ad-  versarial learning stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content=' In ICML, pages 11278–11287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
+page_content='  PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAzT4oBgHgl3EQfiP0X/content/2301.01495v1.pdf'}
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+arXiv:2301.00791v1  [math.NT]  2 Jan 2023
+CONJECTURES FOR DISTRIBUTIONS OF CLASS GROUPS OF
+EXTENSIONS OF NUMBER FIELDS CONTAINING ROOTS OF UNITY
+WILL SAWIN AND MELANIE MATCHETT WOOD
+Abstract. Cohen, Lenstra, and Martinet have given conjectures for the distribution of
+class groups of extensions of number fields, but Achter and Malle have given theoretical
+and numerical evidence that these conjectures are wrong regarding the Sylow p-subgroups
+of the class group when the base number field contains pth roots of unity. We give complete
+conjectures of the distribution of Sylow p-subgroups of class groups of extensions of a number
+field when p does not divide the degree of the Galois closure of the extension.
+These
+conjectures are based on q → ∞ theorems on these distributions in the function field analog
+and use recent work of the authors on explicitly giving a distribution of modules from its
+moments. Our conjecture matches many, but not all, of the previous conjectures that were
+made in special cases taking into account roots of unity.
+1. Introduction
+In 1984, Cohen and Lenstra [CL84] gave conjectures for the distribution of the odd parts
+of class groups of imaginary and real quadratic fields, as well as for any finite abelian group
+A, the prime-to-|A| part of class groups of totally real A-fields. Cohen and Martinet [CM90]
+generalized these conjectures to the situation of an arbitrary number field K0 as a base
+field, and arbitrary group Γ, giving conjectures for distributions of the “good part” of class
+groups of Γ-extensions of a fixed K0 with any fixed behavior at the infinite places of K0. In
+particular, the “good part” includes the product of the Sylow p-subgroups of the class group
+for p ∤ |Γ|. However, these conjectures appear to be wrong at primes dividing the number of
+roots of unity in the base field, as shown by Achter [Ach06] and Malle [Mal08].
+In this paper, we give complete conjectures for the distribution of Sylow p-subgroups of
+class groups of Γ-extensions (for p ∤ |Γ|) of any number field K0 that contains the pth (and
+possibly further) roots of unity.
+Conjecture 1.1. Let Γ be a finite group and p a prime p ∤ |Γ|. Let S = Zp[Γ]/(�
+γ∈Γ γ). Let
+K0 be a number field with u infinite places containing the prth roots of unity but not the pr+1th
+roots of unity, for some r ≥ 1. Let E = E(Γ, K0) be the set of isomorphism classes of Galois
+Γ-extensions K/K0 along with an isomorphism Gal(K/K0) ≃ Γ. Let ClK|K0 := ClK/ClK0.
+As K varies over E, the distribution of S-modules ClK|K0[p∞] is the one with the average
+number of surjective morphisms from ClK|K0[p∞] to V being
+|(∧2
+ZpV )Γ[pr]|
+|V |u
+for any finite S-module V . Theorem 8.3 shows there is a unique distribution on S-modules
+with these moments and gives an explicit formula for it.
+Conjecture 1.1 is motivated by Theorem 3.1, which is based on work of Liu, Zureick-
+Brown, and the second author [LWZ19] and gives the moments of these distributions in a
+1
+
+q → ∞ limit in the function field case. The explicit formulas obtained in Theorem 8.3 are
+based on recent work of the current authors [SW22b] that allows one to explicitly describe
+a distribution of modules (or more general objects) given its moments (see Theorem 5.2).
+The formulas for the conjectural distributions, like all previous such formulas, are given in
+terms of q-series. Some of the formulas are quite involved, and it would be interesting if
+they could be further simplified with ideas from the study of such q-series. We give now, as
+a special case, the formulas for just the distribution on p-torsion that gives the moments of
+Conjecture 1.1.
+Theorem 1.2. Let Γ be a finite group and p a prime such that p ∤ |Γ|. Let V1, . . . , Vc be
+the non-trivial irreducible representations of Γ over Fp. Let κi = EndΓ(Vi) and qi = |κi| and
+dimκi Vi = ni. Let ǫi = −1 if (∧2
+κiVi)Γ ̸= 0, let ǫi = 1 if (∧2
+κiVi)Γ = 0 but (Vi ⊗κi Vi)Γ ̸= 0, and
+let ǫi = 0 if (Vi ⊗κi Vi)Γ = 0. Let u be a positive integer. Let RΓ be the set of isomorphism
+classes of finite dimensional representations V of Γ over Fp with V Γ = 0 (with a trivial
+topology and σ-algebra).
+Then there is a unique measure ν on RΓ such that
+�
+X∈RΓ
+| Sur(X, V )|dν =
+|(∧2
+FpV )Γ|
+|V |u
+for every V ∈ RΓ. For all non-negative integers f1, . . . , fc, we have, for V = �
+i V fi
+i ,
+ν({V }) =
+|(∧2
+FpV )Γ|
+|V |u| Aut(V )|
+�
+Vi
+self-dual
+�
+ℓ≥0
+(1 + q
+−uni− ǫi+1
+2
+−ℓ
+i
+)−1
+�
+i<j
+Vi, Vj dual
+�∞
+ℓ=1(1 − q−ℓ
+i ) �2uni
+ℓ=1 (1 − q−ℓ
+i )
+�uni−fi+fj
+ℓ=1
+(1 − q−ℓ
+i ) �uni+fi−fj
+ℓ=1
+(1 − q−ℓ
+i )
+.
+if |fi − fj| ≤ uni for all pairs of dual representations Vi, Vj, and ν({V }) = 0 otherwise.
+For V = �
+i V fi
+i , we have |V | = �
+i qfini
+i
+, and | Aut(V )| = �
+i q
+f2
+i
+i
+�fi
+ℓ=1(1 − q−ℓ
+i ) and
+|(∧2
+FpV )Γ| =
+�
+Vi self-dual
+q(fi
+2)+(1−ǫi) fi
+2
+i
+�
+i<j,Vi,Vj dual
+q
+fifj
+i
+.
+In the setting of Conjecture 1.1, Conjecture 1.1 and Theorem 1.2 together imply that
+ClK|K0[p], as a representation of Γ over Fp, is distributed as ν from Theorem 1.2.
+1.1. Previous work. In 2006, Achter [Ach06] found that in a large q limit, class groups
+of quadratic extensions of function fields Fq(t) did not always satisfy the analogues of the
+Cohen-Martinet conjectures for function fields. In 2008, Malle [Mal08] found computational
+evidence that the conjectures of Cohen-Lenstra-Martinet were incorrect for the Sylow p-
+subgroups of class groups of number fields when p divides the order of the roots of unity in
+K0. Garton [Gar15] showed that the discrepancies Achter found in the function field case
+could also be explained by the presence of roots of unity in the base field.
+Since 2008, there have been several papers aimed at correcting these conjectures in the
+presence of roots of unity. In his original paper, Malle [Mal08] gave suggested conjectures
+for K0 = Q and Γ = Z/3Z, Z/5Z, as well as K0 = Q(√−3) and Γ = Z/2Z. Malle gave
+further conjectures in a follow-up paper [Mal10], including conjectures for Γ = Z/2Z with
+the Sylow p-subgroup of the class group for any base field containing pth but not p2th
+2
+
+roots of unity. In the Γ = Z/2Z case, Garton [Gar15] studied the distribution Achter saw
+arising in the function field large q limit in terms of a random matrix model, and gave the
+moments of this distribution.
+He was able to give explicit formulas for the distribution
+for base fields containing p2th but not p3th roots of unity. Adam and Malle [AM15] made
+conjectures in further scenarios, including that for Γ = Z/2Z with any roots of unity in
+the base field the class group distribution should match a limit of certain random matrix
+distributions, but they did not give explicit formulas for this distribution for a general base
+field. Lipnowski, Sawin, and Tsimerman [LST20], for the Γ = Z/2Z and K0 = Fq(t) case,
+considered additional pairing data that arises on class groups in the presence of roots of
+unity, proved a function field result about these enriched class groups and gave conjectural
+explicit formulas, expressed in terms of these pairings, for the case when Γ = Z/2Z and K0
+is an arbitrary number field. Liu [Liu22, Conjecture 1.2] gave a moment conjecture which
+overlaps (and agrees) with Conjecture 1.1 in the case that p does not divide the order of any
+unramified extension of K0 (without deriving formulas for the probability distribution). We
+explain the relation of our conjectures to those in [Mal08, Mal10, AM15, LST20] in Section
+12. In summary, our conjectures agree with prior conjectures in many, but not all, cases
+where they overlap.
+Previous work that has been based primarily on empirical computations has faced the
+challenge that there are a limited number of situations that can be experimentally explored,
+and as we see in this paper there are many parameters in these distributions, creating a chal-
+lenge to generalization from limited data. Further, some of the formulas for the distributions
+are quite involved, which makes it challenging to guess a formula from data. Our results in
+the function field case have the benefit of working for an arbitrary Γ and number of roots
+of unity in the base rational function field, which allows us to see many more cases at once
+from this point of view than can be explored with empirical data. Further, our new methods
+of producing distributions explicitly from their moments produce formulas even when those
+formulas are complicated. Still, we expect empirical data in the number field case to be an
+important next step in further understanding of the situation. In Section 11, we explain how
+all of Malle’s previous empirical data on class group distributions supports Conjecture 1.1,
+including data from a case in which Malle did not make any conjecture. We also discuss
+further cases in which it would be interesting to have empirical data.
+Apart from the details of the formulas for the distribution, there are two new qualitative
+phenomena that we conjecture (and in part prove) about the distribution of class groups of
+number fields, that are not present in the work of Cohen, Lenstra, Martinet, and Malle, but
+can be seen in recent work of Breen, Varma, and Voight [BVVE21] in the context of class
+group statistics. First, there are some finite Zp[Γ] modules H with HΓ = 1 that cannot occur
+as ClK|K0[p∞]. These are ones whose multiplicities of dual pairs of irreducible representations
+are too far apart. This can be seen already in Theorem 1.2. In Section 9, we prove that
+these Zp[Γ] modules H that we conjecture occur with density 0 as ClK|K0[p∞] actually never
+occur, generalizing earlier results of Leopoldt [Leo58] and Gras [Gra98].
+Second, we do
+not predict the distribution of the parts of ClK|K0[p∞] corresponding to different irreducible
+representations of Γ are all independent (as all previous conjectures have). The behavior
+that we conjecture involves explicit dependence between an irreducible representation and
+its dual (but otherwise independence between different representations). In Section 2, we
+3
+
+give more details on the example of K0 = Q and Γ = C7, which is the smallest case where
+this dependence occurs.
+1.2. Further remarks.
+Remark 1.3. Conjecture 1.1 is not precise, in that it does not specify an ordering on E so
+that the distribution of class groups is well-defined. Cohen and Martinet [CM90] order fields
+by Nm Disc K, but this is known to not work in general, even when there are not relevant
+roots of unity in the base field [BL20, p. 929]. One possible ordering, as suggested by Bartel
+and Lenstra in [BL20] and also by Theorem 3.1, is by Nm
+�
+Disc(K/K0), where √ denotes
+the radical. We certainly imagine the conjecture only holding for orderings such that the
+proportion of fields in E containing any fixed field K1 ̸⊂ K0 is 0.
+Remark 1.4. Note that these extensions in E in Conjecture 1.1 are split completely at all
+infinite places of K0, because either pr > 2 and all infinite places of K0 are complex, or
+p = 2, which implies |Γ| is odd.
+Remark 1.5. If we consider modules with the distribution from Conjecture 1.1, and then take
+their pk-torsion, we have a distribution on pk-torsion S-modules that does not depend on r
+as long as r ≥ k. So, for example, if we are only asking about the p-torsion (equivalently
+the groups mod p) then the conjectured distribution does not depend on r for r ≥ 1. This
+phenomenon was noticed empirically by Malle for the 2-ranks of cyclic cubic extensions of
+K0 = Q versus K0 = Q(i) [Mal10, Section 6.4]. We can see this phenomenon explicitly in
+the formulas of Theorem 8.3. Even before working out formulas for the distribution, this
+follows from the fact that the moments indexed by pk-torsion groups do not depend on r for
+r ≥ k, and since these moments determine a unique distribution of pk-torsion modules by
+Theorem 5.2, this distribution on pk-torsion modules will not depend on r ≥ k.
+From Theorem 1.2, one can work out, for a V from the distribution ν, the distribution of
+V as an Fp-vector space. It is simple to write down the classical moments of |V |. For a finite
+representation V of Γ over Fp, we have
+|V |k = | HomFp(V, Fk
+p)| = | HomΓ(V, Hom(Fp[Γ], Fk
+p))| = | HomΓ(V, Fp[Γ]k))|
+(by adjointness and the fact that permutation representations are self-dual). Every homo-
+morphism of Γ-representations V → Fp[Γ]k is a surjection onto exactly one subrepresentation
+of Fp[Γ]k. Thus our conjecture implies that the average of |ClK|K0[p]|k is
+�
+V ⊂Fp[Γ]k
+sub-Γ-rep.
+V Γ=1
+|(∧2V )Γ|
+|V |u
+,
+a value straightforward to write down from the representation theory of Γ over Fp.
+If one wants to make a conjecture on the product of the Sylow p-subgroups of ClK|K0 for
+a finite set of primes p not dividing |Γ|, then Theorem 3.1 suggests the distribution with
+moments
+|(∧2
+ZV )Γ[|µ(K0)|]|
+|V |u
+,
+where µ(K0) is the group of roots of unity of K0. (These moments are multiplicative over
+Sylow p-subgroups.) The same argument as in the proof of Theorem 8.3 shows there is a
+4
+
+unique such measure, and the formulas are as in Theorem 8.3 with an additional product
+over p. For infinitely many primes, we would still conjecture that the class group distribu-
+tions weakly converge to the unique distribution with the moments above (see [SW22b] for
+the relevant topology and the uniqueness theorem), but because of the definition of weak
+convergence, that is equivalent to making the conjecture for every finite set of primes.
+As explained in [WW21], the distributions of class groups of Galois extensions, for p ∤ |Γ|,
+can be used to get distributions of class groups of non-Galois extensions. We explain the
+implication of Conjecture 1.1 for distributions of class groups of non-Galois extensions in
+Section 10.
+There are recent developments [Smi22a, Smi22b] and interesting questions about distri-
+butions of the Sylow p-subgroups of ClK for p | |Γ|, including about the interactions with
+roots of unity in the base field, but we will not address those questions here.
+In a forthcoming paper, the authors will extend our function field results and number field
+conjectures to the non-abelian analog of the class group, the Galois group of the maximal
+unramified extension. One feature of this forthcoming treatment is that we will consider
+additional structure on the class group (and its non-abelian analog), as in [SW22a], such
+that the formulas for the distribution with this additional structure will be nicer in certain
+ways.
+1.3. Outline of the paper. In Section 2, we give the example of Γ = C7, for which we
+give the formulas for the distribution in full and one can see the non-independence of dual
+representations. In Section 3, we prove the function field q → ∞ moments that motivate
+Conjecture 1.1. In Section 4, we explain the heuristic assumptions that lead from our function
+field theorem to Conjecture 1.1. In Section 5, we review the results on the moment problem
+for modules that we will need to apply to find formulas for our conjectured distribution from
+our conjectured moments. In Sections 6 and 7, we make some calculations of distributions
+on vector spaces and modules over DVR’s that will be necessary for our final formulas. In
+Section 8, we obtain formulas for the distribution of Conjecture 1.1. In Section 9, we show
+that the modules that we conjecture appear as class groups with density 0 actually never
+appear as class groups. In Section 10, we show how our conjectures for class group of Galois
+extensions lead to explicit conjectures for the distribution of class groups of non-Galois
+extensions. In Section 11, we discuss the match between Malle’s number field data and our
+conjectures (which is excellent). In Section 12, we compare our conjectures to conjectures
+made by Malle, and Adam and Malle.
+1.4. Acknowledgements. We would like to thank Yuan Liu and John Voight for helpful
+conversations. We would like to thank Jürgen Klüners, Yuan Liu, and Gunter Malle for
+helpful comments on an earlier version of this manuscript. The second author would like
+to thank Ben Breen, Ila Varma, and John Voight for interesting conversations on this ques-
+tion during their joint investigation. Will Sawin was supported by NSF grant DMS-2101491
+while working on this paper. Melanie Matchett Wood was partially supported by a Packard
+Fellowship for Science and Engineering, NSF CAREER grant DMS-1652116, and NSF Wa-
+terman Award DMS-2140043 while working on the paper. She was also a Radcliffe Fellow
+during part of this work, and thanks the Radcliffe Institute for Advanced Study for their
+support.
+5
+
+1.5. Notation. We write N for the non-negative integers.
+We write Fq for the finite field with q elements.
+For a group G with an action of Γ, we write GΓ for the invariants. For a a finite abelian
+group A, we write A[p∞] for the Sylow p-subgroup of A.
+For an extension K/K0, we let ClK|K0 be the quotient of the class group ClK of K by the
+image of ClK0 under inclusion of ideals from K0 to K. (Note for a prime p ∤ [K : K0], we
+have that ClK0[p∞] → ClK[p∞] is a injection, since taking the norm map and dividing by
+[K : K0] provides an inverse.)
+A Γ-extension K/K0 is a Galois field extension K/K0, along with a choice of isomorphism
+ι : Gal(K/K0) ≃ Γ.
+An isomorphism of Γ-extensions (K, ι), (K′, ι′) is given by a field
+isomorphism φ : K → K′, fixing each element of K0 and such that the induced map φ∗ :
+Gal(K/K0) → Gal(K′/K0) satisfies ι = ι′ ◦ φ∗. When K/K0 is a Γ-extension, using ι, there
+is an associated action of Γ on ClK|K0. Further, ClK|K0 is a Z[Γ]/(�
+γ∈Γ γ)-module, since
+(�
+γ∈Γ γ)I = NmK/K0 I.
+When K is a function field, and OK is a maximal order in K, we write Cl(OK) for the
+quotient of the ideal group of OK by the principal ideals of OK.
+Given a commutative ring R and an R-module M, we write ∧2
+RM for the R-module
+quotient of M ⊗R M by elements of the form m ⊗ m. We write ∧2M for ∧2
+ZM
+We write Sur(A, B) to denote the set of surjections from A to B in the appropriate
+category.
+We also use Aut(M) to denote automorphisms in the appropriate category, e.g. of R-
+modules if M is an R-module. We will write AutR(M) if we think there is a possibility of
+confusion.
+For a finite set S, we write |S| for the number of elements of S.
+We write Cn for the cyclic group of order n.
+A product �n−1
+i=n an is always 1 by convention.
+We write ηq(k) := �k
+i=1(1 − q−i) and also write η(k) = ηq(k) when q is clear from context,
+and η(0) = 1.
+2. Example: Γ = C7
+In this section, we consider the example K0 = Q and Γ = C7 and p = 2. We have that
+S = Z2[Γ]/(�
+γ∈Γ γ) = R1 × R2, where each Ri is isomorphic to the ring of integers in the
+degree 3 unramified extension of Q2. There are three irreducible representations of C7 over
+F2 (or Q2), the trivial representation, and V1, V2, two three dimensional dual representations
+(that correspond to R1, R2 respectively).
+For K/Q a C7-extension, ClK[2∞] is an S-module. For an S-module V , we let a1 = a1(V )
+denote the R1/2R1-rank of V/(2, R2)V , i.e.
+the multiplicity of V1 in V/2V .
+We define
+a2 = a2(V ) similarly. Conjecture 1.1 predicts that among such K, the S-module ClK[2∞] is
+distributed according to ν, where
+ν({V }) =
+8a1a2
+|V || Aut(V )|
+∞
+�
+i=1
+(1−8−i)
+a1
+�
+i=1
+(1−8−1−i)
+a2
+�
+i=1
+(1−8−1−i)·
+
+
+
+
+
+1 + 8−1
+if a1 = a2
+1
+if |a1 − a2| = 1
+0
+if |a1 − a2| > 1.
+6
+
+(Corollary 8.4 gives this formula.) For comparison to the moments, note |(∧2
+ZpV )Γ[p]| = 8a1a2.
+One can give an explicit formula for V and | Aut(V )| (see proof of Proposition 7.1 or [Mac15,
+II (1.6)]).
+From the above formulas, one can also work out the distribution of V as a Z2-module. As
+an example, we work out the distribution of V/2V , so the predicted distribution of ClK[2],
+from Theorem 1.2. Since Ri/2Ri ≃ F8, for a finite S-module V , we have that V/2 is a
+vector space over F2 with dimension 3k for some k. For V in the support of ν, we see that
+if k is even, we must have a1 = a2 = k/2, and if k is odd, then either a1 = (k − 1)/2 and
+a2 = (k + 1)/2, or a1 = (k + 1)/2 and a2 = (k − 1)/2. Thus we have
+ν({V | |V/2V | = 8k}) =
+�
+8−k2/4−k(1 − 8−2) �k/2
+i=1(1 − 8−i)−2�∞
+i=2(1 − 8−i)
+if k even
+2 · 8−(k2+3)/4−k�(k+1)/2
+i=1
+(1 − 8−i)−1�(k−1)/2
+i=1
+(1 − 8−i)−1�∞
+i=1(1 − 8−i)
+if k odd.
+(We use the formula AutFq(Fn
+q ) = qn2 �n
+i=1(1 − q−i).)
+3. Function field results
+By analyzing components of Hurwitz spaces, Liu, Zureick-Brown, and the second author
+[LWZ19] have found q → ∞ moments of Galois groups of maximal unramified extensions
+of Γ-extensions of Fq(t). In this paper, we focus on the class group distributions (i.e. the
+abelianizations of those Galois groups), and we give such results in this section.
+For a finite group Γ, let EΓ(n, q) be the set of isomorphism classes of extensions K/Fq(t)
+with a choice of isomorphism Gal(K/Fq(t)) ≃ Γ, that are split completely above ∞ and such
+that the radical of the discriminant ideal Disc(K/Fq(t)) has norm qn. For such a K, let OK
+be the maximal order of K over Fq[t] and let Cl(OK) denote its class group.
+Theorem 3.1. Let Γ be a finite group and H be a finite abelian group with action of Γ such
+that (|H|, |Γ|) = 1 and HΓ = 1. Let h be an integer such that h||H|. Then
+lim
+x→∞
+lim
+q→∞
+(q,|Γ||H|)=1
+(q−1,|H|)=h
+�
+n≤x
+�
+K∈EΓ(n,q) | SurΓ(Cl(OK), H)|
+�
+n≤x |EΓ(n, q)|
+= |(∧2H[h])Γ|
+|H|
+,
+where in the limit q is always a prime power.
+Theorem 3.1 follows from [Liu22, Theorem 1.1], which is a more general and refined
+moment theorem, but we give a short argument here for just the statement of Theorem 3.1.
+See also Corollary 8.5 for the implication of these averages on the distribution of Cl(OK).
+Proof. This is [LWZ19, Theorem 1.4], except that we allow (q − 1, |H|) to vary, and there is
+an additional factor of |(∧2H[n])Γ| on the right-hand side, along with some simplifications
+because H is abelian. A finite abelian group with an action of Γ is admissible in the sense
+of [LWZ19] if and only if it has order prime to |Γ| and no Γ-invariants. Also, via class field
+theory, the group Gal(K#/K) considered in [LWZ19] has abelianization Cl(OK).
+The result [LWZ19, Theorem 1.4] follows in a straightforward way from [LWZ19, Theorem
+10.4], and our theorem here follows in exactly the same way from an analog of [LWZ19,
+7
+
+Theorem 10.4] in which the hypothesis that (q − 1, |H|) = 1 is replaced by (q − 1, |H|) = h
+and the component counting result πG,c(q, n) = πΓ(q, n) + OG(ndΓ(q)−2); is replaced by
+πG,c(q, n) = |H2(H, Z)Γ[h]|πΓ(q, n) + OG(ndΓ(q)−2).
+(Recall when H is abelian, H2(H, Z) ≃ ∧2H.)
+This analog of [LWZ19, Theorem 10.4] follows almost entirely the proof of [LWZ19, The-
+orem 10.4], with just a change at the very end. None of the intermediate results require
+that (q − 1, |H|) = 1. However, at the very end of the proof of [LWZ19, Theorem 10.4], it
+is said that “Since the kernel of the map f [: ker(S
+1 → G1) → ker(S
+2 → G2)] on the right
+has order relatively prime to q − 1, any element of ker(S
+2 → G2) has the same number of
+(q−1 − 1)th roots as any preimage in ker(S
+1 → G1).” (We interpret (q−1 − 1) modulo the
+order of group in which we are taking roots.) In the more general setting when q −1 and |H|
+are not relatively prime, we need to account for the difference in the number of (q−1 − 1)th
+roots.
+For any finite group |H| (not necessarily abelian) with (|H|, |Γ|) = 1, we have an exact
+sequence
+1 → H2(H, Z)Γ → H2(H ⋊ Γ, Z) → H2(Γ, Z) → 1
+(e.g. from the Lyndon-Hochschild-Serre spectral sequence and the fact that (|H|, |Γ|) = 1
+implies invariants and co-invariants are the same). Since H2(H, Z) and H2(Γ, Z) are of rela-
+tively prime order, this exact sequence splits canonically. The group ker(S
+1 → G1) mentioned
+above is a quotient of H2(H ⋊ Γ, Z) by a certain subgroup (referred to in [LWZ19, Section
+12] as Qc1) generated by elements that are in the image of a map from H2((Z/|Γ|Z)2, Z)
+(since each x ∈ c1 has order dividing |Γ|). The group ker(S
+2 → G2) mentioned above is a
+quotient of H2(Γ, Z) by the image of this certain subgroup (Qc2, which is the image of Qc1).
+So the kernel of the map f is H2(H, Z)Γ, and in general any element of ker(S
+2 → G2) has
+|H2(H, Z)Γ[q − 1]| times as many (q−1 − 1)th roots as any preimage in ker(S
+1 → G1). This
+is the only change needed to prove the analog of [LWZ19, Theorem 10.4] described above,
+which can then be used to prove this result exactly as in the proof of [LWZ19, Theorem
+1.4].
+□
+4. Heuristic argument
+In this section, we explain the heuristic argument that leads to Conjecture 1.1.
+The Cohen-Martinet conjectures [CM90] include some predictions about Sylow p-subgroups
+of relative class groups of Γ-extensions when p | Γ, but in this paper we only consider the
+case when p ∤ |Γ|. In this case, Cohen and Martinet make conjectures about the distribution
+of ClK|K0[p∞] for a base field K0 and the family of all Γ-extensions K/K0 with specified be-
+havior at the infinite places of K0. In this paper, we only consider families E of Γ-extensions
+K/K0 in which all infinite places of K0 are required to split completely. (See Remark 1.4.)
+In this case, the only feature of K0 that Cohen and Martinet’s conjectures [CM90] take as
+input is the number u of infinite places of K0 (see, e.g., [WW21, Theorem 4.1]). As Malle’s
+computations [Mal08, Mal10] suggest and Theorem 3.1 shows (in the function field case) one
+must also take into account the maximal r such that K0 contains the prth roots of unity.
+One heuristic assumption we make here are that u and r are the only relevant inputs from
+the base field into the conjectures. (We hope that conceptual evidence for this heuristic will
+8
+
+eventually be found in the function field case. If moments are calculated over a function field
+of arbitrary genus, and shown to depend only on u and r, this heuristic assumption would
+be more justified, albeit still depending on the reliability of analogy between number fields
+and function fields.)
+In the function field case, we take the class group of a ring OK whose fraction field is K,
+but unlike the number field case, there is no canonical choice of such a ring. Instead, we pick
+a set of places of K that we consider “infinite places” for the sake of the analogy, and we let
+OK be the elements of K that have non-negative valuation at all non-infinite places of K.
+Via this analogy, we expect there to be uniform behavior across number fields and function
+fields (taking into account their number of infinite places, and the roots of unity that they
+contain).
+However, in the function field case, we can change our notion of “infinite places” by adding
+an additional place v of K0 to the list of infinite places. The new family E′ of Γ-extensions
+will be those from the original family E that are split completely at v. It is generally expected
+that such a restriction at a non-archimedean place does not change class group statistics (e.g.
+see [BV15, Theorem 1], [BV16, Corollary 4], [Woo18, Conjecture 1.4]). (We hope that future
+results will be proven in the function field case that will more fully justify this expectation.)
+Then, ClO′
+K|O′
+K0 (for the new O′
+K, O′
+K0) after the inclusion of v as an infinite place, is the
+quotient, as a Z[Γ]-module, of the original ClOK|OK0 by any prime of K over v. We assume the
+heuristic that this prime is distributed uniformly in ClOK|OK0. (See [Kla17a, Kla17b, Woo18]
+for some conjectures and results in this direction. We hope that future results will be proven
+in the function field case that will more fully justify this heuristic.)
+If X is a random
+Z[Γ]-module with E(| SurΓ(X, G)|) = MG for all finite Z[Γ]-modules G, and if we let X′ be
+the quotient of X by a uniform random element of X, then E(| SurΓ(X′, G)|) = MG/|G|.
+Theorem 3.1 gives (in a large q limit) moments of Cl(OK) assuming only one infinite place,
+and so by the reasoning above, we expect if we modified so as to define OK using u infinite
+places of Fq(t), then we would replace the |H| in the denominator of Theorem 3.1 with |H|u.
+(Again, we hope to see such a result proven in the future. See also [Liu22, Sections 6.2 and
+6.3] for different theoretical heuristics for the |H|u factor.)
+Heuristically further we assume that the limits in Theorem 3.1 (and the extension sug-
+gested above to u infinite places) can be exchanged, as was proven in some special cases in
+[EVW16]. Since we expect every base field K0 with a fixed number u of infinite places and
+prth but not pr+1th roots of unity to give the same class group distributions, if we consider
+all q such that q − 1 has a fixed valuation at p, then we do not expect the limit in q to
+change the average, and in particular we expect Theorem 3.1 (and the extension suggested
+above to u infinite places) to hold without the limit in q. Additionally, we expect the same
+result for number fields with a given u and r, and this leads precisely to the statement of
+Conjecture 1.1.
+5. Results on the moment problem for R-modules
+In [SW22b], we give a result to construct a distribution from its moments in a rather
+general category. In this section, we review that result focusing on the category of finite
+R-modules for a DVR R with maximal ideal m, and very similar categories.
+Let R be either a DVR with finite residue field, a quotient ring of a DVR, or a finite
+product of such rings. Let ˆR denote the product of the completions of R at its maximal
+9
+
+ideals m1, . . . , mr. Let P be the set of isomorphism classes of finitely generated ˆR-modules.
+For k = (k1, . . . , kr) ∈ Nr, let mk = �
+i mki
+i
+and for X an ˆR-module let X≤k = X/mkX.
+We consider a topology on P generated by {X | X≤k ≃ N} for k ∈ Nr and N a finite
+R/mk-module (and we take the associated Borel σ-algebra on P).
+The isomorphism classes of finite R-modules (equivalently, finite ˆR-modules) are in bijec-
+tion with r-tuples λ = (λ1, . . . , λr) of finite partitions λi = (λi
+1 ≥ λi
+2 ≥ . . . ), where the parts
+of λi are bounded above by the minimal positive integer ai such that mai
+i
+= 0. (Here, λ
+corresponds to the module Nλ := ⊕i,jR/mi
+λi
+j. ) We write λ′ for the conjugate of a partition.
+The operation N �→ N≤k simply truncates the ith associated conjugate partitions after the
+first ki terms. We say a module N associated to λ is semi-simple if (λi)′
+2 = 0 for all i, i.e.
+N ≃ �
+i(R/mi)ei for some positive integers ei.
+Let qi = |R/mi|. For two r-tuples of partitions λ, π and a surjection g : Nπ → Nλ we
+define µ(g) to be 0 if ker g is not semi-simple and �
+i(−1)eiq(ei
+2)
+i
+if ker g ≃ �
+i(R/mi)ei. For
+finite R-modules N, P, we define ˆµ(N, P) = �
+g∈Sur(P,N)/ Aut(N) µ(g).
+For a finite R-module N, the N-moment of a measure ν on P is defined to be
+�
+X∈P
+|SurR(X, N)|dν.
+Given values MN for each finite R-module N, for each k ∈ Nr and finite R/mk-module N,
+we define (what will be the formulas for our measures)
+(5.1)
+vk,N :=
+�
+P finite
+R/mk-module
+ˆµ(N, P)
+| Aut(P)|MP,
+and we say the values MN are well-behaved (for R) if the sum (5.1) is absolutely convergent
+for each k ∈ Nr and finite R/mk-module N.
+Theorem 5.2 ([SW22b, Theorems 1.7, 1.8] ). Let R be a finite product of quotients of DVRs
+and, for each finite R-module N, let MN ∈ R such that the values MN are well-behaved. Then
+we have the following.
+(Existence): There is a measure on P with N-moment MN for each finite R-module N if and only
+if the vk,N defined above are non-negative for each k and N.
+(Uniqueness): When such a measure ν exists, it is unique and is given by the formulas
+ν({X | X≤k ≃ N}) = vk,N.
+(Robustness): If νt are measures on P such that for each finite R-module N,
+lim
+t→∞
+�
+X∈P
+|Sur(X, N)|dνt = MN,
+then a measure ν with moments MN exists and the νt weakly converge to ν.
+Remark 5.3. We give some notes on translation from the more general language of [SW22b].
+In [SW22b, Lemma 6.1], it is explained that the category of R-modules is a “diamond cate-
+gory” and the “levels” of [SW22b] correspond to modules over the finite quotient rings R/mk
+of R. The results [SW22b, Lemma 5.7, Lemma 5.10 (2)] show that P is as described here, and
+[SW22b, Lemmas 2.6, 3.8] show that µ is as defined here. The definition of “well-behaved” in
+10
+
+[SW22b] involves an additional factor of Z(π)3, which by [SW22b, Lemma 3.11] is at most
+8r in our case and thus does not affect absolute convergence.
+The measures in this paper will turn out to be supported on finite R-modules, which
+we will show using the formulas provided by Theorem 5.2. Then we can use the following
+statement, weaker than Theorem 5.2, but simpler to think about because it only involves
+finite R-modules.
+Corollary 5.4. Let R be a finite product of quotients of DVRs and, for each finite R-module
+N, let MN ∈ R such that the values MN are well-behaved and are the moments of a measure
+ν supported on finite R-modules. Then if νt are measures on the set of isomorphism classes
+of finite R-modules such that for each finite R-module N,
+lim
+t→∞
+�
+X∈P
+|Sur(X, N)|dνt = MN,
+then the νt weakly converge to ν (for the discrete topology).
+6. Measures on the set of isomorphism classes of Fq vector spaces
+In this section, we compute some distributions from their moments that will be important
+for our eventual class group conjectures.
+If q is a prime power, then R = Fq is a quotient of a DVR, so we can apply Theorem 5.2.
+The category of finite R-modules is just the category of finite Fq-vector spaces (and ˆR = R).
+Proposition 6.1. Let q be a prime power. Let t > 0. The values MFqρ = q
+ρ2+ρ
+2
+−tρ are well-
+behaved (for R) and are the moments of the measure ν (on isomorphism classes of finite
+Fq-vector spaces) such that
+ν(Fq
+λ) =
+q−(λ
+2)−tλ
+�λ
+i=1(1 − q−i) �
+i≥0(1 + q−i−t)
+=
+MFqλ
+| Aut(Fλq)|
+1
+�
+i≥0(1 + q−i−t)
+for all λ ∈ N.
+Recall η(k) := �k
+i=1(1 − q−i).
+Proof. We apply Theorem 5.2. Let N = Fλ
+p and P = Fλ+e
+q
+. There are q(λ+e)λ homomor-
+phisms P → N, a proportion η(λ + e)/η(e) of these are epimorphisms, and for any of these
+epimorphisms we have µ(F, G) = (−1)eq(e
+2). We have | Aut(Fρ
+q)| = qρ2η(ρ). So,
+ˆµ(Fλ
+q, Fλ+e
+q
+)
+| Aut(Fλ+e
+q
+)| = (−1)eq(e
+2)q(λ+e)λη(λ + e)
+η(e)qλ2η(λ)q(λ+e)2η(λ + e) = (−1)eq− e2
+2 − e
+2 −eλ−λ2
+η(e)η(λ)
+.
+Thus
+v1,F :=
+�
+e≥0
+ˆµ(Fλ
+q, Fλ+e
+q
+)
+| Aut(Fλ+e
+q
+)|q
+(λ+e)2+(λ+e)
+2
+−t(λ+e) = q−(λ
+2)−tλ
+η(λ)
+�
+e≥0
+(−1)e
+η(e) q−te.
+We see that the sum is absolutely convergent since t > 0, and thus the moments are well-
+behaved as claimed. By the q-binomial theorem for negative powers,
+�
+e≥0
+ve
+η(e) =
+�
+i≥0
+(1 − vq−i)−1,
+11
+
+and so
+v1,F = q−(λ
+2)−tλ
+η(λ)
+�
+i≥0
+(1 + q−i−t)−1.
+□
+Now consider R = Fq × Fq, and we write R-modules as pairs of Fq-vector spaces.
+If
+one has moments M(N1,N2) that factor as M(N1,N2) = M′
+N1M′′
+N2, then the sums defining well-
+behavedness and the vC,F all factor into two factors, one corresponding to each factor of
+R, reducing the moment problem for the moments M(N1,N2) to understanding the moment
+problems for M′
+N and M′′
+N. However, not all such moments necessarily factor.
+Proposition 6.2. Let q be a prime power and R = Fq × Fq. Let s1 and s2 be integers with
+s1 + s2 ≥ 0. Let M(Fqρ,Fqπ) = qρπ−s1ρ−s2π for all ρ, π ∈ N. If −s1 ≤ λ − φ ≤ s2, let
+Dq(λ, φ, s1, s2) :=
+q−λ2−φ2+λφ−s1λ−s2φη(∞)η(s1 + s2)
+η(λ)η(φ)η(s2 − λ + φ)η(s1 + λ − φ)
+=
+M(Fqλ,Fqφ)
+| Aut(Fq
+λ, Fq
+φ)|
+η(∞)η(s1 + s2)
+η(s2 − λ + φ)η(s1 + λ − φ)
+and let Dq(λ, φ, s1, s2) = 0 otherwise. Then the values MN are well-behaved for R and are
+the moments of the measure ν (on isomorphism classes of pairs of Fq-vector spaces) such
+that for all non-negative integers λ, φ,
+v((Fq
+λ, Fq
+φ)) = Dq(λ, φ, s1, s2).
+If s1 = s2 = 0, then the distribution is supported on the pairs such that λ = φ, and on
+those reduces to the distribution on Fq-vector spaces whose moments are all 1 (see [SW22b,
+Lemma 6.3], a distribution which, at least in the Fp-case, would usually be called the Cohen-
+Lenstra distribution on Fp vector spaces, since it is the distribution conjectured by Cohen and
+Lenstra [CL84, Sec. 9 (C5)] to be the distribution of p-torsion of class groups of imaginary
+quadratic fields (for odd primes p).
+Proof. With reasoning as in Proposition 6.1, we have
+v1,(Fqλ,Fqφ) :=
+�
+e,f≥0
+(−1)e+fq− e2
+2 − e
+2−eλ−λ2− f2
+2 − f
+2 −fφ−φ2
+η(e)η(f)η(λ)η(φ)
+q(λ+e)(φ+f)−s1(λ+e)−s2(φ+f)
+=q−λ2−φ2+λφ−s1λ−s2φ
+η(λ)η(φ)
+�
+e,f≥0
+(−1)e+fq− e2
+2 − e
+2−λe+φe−s1e− f2
+2 − f
+2 −φf+λf−s2f+ef
+η(e)η(f)
+.
+We can check that this sum converges absolutely as follows. If we fix e, we have an inner
+sum �
+f qh(f)/2, where h is a quadratic polynomial with integer coefficients that takes its
+maximum values (on integers) at f = e−φ+λ+s2, e−φ+λ+s2 −1. Thus the inner sum is
+bounded by a constant times the summand for f = e−φ+λ+s2, which then gives an upper
+bound for the entire (absolute) sum by a geometric series that converges if s1 + s2 + 1 > 0.
+12
+
+Now, we evaluate the sum over f in the equation for v1,(Fqλ,Fqφ) above using the q-binomial
+theorem for negative powers and obtain
+v1,(Fqλ,Fqφ) =q−λ2−φ2+λφ−s1λ−s2φ
+η(λ)η(φ)
+�
+e≥0
+(−1)eq− e2
+2 − e
+2−λe+φe−s1e
+η(e)
+�
+k≥0
+(1 − q−φ+λ+e−1−s2−k).
+This final product is 0 if −φ + λ + e − 1 − s2 ≥ 0, and otherwise is η(∞)/η(s2 − λ + φ − e).
+If λ − φ > s2, then we have v1,(Fqλ,Fqφ) = 0 and otherwise
+v1,(Fqλ,Fqφ) =q−λ2−φ2+λφ−s1λ−s2φη(∞)
+η(λ)η(φ)
+s2−λ+φ
+�
+e=0
+(−1)eq− e2
+2 − e
+2−λe+φe−s1e
+η(e)η(s2 − λ + φ − e) .
+This final sum can be evaluated by the q-binomial theorem (for n ≥ 0)
+n
+�
+k=0
+q−(k
+2)
+η(n)
+η(n − k)η(k)tk =
+n−1
+�
+k=0
+(1 + q−kt),
+and we obtain
+v1,(Fqλ,Fqφ) =q−λ2−φ2+λφ−s1λ−s2φη(∞)
+η(λ)η(φ)η(s2 − λ + φ)
+s2−λ+φ−1
+�
+k=0
+(1 − q−k−1−λ+φ−s1).
+If λ − φ < −s1, then we can check that one of the factors in the product above is zero, and
+otherwise we have
+s2−λ+φ−1
+�
+k=0
+(1 − q−k−1−λ+φ−s1) = η(s1 + s2)/η(s1 + λ − φ),
+which gives the proposition.
+□
+7. Distributions on isomorphism classes of modules over an unramified DVR
+In this section, we compute some more distributions from their moments that will be
+important for our eventual class group conjectures.
+Let q be a power of a prime p, and throughout this section let R be the ring of integers
+in the unramified extension of Qp with residue field Fq. Let m be the maximal ideal of R.
+For a partition λ, let w(λ) = �
+i(i − 1)λi = �
+j
+�λ′
+j
+2
+�
+= | ∧2
+R Nλ|. We write (1e) to denote the
+partition with e 1’s. We define |λ| := �
+i λi. We write N≤k := N/mk
+Proposition 7.1. Let q, R be as above. Let s, ǫ be real numbers such that s ≥ 0 and s−ǫ ≥ 0,
+and r a positive integer, and let MNρ′ = |∧2
+R Nρ′[pr]||Nρ′[pr]|ǫ|Nρ′|−s = q
+�r
+j=1((
+ρj
+2 )+ǫρj)−s|ρ| for
+all partitions ρ. We define
+Eq(t, m) :=
+m
+�
+e=0
+(−1)eq−(t+1)eη(m)
+η(e)η(m − e)
+.
+13
+
+Then Eq(t, m) is positive for all real t ≥ 0 and m ∈ N. Further, the values MN are well-
+behaved for R and are the moments of the measure ν such that
+ν({P | P ≤k ≃ Nλ′}) = q
+�
+j(−(
+λj
+2 )−(s−ǫ+1)λj)+�
+j≥r+1(−(
+λj
+2 )−ǫλj)
+�
+j≥2 η(λj−1 − λj)
+��
+i≥0
+(1 + q−s+ǫ−1−i)−1
+�
+×
+min(k,r)
+�
+j=2
+Eq(s − ǫ, λj−1 − λj)
+λmin(k,r)
+�
+i=λk+1
+(1 − q−s−i)
+=
+MNλ′
+| Aut(Nλ′)|
+�
+i≥0
+(1 + q−s+ǫ−1−i)−1
+min(k,r)
+�
+j=2
+Eq(s − ǫ, λj−1 − λj)
+λmin(k,r)
+�
+i=λk+1
+(1 − q−s−i)
+for all k ≥ 1 and λ with λk+1 = 0. This measure is supported on finite R-modules. Finally,
+ν({Nλ′}) is given by the above expression for any k > λ′
+1.
+By convention, the product from λk + 1 to λmin(k,r) is 1 if λk = λmin(k,r). The result also is
+proven below for the case r = ∞, with the natural interpretations.
+Proof. If we let ae =
+q−(t+1)e
+η(e)η(m−e), we note that
+ae+1
+ae
+= q−t−1(1 − q−m+e)
+(1 − q−e−1)
+< q−1
+1
+1 − q−1 ≤ 1,
+since t ≥ 0 and e ≥ 0 and q ≥ 2. So Eq(t, m) is an alternating series whose terms are
+decreasing in absolute value and whose first term is positive, which implies Eq(t, m) > 0.
+We now apply Theorem 5.2.
+We let N = Nλ′.
+Given a partition ρ, the number of
+surjections Nρ′ → Nλ′ with simple kernel of type (1e), up to automorphisms of Nλ′, is
+q
+�
+j (
+ρj
+2 )−�
+j (
+λj
+2 )−(e
+2) �
+j≥1
+η(ρj − ρj+1)
+η(ρj − λj)η(λj − ρj+1)
+if |ρ| = |λ| + e and ρj − λj, λj − ρj+1 ≥ 0 for all j ≥ 1; and 0 otherwise [Mac15, Ch. II
+(4.6)]. Each of these surjections has µ(Nλ′, Nρ′) = (−1)eq(e
+2). Let ρ′ contain mj j’s (so
+mj = ρj − ρj+1 and ρj = mj + mj+1 + . . . ). We have | Aut(Nρ′)| = q
+�
+j ρ2
+j �
+j η(mj) ([Mac15,
+14
+
+II (1.6)]). We have | ∧2
+R Nρ′[pr]| = q
+�r
+j=1 (
+ρj
+2 ). Letting ρj = λj + ej, we have
+vk,Nλ′ =
+�
+e1...,ek≥0
+∀i≥1:ei+1≤λi−λi+1
+q
+�
+j (
+ρj
+2 )−�
+j (
+λj
+2 )−(e
+2) �
+j≥1
+η(ρj−ρj+1)
+η(ρj−λj)η(λj−ρj+1)(−1)ejq(
+ej
+2 )q
+�r
+j=1((
+ρj
+2 )+ǫρj)−s|ρ|
+q
+�
+j(ρj)2 �
+j≥1 η(mj)
+=
+�
+e1...,ek≥0
+∀i≥1:ei+1≤λi−λi+1
+(−1)
+�
+j ejq
+�
+j(2(
+λj+ej
+2 )−(
+λj
+2 )−(s−ǫ)(λj+ej))−�
+j>r((
+λj+ej
+2 )+ǫ(λj+ej))
+q
+�
+j(λj+ej)2 �
+j≥1 η(ej)η(λj − λj+1 − ej+1)
+=
+�
+e1...,ek≥0
+∀i≥1:ei+1≤λi−λi+1
+(−1)
+�
+j ejq
+�
+j(−(
+λj
+2 )−(s−ǫ+1)λj−(s−ǫ+1)ej)−�
+j>r((
+λj+ej
+2 )+ǫ(λj+ej))
+�
+j≥1 η(ej)η(λj − λj+1 − ej+1)
+= q
+�
+j(−(
+λj
+2 )−(s−ǫ+1)λj)
+η(λk)
+��
+e1≥0
+(−1)e1 q−(s−ǫ+1)e1
+η(e1)
+� min(k,r)
+�
+j=2
+λj−1−λj
+�
+ej=0
+(−1)ejq−(s−ǫ+1)ej
+η(ej)η(λj−1 − λj − ej)
+×
+k
+�
+j=min(k,r)+1
+λj−1−λj
+�
+ej=0
+(−1)ejq−(s−ǫ+1)ej−(
+λj+ej
+2 )−ǫ(λj+ej)
+η(ej)η(λj−1 − λj − ej)
+.
+The above sums are absolutely convergent when s − ǫ + 1 > 0, so we have well-behavedness.
+We use the q-binomial theorem for negative powers for the e1 sum. For 2 ≤ j ≤ min(k, r),
+the sum over ej is equal to Eq(s − ǫ, λj−1 − λj)/η(λj−1 − λj). Considering the sum over ej
+for j > min(k, r), we have
+λj−1−λj
+�
+ej=0
+(−1)ejq−(s−ǫ+1)ej−(
+λj+ej
+2 )−ǫ(λj+ej)
+η(ej)η(λj−1 − λj − ej)
+= q−(
+λj
+2 )−ǫλj
+λj−1−λj
+�
+ej=0
+(−q−s−1−λj)ejq−(
+ej
+2 )
+η(ej)η(λj−1 − λj − ej)
+=
+q−(
+λj
+2 )−ǫλj
+η(λj−1 − λj)
+λj−1−λj−1
+�
+i=0
+(1 − q−s−1−λj−i)
+by the q-binomial theorem. We can combine the products above over different j and have
+k
+�
+j=min(k,r)+1
+λj−1−λj−1
+�
+i=0
+(1 − q−s−1−λj−i) =
+λmin(k,r)
+�
+i=λk+1
+(1 − q−s−i).
+We put these identities all together, pull out all the powers of q, and collect the η(λj−1 − λj)
+terms in the denominator to obtain the equation for ν({P | P ≤k ≃ Nλ′}) in the proposition.
+There are countably many finitely generated ˆR-modules, and we will show each infinite
+one occurs with probability 0. Let N be a finitely generated ˆR-module and define λ(k) so
+that Nλ(k)′ = N≤k. In particular, if N = ˆRℓ × T for some T with mk0T = 0, then for k ≥ k0,
+we have that λ(k)j does not depend on k for j ≤ k0, and λ(k)j = ℓ for k0 < j ≤ k. So, if
+ℓ ≥ 1, as k → ∞, we have ν({P | P ≤k ≃ Nλ(k)′}) → 0, which implies ν({N}) = 0. Thus the
+measure is supported on finite modules.
+If λ′
+1 < k, then the only finitely generated ˆR-module P with P ≤k ≃ Nλ′ is P = Nλ′, which
+gives the final statement.
+□
+15
+
+We write E(t, m) := Eq(t, m) when the value of q is clear. While we do not have a general
+product formula for E(t, m), we do have one when t = 0.
+Lemma 7.2. For m a non-negative integer,
+E(0, m) =
+⌈m/2⌉
+�
+j=1
+(1 − q−(2j−1)).
+Proof. There are two forms of Pascal’s identity for the q-binomial coefficients.
+These are, for 1 ≤ e ≤ m − 1,
+η(m)
+η(e)η(m − e) = q−e
+η(m − 1)
+η(e)η(m − 1 − e) +
+η(m − 1)
+η(e − 1)η(m − e)
+and
+η(m)
+η(e)η(m − e) =
+η(m − 1)
+η(e)η(m − 1 − e) + q−(m−e)
+η(m − 1)
+η(e − 1)η(m − e).
+They can both be verified immediately by expanding the products. Plugged into
+E(t, m) =
+m
+�
+e=0
+(−1)eq−(t+1)e
+η(m)
+η(e)η(m − e)
+these give the recurrences
+E(t, m) = E(t + 1, m − 1) − q−(t+1)E(t, m − 1)
+and
+E(t, m) = E(t, m − 1) − q−m−tE(t − 1, m − 1)
+Specializing we get
+(7.3)
+E(−1, m) = E(0, m − 1) − E(−1, m − 1)
+(7.4)
+E(0, m) = E(0, m − 1) − q−mE(−1, m − 1).
+We have E(−1, m) = 0 for m odd since the e term cancels the m−e term. That, together
+with (7.3), gives
+E(0, 2k) = E(−1, 2k)+E(−1, 2k+1) = E(−1, 2k) = E(−1, 2k)+E(−1, 2k−1) = E(0, 2k−1)
+which together with (7.4) for m = 2k − 1 gives
+E(−1, 2k) = E(0, 2k − 1) = E(0, 2k − 2) − q−(2k−1)E(−1, 2k − 2)
+=E(−1, 2k − 2) − q−(2k−1)E(−1, 2k − 2) = (1 − q−(2k−1))E(−1, 2k − 2)
+so by induction and E(−1, 0) = 1 we have
+E(0, 2k) = E(0, 2k − 1) = E(−1, 2k) =
+k
+�
+j=1
+(1 − q−(2j−1))
+which gives the stated formula.
+□
+Before we consider our next moments, we verify that a certain q-series is always non-
+negative.
+16
+
+Lemma 7.5. For integers λ−1, λ, φ−1, φ, s1, s2, with λ−1 ≥ λ and φ−1 ≥ φ, we define
+Fq(λ−1, λ, φ−1, φ, s1, s2) :=
+λ−1−λ
+�
+e=0
+φ−1−φ
+�
+f=0
+(−1)e+fq(λ+e
+2 )−(λ
+2)−s1(λ+e)+(φ+f
+2 )−(φ
+2)−s2(φ+f)+(λ+e)(φ+f)
+q(λ+e)2+(φ+f)2η(e)η(λ−1 − λ − e)η(f)η(φ−1 − φ − f) .
+If φ−1 − λ−1 ≤ s1 and λ−1 − φ−1 ≤ s2, then
+(1) Fq(λ−1, λ, φ−1, φ, s1, s2) ≥ 0, and
+(2) Fq(λ−1, λ, φ−1, φ, s1, s2) is positive if and only if φ − λ ≤ s1 and λ − φ ≤ s2.
+Proof. We rearrange to obtain
+Fq(λ−1, λ, φ−1, φ, s1, s2)
+=
+λ−1−λ
+�
+e=0
+φ−1−φ
+�
+f=0
+(−1)e+fq(λ+e
+2 )−(λ
+2)−s1(λ+e)+(φ+f
+2 )−(φ
+2)−s2(φ+f)+(λ+e)(φ+f)
+q(λ+e)2+(φ+f)2η(e)η(λ−1 − λ − e)η(f)η(φ−1 − φ − f)
+= q−λ2−s1λ−φ2−s2φ+λφ
+λ−1−λ
+�
+e=0
+φ−1−φ
+�
+f=0
+(−1)e+fq−(e
+2)+e(−λ−s1+φ−1)−(f
+2)+f(−φ−1−s2+λ+e)
+η(e)η(λ−1 − λ − e)η(f)η(φ−1 − φ − f)
+.
+Now we apply the q-binomial theorem to the sum in the f, and obtain that the above sum
+is equal to
+q−λ2−s1λ−φ2−s2φ+λφ
+λ−1−λ
+�
+e=0
+(−1)eq−(e
+2)+e(−λ−s1+φ−1)
+η(e)η(λ−1 − λ − e)η(φ−1 − φ)
+φ−1−φ−1
+�
+i=0
+(1 − q−φ−1−s2+λ+e−i).
+If e ≥ s2 − λ + φ + 1, then the i = −φ − 1 − s2 + λ + e term in the product exists because
+0 ≤ −φ − 1 − s2 + λ + e ≤ −φ − 1 − s2 + λ−1 ≤ φ−1 − φ − 1.
+This term is (1 − q0) = 0, and thus the summand for e ≥ s2 − λ + φ + 1 is 0.
+If λ − φ > s2, in then follows that every summand is 0 and Fq(λ−1, λ, φ−1, φ, s1, s2) = 0.
+Similarly, if φ − λ > s1 we also have Fq(λ−1, λ, φ−1, φ, s1, s2) = 0.
+For the rest of the proof we assume that φ − λ ≤ s1 and λ − φ ≤ s2, and we only consider
+summands with e ≤ s2 − λ + φ as the others are 0. In this case the exponents of q in the
+product are
+−φ − 1 − s2 + λ + e − i ≤ −1
+and so the only non-positive factor in each summand is the (−1)e.
+If we let ae be the e summand in our sum, then we can compute
+|ae+1|
+|ae|
+= q−e−λ−s1+φ−1(1 − qλ−1−λ−e)
+(1 − q−e−1)
+(1 − q−φ−1−s2+λ+e+1)
+(1 − q−s2+λ+e−φ−1) .
+For 0 ≤ e ≤ min(s2 − λ + φ, λ−1 − λ) − 1, we then have
+|ae+1|
+|ae|
+< q−e−λ−s1+φ−1
+(1 − q−1)2 .
+If −e − λ − s1 + φ − 1 ≤ −2, then the above is ≤ 1, then then our sum for Fq is an
+alternating sum with first term positive and terms decreasing in absolute value, and thus
+(strictly) positive. Otherwise, if −e − λ − s1 + φ − 1 ≥ −1, then e ≤ −λ − s1 + φ ≤ 0, and
+17
+
+so −λ − s1 + φ = 0. Similarly, Fq is positive or −φ − s2 + λ = 0. If −φ − s2 + λ = 0, then
+there is only 1 summand we consider, e = 0, and thus Fq is positive.
+□
+Definition 7.6. We define ¯Fq(λ−1, λ, φ−1, φ, s1, s2) so that
+Fq(λ−1, λ, φ−1, φ, s1, s2) =
+q−λ2−s1λ−φ2−s2φ+λφ
+η(λ−1 − λ)η(φ−1 − φ)
+¯Fq(λ−1, λ, φ−1, φ, s1, s2),
+where Fq is defined in Lemma 7.5.
+Now we consider the ring R × R, whose modules we view as pairs of R-modules.
+Proposition 7.7. Let q, R be as above. Let s1, s2 be integers such that s1, s2 ≥ 0, and let
+r ≥ 1 be an integer. Let MNρ′,Nπ′ = q(
+�r
+j=1 ρjπj)−s1|ρ|−s2|π| for all partitions ρ, π. Then the
+values MN are well-behaved for R × R and are the moments of a measure ν such that for all
+k1, k2 ≥ 1 and all partitions λ, φ with λk1+1 = φk2+1 = 0, we have
+ν({P | P ≤k ≃ (Nλ′, Nφ′)}) = q(
+�r
+j=1 λjφj)+�
+j −(λj)2−s1λj−(φj)2−s2φj
+�
+j≥2 η(λj−1 − λj)η(φj−1 − φj)
+×
+η(∞)η(s1 + s2)
+η(s2 − λ1 + φ1)η(s1 + λ1 − φ1)
+min(r,k1,k2)
+�
+j=2
+¯Fq(λj−1, λj, φj−1, φj, s1, s2)
+×
+λmin(r,k1,k2)
+�
+i=λk1+1
+(1 − q−s1−i)
+λmin(r,k1,k2)
+�
+i=φk2+1
+(1 − q−s2−i)
+if −s1 ≤ λ1 − φ1 ≤ s2, and ν({P | P ≤k ≃ (Nλ′, Nφ′)}) = 0 otherwise. The above expression
+is positive if and only if −s1 ≤ λj − φj ≤ s2 for all 1 ≤ j ≤ min(r, k1, k2). The measure ν is
+supported on pairs of finite R-modules.
+Note the factor
+q(
+�r
+j=1 λjφj)+�
+j −(λj)2−s1λj−(φj)2−s2φj
+�
+j≥2 η(λj−1 − λj)η(φj−1 − φj)
+=
+MNλ′,Nφ′
+| Aut(Nλ′, Nφ′)|.
+Proof. By considerations as in the proof of Proposition 7.1 (in particular the second line of
+the large display) we have
+vk,(Nλ′,Nφ′)
+=
+�
+e1...,ek1≥0
+f1...,fk2≥0
+∀i≥1:ei+1≤λi−λi+1
+∀i≥1:fi+1≤φi−φi+1
+(−1)
+�
+j ej+fjq
+�
+j((
+λj+ej
+2 )−(
+λj
+2 )−s1(λj+ej)+(
+φj +fj
+2 )−(
+φj
+2 )−s2(φj+fj))+�min(r,k1,k2)
+j=1
+(λj+ej)(φj+fj)
+q
+�
+j(λj+ej)2+(φj+fj)2 �
+j η(ej)η(λj − λj+1 − ej+1)η(fj)η(φj − φj+1 − fj+1)
+(By convention, λj = ej = 0 for j > k1 and φj = fj = 0 for j > k2.)
+If k1 or k2 = 0, then vk,(Nλ′,Nφ′) can be computed just in the category of finite R-modules,
+and this case is treated in [SW22b, Lemma 6.3] for moments MN = |N|−s, and in particular
+vk,(Nλ′,Nφ′) ≥ 0.
+As in the proof of Proposition 7.1, the above sum factors over j, and the only factor that
+is an infinite sum is the j = 1 sum, which is the same as the sum in Proposition 6.2 (the
+18
+
+λ, φ, e, f there are the λ1, φ1, e1, f1 here, respectively), as long as k1, k2 ≥ 1. By the argument
+in the proof of Proposition 6.2 we then have that the above sum converges absolutely as long
+as s1 + s2 + 1 > 0. The j = 1 factor
+�
+e1≥0
+f1≥0
+(−1)e1+f1q((λ1+e1
+2 )−(λ1
+2 )−s1(λ1+e1)+(φ1+f1
+2 )−(φ1
+2 )−s2(φ1+f1))+(λ1+e1)(φ1+f1)
+q(λ1+e1)2+(φ1+f1)2η(e1)η(f1)
+evaluates to η(λ1)η(φ1)Dq(λ1, φ1, s1, s2) as in the proof of Proposition 6.2, and in particular
+is non-negative and is positive if and only if φ1 − λ1 ≤ s1 and λ1 − φ1 ≤ s2
+The factor for 2 ≤ j ≤ min(r, k1, k2) is Fq(λj−1, λj, φj−1, φj, s1, s2), and Lemma 7.5 ad-
+dresses exactly when this is positive or 0. Inductively, it follows that
+Dq(λ1, φ1, s1, s2)
+min(r,k1,k2)
+�
+j=2
+Fq(λj−1, λj, φj−1, φj, s1, s2) ≥ 0,
+and is positive if and only if φj − λj ≤ s1 and λj − φj ≤ s2 for all 1 ≤ j ≤ min(r, k1, k2).
+The λ part of the factor for min(r, k1, k2) < j ≤ k1 further factors into
+λj−1−λj
+�
+ej=0
+(−1)ejq((
+λj+ej
+2 )−(
+λj
+2 )−s1(λj+ej))
+q(λj+ej)2η(ej)η(λj−1 − λj − ej) =q−(λj)2−s1λj
+λj−1−λj
+�
+ej=0
+(−1)ejq−(
+ej
+2 )+ej(−λj−s1+1)
+η(ej)η(λj−1 − λj − ej)
+= q−(λj)2−s1λj
+η(λj−1 − λj)
+λj−1−λj−1
+�
+i=0
+(1 − q−λj−s1−1−i).
+for each j ≤ k1, and an analogous expression for j ≤ k2 involving φj, φj−1, s2. Since si ≥ 0, we
+see the exponents of q in the products above are all negative, and thus the above expression
+is always positive.
+We pull out all the powers of q and expressions η(λj−1 − λj) and η(φj−1 − φj), and
+we combine the remaining products for the j > min(r, k1, k2) terms as in the proof of
+Proposition 7.1. Putting this all together, we obtain the formula for the measures.
+There are countably many finitely generated ˆR× ˆR-modules, and we will show each infinite
+one occurs with probability 0. Let N = (N1, N2) be an ordered pair of finitely generated ˆR-
+modules. We define partitions λ(k) and φ(k) such that (Nλ(k)′, Nφ(k)′) = N≤k. In particular,
+if Ni = ˆRℓi × Ti for some Ti with mk0Ti = 0, then for k ≥ k0, we have that λ(k)j and φ(k)j
+do not depend on k for j ≤ k0, and λ(k)j = ℓ1 and φ(k)j = ℓ2 for k0 < j ≤ k. If either ℓ1 ≥ 1
+or ℓ2 ≥ 1, then ν({P | P ≤k ≃ N≤k}) → 0, which implies ν({N}) = 0. Thus the measure is
+supported on pairs of finite modules.
+□
+8. Representations of Γ and formulas for our conjectured distributions
+Let Γ be a finite group. In this section, we explain the translation between the moment
+problem for Zp[Γ]-modules and the one for the kind of R we considered in Sections 5, 6 and
+7.
+For a prime p ∤ |Γ|, we let T = Zp[Γ]. Since p ∤ |Γ|, the primitive central idempotents
+e1, . . . , em in Qp[Γ] are all in in Zp[Γ]. The isomorphism classes of indecomposable projective
+T-modules Vi are in bijection with the ei such that the action of T on Vi factors through
+eiT. Then the (distinct isomorphism classes of) irreducible representations of Γ over Qp are
+19
+
+Vi ⊗ Qp and the (distinct isomorphism classes of) irreducible representations of Γ over Fp
+are Vi ⊗ Fp. We let e1 = |Γ|−1 �
+γ∈Γ γ, corresponding to the trivial representation. Since
+T = �m
+i=1 eiT, the category of finite T-modules is the product of the categories of finite eiT
+modules. Each eiT is isomorphic to the ring of ni × ni matrices Mni×ni(Rdi) over Rdi, the
+ring of integers in the degree di unramified extension of Qp, where nidi is the rank of Vi over
+Zp, and di is the number of irreducible representations that Vi⊗Qp (or Vi ⊗Fp) is the sum of.
+(See, e.g. [Ser77, Section 12.2, Section 15.5], for some of this background on representations
+when p ∤ |Γ|.)
+By the Morita theorem, the category of Rdi-modules is equivalent to the category of
+eiT-modules, by an equivalence Fi that takes an Rdi-module H to an eiT-module whose
+underlying Rdi-module is Hni. Thus, the computations that we have done above in the
+categories of Rdi-modules can be applied to eiT-modules.
+There is an involution σ on the primitive central idempotents ei, fixing those corresponding
+to self-dual representations (over Qp or Fp), and exchanging the ei, ei′ corresponding to pairs
+of dual representations. For an ei fixed by σ corresponding to a representation Vi over Fp,
+let κi = Fpdi = EndΓ(Vi), and let ǫi = −1 if (∧2
+κiVi)Γ ̸= 0, let ǫi = 1 if (∧2
+κiVi)Γ = 0 but
+(Vi ⊗κi Vi)Γ ̸= 0, and let ǫi = 0 if (Vi ⊗κi Vi)Γ = 0.
+The following lemmas are straightforward calculations from the definitions.
+Lemma 8.1. Let Γ, p, ei, di, ni, ǫi be as above such that ei corresponds to a self-dual repre-
+sentation Vi over Fp. Let λ be a partition, and let Vλ be the eiT-module corresponding (via
+the Morita theorem) to the Rdi-module Nλ. Let q = pdi. Let r ≥ 0 be an integer. Then
+|(∧2
+ZpVλ′[pr])Γ||Vλ′|−u = q
+�r
+j=1 (
+λj
+2 )+(1−ǫi)
+λj
+2 +�
+j≥0 −uniλj.
+Lemma 8.2. Let Γ, p, ei, di, ni be as above such that ei, ei′ correspond to non-isomorphic
+dual representations Vi, Vi′ over Fp. Let λ, φ be partitions, and let Vλ be the eiT-module
+corresponding (via the Morita theorem) to the Rdi-module Nλ, and Wφ be the ei′T-module
+corresponding to the Rdi-module Nφ. Let q = pdi. Let r ≥ 0 be an integer. Then
+|(∧2
+Zp(Vλ′ × Wφ′)[pr])Γ||Vλ′ × Wφ′|−u = q
+�r
+j=1 λjφj+�
+j≥0 −uniλj−uniφj.
+Let mi be the maximal ideal of Rdi (which is generated by p, but we write like this
+since we will consider modules for the product of Rdi.) For k = (k2, . . . , km) ∈ Nm−1, we
+write mk = �
+i≥2 mki
+i
+for the ideal of T.
+The finite T/e1T-modules are given by tuples
+λ = (λ2, . . . , λm) of partitions such that λ corresponds to the module Vλ := ⊕iFi(Nλi) (and
+Nλi is the Rdi-module described in Section 5).1 For a T/e1T-module V , we let V ≤k = V/mkV.
+For a tuple λ of partitions, we write λ′ for the tuple in which each partition is the conjugate
+of the corresponding partition from λ.
+We can now give an explicit description of the measure of Conjecture 1.1. Recall that Eq
+and ¯Fq are defined in Proposition 7.1 and Definition 7.6 respectively.
+Theorem 8.3. Let Γ be a finite group and p a prime such that p ∤ |Γ|. Let m, T, ei, ni, di, ǫi
+be as above. Let qi = pdi. Let r, u be positive integers. Let S = T/e1T. Then there is a
+unique measure ν on the set of isomorphism classes of finite S-modules (with the discrete
+1We apologize for the slight abuse of notation, since the ring R is implicit in the notation Nλ.
+20
+
+topology and σ-algebra) such that
+�
+X
+| Sur(X, V )|dν =
+|(∧2
+ZpV )Γ[pr]|
+|V |u
+for every finite S-module V . For all k ∈ Nm−1 and tuples of partitions λ such that λi
+ki+1 = 0
+for all i ≥ 2,
+ν({V | V ≤k ≃ Vλ′}) =
+|(∧2
+ZpVλ′)Γ[pr]|
+|Vλ′|u| Aut(Vλ′)|
+×
+�
+i≥2
+σ(ei)=ei
+
+
+
+�
+ℓ≥0
+(1 + q
+−uni− ǫi+1
+2
+−ℓ
+i
+)−1
+min(r,ki)
+�
+ℓ=2
+Eqi(uni − 1 − ǫi
+2
+, λi
+ℓ−1 − λi
+ℓ)
+λi
+min(r,ki)
+�
+ℓ=λi
+ki+1
+(1 − q−uni−ℓ
+i
+)
+
+
+
+×
+�
+2≤i<j
+σ(ei)=ej
+�
+ηqi(∞)ηqi(2uni)
+ηqi(uni − λi
+1 + λj
+1)ηqi(uni + λi
+1 − λj
+1)
+min(r,ki,kj)
+�
+ℓ=2
+¯Fqi(λi
+ℓ−1, λi
+ℓ, λj
+ℓ−1, λj
+ℓ, uni, uni)
+λi
+min(r,ki,kj)
+�
+ℓ=λi
+ki+1
+(1 − q−uni−ℓ
+i
+)
+λj
+min(r,ki,kj)
+�
+ℓ=λj
+kj +1
+(1 − q−uni−ℓ
+i
+)
+
+
+ ,
+if |λi
+1 − λj
+1| ≤ uni for all 2 ≤ i < j with σ(ei) = ej, and ν({V | V ≤k ≃ Vλ′}) = 0 otherwise.
+The expression above is positive if and only if |λi
+ℓ−λj
+ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej
+and all 1 ≤ ℓ ≤ min(r, ki, kj). Moreover, any sequence νt of measures of isomorphism classes
+of finite S-modules such that
+lim
+t→∞
+�
+X
+|Sur(X, V )|dνt =
+|(∧2
+ZpV )Γ[pr]|
+|V |u
+,
+for every finite S-module V , has the property that limt→∞ νt(V ) = ν(V ) for every finite
+S-module V .
+If one wants a completely explicit formula, |(∧2
+ZpV )Γ[pr]| is multiplicative over σ-orbits of
+the ei, and explicit formulas for each factor are given in Lemmas 8.1 and 8.2. We also have
+| Aut(Vλ′)| = q
+�
+i,j(λi
+j)2 �
+i,j ηqi(λi
+j − λi
+j+1).
+Proof of Theorem 8.3. Using the Morita Theorem, we can translate this to an analogous
+statement on finite �
+i≥2 Rdi-modules. We then apply Theorem 5.2. Here note that the
+moments factor over the σ orbits of e2, . . . em. Thus the sum defining well-behavedness and
+giving formulas for the measure similarly factors. In each factor, we have moments given by
+Lemma 8.1 or Lemma 8.2, to which we can apply Proposition 7.1 or Proposition 7.7 to obtain
+not only that the moments are well-behaved, but that the resulting measures are supported
+on finite modules and given by the corresponding factor of the formula in this result. When
+we apply Proposition 7.1, we do so with s = uni and ǫ = 1−ǫi
+2 , and so we use u ≥ 1 so that
+s − ǫ ≥ 0. Noting that ni = nj, qi = qj gives the result.
+□
+We can prove Theorem 1.2 similarly. The formulas for Theorem 1.2 can be obtained from
+Theorem 8.3 by taking k = (1, 1, . . . ).
+21
+
+Proof of Theorem 1.2. The proof is very similar to that of Theorem 8.3 except that we apply
+Theorem 5.2 for the category of representations of Γ over Fp instead of the category Zp[Γ]-
+modules. The well-behavedness required to apply Theorem 5.2 is just the k = (1, 1, . . . )
+special case of the well-behavedness used in the proof of Theorem 8.3.
+The formulas for the probabilities given by Theorem 5.2 are then precisely the k =
+(1, 1, . . . ) formulas from Theorem 8.3.
+This simplifies the formulas substantially.
+Each
+of the partitions λi become integers λi
+1. For λ′ such that (λ′)i = (1, . . . , 1) for each i and (λi
+1
+gives the number of 1), we have
+ν({V | V/pV ≃ Vλ′}) =
+|(∧2
+ZpVλ′)Γ[p]|
+|Vλ′|u| Aut(Vλ′)|×
+�
+i≥2
+σ(ei)=ei
+�
+ℓ≥0
+(1 + q
+−uni− ǫi+1
+2
+−ℓ
+i
+)−1
+�
+2≤i<j
+σ(ei)=ej
+�
+ηqi(∞)ηqi(2uni)
+ηqi(uni − λi
+1 + λj
+1)ηqi(uni + λi
+1 − λj
+1)
+�
+.
+if |λi
+1−λj
+1| ≤ uni for all 2 ≤ i < j with σ(ei) = ej and ν({V | V/pV ≃ Vλ′}) = 0 otherwise.
+□
+Similarly, to obtain an analogue of Theorem 1.2 describing the pk-torsion, we would take
+k = (k, k, . . . ).
+Since a finite S-module can be detected mod mk for sufficiently large ki, we can use
+Theorem 8.3 to give the measure on any particular S-module.
+Corollary 8.4. For the measure ν in Theorem 8.3 and any tuple λ of partitions, we have
+ν({Vλ′}) =
+|(∧2
+ZpVλ′)Γ[pr]|
+|Vλ′|u| Aut(Vλ′)|
+×
+�
+i≥2
+σ(ei)=ei
+
+�
+ℓ≥0
+(1 + q
+−uni− ǫi+1
+2
+−ℓ
+i
+)−1
+r�
+ℓ=2
+Eqi(uni − 1 − ǫi
+2
+, λi
+ℓ−1 − λi
+ℓ)
+λi
+r
+�
+ℓ=1
+(1 − q−uni−ℓ
+i
+)
+
+
+×
+�
+2≤i<j
+σ(ei)=ej
+�
+ηqi(∞)ηqi(2uni)
+ηqi(uni − λi
+1 + λj
+1)ηqi(uni + λi
+1 − λj
+1)
+r�
+ℓ=2
+¯Fqi(λi
+ℓ−1, λi
+ℓ, λj
+ℓ−1, λj
+ℓ, uni, uni)
+λi
+r
+�
+ℓ=1
+(1 − q−uni−ℓ
+i
+)
+λj
+r
+�
+ℓ=1
+(1 − q−uni−ℓ
+i
+)
+
+ ,
+if |λi
+1 − λj
+1| ≤ uni for all 2 ≤ i < j with σ(ei) = ej, and ν({Vλ′}) = 0 otherwise. The
+expression above is positive if and only if |λi
+ℓ − λj
+ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej
+and all 1 ≤ ℓ ≤ r.
+Proof. We apply Theorem 8.3 with ki sufficiently large such that {V | V ≤k ≃ Vλ′} = {Vλ′}
+(so in particular, λi
+ki = 0) and also such that ki ≥ r.
+□
+We now have a function field corollary of Theorem 3.1 on the distribution of class groups
+of function fields. Recall EΓ(n, q) is the set of isomorphism classes of extensions K/Fq(t)
+with a choice of isomorphism Gal(K/Fq(t)) ≃ Γ, that are split completely above ∞ and
+22
+
+such that the radical of the discriminant ideal Disc(K/Fq(t)) has norm qn. For a finite set
+of primes P, an abelian group A, let AP be the product of the Sylow p-subgroups of A for
+p ∈ P. An abelian P-group is a finite product of abelian p-groups for p ∈ P.
+Corollary 8.5. Let Γ be a finite group and P a finite set of primes not dividing |Γ|. Let H
+be a finite abelian P-group Γ-module such that HΓ = 1. For each p ∈ P, let rp be a positive
+integer. Let q1, . . . be any sequence of prime powers, all relatively prime to |Γ| and all primes
+in P, and such that for all i, we have that Fqi(t) contains the prpth roots of unity, but not
+the prp+1st roots of unity.
+Then, as long as qx grows sufficiently fast with x, we have
+lim
+x→∞
+�
+n≤x |{K ∈ EΓ(n, qx) | Cl(OK)P ≃Γ H)}
+�
+n≤x |EΓ(n, qx)|
+=
+�
+p∈P
+νp(H{p}),
+where νp is the measure from Theorem 8.3 (given Γ, p, r = rp and u = 1).
+Proof. We start with Theorem 3.1. It is a formal consequence from the statement of Theo-
+rem 3.1 with two limits to obtain the same statement with just limx→∞ and q replaced by qx
+(growing sufficiently fast with x). Now we apply Theorem 5.2, and in particular uniqueness
+and robustness. The proof of Theorem 8.3 shows that these moments are well-behaved. Thus
+the corollary follows from Theorem 5.2.
+□
+9. Class groups cannot be certain modules
+Using Theorem 8.3, Conjecture 1.1 predicts that the density of fields for which Vλ is the
+class group is zero unless |λi
+ℓ − λj
+ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej, and 1 ≤ ℓ ≤ r.
+We can verify this prediction by showing that, in fact, fields violating this condition do not
+exist.
+Theorem 9.1. Let Γ be a finite group and p a prime such that p ∤ |Γ|. Let m, T, ei, ni, di be
+as at the start of Section 8. Let r and u be positive integers.
+Let K0 be a number field with u infinite places containing the prth roots of unity. Let K
+be a Galois extension of K0 with Galois group Γ. Let λ be the unique tuple of partitions such
+that ClK[p∞]/ClK0[p∞] ∼= Vλ′.
+Then |λi
+ℓ − λj
+ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej, and 1 ≤ ℓ ≤ r.
+The case r = 1 of Theorem 9.1 was shown previously by Gras [Gra98, Theorem 7.7(α)(i)
+and Theorem 8.8(α)(i)], with the special case where K is Galois over Q obtained earlier
+by Leopoldt [Leo58, Satz 2 on p. 170]. See also [BVVE21, Proposition 4.2.2] for the case
+K0 = Q and p = 2.
+Proof. As in [LST20, Definition 4.1], we can define a map ψK
+Cl(K)∨[pr] ∼= H1(Spec OK, Z/prZ) → H1(Spec OK, µpr) → Cl(K)[pr]
+where the first isomorphism arises from class field theory, the middle arrow is cup product
+with a generator of the prth roots of unity of K0, and the last arrow arises from the Kummer
+exact sequence and the identification Cl(K) ∼= H1(Spec OK, Gm). Each step is clearly Γ-
+invariant (because, in particular, the roots of unity of K0 are Γ-invariant) so this gives a
+map of Γ-modules.
+23
+
+By the argument in the first two paragraphs of the proof of [LST20, Lemma 6.21], the
+kernel of ψK is isomorphic as a Zp[Γ]-module to a submodule of O×
+K ⊗ Z/pr.
+Let 2 ≤ i < j with σ(ei) = ej. Applying the left-exact functor HomΓ(Vi, ·), we see that
+there is a map
+HomΓ(Vi, Cl(K)∨)[pr] → HomΓ(Vi, Cl(K))[pr]
+whose kernel is a subgroup of HomΓ(Vi, O×
+K⊗Z/pr). We have HomΓ(Vi, Cl(K)) ∼= �∞
+ℓ=1(Rdi/pℓ)λi
+ℓ−λi
+ℓ+1
+so HomΓ(Vi, Cl(K))[pr] ∼= �r−1
+ℓ=1(Rdi/pℓ)λi
+ℓ−λi
+ℓ+1 × (Rdi/pr)λi
+r. Similarly HomΓ(Vi, Cl(K)∨) ∼=
+�∞
+ℓ=1(Rdi/pℓ)λj
+ℓ−λj
+ℓ+1 so HomΓ(Vi, Cl(K)∨)[pr] ∼= �r−1
+ℓ=1(Rdi/pℓ)λj
+ℓ−λj
+ℓ+1 × (Rdi/pr)λj
+r.
+Since our assumptions imply K/K0 is split at ∞, each infinite place of K0 lies under a
+simply transitive Γ-orbit of infinite places of K. Thus, by Dirichlet’s unit theorem, O×
+K ⊗ R
+is isomorphic to R[Γ]u/R as an R[Γ]-module. Since p is prime to the order of Γ, this implies
+that O×
+K ⊗Zp, modulo torsion, is isomorphic to Zp[Γ]u/Zp as a Zp[Γ]-module. Further, using
+p ∤ |Γ|, we then have that, as a Zp[Γ]-module, O×
+K ⊗ Zp ≃ Zp[Γ]u/Zp × (µ∞(K) ⊗ Zp). Since
+the pth roots of unity are in K0, the cyclic group µ∞(K) ⊗ Zp has a trivial Γ-submodule,
+and hence has trivial Γ-action since p ∤ |Γ|. Thus Hom(Vi, O×
+K ⊗ Z/pr) ∼= (Rdi/pr)uni.
+It follows that we have a map
+r−1
+�
+ℓ=1
+(Rdi/pℓ)λj
+ℓ−λj
+ℓ+1 × (Rdi/pr)λj
+r →
+r−1
+�
+ℓ=1
+(Rdi/pℓ)λi
+ℓ−λi
+ℓ+1 × (Rdi/pr)λi
+r
+whose kernel has Rdi/p-rank ≤ uni. We say the Rdi-module �
+ℓ≥1(Rdi/pℓ)ρℓ has Rdi/pℓ-rank
+ρℓ + ρℓ+1 + · · ·. Since in a map of p-group Rdi-modules, the Rdi/pℓ-rank of the source minus
+the Rdi/pℓ rank of the target is at most the Rdi/p-rank of the kernel (e.g.
+see [Mac15,
+II (4.3)(i)]), this gives λj
+ℓ − λi
+ℓ ≤ uni for each 1 ≤ ℓ ≤ r A symmetric argument gives
+λi
+ℓ − λj
+ℓ ≤ uni.
+□
+10. Non-Galois extensions
+In this section, we explain how our conjecture also gives, as a consequence, the distribution
+of Sylow p-subgroups of class groups of non-Galois fields in the presence of roots of unity.
+Let Γ be a finite group and Γ′ a subgroup of Γ. If K0 is a number field, and K/K0 is
+an extension with an isomorphism Gal(K/K0) ≃ Γ, we can also consider the (relative) class
+group ClL|K0 of the fixed field L = KΓ′. Let p be a prime not dividing |Γ|, and let Vi,
+ei,ni,di,T be as at the start of Section 8., and let S = T/e1T. Let midi = rankZp V Γ′
+i .
+Since p ∤ |Γ|, we have that the natural inclusion map
+ClL|K0[p∞] → ClK|K0[p∞]Γ′
+is an isomorphism [CM90, Corollary 7.7]. Thus, a conjecture on the distribution of ClK|K0[p∞]
+as Zp[Γ]-modules formally implies a conjecture on the distribution of ClL|K0[p∞] as Zp-
+modules. However, we will see that one can make the implication much more explicit.
+Let eΓ′ = |Γ′|−1 �
+γ∈Γ′ γ. We have that eΓ′ is a (not necessarily central) idempotent of
+Zp[Γ]. For any Zp[Γ]-module V , we have that eΓ′V = V Γ′. Let o := eΓ′SeΓ′. For any S-
+module V , we have that V Γ′ is naturally an o-module, allowing the action to take place in
+V .
+Let eΓ|Γ′ be the central idempotent that is the sum of all the idempotents ei corresponding
+to non-trivial projective Zp[Γ]-modules that have non-trivial Γ′-invariants, i.e. the ei such
+24
+
+that eΓ′ei ̸= 0. Let M be the set of the corresponding i. The Vi for i ∈ M are exactly the
+non-trivial modules contained in IndΓ
+Γ′ Zp [WW21, Lemma 7.10]. For any S-module V , we
+have that V Γ′ = (eΓ|Γ′V )Γ′. We have eΓ′eΓ|Γ′ = eΓ′.
+The ring o is a maximal order in a semi-simple Qp algebra with primitive central idem-
+potents eΓ′ei = eieΓ′ for i ∈ M (over Q, this can be found in [WW21, Proposition 8.3,
+Corollary 8.10], and the argument over Qp is the same).
+There is a Morita equivalence
+between eΓ|Γ′S-modules and o-modules taking V �→ V Γ′ (see [WW21, Theorem 8.9]). The
+inverse functor takes U to I(U) := eΓ|Γ′SeΓ′ ⊗o U. If U = ⊕i∈MeΓ′eiU is an o-module, and
+V is a eΓ|Γ′S-module with V Γ′ = U (so V ≃ I(U)), then V = ⊕i∈MeiV , with
+(10.1)
+|eiV | = |eΓ′eiU|ni/mi
+by [CM90, Theorem 7.3].
+The finite o-modules are thus indexed by tuples of partitions
+(λi)i∈M, given as (Vλ)Γ′. If λ is a tuple of partitions indexed by elements of M, by abuse of
+notation, we also use λ to denote the extension by zero, i.e. the tuple (λ2, . . . , λm), where
+λi = 0 for i ̸∈ M. The following is an immediate consequence of the Morita equivalence.
+Proposition 10.2. The measure ν on S-modules from Theorem 8.3, pushed forward to a
+measure on finite o-modules via the map V �→ V Γ′ from S-modules to o-modules, gives a
+measure νΓ′ on o-modules as follows. Let k ∈ NM and let λ be a tuple of partitions indexed
+by the elements of M such that λi
+ki+1 = 0 for all i ∈ M. Then
+νΓ′({U | I(U)≤k ≃ Vλ′}) = ν({V | V ≤k ≃ Vλ′}).
+(10.3)
+Also, the U-moment of νΓ′ is |(∧2
+ZpV )Γ[pr]||V |−u for V = I(U).
+The right-hand side of (10.3) is given by an explicit formula in Theorem 8.3 (this is just
+the special case of that formula when ki = 0 for k ̸∈ M). It then follows that the probability
+of an individual module νΓ′({(Vλ′)Γ′}) is given by a formula that is the same as that given
+in Corollary 8.4, but with the product restricted to i ∈ M. Note that the terms
+|(∧2
+ZpVλ′)Γ[pr]|
+|Vλ′|u| Aut(Vλ′)|
+can all be given by explicit formulas (e.g. using Lemmas 8.1 and 8.2). Also, by the categorical
+equivalence, we have | Aut(Vλ′)| = | Auto((Vλ′)Γ′)|.
+This shows that Conjecture 1.1 gives explicit conjectures for the distribution of Sylow
+p-subgroups of class groups of non-Galois fields in the presence of roots of unity, for p not
+dividing the order of the Galois closure, in particular that they are given by the measure νΓ′
+above.
+11. Experimental evidence
+Malle [Mal08, Mal10] has done extensive computations on class groups over extensions of
+a base field with roots of unity. Most of his tables of computed class group data strongly
+support his conjectures, showing remarkably good convergence of the probabilities of various
+Sylow p-subgroups to the conjectured amount. Since we see in Section 12 that his conjectures
+are (in all cases we shall mention in this section) special cases of our conjecture, these all
+provide good evidence for our conjecture. The cases for which this provides data are as
+follows.
+25
+
+(1) Γ = C2, p = 3, r = 1, u = 1 (K0 = Q(√−3)): [Mal08, Table 9] and [Mal10, Table 1]
+on 3-ranks, [Mal10, Table 2] on 3-Sylow
+(2) Γ = C2, p = 5, r = 1, u = 1 (K0 = Q(ζ5)): [Mal10, Table 3] on 5-ranks
+(3) Γ = C3, p = 2, r = 1, u = 1 (K0 = Q, Q(√−3)): [Mal08, Tables 3,4] on 2-ranks and
+[Mal10, Tables 5,6,7] and [Mal10, Tables 7,8,9] on 2-Sylow
+(4) Γ = C3, p = 2, r = 1, u = 2 (K0 = Q(
+√
+5)): [Mal10, Table 10] on 2-ranks and [Mal10,
+Table 11] on 2-Sylow
+(5) Γ = C3, p = 2, r = 2, u = 1 (K0 = Q(i)): [Mal10, Table 12] on 2-ranks
+(6) Γ = C5, p = 2, r = 1, u = 1 (K0 = Q): [Mal08, Table 8] on 2-ranks
+Further, Malle provides a chart of data for Sylow 2-subgroups of class groups of C3-
+extensions of Q(i) (so r = 2, u = 1) for which he makes no conjecture to explain, only
+remarking they do not seem to fit well with any known formulas. Our conjecture provides
+an excellent match for this data, which we present in Table 1.
+(λ1, . . . ):
+(0)
+(1)
+(1,1)
+(2)
+(1,1,1)
+(2,1)
+(3)
+(1,1,1,1)
+(2,1,1)
+D
+1
+22
+42
+24
+82
+42 × 22
+26
+162
+82 × 22
+≤ 1015
+0.864
+0.115
+0.016
+0.25E-2
+0.12E-2
+0.50E-3
+0
+0.7E-4
+0.6E-4
+≥ 1016
+0.859
+0.120
+0.016
+0.30E-2
+0.13E-2
+0.42E-3
+0
+0.2E-3
+0.4E-4
+≥ 1024
+0.854
+0.123
+0.017
+0.40E-2
+0.11E-2
+0.64E-3
+0.1E-4
+0.7E-4
+0.3E-4
+≥ 1032
+0.853
+0.125
+0.017
+0.40E-2
+0.10E-2
+0.62E-3
+0.4E-4
+0.6E-4
+0.4E-4
+Conj. 1.1
+0.853
+0.124
+0.0166
+0.38E-2
+0.10E-2
+0.60E-3
+0.38E-4
+0.66E-4
+0.37E-4
+[Mal10, (6-2)]
+0.853
+0.133
+0.0083
+0.44E-2
+0.52E-3
+0.34E-3
+0.35E-4
+0.33E-4
+0.21E-4
+Cohen-Martinet
+0.918
+0.076
+0.0048
+0.03E-2
+0.30E-3
+0.25E-4
+0.79E-7
+0.19E-4
+0.16E-4
+Table 1. Malle’s data on 2-Sylow for C3 extensions K/Q(i) from [Mal10,
+Table 13] compared to various conjectures
+If K/Q(i) is a C3-extension, then ClK|Q(i)[2∞] is a module for Z2[ζ3], where ζ3 is a primitive
+3rd root of unity. In Table 1, the columns correspond to the first few finite Z2[ζ3]-modules,
+with the top row labeling them in our notation assuming the module is Vλ′, and the second
+row labeling them with the notation of [Mal10] which reflects their underlying structure as
+abelian groups. Malle computed all C3 extensions of Q(i) with discriminant at most 1015,
+and for some other values of D the first S such extensions with discriminant at least D (it
+appears S is 5 · 105 or 4 · 105 in each case). The numbers in the middle part of the table
+show, for each collection of extensions K/Q(i), the proportion with ClK|Q(i)[2∞] which is the
+given Z2[ζ3]-module. In the bottom part of the chart, one sees the proportions predicted
+by Conjecture 1.1, Malle’s formula [Mal10, (6-2)], and the original conjecture of Cohen and
+Martinet [CM90].
+We see quite good agreement with Conjecture 1.1.
+Malle recognized
+that the formula [Mal10, (6-2)] was not a good match for these values, but he included this
+comparison so we do as well.
+It would be very interesting to have more experimental evidence for Conjecture 1.1. The
+case of Γ = C7, with p = 2, and K0 = Q is particularly interesting, as it is one of the smallest
+cases (being about fields of absolute degree 7) in which we have no current experimental
+evidence, and also the first in which there are dual pairs of representations, and so our
+26
+
+conjecture predicts a lack of independence between representations (a new phenomenon
+worthy of backing up with experimental evidence). Also Γ = C7 ⋊C3 has a transitive degree
+7 permutation action, and letting Γ′ be the stabilizer of a point, one could consider the
+non-Galois K that are the Γ′ fixed fields of Γ-extensions, with p = 2, and K0 = Q, to check
+a non-Galois implication of our conjecture for degree 7 fields. Other relatively small degree
+cases are degree 4 abelian or D4 extensions with p = 3 and K = Q(ζ3). The case of Γ = C11,
+with p = 2, and K0 = Q, is also of interest, because as we will see below in Section 12.3, our
+conjecture differs from a suggestion of Malle’s in this case. Other interesting cases include
+K0 = Q(ζ5) with Γ = C4 and p = 5, or K0 = Q(ζ3) with Γ = C8 and p = 3, as they have
+dual irreducible representations and p odd.
+12. Comparison to previous conjectures in special cases
+Malle [Mal08, Mal10] and Adam and Malle [AM15] have made conjectures about class
+group distributions in the presence of roots of unity in particular situations. In this section,
+we explain how these conjectures relate to Conjecture 1.1.
+Let Γ be a finite group and Γ′ be a subgroup with �
+γ∈Γ γΓ′γ−1 = 1. Let p be a prime
+p ∤ |Γ|. Throughout this section, we will consider the distribution of ClKΓ′/K0[p∞], as K
+ranges over Γ extensions of K0 for some fixed number field K0 containing prth roots of unity
+but not pr+1th roots of unity, and with u infinite places. (So we can take Γ′ = 1 to get
+the case of relative class groups of Galois extensions, or other Γ′ to consider non-Galois
+extensions.) We say (Γ, Γ′) is an absolutely irreducible pair if the representation IndΓ
+Γ′ C is
+the sum of one trivial representation and an irreducible representation.
+The conjectures of Malle and Adam are mostly in the case when (Γ, Γ′) is an absolutely
+irreducible pair. We have that (Sn, Sn−1) is an absolutely irreducible pair, so this covers the
+case of degree n extensions with Galois closure with Galois group the symmetric group Sn,
+including quadratic extensions. However, such pairs are rare, with only 14 examples with
+|Γ| ≤ 120, and the only example giving Galois fields (i.e. with Γ′ = 1) being Γ = S2.
+When (Γ, Γ′) is an absolutely irreducible pair and r = 1, Malle [Mal10, Conjecture 2.1]
+gives a conjecture for the distribution of ClKΓ′/K0[p∞] and we show below that in this case, our
+conjecture specializes to [Mal10, Conjecture 2.1]. 2 When (Γ, Γ′) is an absolutely irreducible
+pair and r ≥ 1, Adam and Malle make a conjecture [AM15, Conjecture 5.1] that does not
+give formulas for their conjectured distribution of ClKΓ′/K0[p∞], but instead conjectures that
+the distribution of ClKΓ′/K0[p∞] agrees with a limit of certain distributions from finite matrix
+groups. In Section 12.1, we show that the limit of these distributions from matrix groups
+exists and, indeed, agrees with our Conjecture 1.1 and the explicit formulas we give.
+Malle also makes some conjectures in the cases Γ = C3, C5 and p = 2 ([Mal08, (1),(2)],[Mal10,
+(6-1),(6-2),(6-3),(6-4)]), and we check in Section 12.2 that Conjecture 1.1 agrees with all of
+these conjectures in these specific cases.
+However, Malle makes a speculation for the situation in which IndΓ
+Γ′ Q is the sum of one
+trivial representation and an irreducible representation (over Q) [Mal10, Section 2], and we
+show in Section 12.3 that in general his speculation does not agree with Conjecture 1.1. We
+show below that the only Galois cases in which Malle’s speculation agrees with Conjecture 1.1
+are when Γ is C3 or C5 and p = 2. Luckily, these seem to be precisely the cases in which Malle
+2Malle also makes some conjectures when p | |Γ|, but we don’t discuss those here.
+27
+
+has computational evidence for his speculation (for p ∤ |Γ|). In many cases of this situation,
+our conjecture is qualitatively different from [Mal10, Section 2] because we conjecture (and
+prove in Theorem 9.1) that certain modules occur with probability 0 while [Mal10, Section
+2] assigns positive probability to each module.
+12.1. Absolutely irreducible pairs. Let (Γ, Γ′) be an absolutely irreducible pair and p a
+prime p ∤ |Γ|. In the notation of Section 10, we then have |M| = 1 (without loss of generality
+we say M = {2}) and d2 = 1, and m2 = 1, and q2 = p. We have that o = Zp ([WW21,
+Proposition 8.16]). The argument in [WW21, Proposition 8.16] shows that o = Zp if and
+only if (Γ, Γ′) is an absolutely irreducible pair. This highlights the appeal of this condition
+in this context, for when it fails, the groups ClKΓ′/K0[p∞] have an o-module structure, for
+some o larger than Zp, not just abelian group structure.
+Thus in the absolutely irreducible pair situation, we are looking simply for a distribution
+of finite Zp-modules.
+For a partition λ, which we extend by 0 to an (m − 1)-tuple of
+partitions, we have the o-module (Vλ)Γ′ = �
+i Z/pλiZ. We also have |Vλ| = | �
+i Z/pλiZ|n2 by
+Equation (10.1) (and n2 = [Γ : Γ′] − 1).
+Lemma 12.1. Let (Γ, Γ′) be an absolutely irreducible pair and p a prime p ∤ |Γ|, then Γ has
+even order, σ(2) = 2, and ǫ2 = 1.
+Proof. For a permutation representation, the standard pairing is an invariant symmetric
+perfect pairing (since permutation matrices are orthogonal), and thus IndΓ
+Γ′ C is self-dual.
+If IndΓ
+Γ′ C = C × W for an irreducible representation W, since W is non-trivial, it must be
+self-dual. Odd order groups have no non-trivial self-dual irreducible representations, which
+implies |Γ| is even. Let Wp be the mod p representation corresponding to W. We have that
+(Sym2 IndΓ
+Γ′ Fp)Γ is at least two dimensional, since it includes the pairing on IndΓ
+Γ′ Fp that
+first takes the quotient to the trivial representation, as well as the perfect pairing mentioned
+above, which implies that (Sym2 Wp)Γ ̸= 0. Since p is odd, this implies Wp is self-dual (i.e.
+σ(2) = 2) and ǫ2 = 1.
+□
+12.1.1. r = 1. First,we assume r = 1. We then see that in this case (absolutely irreducible
+pair and r = 1) the formula for the measure νΓ′ of Proposition 10.2 simplifies significantly.
+We have, for a partition λ, and group U = �
+i Z/pλ′
+iZ, our conjecture for the distribution of
+ClKΓ′/K0[p∞] is given by
+νΓ′({U}) =
+p(λ1
+2 )
+|U|un2| Aut(U)|
+�
+ℓ≥0
+(1 + p−un2−1−ℓ)−1
+λ1
+�
+ℓ=1
+(1 − p−un2−ℓ).
+(12.2)
+Now we compare this to Malle’s [Mal10, Conjecture 2.1]. (We assume that [Mal10, Con-
+jecture 2.1] has a typo and the product in the numerator should start at i = u + 1 since
+this modification is what is shown in the paper of Adam and Malle [Mal10, Example 5.3],
+where it is said to be exactly [Mal10, Conjecture 2.1].) We need to compute the u in Malle’s
+formula (which is not quite our u). Since we have that K/K0 is split at all infinite places by
+Remark 1.4, we have that Malle’s χE is the character that is the sum of a copy of the char-
+acter of the regular representation for each infinite place of K0 minus one trivial character.
+It follows that Malle’s u is our un2 (in the absolutely irreducible pair case). Then we see
+that Equation (12.2) exactly agrees with [Mal10, Conjecture 2.1] (with the typo corrected).
+28
+
+Even though both these conjectures imply a conjecture on p-ranks, as a double check, we
+compare our conjectured formula for the distribution of p-ranks to Malle’s [Mal10, Propo-
+sition 2.2]. To obtain our formula, we apply Proposition 10.2 with k = 1 (leading us to
+Theorem 8.3 with k = (1, 0, 0, . . . ).
+νΓ′({U|rkpU = λ}) =
+p(λ
+2)
+|(Z/pZ)λ|un2| Aut((Z/pZ)λ)|
+�
+ℓ≥0
+(1 + p−un2−1−ℓ)−1.
+(12.3)
+This agrees with the distribution of [Mal10, Proposition 2.2].
+12.1.2. r ≥ 1. Now we consider a general r ≥ 1 (still with an absolutely irreducible pair
+(Γ, Γ′) and p ∤ |Γ|). In this case, Adam and Malle [AM15, Conjecture 5.1] conjecture that
+ClKΓ′/K0[p∞] is distributed as the limit of certain distributions coming from matrix groups,
+and we describe those distributions here. Given a prime p, and positive integers r ≤ k, g, let
+Sp(r)
+2g (Z/pkZ) be the subgroup of matrices of GL2g(Z/pkZ), that when reduced mod pr are
+in Sp2g(Z/prZ). In other words, these are matrices that are symplectic mod pr. Adam and
+Malle define a distribution νr
+AM of finite abelian p-groups such that
+(12.4)
+νr
+AM({G}) := lim
+g→∞
+|{A ∈ Sp(r)
+2g (Z/pkZ) | ker(A − I2g) ≃ G|
+|Sp(r)
+2g (Z/pkZ)|
+,
+where I2g is the 2g × 2g identity matrix, and we take the kernel of A − I2g as a map
+(Z/pkZ)2g → (Z/pkZ)2g and k is any positive integer such that pk−1G = 0. As far as we can
+tell, they do not show that these limits exist or that they give a probability distribution, but
+we will show that here.
+Lemma 12.5. For r ≥ 1, the limits defining νr
+AM exist and define a probability measure on
+the set of (isomorphism classes of) finite abelian p-groups such that
+�
+G | Sur(G, H)|dνr
+AM =
+| ∧2 H[pr]| for every finite abelian p-group H.
+Proof. First, we note that by Proposition 7.1 there is a probability measure νr on the set
+of finite abelian p-groups with these moments, and the moments are well-behaved. For B a
+2g×2g matrix over Z/pkZ, we have ker(B) ≃ cok(B), and so we can replace ker(A−Ig) with
+coker(A−Ig). When pk−1G = 0, the cokernel of a Zp linear map B : Z2g
+p → Z2g
+p is isomorphic
+to G if and only if the same is true for the reduction of ¯B : (Z/pkZ)2g → (Z/pkZ)2g mod pk.
+So we replace Z/pkZ with Zp (using the analogous definition for Sp(r)
+2g (Zp)), and counting
+measure on Sp(r)
+2g (Z/pkZ) with Haar measure on Sp(r)
+2g (Zp).
+Let νg be the distribution of coker(A−Ig) for a Haar random A ∈ Sp(r)
+2g (Zp). By Theorem
+5.2, if we show that
+(12.6)
+lim
+g→∞
+�
+G
+| Sur(G, H)|dνg = | ∧2 H[pr]|,
+for every finite abelian p-group H, then the νg weakly converge to νr, and since the delta
+function on a single group is continuous, the limits defining νr
+AM({G}) exist and we have
+νr
+AM({G}) = νr({G}), which would complete the proof of the lemma.
+To show (12.6), we have that Sp(r)
+2g (Zp) acts on Sur(Z2g
+p , H), and the fixed points of A ∈
+Sp(r)
+2g (Zp) are precisely the elements of Sur(coker(A−Ig), H). By Burnside’s Lemma, we then
+29
+
+have that the average of Sur(coker(A − Ig), H) over A ∈ Sur(Z2g
+p , H) is the number of orbits
+of Sp(r)
+2g (Zp) on Sur(Z2g
+p , H). By [Mic06, Theorem 2.14], we have that the orbits of Sp2g(Zp)
+on Sur(Z2g
+p , H), for g sufficiently large given H, are precisely the fibers of a map
+F : Sur(Z2g
+p , H) → ∧2H
+that we now describe. Let x1, . . . , x2g be a standard basis of Z2g
+p , and then F(φ) = �g
+i=1 φ(xi)∧
+φ(xi+g). The same statement for H/prH, tells us that
+¯F : Sur(Z2g
+p , H) → ∧2(H/prH)
+is invariant on Sp(r)
+2g (Zp) orbits.
+Since |∧2 (H/prH)| = |∧2 (H)[pr]|, we will be done if we can show that each fiber of ¯F is a
+single Sp(r)
+2g (Zp) orbit. Consider two elements a, b in the same fiber of ¯F. We pick a minimal
+generating set h1, . . . , hs of H. We write F(a) = �
+i<j ai,jhi ∧ hj and F(b) = �
+i<j bi,jhi ∧ hj
+for ai,j, bi,j ∈ Zp such that ai,j ≡ bi,j (mod pr) for all i, j. We chose a bijection u : {i | 1 ≤
+i ≤
+�s
+2
+�
+} → {(i, j) | 1 ≤ i < j ≤ s} and write u1(i), u2(i) for the two components. For g large
+enough, given H (as above and so g ≥ s +
+�s
+2
+�
+), we can choose a surjection φa ∈ Sur(Z2g
+p , H)
+such that
+• φa(xi) = hi for 1 ≤ i ≤ s,
+• φa(xs+i) = au(i)hu1(i) for 1 ≤ i ≤
+�s
+2
+�
+,
+• φa(xi) = 0 for s +
+�s
+2
+�
+< i ≤ g,
+• φa(xg+i) = 0 for 1 ≤ i ≤ s and s +
+�s
+2
+�
+< i ≤ g, and
+• φa(xg+s+i) = hu2(i) for 1 ≤ i ≤
+�s
+2
+�
+.
+We have F(φa) = F(a), and so φa and a are in the same Sp2g(Zp) orbit and hence the
+same Sp(r)
+2g (Zp) orbit. We choose φb similarly. There is a Zp-module map sending xs+i �→
+xs+i + (bu(i) − au(i))xu1(i) for 1 ≤ i ≤
+�s
+2
+�
+and mapping all other xi �→ xi. This map is the
+identity mod pr and takes φa to φb. Thus we conclude φa and φb (and hence a and b) are in
+the same Sp(r)
+2g (Zp) orbit, which concludes the proof.
+□
+Proposition 7.1 gives explicit formulas for νr
+AM, and since we apply the proposition in the
+case that ǫ = s = 0, we even have nice product formulas for this measure using Lemma 7.2,
+for any r ≥ 1. Adam and Malle noted it was possible to calculate νr
+AM(G) explicitly for any
+given r and G [AM15, Remark 3.5], but did not have a closed formula for r ≥ 3.
+Finally, we are ready to state Adam and Malle’s conjecture [AM15, Conjecture 5.1]. For
+an absolutely irreducible pair (Γ, Γ′), Adam and Malle [AM15, Conjecture 5.1] conjecture
+that ClKΓ′/K0[p∞] is distributed like the quotient of a random group from νr
+AM(G) by un2
+uniform, independent random elements (as above Malle’s u is our un2). Such a random
+group has H-moment | ∧2 H[pr]||H|−un2 for every H, and by Proposition 7.1 there is a
+unique distribution with these moments. By Proposition 10.2 and Lemma 8.1, we see that
+Conjecture 1.1 predicts the distribution with the same moments, and so our conjectures
+agree in this case.
+In the special case when Γ = Z/2Z, there is also a conjecture [LST20, Conjecture 1.2]
+which gives a distribution for the class group together with extra invariants ω, ψ. Ignoring
+the extra invariants, [LST20, Conjecture 1.2] implies a conjecture simply for the distribution
+of the class group. This agrees with Conjecture 1.1. To check this, it suffices to check that
+30
+
+the moments of the distribution Qtµ of [LST20, Conjecture 1.2] agree with the moments
+described in Conjecture 1.1, since we have seen that these moments uniquely determine
+a distribution. The (G, ωG, ψG)-moment of Qtµ is calculated in [LST20, Lemma 8.12 and
+Theorem 2.3] as
+1
+|G|t| Sym2(G)[ℓn]| for each possible value of the invariants ωG, ψG. It is straight-
+forward to check by writing G as a product of cyclic groups that there are | Sym2(G)[ℓn]|
+possible values of ψG for each ωG and | ∧2 (G)[ℓn]| possible values of ωG, so summing over
+these possibilities, the G-moment is |∧2(G)[ℓn|
+|G|t
+. This agrees with Conjecture 1.1 since t and u
+are both defined to equal the number of infinite places.
+12.2. C3, C5. Now let Γ be Cℓ for ℓ = 3, 5 (and Γ′ = 1) and p = 2. In this case, there is
+only one non-trivial indecomposable module V2. We have n2 = 1 and d2 = ℓ − 1 and ǫ2 = 0.
+When r = 1, our conjecture for the distribution of ClKΓ′/K0[p∞] (from Conjecture 1.1, using
+Corollary 8.4, Lemma 8.1 for the formulas) is
+ν({Vλ′}) =
+2
+ℓ−1
+2 λ2
+1
+|Vλ′|u| Aut(Vλ′)|
+�
+i≥0
+(1 + (2ℓ−1)−u− 1
+2−i)−1
+λ1
+�
+i=1
+(1 − (2ℓ−1)−u−i).
+Note that
+�
+i≥0
+(1+(x2)− 1
+2 −i) =
+�
+i≥0
+(1+x−2i−1) =
+�
+i≥0(1 − (x2)−2i−1)
+�
+i≥0(1 − x−2i−1)
+=
+�
+i≥0(1 − (x2)−i)
+�
+i≥0(1 − (x2)−2i)·
+�
+i≥0(1 − x−2i)
+�
+i≥0(1 − x−i) .
+From this one can check that Malle’s conjectures [Mal10, (6-2),(6-4)] in the cases u = 1, 2
+agree with our conjecture for C3 and r = 1. It then formally follows that Malle’s conjectures
+[Mal10, (6-1),(6-3)] for the 2-ranks agree with ours (and it is also easily verified using Theo-
+rem 1.2). One can also check that Malle’s conjecture [Mal08, Equation (2)] agrees with our
+conjecture for C5 and u = r = 1. Malle [Mal10, Section 6.4] suggests that the 2-rank pre-
+diction for u = 1 [Mal10, (6-1)] should still hold for r > 1, which agrees with our conjecture
+(see Remark 1.5).
+12.3. Irreducible pairs. We say (Γ, Γ′) is an irreducible pair if the representation IndΓ
+Γ′ Q
+is the sum of one trivial representation and an irreducible representation W (over Q), and
+moreover that W has Schur index 1 (so EndΓ(W) is a field and not just a division algebra).
+12.3.1. Irreducible over Qp. Let (Γ, Γ′) be an irreducible pair. We first consider the case
+when Wp := W ⊗Q Qp is irreducible over Qp. In the notation of Section 10, we then have
+|M| = 1 (without loss of generality we say M = {2}). Let d = d2 and q = q2 = pd. We have
+m2 = 1. As in the proof of Lemma 12.1, we have that Wp is self-dual. By [WW21, Section
+8.3], we have that o is the maximal order in the field EndΓ Wp. Then our conjecture for the
+distribution of ClKΓ′/K0[p∞], in the case r = 1, is given by, for U = V Γ′
+λ′ ,
+νΓ′({U}) =
+|(∧2
+ZpVλ′)Γ[p]|
+|Vλ′|u| Auto(U)|
+�
+ℓ≥0
+(1 + q−un2− ǫ2+1
+2
+−ℓ)−1
+λ1
+�
+ℓ=1
+(1 − q−un2−ℓ)
+=
+q(λ1
+2 )+(1−ǫ2) λ1
+2
+|U|n2u| Auto(U)|
+�
+ℓ≥0
+(1 + q−un2− ǫ2+1
+2
+−ℓ)−1
+λ1
+�
+ℓ=1
+(1 − q−un2−ℓ).
+(12.7)
+31
+
+using Proposition 10.2 and Theorem 8.3 for the first line, and Lemma 8.1 and (10.1) for the
+second line. In [Mal10, Section 2], for r = 1 Malle suggests that the probability for U should
+be
+(12.8)
+c′dλ1
+q(λ1
+2 )
+|U|n2u| Auto(U)|
+λ1
+�
+ℓ=1
+(1 − q−un2−ℓ),
+for some unspecified c′ not depending on U. (We have absorbed some other constant parts
+of the formula into the unspecified constant from [Mal10]. Also, one can check that given
+that Wp is irreducible, our o is O ⊗Z Zp for Malle’s O, and so our d agrees with Malle’s d.)
+In order for (12.7) and (12.8) to agree, we must have
+p
+1−ǫ2
+2
+d = d.
+Since d is a positive integer, and p is a prime, the above equality holds (i.e. our conjectures
+agree) exactly when
+• ǫ2 = 1 and d = 1 (so Wp is absolutely irreducible over Qp and hence Q and we have
+an absolutely irreducible pair, as discussed above), or
+• ǫ2 = 0 and p = 2 and d = 2, 4.
+If we are interested in the Galois case (so Γ′ = 1), then d = |Γ|−1, and the only possibilities
+where d = 2, 4 are the Γ = C3, C5 examples discussed above.
+12.3.2. Reducible over Qp. Let (Γ, Γ′) be an irreducible pair. If Wp is reducible over Qp,
+then that means p splits in the field End(W). Let O be the maximal order End(W). Malle
+[Mal10, Section 2] gives a suggestion in terms of the “O-rank” of the p-group H. Since O/p
+is not a field, it is not clear to the authors what this rank means. Let us consider Γ = C7 and
+Γ′ = 1 and p = 2. We have O = o = Z[ζ7], but O ⊗Z Z2 = R1 × R2 for two extensions Ri of
+degree 3 over Z2 (both isomorphic to the same unramified extension). We have q = q2 = 23.
+In the case r = 1, our conjecture now gives probabilities
+ν({Vλ′}) =
+qλ2
+1λ3
+1
+|Vλ′|u| Aut(Vλ′)|
+ηq(∞)ηq(2u)
+ηq(u − λ2
+1 + λ3
+1)ηq(u + λ2
+1 − λ3
+1)
+λ2
+1
+�
+ℓ=1
+(1 − q−u−ℓ)
+λ3
+1
+�
+ℓ=1
+(1 − q−u−ℓ),
+if |λ2
+1 − λ3
+1| ≤ u and ν({Vλ′}) = 0 otherwise. We consider 4 2-torsion O-modules: 1, R1/2,
+R2/2, R1/2×R2/2. These correspond to the tuples of partitions (0, 0), ((1), 0)(0, (1)), ((1), (1)).
+If we look at the quantities ν({V })|V|u| Aut(V)| for our four modules, they are in ratios
+1 : 1 − 2−3u : 1 − 2−3u : 23(1 − 23(−u−1))2.
+while in [Mal10, Section 2], the corresponding ratio between a rank 0 group and a rank 1
+group is
+1 : 6(1 − 26(−u−1)).
+We do not see a way to interpret “O-rank” as to make our conjecture align with Malle’s
+suggestion.
+More qualitatively, our conjecture assigns probability 0 to any Vλ′ with |λ2
+1 − λ3
+1| > u, e.g.
+if u = 1, we assign probability 0 to any group which is rank 0 for R1 and rank 2 for R2.
+In contrast, [Mal10, Section 2] suggests that every 2-group O-module occurs with positive
+probability. So in this setting our Conjecture 1.1, and Theorem 9.1, do not agree with Malle’s
+suggestion.
+32
+
+References
+[Ach06]
+Jeffrey D. Achter. The distribution of class groups of function fields. Journal of Pure and Applied
+Algebra, 204(2):316–333, February 2006.
+[AM15]
+Michael Adam and Gunter Malle. A class group heuristic based on the distribution of 1-
+eigenspaces in matrix groups. Journal of Number Theory, 149:225–235, April 2015.
+[BL20]
+Alex Bartel and Hendrik W. Lenstra. On class groups of random number fields. Proceedings of
+the London Mathematical Society, 121(4):927–953, 2020.
+[BV15]
+Manjul Bhargava and Ila Varma. On the mean number of 2-torsion elements in the class groups,
+narrow class groups, and ideal groups of cubic orders and fields. Duke Mathematical Journal,
+164(10):1911–1933, 2015.
+[BV16]
+Manjul Bhargava and Ila Varma. The mean number of 3-torsion elements in the class groups and
+ideal groups of quadratic orders. Proceedings of the London Mathematical Society, 112(2):235–266,
+March 2016.
+[BVVE21] Benjamin Breen, Ila Varma, John Voight, and appendix with Noam Elkies. On unit signatures
+and narrow class groups of odd degree abelian number fields. (arXiv:1910.00449), April 2021.
+[CL84]
+Henri Cohen and Hendrik W. Lenstra, Jr. Heuristics on class groups of number fields. In Number
+Theory (Noordwijkerhout, 1983), volume 1068 of Lecture Notes in Math., pages 33–62. Springer,
+Berlin, 1984.
+[CM90]
+Henri Cohen and Jacques Martinet. Étude heuristique des groupes de classes des corps de nom-
+bres. Journal für die Reine und Angewandte Mathematik, 404:39–76, 1990.
+[EVW16]
+Jordan S. Ellenberg, Akshay Venkatesh, and Craig Westerland. Homological stability for Hurwitz
+spaces and the Cohen-Lenstra conjecture over function fields. Annals of Mathematics. Second
+Series, 183(3):729–786, 2016.
+[Gar15]
+Derek Garton. Random matrices, the Cohen–Lenstra heuristics, and roots of unity. Algebra &
+Number Theory, 9(1):149–171, February 2015.
+[Gra98]
+Georges Gras. Théorèmes de réflexion. Journal de théorie des nombres de Bordeaux, 10(2):399–
+499, 1998.
+[Kla17a]
+Zev Klagsbrun. The average sizes of two-torsion subgroups in quotients of class groups of cubic
+fields. arXiv:1701.02838 [math], January 2017.
+[Kla17b]
+Zev Klagsbrun. Davenport-Heilbronn Theorems for Quotients of Class Groups. arXiv:1701.02834
+[math], January 2017.
+[Leo58]
+Heinrich-Wolfgang Leopoldt. Zur Struktur der l-Klassengruppe galoisscher Zahlkörper. Journal
+für die reine und angewandte Mathematik, 199:165–174, 1958.
+[Liu22]
+Yuan
+Liu.
+Non-abelian
+Cohen–Lenstra
+Heuristics
+in
+the
+presence
+of
+roots
+of
+unity.
+(arXiv:2202.09471), February 2022.
+[LST20]
+Michael Lipnowski, Will Sawin, and Jacob Tsimerman. Cohen-Lenstra heuristics and bilinear
+pairings in the presence of roots of unity. arXiv:2007.12533 [math], July 2020.
+[LWZ19]
+Yuan Liu, Melanie Matchett Wood, and David Zureick-Brown. A predicted distribution for Galois
+groups of maximal unramified extensions. arXiv:1907.05002 [math], July 2019.
+[Mac15]
+I. G. Macdonald. Symmetric Functions and Hall Polynomials. Oxford Classic Texts in the Physical
+Sciences. The Clarendon Press, Oxford University Press, New York, second edition, 2015.
+[Mal08]
+Gunter Malle. Cohen–Lenstra heuristic and roots of unity. Journal of Number Theory,
+128(10):2823–2835, October 2008.
+[Mal10]
+Gunter Malle. On the distribution of class groups of number fields. Experimental Mathematics,
+19(4):465–474, 2010.
+[Mic06]
+A. A. George Michael. Finite Abelian actions on surfaces. Topology and its Applications,
+153(14):2591–2612, August 2006.
+[Ser77]
+Jean-Pierre Serre. Linear Representations of Finite Groups. Graduate Texts in Mathematics.
+Springer-Verlag, New York, 1977.
+[Smi22a]
+Alexander
+Smith.
+The
+distribution
+of
+ℓ∞-Selmer
+groups
+in
+degree
+ℓ
+twist
+families.
+arXiv.2207.05674, July 2022.
+33
+
+[Smi22b]
+Alexander
+Smith.
+The
+distribution
+of
+fixed
+point
+Selmer
+groups
+in
+twist
+families.
+arXiv.2207.05143, July 2022.
+[SW22a]
+Will Sawin and Melanie Matchett Wood. Finite quotients of 3-manifold groups. arXiv:2203.01140,
+March 2022.
+[SW22b]
+Will Sawin and Melanie Matchett Wood. The moment problem for random objects in a category.
+arXiv:2210.06279, October 2022.
+[Woo18]
+Melanie Matchett Wood. Cohen-Lenstra heuristics and local conditions. Research in Number
+Theory, 4(4):41, September 2018.
+[WW21]
+Weitong Wang and Melanie Matchett Wood. Moments and interpretations of the Co-
+hen–Lenstra–Martinet heuristics. Commentarii Mathematici Helvetici, 96(2):339–387, June 2021.
+Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027
+USA
+Email address: sawin@math.columbia.edu
+Department of Mathematics, Harvard University, Science Center Room 325, 1 Oxford
+Street, Cambridge, MA 02138 USA
+Email address: mmwood@math.harvard.edu
+34
+
diff --git a/DtAyT4oBgHgl3EQf4vpJ/content/tmp_files/load_file.txt b/DtAyT4oBgHgl3EQf4vpJ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e277a1c111c8b72501ba6ee136348db504acd844
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+++ b/DtAyT4oBgHgl3EQf4vpJ/content/tmp_files/load_file.txt
@@ -0,0 +1,1351 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf,len=1350
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='00791v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='NT]  2 Jan 2023  CONJECTURES FOR DISTRIBUTIONS OF CLASS GROUPS OF  EXTENSIONS OF NUMBER FIELDS CONTAINING ROOTS OF UNITY  WILL SAWIN AND MELANIE MATCHETT WOOD  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Cohen, Lenstra, and Martinet have given conjectures for the distribution of  class groups of extensions of number fields, but Achter and Malle have given theoretical  and numerical evidence that these conjectures are wrong regarding the Sylow p-subgroups  of the class group when the base number field contains pth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We give complete  conjectures of the distribution of Sylow p-subgroups of class groups of extensions of a number  field when p does not divide the degree of the Galois closure of the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  These  conjectures are based on q → ∞ theorems on these distributions in the function field analog  and use recent work of the authors on explicitly giving a distribution of modules from its  moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Our conjecture matches many, but not all, of the previous conjectures that were  made in special cases taking into account roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Introduction  In 1984, Cohen and Lenstra [CL84] gave conjectures for the distribution of the odd parts  of class groups of imaginary and real quadratic fields, as well as for any finite abelian group  A, the prime-to-|A| part of class groups of totally real A-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Cohen and Martinet [CM90]  generalized these conjectures to the situation of an arbitrary number field K0 as a base  field, and arbitrary group Γ, giving conjectures for distributions of the “good part” of class  groups of Γ-extensions of a fixed K0 with any fixed behavior at the infinite places of K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In  particular, the “good part” includes the product of the Sylow p-subgroups of the class group  for p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' However, these conjectures appear to be wrong at primes dividing the number of  roots of unity in the base field, as shown by Achter [Ach06] and Malle [Mal08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  In this paper, we give complete conjectures for the distribution of Sylow p-subgroups of  class groups of Γ-extensions (for p ∤ |Γ|) of any number field K0 that contains the pth (and  possibly further) roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ be a finite group and p a prime p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let S = Zp[Γ]/(�  γ∈Γ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let  K0 be a number field with u infinite places containing the prth roots of unity but not the pr+1th  roots of unity, for some r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let E = E(Γ, K0) be the set of isomorphism classes of Galois  Γ-extensions K/K0 along with an isomorphism Gal(K/K0) ≃ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let ClK|K0 := ClK/ClK0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  As K varies over E, the distribution of S-modules ClK|K0[p∞] is the one with the average  number of surjective morphisms from ClK|K0[p∞] to V being  |(∧2  ZpV )Γ[pr]|  |V |u  for any finite S-module V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 shows there is a unique distribution on S-modules  with these moments and gives an explicit formula for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 is motivated by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, which is based on work of Liu, Zureick-  Brown, and the second author [LWZ19] and gives the moments of these distributions in a  1  q → ∞ limit in the function field case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The explicit formulas obtained in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 are  based on recent work of the current authors [SW22b] that allows one to explicitly describe  a distribution of modules (or more general objects) given its moments (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The formulas for the conjectural distributions, like all previous such formulas, are given in  terms of q-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Some of the formulas are quite involved, and it would be interesting if  they could be further simplified with ideas from the study of such q-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We give now, as  a special case, the formulas for just the distribution on p-torsion that gives the moments of  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ be a finite group and p a prime such that p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , Vc be  the non-trivial irreducible representations of Γ over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let κi = EndΓ(Vi) and qi = |κi| and  dimκi Vi = ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let ǫi = −1 if (∧2  κiVi)Γ ̸= 0, let ǫi = 1 if (∧2  κiVi)Γ = 0 but (Vi ⊗κi Vi)Γ ̸= 0, and  let ǫi = 0 if (Vi ⊗κi Vi)Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let u be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let RΓ be the set of isomorphism  classes of finite dimensional representations V of Γ over Fp with V Γ = 0 (with a trivial  topology and σ-algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Then there is a unique measure ν on RΓ such that  �  X∈RΓ  | Sur(X, V )|dν =  |(∧2  FpV )Γ|  |V |u  for every V ∈ RΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For all non-negative integers f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , fc, we have, for V = �  i V fi  i ,  ν({V }) =  |(∧2  FpV )Γ|  |V |u| Aut(V )|  �  Vi  self-dual  �  ℓ≥0  (1 + q  −uni− ǫi+1  2  −ℓ  i  )−1  �  i<j  Vi, Vj dual  �∞  ℓ=1(1 − q−ℓ  i ) �2uni  ℓ=1 (1 − q−ℓ  i )  �uni−fi+fj  ℓ=1  (1 − q−ℓ  i ) �uni+fi−fj  ℓ=1  (1 − q−ℓ  i )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  if |fi − fj| ≤ uni for all pairs of dual representations Vi, Vj, and ν({V }) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For V = �  i V fi  i , we have |V | = �  i qfini  i  , and | Aut(V )| = �  i q  f2  i  i  �fi  ℓ=1(1 − q−ℓ  i ) and  |(∧2  FpV )Γ| =  �  Vi self-dual  q(fi  2)+(1−ǫi) fi  2  i  �  i<j,Vi,Vj dual  q  fifj  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  In the setting of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 together imply that  ClK|K0[p], as a representation of Γ over Fp, is distributed as ν from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In 2006, Achter [Ach06] found that in a large q limit, class groups  of quadratic extensions of function fields Fq(t) did not always satisfy the analogues of the  Cohen-Martinet conjectures for function fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In 2008, Malle [Mal08] found computational  evidence that the conjectures of Cohen-Lenstra-Martinet were incorrect for the Sylow p-  subgroups of class groups of number fields when p divides the order of the roots of unity in  K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Garton [Gar15] showed that the discrepancies Achter found in the function field case  could also be explained by the presence of roots of unity in the base field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Since 2008, there have been several papers aimed at correcting these conjectures in the  presence of roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In his original paper, Malle [Mal08] gave suggested conjectures  for K0 = Q and Γ = Z/3Z, Z/5Z, as well as K0 = Q(√−3) and Γ = Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Malle gave  further conjectures in a follow-up paper [Mal10], including conjectures for Γ = Z/2Z with  the Sylow p-subgroup of the class group for any base field containing pth but not p2th  2  roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In the Γ = Z/2Z case, Garton [Gar15] studied the distribution Achter saw  arising in the function field large q limit in terms of a random matrix model, and gave the  moments of this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  He was able to give explicit formulas for the distribution  for base fields containing p2th but not p3th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Adam and Malle [AM15] made  conjectures in further scenarios, including that for Γ = Z/2Z with any roots of unity in  the base field the class group distribution should match a limit of certain random matrix  distributions, but they did not give explicit formulas for this distribution for a general base  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Lipnowski, Sawin, and Tsimerman [LST20], for the Γ = Z/2Z and K0 = Fq(t) case,  considered additional pairing data that arises on class groups in the presence of roots of  unity, proved a function field result about these enriched class groups and gave conjectural  explicit formulas, expressed in terms of these pairings, for the case when Γ = Z/2Z and K0  is an arbitrary number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Liu [Liu22, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2] gave a moment conjecture which  overlaps (and agrees) with Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 in the case that p does not divide the order of any  unramified extension of K0 (without deriving formulas for the probability distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We  explain the relation of our conjectures to those in [Mal08, Mal10, AM15, LST20] in Section  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In summary, our conjectures agree with prior conjectures in many, but not all, cases  where they overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Previous work that has been based primarily on empirical computations has faced the  challenge that there are a limited number of situations that can be experimentally explored,  and as we see in this paper there are many parameters in these distributions, creating a chal-  lenge to generalization from limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Further, some of the formulas for the distributions  are quite involved, which makes it challenging to guess a formula from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Our results in  the function field case have the benefit of working for an arbitrary Γ and number of roots  of unity in the base rational function field, which allows us to see many more cases at once  from this point of view than can be explored with empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Further, our new methods  of producing distributions explicitly from their moments produce formulas even when those  formulas are complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Still, we expect empirical data in the number field case to be an  important next step in further understanding of the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 11, we explain how  all of Malle’s previous empirical data on class group distributions supports Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1,  including data from a case in which Malle did not make any conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We also discuss  further cases in which it would be interesting to have empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Apart from the details of the formulas for the distribution, there are two new qualitative  phenomena that we conjecture (and in part prove) about the distribution of class groups of  number fields, that are not present in the work of Cohen, Lenstra, Martinet, and Malle, but  can be seen in recent work of Breen, Varma, and Voight [BVVE21] in the context of class  group statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' First, there are some finite Zp[Γ] modules H with HΓ = 1 that cannot occur  as ClK|K0[p∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' These are ones whose multiplicities of dual pairs of irreducible representations  are too far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This can be seen already in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 9, we prove that  these Zp[Γ] modules H that we conjecture occur with density 0 as ClK|K0[p∞] actually never  occur, generalizing earlier results of Leopoldt [Leo58] and Gras [Gra98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Second, we do  not predict the distribution of the parts of ClK|K0[p∞] corresponding to different irreducible  representations of Γ are all independent (as all previous conjectures have).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The behavior  that we conjecture involves explicit dependence between an irreducible representation and  its dual (but otherwise independence between different representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 2, we  3  give more details on the example of K0 = Q and Γ = C7, which is the smallest case where  this dependence occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Further remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 is not precise, in that it does not specify an ordering on E so  that the distribution of class groups is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Cohen and Martinet [CM90] order fields  by Nm Disc K, but this is known to not work in general, even when there are not relevant  roots of unity in the base field [BL20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' 929].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' One possible ordering, as suggested by Bartel  and Lenstra in [BL20] and also by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, is by Nm  �  Disc(K/K0), where √ denotes  the radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We certainly imagine the conjecture only holding for orderings such that the  proportion of fields in E containing any fixed field K1 ̸⊂ K0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Note that these extensions in E in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 are split completely at all  infinite places of K0, because either pr > 2 and all infinite places of K0 are complex, or  p = 2, which implies |Γ| is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If we consider modules with the distribution from Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, and then take  their pk-torsion, we have a distribution on pk-torsion S-modules that does not depend on r  as long as r ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' So, for example, if we are only asking about the p-torsion (equivalently  the groups mod p) then the conjectured distribution does not depend on r for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This  phenomenon was noticed empirically by Malle for the 2-ranks of cyclic cubic extensions of  K0 = Q versus K0 = Q(i) [Mal10, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We can see this phenomenon explicitly in  the formulas of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Even before working out formulas for the distribution, this  follows from the fact that the moments indexed by pk-torsion groups do not depend on r for  r ≥ k, and since these moments determine a unique distribution of pk-torsion modules by  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, this distribution on pk-torsion modules will not depend on r ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, one can work out, for a V from the distribution ν, the distribution of  V as an Fp-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' It is simple to write down the classical moments of |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For a finite  representation V of Γ over Fp, we have  |V |k = | HomFp(V, Fk  p)| = | HomΓ(V, Hom(Fp[Γ], Fk  p))| = | HomΓ(V, Fp[Γ]k))|  (by adjointness and the fact that permutation representations are self-dual).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Every homo-  morphism of Γ-representations V → Fp[Γ]k is a surjection onto exactly one subrepresentation  of Fp[Γ]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus our conjecture implies that the average of |ClK|K0[p]|k is  �  V ⊂Fp[Γ]k  sub-Γ-rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  V Γ=1  |(∧2V )Γ|  |V |u  ,  a value straightforward to write down from the representation theory of Γ over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If one wants to make a conjecture on the product of the Sylow p-subgroups of ClK|K0 for  a finite set of primes p not dividing |Γ|, then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 suggests the distribution with  moments  |(∧2  ZV )Γ[|µ(K0)|]|  |V |u  ,  where µ(K0) is the group of roots of unity of K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (These moments are multiplicative over  Sylow p-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=') The same argument as in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 shows there is a  4  unique such measure, and the formulas are as in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 with an additional product  over p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For infinitely many primes, we would still conjecture that the class group distribu-  tions weakly converge to the unique distribution with the moments above (see [SW22b] for  the relevant topology and the uniqueness theorem), but because of the definition of weak  convergence, that is equivalent to making the conjecture for every finite set of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  As explained in [WW21], the distributions of class groups of Galois extensions, for p ∤ |Γ|,  can be used to get distributions of class groups of non-Galois extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We explain the  implication of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 for distributions of class groups of non-Galois extensions in  Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  There are recent developments [Smi22a, Smi22b] and interesting questions about distri-  butions of the Sylow p-subgroups of ClK for p | |Γ|, including about the interactions with  roots of unity in the base field, but we will not address those questions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  In a forthcoming paper, the authors will extend our function field results and number field  conjectures to the non-abelian analog of the class group, the Galois group of the maximal  unramified extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' One feature of this forthcoming treatment is that we will consider  additional structure on the class group (and its non-abelian analog), as in [SW22a], such  that the formulas for the distribution with this additional structure will be nicer in certain  ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 2, we give the example of Γ = C7, for which we  give the formulas for the distribution in full and one can see the non-independence of dual  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 3, we prove the function field q → ∞ moments that motivate  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 4, we explain the heuristic assumptions that lead from our function  field theorem to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 5, we review the results on the moment problem  for modules that we will need to apply to find formulas for our conjectured distribution from  our conjectured moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Sections 6 and 7, we make some calculations of distributions  on vector spaces and modules over DVR’s that will be necessary for our final formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In  Section 8, we obtain formulas for the distribution of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 9, we show  that the modules that we conjecture appear as class groups with density 0 actually never  appear as class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 10, we show how our conjectures for class group of Galois  extensions lead to explicit conjectures for the distribution of class groups of non-Galois  extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 11, we discuss the match between Malle’s number field data and our  conjectures (which is excellent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 12, we compare our conjectures to conjectures  made by Malle, and Adam and Malle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We would like to thank Yuan Liu and John Voight for helpful  conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We would like to thank Jürgen Klüners, Yuan Liu, and Gunter Malle for  helpful comments on an earlier version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The second author would like  to thank Ben Breen, Ila Varma, and John Voight for interesting conversations on this ques-  tion during their joint investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Will Sawin was supported by NSF grant DMS-2101491  while working on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Melanie Matchett Wood was partially supported by a Packard  Fellowship for Science and Engineering, NSF CAREER grant DMS-1652116, and NSF Wa-  terman Award DMS-2140043 while working on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' She was also a Radcliffe Fellow  during part of this work, and thanks the Radcliffe Institute for Advanced Study for their  support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We write N for the non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We write Fq for the finite field with q elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a group G with an action of Γ, we write GΓ for the invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For a a finite abelian  group A, we write A[p∞] for the Sylow p-subgroup of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For an extension K/K0, we let ClK|K0 be the quotient of the class group ClK of K by the  image of ClK0 under inclusion of ideals from K0 to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (Note for a prime p ∤ [K : K0], we  have that ClK0[p∞] → ClK[p∞] is a injection, since taking the norm map and dividing by  [K : K0] provides an inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  A Γ-extension K/K0 is a Galois field extension K/K0, along with a choice of isomorphism  ι : Gal(K/K0) ≃ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  An isomorphism of Γ-extensions (K, ι), (K′, ι′) is given by a field  isomorphism φ : K → K′, fixing each element of K0 and such that the induced map φ∗ :  Gal(K/K0) → Gal(K′/K0) satisfies ι = ι′ ◦ φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' When K/K0 is a Γ-extension, using ι, there  is an associated action of Γ on ClK|K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Further, ClK|K0 is a Z[Γ]/(�  γ∈Γ γ)-module, since  (�  γ∈Γ γ)I = NmK/K0 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  When K is a function field, and OK is a maximal order in K, we write Cl(OK) for the  quotient of the ideal group of OK by the principal ideals of OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Given a commutative ring R and an R-module M, we write ∧2  RM for the R-module  quotient of M ⊗R M by elements of the form m ⊗ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We write ∧2M for ∧2  ZM  We write Sur(A, B) to denote the set of surjections from A to B in the appropriate  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We also use Aut(M) to denote automorphisms in the appropriate category, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' of R-  modules if M is an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We will write AutR(M) if we think there is a possibility of  confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a finite set S, we write |S| for the number of elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We write Cn for the cyclic group of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  A product �n−1  i=n an is always 1 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We write ηq(k) := �k  i=1(1 − q−i) and also write η(k) = ηq(k) when q is clear from context,  and η(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Example: Γ = C7  In this section, we consider the example K0 = Q and Γ = C7 and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have that  S = Z2[Γ]/(�  γ∈Γ γ) = R1 × R2, where each Ri is isomorphic to the ring of integers in the  degree 3 unramified extension of Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' There are three irreducible representations of C7 over  F2 (or Q2), the trivial representation, and V1, V2, two three dimensional dual representations  (that correspond to R1, R2 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For K/Q a C7-extension, ClK[2∞] is an S-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For an S-module V , we let a1 = a1(V )  denote the R1/2R1-rank of V/(2, R2)V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  the multiplicity of V1 in V/2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We define  a2 = a2(V ) similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 predicts that among such K, the S-module ClK[2∞] is  distributed according to ν, where  ν({V }) =  8a1a2  |V || Aut(V )|  ∞  �  i=1  (1−8−i)  a1  �  i=1  (1−8−1−i)  a2  �  i=1  (1−8−1−i)·  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1 + 8−1  if a1 = a2  1  if |a1 − a2| = 1  0  if |a1 − a2| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  6  (Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4 gives this formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=') For comparison to the moments, note |(∧2  ZpV )Γ[p]| = 8a1a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  One can give an explicit formula for V and | Aut(V )| (see proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 or [Mac15,  II (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  From the above formulas, one can also work out the distribution of V as a Z2-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' As  an example, we work out the distribution of V/2V , so the predicted distribution of ClK[2],  from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since Ri/2Ri ≃ F8, for a finite S-module V , we have that V/2 is a  vector space over F2 with dimension 3k for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For V in the support of ν, we see that  if k is even, we must have a1 = a2 = k/2, and if k is odd, then either a1 = (k − 1)/2 and  a2 = (k + 1)/2, or a1 = (k + 1)/2 and a2 = (k − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus we have  ν({V | |V/2V | = 8k}) =  �  8−k2/4−k(1 − 8−2) �k/2  i=1(1 − 8−i)−2�∞  i=2(1 − 8−i)  if k even  2 · 8−(k2+3)/4−k�(k+1)/2  i=1  (1 − 8−i)−1�(k−1)/2  i=1  (1 − 8−i)−1�∞  i=1(1 − 8−i)  if k odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (We use the formula AutFq(Fn  q ) = qn2 �n  i=1(1 − q−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Function field results  By analyzing components of Hurwitz spaces, Liu, Zureick-Brown, and the second author  [LWZ19] have found q → ∞ moments of Galois groups of maximal unramified extensions  of Γ-extensions of Fq(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this paper, we focus on the class group distributions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' the  abelianizations of those Galois groups), and we give such results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a finite group Γ, let EΓ(n, q) be the set of isomorphism classes of extensions K/Fq(t)  with a choice of isomorphism Gal(K/Fq(t)) ≃ Γ, that are split completely above ∞ and such  that the radical of the discriminant ideal Disc(K/Fq(t)) has norm qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For such a K, let OK  be the maximal order of K over Fq[t] and let Cl(OK) denote its class group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ be a finite group and H be a finite abelian group with action of Γ such  that (|H|, |Γ|) = 1 and HΓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let h be an integer such that h||H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then  lim  x→∞  lim  q→∞  (q,|Γ||H|)=1  (q−1,|H|)=h  �  n≤x  �  K∈EΓ(n,q) | SurΓ(Cl(OK), H)|  �  n≤x |EΓ(n, q)|  = |(∧2H[h])Γ|  |H|  ,  where in the limit q is always a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 follows from [Liu22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1], which is a more general and refined  moment theorem, but we give a short argument here for just the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  See also Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5 for the implication of these averages on the distribution of Cl(OK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This is [LWZ19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4], except that we allow (q − 1, |H|) to vary, and there is  an additional factor of |(∧2H[n])Γ| on the right-hand side, along with some simplifications  because H is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' A finite abelian group with an action of Γ is admissible in the sense  of [LWZ19] if and only if it has order prime to |Γ| and no Γ-invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Also, via class field  theory, the group Gal(K#/K) considered in [LWZ19] has abelianization Cl(OK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The result [LWZ19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4] follows in a straightforward way from [LWZ19, Theorem  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4], and our theorem here follows in exactly the same way from an analog of [LWZ19,  7  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4] in which the hypothesis that (q − 1, |H|) = 1 is replaced by (q − 1, |H|) = h  and the component counting result πG,c(q, n) = πΓ(q, n) + OG(ndΓ(q)−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' is replaced by  πG,c(q, n) = |H2(H, Z)Γ[h]|πΓ(q, n) + OG(ndΓ(q)−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (Recall when H is abelian, H2(H, Z) ≃ ∧2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  This analog of [LWZ19, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4] follows almost entirely the proof of [LWZ19, The-  orem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4], with just a change at the very end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' None of the intermediate results require  that (q − 1, |H|) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' However, at the very end of the proof of [LWZ19, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4], it  is said that “Since the kernel of the map f [: ker(S  1 → G1) → ker(S  2 → G2)] on the right  has order relatively prime to q − 1, any element of ker(S  2 → G2) has the same number of  (q−1 − 1)th roots as any preimage in ker(S  1 → G1).” (We interpret (q−1 − 1) modulo the  order of group in which we are taking roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=') In the more general setting when q −1 and |H|  are not relatively prime, we need to account for the difference in the number of (q−1 − 1)th  roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For any finite group |H| (not necessarily abelian) with (|H|, |Γ|) = 1, we have an exact  sequence  1 → H2(H, Z)Γ → H2(H ⋊ Γ, Z) → H2(Γ, Z) → 1  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' from the Lyndon-Hochschild-Serre spectral sequence and the fact that (|H|, |Γ|) = 1  implies invariants and co-invariants are the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since H2(H, Z) and H2(Γ, Z) are of rela-  tively prime order, this exact sequence splits canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The group ker(S  1 → G1) mentioned  above is a quotient of H2(H ⋊ Γ, Z) by a certain subgroup (referred to in [LWZ19, Section  12] as Qc1) generated by elements that are in the image of a map from H2((Z/|Γ|Z)2, Z)  (since each x ∈ c1 has order dividing |Γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The group ker(S  2 → G2) mentioned above is a  quotient of H2(Γ, Z) by the image of this certain subgroup (Qc2, which is the image of Qc1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  So the kernel of the map f is H2(H, Z)Γ, and in general any element of ker(S  2 → G2) has  |H2(H, Z)Γ[q − 1]| times as many (q−1 − 1)th roots as any preimage in ker(S  1 → G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This  is the only change needed to prove the analog of [LWZ19, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4] described above,  which can then be used to prove this result exactly as in the proof of [LWZ19, Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Heuristic argument  In this section, we explain the heuristic argument that leads to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The Cohen-Martinet conjectures [CM90] include some predictions about Sylow p-subgroups  of relative class groups of Γ-extensions when p | Γ, but in this paper we only consider the  case when p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this case, Cohen and Martinet make conjectures about the distribution  of ClK|K0[p∞] for a base field K0 and the family of all Γ-extensions K/K0 with specified be-  havior at the infinite places of K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this paper, we only consider families E of Γ-extensions  K/K0 in which all infinite places of K0 are required to split completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (See Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  In this case, the only feature of K0 that Cohen and Martinet’s conjectures [CM90] take as  input is the number u of infinite places of K0 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=', [WW21, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' As Malle’s  computations [Mal08, Mal10] suggest and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 shows (in the function field case) one  must also take into account the maximal r such that K0 contains the prth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  One heuristic assumption we make here are that u and r are the only relevant inputs from  the base field into the conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (We hope that conceptual evidence for this heuristic will  8  eventually be found in the function field case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If moments are calculated over a function field  of arbitrary genus, and shown to depend only on u and r, this heuristic assumption would  be more justified, albeit still depending on the reliability of analogy between number fields  and function fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  In the function field case, we take the class group of a ring OK whose fraction field is K,  but unlike the number field case, there is no canonical choice of such a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Instead, we pick  a set of places of K that we consider “infinite places” for the sake of the analogy, and we let  OK be the elements of K that have non-negative valuation at all non-infinite places of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Via this analogy, we expect there to be uniform behavior across number fields and function  fields (taking into account their number of infinite places, and the roots of unity that they  contain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  However, in the function field case, we can change our notion of “infinite places” by adding  an additional place v of K0 to the list of infinite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The new family E′ of Γ-extensions  will be those from the original family E that are split completely at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' It is generally expected  that such a restriction at a non-archimedean place does not change class group statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  see [BV15, Theorem 1], [BV16, Corollary 4], [Woo18, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (We hope that future  results will be proven in the function field case that will more fully justify this expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  Then, ClO′  K|O′  K0 (for the new O′  K, O′  K0) after the inclusion of v as an infinite place, is the  quotient, as a Z[Γ]-module, of the original ClOK|OK0 by any prime of K over v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We assume the  heuristic that this prime is distributed uniformly in ClOK|OK0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (See [Kla17a, Kla17b, Woo18]  for some conjectures and results in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We hope that future results will be proven  in the function field case that will more fully justify this heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  If X is a random  Z[Γ]-module with E(| SurΓ(X, G)|) = MG for all finite Z[Γ]-modules G, and if we let X′ be  the quotient of X by a uniform random element of X, then E(| SurΓ(X′, G)|) = MG/|G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 gives (in a large q limit) moments of Cl(OK) assuming only one infinite place,  and so by the reasoning above, we expect if we modified so as to define OK using u infinite  places of Fq(t), then we would replace the |H| in the denominator of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 with |H|u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (Again, we hope to see such a result proven in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' See also [Liu22, Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 and  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3] for different theoretical heuristics for the |H|u factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  Heuristically further we assume that the limits in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 (and the extension sug-  gested above to u infinite places) can be exchanged, as was proven in some special cases in  [EVW16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since we expect every base field K0 with a fixed number u of infinite places and  prth but not pr+1th roots of unity to give the same class group distributions, if we consider  all q such that q − 1 has a fixed valuation at p, then we do not expect the limit in q to  change the average, and in particular we expect Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 (and the extension suggested  above to u infinite places) to hold without the limit in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Additionally, we expect the same  result for number fields with a given u and r, and this leads precisely to the statement of  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Results on the moment problem for R-modules  In [SW22b], we give a result to construct a distribution from its moments in a rather  general category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this section, we review that result focusing on the category of finite  R-modules for a DVR R with maximal ideal m, and very similar categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let R be either a DVR with finite residue field, a quotient ring of a DVR, or a finite  product of such rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let ˆR denote the product of the completions of R at its maximal  9  ideals m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let P be the set of isomorphism classes of finitely generated ˆR-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For k = (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , kr) ∈ Nr, let mk = �  i mki  i  and for X an ˆR-module let X≤k = X/mkX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We consider a topology on P generated by {X | X≤k ≃ N} for k ∈ Nr and N a finite  R/mk-module (and we take the associated Borel σ-algebra on P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The isomorphism classes of finite R-modules (equivalently, finite ˆR-modules) are in bijec-  tion with r-tuples λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , λr) of finite partitions λi = (λi  1 ≥ λi  2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ), where the parts  of λi are bounded above by the minimal positive integer ai such that mai  i  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (Here, λ  corresponds to the module Nλ := ⊕i,jR/mi  λi  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ) We write λ′ for the conjugate of a partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The operation N �→ N≤k simply truncates the ith associated conjugate partitions after the  first ki terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We say a module N associated to λ is semi-simple if (λi)′  2 = 0 for all i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  N ≃ �  i(R/mi)ei for some positive integers ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let qi = |R/mi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For two r-tuples of partitions λ, π and a surjection g : Nπ → Nλ we  define µ(g) to be 0 if ker g is not semi-simple and �  i(−1)eiq(ei  2)  i  if ker g ≃ �  i(R/mi)ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For  finite R-modules N, P, we define ˆµ(N, P) = �  g∈Sur(P,N)/ Aut(N) µ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a finite R-module N, the N-moment of a measure ν on P is defined to be  �  X∈P  |SurR(X, N)|dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Given values MN for each finite R-module N, for each k ∈ Nr and finite R/mk-module N,  we define (what will be the formulas for our measures)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1)  vk,N :=  �  P finite  R/mk-module  ˆµ(N, P)  | Aut(P)|MP,  and we say the values MN are well-behaved (for R) if the sum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1) is absolutely convergent  for each k ∈ Nr and finite R/mk-module N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 ([SW22b, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='8] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let R be a finite product of quotients of DVRs  and, for each finite R-module N, let MN ∈ R such that the values MN are well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then  we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (Existence): There is a measure on P with N-moment MN for each finite R-module N if and only  if the vk,N defined above are non-negative for each k and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (Uniqueness): When such a measure ν exists, it is unique and is given by the formulas  ν({X | X≤k ≃ N}) = vk,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (Robustness): If νt are measures on P such that for each finite R-module N,  lim  t→∞  �  X∈P  |Sur(X, N)|dνt = MN,  then a measure ν with moments MN exists and the νt weakly converge to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We give some notes on translation from the more general language of [SW22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  In [SW22b, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1], it is explained that the category of R-modules is a “diamond cate-  gory” and the “levels” of [SW22b] correspond to modules over the finite quotient rings R/mk  of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The results [SW22b, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='10 (2)] show that P is as described here, and  [SW22b, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='8] show that µ is as defined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The definition of “well-behaved” in  10  [SW22b] involves an additional factor of Z(π)3, which by [SW22b, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='11] is at most  8r in our case and thus does not affect absolute convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The measures in this paper will turn out to be supported on finite R-modules, which  we will show using the formulas provided by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then we can use the following  statement, weaker than Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, but simpler to think about because it only involves  finite R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let R be a finite product of quotients of DVRs and, for each finite R-module  N, let MN ∈ R such that the values MN are well-behaved and are the moments of a measure  ν supported on finite R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then if νt are measures on the set of isomorphism classes  of finite R-modules such that for each finite R-module N,  lim  t→∞  �  X∈P  |Sur(X, N)|dνt = MN,  then the νt weakly converge to ν (for the discrete topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Measures on the set of isomorphism classes of Fq vector spaces  In this section, we compute some distributions from their moments that will be important  for our eventual class group conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If q is a prime power, then R = Fq is a quotient of a DVR, so we can apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The category of finite R-modules is just the category of finite Fq-vector spaces (and ˆR = R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let q be a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The values MFqρ = q  ρ2+ρ  2  −tρ are well-  behaved (for R) and are the moments of the measure ν (on isomorphism classes of finite  Fq-vector spaces) such that  ν(Fq  λ) =  q−(λ  2)−tλ  �λ  i=1(1 − q−i) �  i≥0(1 + q−i−t)  =  MFqλ  | Aut(Fλq)|  1  �  i≥0(1 + q−i−t)  for all λ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Recall η(k) := �k  i=1(1 − q−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let N = Fλ  p and P = Fλ+e  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' There are q(λ+e)λ homomor-  phisms P → N, a proportion η(λ + e)/η(e) of these are epimorphisms, and for any of these  epimorphisms we have µ(F, G) = (−1)eq(e  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have | Aut(Fρ  q)| = qρ2η(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' So,  ˆµ(Fλ  q, Fλ+e  q  )  | Aut(Fλ+e  q  )| = (−1)eq(e  2)q(λ+e)λη(λ + e)  η(e)qλ2η(λ)q(λ+e)2η(λ + e) = (−1)eq− e2  2 − e  2 −eλ−λ2  η(e)η(λ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Thus  v1,F :=  �  e≥0  ˆµ(Fλ  q, Fλ+e  q  )  | Aut(Fλ+e  q  )|q  (λ+e)2+(λ+e)  2  −t(λ+e) = q−(λ  2)−tλ  η(λ)  �  e≥0  (−1)e  η(e) q−te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We see that the sum is absolutely convergent since t > 0, and thus the moments are well-  behaved as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By the q-binomial theorem for negative powers,  �  e≥0  ve  η(e) =  �  i≥0  (1 − vq−i)−1,  11  and so  v1,F = q−(λ  2)−tλ  η(λ)  �  i≥0  (1 + q−i−t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  Now consider R = Fq × Fq, and we write R-modules as pairs of Fq-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If  one has moments M(N1,N2) that factor as M(N1,N2) = M′  N1M′′  N2, then the sums defining well-  behavedness and the vC,F all factor into two factors, one corresponding to each factor of  R, reducing the moment problem for the moments M(N1,N2) to understanding the moment  problems for M′  N and M′′  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' However, not all such moments necessarily factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let q be a prime power and R = Fq × Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let s1 and s2 be integers with  s1 + s2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let M(Fqρ,Fqπ) = qρπ−s1ρ−s2π for all ρ, π ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If −s1 ≤ λ − φ ≤ s2, let  Dq(λ, φ, s1, s2) :=  q−λ2−φ2+λφ−s1λ−s2φη(∞)η(s1 + s2)  η(λ)η(φ)η(s2 − λ + φ)η(s1 + λ − φ)  =  M(Fqλ,Fqφ)  | Aut(Fq  λ, Fq  φ)|  η(∞)η(s1 + s2)  η(s2 − λ + φ)η(s1 + λ − φ)  and let Dq(λ, φ, s1, s2) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then the values MN are well-behaved for R and are  the moments of the measure ν (on isomorphism classes of pairs of Fq-vector spaces) such  that for all non-negative integers λ, φ,  v((Fq  λ, Fq  φ)) = Dq(λ, φ, s1, s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If s1 = s2 = 0, then the distribution is supported on the pairs such that λ = φ, and on  those reduces to the distribution on Fq-vector spaces whose moments are all 1 (see [SW22b,  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3], a distribution which, at least in the Fp-case, would usually be called the Cohen-  Lenstra distribution on Fp vector spaces, since it is the distribution conjectured by Cohen and  Lenstra [CL84, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' 9 (C5)] to be the distribution of p-torsion of class groups of imaginary  quadratic fields (for odd primes p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' With reasoning as in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, we have  v1,(Fqλ,Fqφ) :=  �  e,f≥0  (−1)e+fq− e2  2 − e  2−eλ−λ2− f2  2 − f  2 −fφ−φ2  η(e)η(f)η(λ)η(φ)  q(λ+e)(φ+f)−s1(λ+e)−s2(φ+f)  =q−λ2−φ2+λφ−s1λ−s2φ  η(λ)η(φ)  �  e,f≥0  (−1)e+fq− e2  2 − e  2−λe+φe−s1e− f2  2 − f  2 −φf+λf−s2f+ef  η(e)η(f)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We can check that this sum converges absolutely as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If we fix e, we have an inner  sum �  f qh(f)/2, where h is a quadratic polynomial with integer coefficients that takes its  maximum values (on integers) at f = e−φ+λ+s2, e−φ+λ+s2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus the inner sum is  bounded by a constant times the summand for f = e−φ+λ+s2, which then gives an upper  bound for the entire (absolute) sum by a geometric series that converges if s1 + s2 + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12  Now, we evaluate the sum over f in the equation for v1,(Fqλ,Fqφ) above using the q-binomial  theorem for negative powers and obtain  v1,(Fqλ,Fqφ) =q−λ2−φ2+λφ−s1λ−s2φ  η(λ)η(φ)  �  e≥0  (−1)eq− e2  2 − e  2−λe+φe−s1e  η(e)  �  k≥0  (1 − q−φ+λ+e−1−s2−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  This final product is 0 if −φ + λ + e − 1 − s2 ≥ 0, and otherwise is η(∞)/η(s2 − λ + φ − e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If λ − φ > s2, then we have v1,(Fqλ,Fqφ) = 0 and otherwise  v1,(Fqλ,Fqφ) =q−λ2−φ2+λφ−s1λ−s2φη(∞)  η(λ)η(φ)  s2−λ+φ  �  e=0  (−1)eq− e2  2 − e  2−λe+φe−s1e  η(e)η(s2 − λ + φ − e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  This final sum can be evaluated by the q-binomial theorem (for n ≥ 0)  n  �  k=0  q−(k  2)  η(n)  η(n − k)η(k)tk =  n−1  �  k=0  (1 + q−kt),  and we obtain  v1,(Fqλ,Fqφ) =q−λ2−φ2+λφ−s1λ−s2φη(∞)  η(λ)η(φ)η(s2 − λ + φ)  s2−λ+φ−1  �  k=0  (1 − q−k−1−λ+φ−s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If λ − φ < −s1, then we can check that one of the factors in the product above is zero, and  otherwise we have  s2−λ+φ−1  �  k=0  (1 − q−k−1−λ+φ−s1) = η(s1 + s2)/η(s1 + λ − φ),  which gives the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Distributions on isomorphism classes of modules over an unramified DVR  In this section, we compute some more distributions from their moments that will be  important for our eventual class group conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let q be a power of a prime p, and throughout this section let R be the ring of integers  in the unramified extension of Qp with residue field Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let m be the maximal ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a partition λ, let w(λ) = �  i(i − 1)λi = �  j  �λ′  j  2  �  = | ∧2  R Nλ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We write (1e) to denote the  partition with e 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We define |λ| := �  i λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We write N≤k := N/mk  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let q, R be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let s, ǫ be real numbers such that s ≥ 0 and s−ǫ ≥ 0,  and r a positive integer, and let MNρ′ = |∧2  R Nρ′[pr]||Nρ′[pr]|ǫ|Nρ′|−s = q  �r  j=1((  ρj  2 )+ǫρj)−s|ρ| for  all partitions ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We define  Eq(t, m) :=  m  �  e=0  (−1)eq−(t+1)eη(m)  η(e)η(m − e)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  13  Then Eq(t, m) is positive for all real t ≥ 0 and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Further, the values MN are well-  behaved for R and are the moments of the measure ν such that  ν({P | P ≤k ≃ Nλ′}) = q  �  j(−(  λj  2 )−(s−ǫ+1)λj)+�  j≥r+1(−(  λj  2 )−ǫλj)  �  j≥2 η(λj−1 − λj)  ��  i≥0  (1 + q−s+ǫ−1−i)−1  �  ×  min(k,r)  �  j=2  Eq(s − ǫ, λj−1 − λj)  λmin(k,r)  �  i=λk+1  (1 − q−s−i)  =  MNλ′  | Aut(Nλ′)|  �  i≥0  (1 + q−s+ǫ−1−i)−1  min(k,r)  �  j=2  Eq(s − ǫ, λj−1 − λj)  λmin(k,r)  �  i=λk+1  (1 − q−s−i)  for all k ≥ 1 and λ with λk+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This measure is supported on finite R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Finally,  ν({Nλ′}) is given by the above expression for any k > λ′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  By convention, the product from λk + 1 to λmin(k,r) is 1 if λk = λmin(k,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The result also is  proven below for the case r = ∞, with the natural interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If we let ae =  q−(t+1)e  η(e)η(m−e), we note that  ae+1  ae  = q−t−1(1 − q−m+e)  (1 − q−e−1)  < q−1  1  1 − q−1 ≤ 1,  since t ≥ 0 and e ≥ 0 and q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' So Eq(t, m) is an alternating series whose terms are  decreasing in absolute value and whose first term is positive, which implies Eq(t, m) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We now apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We let N = Nλ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Given a partition ρ, the number of  surjections Nρ′ → Nλ′ with simple kernel of type (1e), up to automorphisms of Nλ′, is  q  �  j (  ρj  2 )−�  j (  λj  2 )−(e  2) �  j≥1  η(ρj − ρj+1)  η(ρj − λj)η(λj − ρj+1)  if |ρ| = |λ| + e and ρj − λj, λj − ρj+1 ≥ 0 for all j ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' and 0 otherwise [Mac15, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' II  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Each of these surjections has µ(Nλ′, Nρ′) = (−1)eq(e  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let ρ′ contain mj j’s (so  mj = ρj − ρj+1 and ρj = mj + mj+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have | Aut(Nρ′)| = q  �  j ρ2  j �  j η(mj) ([Mac15,  14  II (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have | ∧2  R Nρ′[pr]| = q  �r  j=1 (  ρj  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Letting ρj = λj + ej, we have  vk,Nλ′ =  �  e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=',ek≥0  ∀i≥1:ei+1≤λi−λi+1  q  �  j (  ρj  2 )−�  j (  λj  2 )−(e  2) �  j≥1  η(ρj−ρj+1)  η(ρj−λj)η(λj−ρj+1)(−1)ejq(  ej  2 )q  �r  j=1((  ρj  2 )+ǫρj)−s|ρ|  q  �  j(ρj)2 �  j≥1 η(mj)  =  �  e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=',ek≥0  ∀i≥1:ei+1≤λi−λi+1  (−1)  �  j ejq  �  j(2(  λj+ej  2 )−(  λj  2 )−(s−ǫ)(λj+ej))−�  j>r((  λj+ej  2 )+ǫ(λj+ej))  q  �  j(λj+ej)2 �  j≥1 η(ej)η(λj − λj+1 − ej+1)  =  �  e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=',ek≥0  ∀i≥1:ei+1≤λi−λi+1  (−1)  �  j ejq  �  j(−(  λj  2 )−(s−ǫ+1)λj−(s−ǫ+1)ej)−�  j>r((  λj+ej  2 )+ǫ(λj+ej))  �  j≥1 η(ej)η(λj − λj+1 − ej+1)  = q  �  j(−(  λj  2 )−(s−ǫ+1)λj)  η(λk)  ��  e1≥0  (−1)e1 q−(s−ǫ+1)e1  η(e1)  � min(k,r)  �  j=2  λj−1−λj  �  ej=0  (−1)ejq−(s−ǫ+1)ej  η(ej)η(λj−1 − λj − ej)  ×  k  �  j=min(k,r)+1  λj−1−λj  �  ej=0  (−1)ejq−(s−ǫ+1)ej−(  λj+ej  2 )−ǫ(λj+ej)  η(ej)η(λj−1 − λj − ej)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The above sums are absolutely convergent when s − ǫ + 1 > 0, so we have well-behavedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We use the q-binomial theorem for negative powers for the e1 sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For 2 ≤ j ≤ min(k, r),  the sum over ej is equal to Eq(s − ǫ, λj−1 − λj)/η(λj−1 − λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Considering the sum over ej  for j > min(k, r), we have  λj−1−λj  �  ej=0  (−1)ejq−(s−ǫ+1)ej−(  λj+ej  2 )−ǫ(λj+ej)  η(ej)η(λj−1 − λj − ej)  = q−(  λj  2 )−ǫλj  λj−1−λj  �  ej=0  (−q−s−1−λj)ejq−(  ej  2 )  η(ej)η(λj−1 − λj − ej)  =  q−(  λj  2 )−ǫλj  η(λj−1 − λj)  λj−1−λj−1  �  i=0  (1 − q−s−1−λj−i)  by the q-binomial theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We can combine the products above over different j and have  k  �  j=min(k,r)+1  λj−1−λj−1  �  i=0  (1 − q−s−1−λj−i) =  λmin(k,r)  �  i=λk+1  (1 − q−s−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We put these identities all together, pull out all the powers of q, and collect the η(λj−1 − λj)  terms in the denominator to obtain the equation for ν({P | P ≤k ≃ Nλ′}) in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  There are countably many finitely generated ˆR-modules, and we will show each infinite  one occurs with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let N be a finitely generated ˆR-module and define λ(k) so  that Nλ(k)′ = N≤k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In particular, if N = ˆRℓ × T for some T with mk0T = 0, then for k ≥ k0,  we have that λ(k)j does not depend on k for j ≤ k0, and λ(k)j = ℓ for k0 < j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' So, if  ℓ ≥ 1, as k → ∞, we have ν({P | P ≤k ≃ Nλ(k)′}) → 0, which implies ν({N}) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus the  measure is supported on finite modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If λ′  1 < k, then the only finitely generated ˆR-module P with P ≤k ≃ Nλ′ is P = Nλ′, which  gives the final statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  15  We write E(t, m) := Eq(t, m) when the value of q is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' While we do not have a general  product formula for E(t, m), we do have one when t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For m a non-negative integer,  E(0, m) =  ⌈m/2⌉  �  j=1  (1 − q−(2j−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' There are two forms of Pascal’s identity for the q-binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  These are, for 1 ≤ e ≤ m − 1,  η(m)  η(e)η(m − e) = q−e  η(m − 1)  η(e)η(m − 1 − e) +  η(m − 1)  η(e − 1)η(m − e)  and  η(m)  η(e)η(m − e) =  η(m − 1)  η(e)η(m − 1 − e) + q−(m−e)  η(m − 1)  η(e − 1)η(m − e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  They can both be verified immediately by expanding the products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Plugged into  E(t, m) =  m  �  e=0  (−1)eq−(t+1)e  η(m)  η(e)η(m − e)  these give the recurrences  E(t, m) = E(t + 1, m − 1) − q−(t+1)E(t, m − 1)  and  E(t, m) = E(t, m − 1) − q−m−tE(t − 1, m − 1)  Specializing we get  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3)  E(−1, m) = E(0, m − 1) − E(−1, m − 1)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4)  E(0, m) = E(0, m − 1) − q−mE(−1, m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We have E(−1, m) = 0 for m odd since the e term cancels the m−e term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' That, together  with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3), gives  E(0, 2k) = E(−1, 2k)+E(−1, 2k+1) = E(−1, 2k) = E(−1, 2k)+E(−1, 2k−1) = E(0, 2k−1)  which together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4) for m = 2k − 1 gives  E(−1, 2k) = E(0, 2k − 1) = E(0, 2k − 2) − q−(2k−1)E(−1, 2k − 2)  =E(−1, 2k − 2) − q−(2k−1)E(−1, 2k − 2) = (1 − q−(2k−1))E(−1, 2k − 2)  so by induction and E(−1, 0) = 1 we have  E(0, 2k) = E(0, 2k − 1) = E(−1, 2k) =  k  �  j=1  (1 − q−(2j−1))  which gives the stated formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  Before we consider our next moments, we verify that a certain q-series is always non-  negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  16  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For integers λ−1, λ, φ−1, φ, s1, s2, with λ−1 ≥ λ and φ−1 ≥ φ, we define  Fq(λ−1, λ, φ−1, φ, s1, s2) :=  λ−1−λ  �  e=0  φ−1−φ  �  f=0  (−1)e+fq(λ+e  2 )−(λ  2)−s1(λ+e)+(φ+f  2 )−(φ  2)−s2(φ+f)+(λ+e)(φ+f)  q(λ+e)2+(φ+f)2η(e)η(λ−1 − λ − e)η(f)η(φ−1 − φ − f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If φ−1 − λ−1 ≤ s1 and λ−1 − φ−1 ≤ s2, then  (1) Fq(λ−1, λ, φ−1, φ, s1, s2) ≥ 0, and  (2) Fq(λ−1, λ, φ−1, φ, s1, s2) is positive if and only if φ − λ ≤ s1 and λ − φ ≤ s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We rearrange to obtain  Fq(λ−1, λ, φ−1, φ, s1, s2)  =  λ−1−λ  �  e=0  φ−1−φ  �  f=0  (−1)e+fq(λ+e  2 )−(λ  2)−s1(λ+e)+(φ+f  2 )−(φ  2)−s2(φ+f)+(λ+e)(φ+f)  q(λ+e)2+(φ+f)2η(e)η(λ−1 − λ − e)η(f)η(φ−1 − φ − f)  = q−λ2−s1λ−φ2−s2φ+λφ  λ−1−λ  �  e=0  φ−1−φ  �  f=0  (−1)e+fq−(e  2)+e(−λ−s1+φ−1)−(f  2)+f(−φ−1−s2+λ+e)  η(e)η(λ−1 − λ − e)η(f)η(φ−1 − φ − f)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Now we apply the q-binomial theorem to the sum in the f, and obtain that the above sum  is equal to  q−λ2−s1λ−φ2−s2φ+λφ  λ−1−λ  �  e=0  (−1)eq−(e  2)+e(−λ−s1+φ−1)  η(e)η(λ−1 − λ − e)η(φ−1 − φ)  φ−1−φ−1  �  i=0  (1 − q−φ−1−s2+λ+e−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If e ≥ s2 − λ + φ + 1, then the i = −φ − 1 − s2 + λ + e term in the product exists because  0 ≤ −φ − 1 − s2 + λ + e ≤ −φ − 1 − s2 + λ−1 ≤ φ−1 − φ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  This term is (1 − q0) = 0, and thus the summand for e ≥ s2 − λ + φ + 1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If λ − φ > s2, in then follows that every summand is 0 and Fq(λ−1, λ, φ−1, φ, s1, s2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Similarly, if φ − λ > s1 we also have Fq(λ−1, λ, φ−1, φ, s1, s2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For the rest of the proof we assume that φ − λ ≤ s1 and λ − φ ≤ s2, and we only consider  summands with e ≤ s2 − λ + φ as the others are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this case the exponents of q in the  product are  −φ − 1 − s2 + λ + e − i ≤ −1  and so the only non-positive factor in each summand is the (−1)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If we let ae be the e summand in our sum, then we can compute  |ae+1|  |ae|  = q−e−λ−s1+φ−1(1 − qλ−1−λ−e)  (1 − q−e−1)  (1 − q−φ−1−s2+λ+e+1)  (1 − q−s2+λ+e−φ−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For 0 ≤ e ≤ min(s2 − λ + φ, λ−1 − λ) − 1, we then have  |ae+1|  |ae|  < q−e−λ−s1+φ−1  (1 − q−1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If −e − λ − s1 + φ − 1 ≤ −2, then the above is ≤ 1, then then our sum for Fq is an  alternating sum with first term positive and terms decreasing in absolute value, and thus  (strictly) positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Otherwise, if −e − λ − s1 + φ − 1 ≥ −1, then e ≤ −λ − s1 + φ ≤ 0, and  17  so −λ − s1 + φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Similarly, Fq is positive or −φ − s2 + λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If −φ − s2 + λ = 0, then  there is only 1 summand we consider, e = 0, and thus Fq is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We define ¯Fq(λ−1, λ, φ−1, φ, s1, s2) so that  Fq(λ−1, λ, φ−1, φ, s1, s2) =  q−λ2−s1λ−φ2−s2φ+λφ  η(λ−1 − λ)η(φ−1 − φ)  ¯Fq(λ−1, λ, φ−1, φ, s1, s2),  where Fq is defined in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Now we consider the ring R × R, whose modules we view as pairs of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let q, R be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let s1, s2 be integers such that s1, s2 ≥ 0, and let  r ≥ 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let MNρ′,Nπ′ = q(  �r  j=1 ρjπj)−s1|ρ|−s2|π| for all partitions ρ, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then the  values MN are well-behaved for R × R and are the moments of a measure ν such that for all  k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' k2 ≥ 1 and all partitions λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' φ with λk1+1 = φk2+1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' we have  ν({P | P ≤k ≃ (Nλ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Nφ′)}) = q(  �r  j=1 λjφj)+�  j −(λj)2−s1λj−(φj)2−s2φj  �  j≥2 η(λj−1 − λj)η(φj−1 − φj)  ×  η(∞)η(s1 + s2)  η(s2 − λ1 + φ1)η(s1 + λ1 − φ1)  min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='k2)  �  j=2  ¯Fq(λj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' φj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' φj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' s2)  ×  λmin(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='k2)  �  i=λk1+1  (1 − q−s1−i)  λmin(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='k2)  �  i=φk2+1  (1 − q−s2−i)  if −s1 ≤ λ1 − φ1 ≤ s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' and ν({P | P ≤k ≃ (Nλ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Nφ′)}) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The above expression  is positive if and only if −s1 ≤ λj − φj ≤ s2 for all 1 ≤ j ≤ min(r, k1, k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The measure ν is  supported on pairs of finite R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Note the factor  q(  �r  j=1 λjφj)+�  j −(λj)2−s1λj−(φj)2−s2φj  �  j≥2 η(λj−1 − λj)η(φj−1 − φj)  =  MNλ′,Nφ′  | Aut(Nλ′, Nφ′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By considerations as in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 (in particular the second line of  the large display) we have  vk,(Nλ′,Nφ′)  =  �  e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=',ek1≥0  f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=',fk2≥0  ∀i≥1:ei+1≤λi−λi+1  ∀i≥1:fi+1≤φi−φi+1  (−1)  �  j ej+fjq  �  j((  λj+ej  2 )−(  λj  2 )−s1(λj+ej)+(  φj +fj  2 )−(  φj  2 )−s2(φj+fj))+�min(r,k1,k2)  j=1  (λj+ej)(φj+fj)  q  �  j(λj+ej)2+(φj+fj)2 �  j η(ej)η(λj − λj+1 − ej+1)η(fj)η(φj − φj+1 − fj+1)  (By convention, λj = ej = 0 for j > k1 and φj = fj = 0 for j > k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  If k1 or k2 = 0, then vk,(Nλ′,Nφ′) can be computed just in the category of finite R-modules,  and this case is treated in [SW22b, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3] for moments MN = |N|−s, and in particular  vk,(Nλ′,Nφ′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  As in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, the above sum factors over j, and the only factor that  is an infinite sum is the j = 1 sum, which is the same as the sum in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 (the  18  λ, φ, e, f there are the λ1, φ1, e1, f1 here, respectively), as long as k1, k2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By the argument  in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 we then have that the above sum converges absolutely as long  as s1 + s2 + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The j = 1 factor  �  e1≥0  f1≥0  (−1)e1+f1q((λ1+e1  2 )−(λ1  2 )−s1(λ1+e1)+(φ1+f1  2 )−(φ1  2 )−s2(φ1+f1))+(λ1+e1)(φ1+f1)  q(λ1+e1)2+(φ1+f1)2η(e1)η(f1)  evaluates to η(λ1)η(φ1)Dq(λ1, φ1, s1, s2) as in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, and in particular  is non-negative and is positive if and only if φ1 − λ1 ≤ s1 and λ1 − φ1 ≤ s2  The factor for 2 ≤ j ≤ min(r, k1, k2) is Fq(λj−1, λj, φj−1, φj, s1, s2), and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5 ad-  dresses exactly when this is positive or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Inductively, it follows that  Dq(λ1, φ1, s1, s2)  min(r,k1,k2)  �  j=2  Fq(λj−1, λj, φj−1, φj, s1, s2) ≥ 0,  and is positive if and only if φj − λj ≤ s1 and λj − φj ≤ s2 for all 1 ≤ j ≤ min(r, k1, k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The λ part of the factor for min(r, k1, k2) < j ≤ k1 further factors into  λj−1−λj  �  ej=0  (−1)ejq((  λj+ej  2 )−(  λj  2 )−s1(λj+ej))  q(λj+ej)2η(ej)η(λj−1 − λj − ej) =q−(λj)2−s1λj  λj−1−λj  �  ej=0  (−1)ejq−(  ej  2 )+ej(−λj−s1+1)  η(ej)η(λj−1 − λj − ej)  = q−(λj)2−s1λj  η(λj−1 − λj)  λj−1−λj−1  �  i=0  (1 − q−λj−s1−1−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  for each j ≤ k1, and an analogous expression for j ≤ k2 involving φj, φj−1, s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since si ≥ 0, we  see the exponents of q in the products above are all negative, and thus the above expression  is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We pull out all the powers of q and expressions η(λj−1 − λj) and η(φj−1 − φj), and  we combine the remaining products for the j > min(r, k1, k2) terms as in the proof of  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Putting this all together, we obtain the formula for the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  There are countably many finitely generated ˆR× ˆR-modules, and we will show each infinite  one occurs with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let N = (N1, N2) be an ordered pair of finitely generated ˆR-  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We define partitions λ(k) and φ(k) such that (Nλ(k)′, Nφ(k)′) = N≤k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In particular,  if Ni = ˆRℓi × Ti for some Ti with mk0Ti = 0, then for k ≥ k0, we have that λ(k)j and φ(k)j  do not depend on k for j ≤ k0, and λ(k)j = ℓ1 and φ(k)j = ℓ2 for k0 < j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If either ℓ1 ≥ 1  or ℓ2 ≥ 1, then ν({P | P ≤k ≃ N≤k}) → 0, which implies ν({N}) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus the measure is  supported on pairs of finite modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Representations of Γ and formulas for our conjectured distributions  Let Γ be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this section, we explain the translation between the moment  problem for Zp[Γ]-modules and the one for the kind of R we considered in Sections 5, 6 and  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a prime p ∤ |Γ|, we let T = Zp[Γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since p ∤ |Γ|, the primitive central idempotents  e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , em in Qp[Γ] are all in in Zp[Γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The isomorphism classes of indecomposable projective  T-modules Vi are in bijection with the ei such that the action of T on Vi factors through  eiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then the (distinct isomorphism classes of) irreducible representations of Γ over Qp are  19  Vi ⊗ Qp and the (distinct isomorphism classes of) irreducible representations of Γ over Fp  are Vi ⊗ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We let e1 = |Γ|−1 �  γ∈Γ γ, corresponding to the trivial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since  T = �m  i=1 eiT, the category of finite T-modules is the product of the categories of finite eiT  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Each eiT is isomorphic to the ring of ni × ni matrices Mni×ni(Rdi) over Rdi, the  ring of integers in the degree di unramified extension of Qp, where nidi is the rank of Vi over  Zp, and di is the number of irreducible representations that Vi⊗Qp (or Vi ⊗Fp) is the sum of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' [Ser77, Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5], for some of this background on representations  when p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  By the Morita theorem, the category of Rdi-modules is equivalent to the category of  eiT-modules, by an equivalence Fi that takes an Rdi-module H to an eiT-module whose  underlying Rdi-module is Hni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus, the computations that we have done above in the  categories of Rdi-modules can be applied to eiT-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  There is an involution σ on the primitive central idempotents ei, fixing those corresponding  to self-dual representations (over Qp or Fp), and exchanging the ei, ei′ corresponding to pairs  of dual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For an ei fixed by σ corresponding to a representation Vi over Fp,  let κi = Fpdi = EndΓ(Vi), and let ǫi = −1 if (∧2  κiVi)Γ ̸= 0, let ǫi = 1 if (∧2  κiVi)Γ = 0 but  (Vi ⊗κi Vi)Γ ̸= 0, and let ǫi = 0 if (Vi ⊗κi Vi)Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The following lemmas are straightforward calculations from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ, p, ei, di, ni, ǫi be as above such that ei corresponds to a self-dual repre-  sentation Vi over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let λ be a partition, and let Vλ be the eiT-module corresponding (via  the Morita theorem) to the Rdi-module Nλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let q = pdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let r ≥ 0 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then  |(∧2  ZpVλ′[pr])Γ||Vλ′|−u = q  �r  j=1 (  λj  2 )+(1−ǫi)  λj  2 +�  j≥0 −uniλj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ, p, ei, di, ni be as above such that ei, ei′ correspond to non-isomorphic  dual representations Vi, Vi′ over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let λ, φ be partitions, and let Vλ be the eiT-module  corresponding (via the Morita theorem) to the Rdi-module Nλ, and Wφ be the ei′T-module  corresponding to the Rdi-module Nφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let q = pdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let r ≥ 0 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then  |(∧2  Zp(Vλ′ × Wφ′)[pr])Γ||Vλ′ × Wφ′|−u = q  �r  j=1 λjφj+�  j≥0 −uniλj−uniφj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let mi be the maximal ideal of Rdi (which is generated by p, but we write like this  since we will consider modules for the product of Rdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=') For k = (k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , km) ∈ Nm−1, we  write mk = �  i≥2 mki  i  for the ideal of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The finite T/e1T-modules are given by tuples  λ = (λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , λm) of partitions such that λ corresponds to the module Vλ := ⊕iFi(Nλi) (and  Nλi is the Rdi-module described in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 For a T/e1T-module V , we let V ≤k = V/mkV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a tuple λ of partitions, we write λ′ for the tuple in which each partition is the conjugate  of the corresponding partition from λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We can now give an explicit description of the measure of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Recall that Eq  and ¯Fq are defined in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 and Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ be a finite group and p a prime such that p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let m, T, ei, ni, di, ǫi  be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let qi = pdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let r, u be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let S = T/e1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then there is a  unique measure ν on the set of isomorphism classes of finite S-modules (with the discrete  1We apologize for the slight abuse of notation, since the ring R is implicit in the notation Nλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  20  topology and σ-algebra) such that  �  X  | Sur(X, V )|dν =  |(∧2  ZpV )Γ[pr]|  |V |u  for every finite S-module V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For all k ∈ Nm−1 and tuples of partitions λ such that λi  ki+1 = 0  for all i ≥ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  ν({V | V ≤k ≃ Vλ′}) =  |(∧2  ZpVλ′)Γ[pr]|  |Vλ′|u| Aut(Vλ′)|  ×  �  i≥2  σ(ei)=ei  \uf8eb  \uf8ec  \uf8ed  �  ℓ≥0  (1 + q  −uni− ǫi+1  2  −ℓ  i  )−1  min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='ki)  �  ℓ=2  Eqi(uni − 1 − ǫi  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λi  ℓ−1 − λi  ℓ)  λi  min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='ki)  �  ℓ=λi  ki+1  (1 − q−uni−ℓ  i  )  \uf8f6  \uf8f7  \uf8f8  ×  �  2≤i<j  σ(ei)=ej  �  ηqi(∞)ηqi(2uni)  ηqi(uni − λi  1 + λj  1)ηqi(uni + λi  1 − λj  1)  min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='kj)  �  ℓ=2  ¯Fqi(λi  ℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λi  ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λj  ℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λj  ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' uni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' uni)  λi  min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='kj)  �  ℓ=λi  ki+1  (1 − q−uni−ℓ  i  )  λj  min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='kj)  �  ℓ=λj  kj +1  (1 − q−uni−ℓ  i  )  \uf8f6  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  if |λi  1 − λj  1| ≤ uni for all 2 ≤ i < j with σ(ei) = ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' and ν({V | V ≤k ≃ Vλ′}) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The expression above is positive if and only if |λi  ℓ−λj  ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej  and all 1 ≤ ℓ ≤ min(r, ki, kj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Moreover, any sequence νt of measures of isomorphism classes  of finite S-modules such that  lim  t→∞  �  X  |Sur(X, V )|dνt =  |(∧2  ZpV )Γ[pr]|  |V |u  ,  for every finite S-module V , has the property that limt→∞ νt(V ) = ν(V ) for every finite  S-module V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If one wants a completely explicit formula, |(∧2  ZpV )Γ[pr]| is multiplicative over σ-orbits of  the ei, and explicit formulas for each factor are given in Lemmas 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We also have  | Aut(Vλ′)| = q  �  i,j(λi  j)2 �  i,j ηqi(λi  j − λi  j+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Using the Morita Theorem, we can translate this to an analogous  statement on finite �  i≥2 Rdi-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We then apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Here note that the  moments factor over the σ orbits of e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus the sum defining well-behavedness and  giving formulas for the measure similarly factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In each factor, we have moments given by  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 or Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, to which we can apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 or Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7 to obtain  not only that the moments are well-behaved, but that the resulting measures are supported  on finite modules and given by the corresponding factor of the formula in this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' When  we apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, we do so with s = uni and ǫ = 1−ǫi  2 , and so we use u ≥ 1 so that  s − ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Noting that ni = nj, qi = qj gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  We can prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The formulas for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 can be obtained from  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 by taking k = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  21  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The proof is very similar to that of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 except that we apply  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 for the category of representations of Γ over Fp instead of the category Zp[Γ]-  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The well-behavedness required to apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 is just the k = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' )  special case of the well-behavedness used in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The formulas for the probabilities given by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 are then precisely the k =  (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ) formulas from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  This simplifies the formulas substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Each  of the partitions λi become integers λi  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For λ′ such that (λ′)i = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , 1) for each i and (λi  1  gives the number of 1), we have  ν({V | V/pV ≃ Vλ′}) =  |(∧2  ZpVλ′)Γ[p]|  |Vλ′|u| Aut(Vλ′)|×  �  i≥2  σ(ei)=ei  �  ℓ≥0  (1 + q  −uni− ǫi+1  2  −ℓ  i  )−1  �  2≤i<j  σ(ei)=ej  �  ηqi(∞)ηqi(2uni)  ηqi(uni − λi  1 + λj  1)ηqi(uni + λi  1 − λj  1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  if |λi  1−λj  1| ≤ uni for all 2 ≤ i < j with σ(ei) = ej and ν({V | V/pV ≃ Vλ′}) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  Similarly, to obtain an analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 describing the pk-torsion, we would take  k = (k, k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Since a finite S-module can be detected mod mk for sufficiently large ki, we can use  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 to give the measure on any particular S-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For the measure ν in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 and any tuple λ of partitions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' we have  ν({Vλ′}) =  |(∧2  ZpVλ′)Γ[pr]|  |Vλ′|u| Aut(Vλ′)|  ×  �  i≥2  σ(ei)=ei  \uf8eb  \uf8ed�  ℓ≥0  (1 + q  −uni− ǫi+1  2  −ℓ  i  )−1  r�  ℓ=2  Eqi(uni − 1 − ǫi  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λi  ℓ−1 − λi  ℓ)  λi  r  �  ℓ=1  (1 − q−uni−ℓ  i  )  \uf8f6  \uf8f8  ×  �  2≤i<j  σ(ei)=ej  �  ηqi(∞)ηqi(2uni)  ηqi(uni − λi  1 + λj  1)ηqi(uni + λi  1 − λj  1)  r�  ℓ=2  ¯Fqi(λi  ℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λi  ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λj  ℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' λj  ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' uni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' uni)  λi  r  �  ℓ=1  (1 − q−uni−ℓ  i  )  λj  r  �  ℓ=1  (1 − q−uni−ℓ  i  )  \uf8f6  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  if |λi  1 − λj  1| ≤ uni for all 2 ≤ i < j with σ(ei) = ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' and ν({Vλ′}) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The  expression above is positive if and only if |λi  ℓ − λj  ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej  and all 1 ≤ ℓ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We apply Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 with ki sufficiently large such that {V | V ≤k ≃ Vλ′} = {Vλ′}  (so in particular, λi  ki = 0) and also such that ki ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  We now have a function field corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 on the distribution of class groups  of function fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Recall EΓ(n, q) is the set of isomorphism classes of extensions K/Fq(t)  with a choice of isomorphism Gal(K/Fq(t)) ≃ Γ, that are split completely above ∞ and  22  such that the radical of the discriminant ideal Disc(K/Fq(t)) has norm qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For a finite set  of primes P, an abelian group A, let AP be the product of the Sylow p-subgroups of A for  p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' An abelian P-group is a finite product of abelian p-groups for p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ be a finite group and P a finite set of primes not dividing |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let H  be a finite abelian P-group Γ-module such that HΓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For each p ∈ P, let rp be a positive  integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' be any sequence of prime powers, all relatively prime to |Γ| and all primes  in P, and such that for all i, we have that Fqi(t) contains the prpth roots of unity, but not  the prp+1st roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Then, as long as qx grows sufficiently fast with x, we have  lim  x→∞  �  n≤x |{K ∈ EΓ(n, qx) | Cl(OK)P ≃Γ H)}  �  n≤x |EΓ(n, qx)|  =  �  p∈P  νp(H{p}),  where νp is the measure from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 (given Γ, p, r = rp and u = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We start with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' It is a formal consequence from the statement of Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 with two limits to obtain the same statement with just limx→∞ and q replaced by qx  (growing sufficiently fast with x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Now we apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, and in particular uniqueness  and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 shows that these moments are well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus  the corollary follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Class groups cannot be certain modules  Using Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 predicts that the density of fields for which Vλ is the  class group is zero unless |λi  ℓ − λj  ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej, and 1 ≤ ℓ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We can verify this prediction by showing that, in fact, fields violating this condition do not  exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Γ be a finite group and p a prime such that p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let m, T, ei, ni, di be  as at the start of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let r and u be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let K0 be a number field with u infinite places containing the prth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let K  be a Galois extension of K0 with Galois group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let λ be the unique tuple of partitions such  that ClK[p∞]/ClK0[p∞] ∼= Vλ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Then |λi  ℓ − λj  ℓ| ≤ uni for all 2 ≤ i < j with σ(ei) = ej, and 1 ≤ ℓ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The case r = 1 of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 was shown previously by Gras [Gra98, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7(α)(i)  and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='8(α)(i)], with the special case where K is Galois over Q obtained earlier  by Leopoldt [Leo58, Satz 2 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' See also [BVVE21, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2] for the case  K0 = Q and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' As in [LST20, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1], we can define a map ψK  Cl(K)∨[pr] ∼= H1(Spec OK, Z/prZ) → H1(Spec OK, µpr) → Cl(K)[pr]  where the first isomorphism arises from class field theory, the middle arrow is cup product  with a generator of the prth roots of unity of K0, and the last arrow arises from the Kummer  exact sequence and the identification Cl(K) ∼= H1(Spec OK, Gm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Each step is clearly Γ-  invariant (because, in particular, the roots of unity of K0 are Γ-invariant) so this gives a  map of Γ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  23  By the argument in the first two paragraphs of the proof of [LST20, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='21], the  kernel of ψK is isomorphic as a Zp[Γ]-module to a submodule of O×  K ⊗ Z/pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let 2 ≤ i < j with σ(ei) = ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Applying the left-exact functor HomΓ(Vi, ·), we see that  there is a map  HomΓ(Vi, Cl(K)∨)[pr] → HomΓ(Vi, Cl(K))[pr]  whose kernel is a subgroup of HomΓ(Vi, O×  K⊗Z/pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have HomΓ(Vi, Cl(K)) ∼= �∞  ℓ=1(Rdi/pℓ)λi  ℓ−λi  ℓ+1  so HomΓ(Vi, Cl(K))[pr] ∼= �r−1  ℓ=1(Rdi/pℓ)λi  ℓ−λi  ℓ+1 × (Rdi/pr)λi  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Similarly HomΓ(Vi, Cl(K)∨) ∼=  �∞  ℓ=1(Rdi/pℓ)λj  ℓ−λj  ℓ+1 so HomΓ(Vi, Cl(K)∨)[pr] ∼= �r−1  ℓ=1(Rdi/pℓ)λj  ℓ−λj  ℓ+1 × (Rdi/pr)λj  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Since our assumptions imply K/K0 is split at ∞, each infinite place of K0 lies under a  simply transitive Γ-orbit of infinite places of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus, by Dirichlet’s unit theorem, O×  K ⊗ R  is isomorphic to R[Γ]u/R as an R[Γ]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since p is prime to the order of Γ, this implies  that O×  K ⊗Zp, modulo torsion, is isomorphic to Zp[Γ]u/Zp as a Zp[Γ]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Further, using  p ∤ |Γ|, we then have that, as a Zp[Γ]-module, O×  K ⊗ Zp ≃ Zp[Γ]u/Zp × (µ∞(K) ⊗ Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since  the pth roots of unity are in K0, the cyclic group µ∞(K) ⊗ Zp has a trivial Γ-submodule,  and hence has trivial Γ-action since p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus Hom(Vi, O×  K ⊗ Z/pr) ∼= (Rdi/pr)uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  It follows that we have a map  r−1  �  ℓ=1  (Rdi/pℓ)λj  ℓ−λj  ℓ+1 × (Rdi/pr)λj  r →  r−1  �  ℓ=1  (Rdi/pℓ)λi  ℓ−λi  ℓ+1 × (Rdi/pr)λi  r  whose kernel has Rdi/p-rank ≤ uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We say the Rdi-module �  ℓ≥1(Rdi/pℓ)ρℓ has Rdi/pℓ-rank  ρℓ + ρℓ+1 + · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since in a map of p-group Rdi-modules, the Rdi/pℓ-rank of the source minus  the Rdi/pℓ rank of the target is at most the Rdi/p-rank of the kernel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  see [Mac15,  II (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3)(i)]), this gives λj  ℓ − λi  ℓ ≤ uni for each 1 ≤ ℓ ≤ r A symmetric argument gives  λi  ℓ − λj  ℓ ≤ uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Non-Galois extensions  In this section, we explain how our conjecture also gives, as a consequence, the distribution  of Sylow p-subgroups of class groups of non-Galois fields in the presence of roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let Γ be a finite group and Γ′ a subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If K0 is a number field, and K/K0 is  an extension with an isomorphism Gal(K/K0) ≃ Γ, we can also consider the (relative) class  group ClL|K0 of the fixed field L = KΓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let p be a prime not dividing |Γ|, and let Vi,  ei,ni,di,T be as at the start of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=', and let S = T/e1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let midi = rankZp V Γ′  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Since p ∤ |Γ|, we have that the natural inclusion map  ClL|K0[p∞] → ClK|K0[p∞]Γ′  is an isomorphism [CM90, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus, a conjecture on the distribution of ClK|K0[p∞]  as Zp[Γ]-modules formally implies a conjecture on the distribution of ClL|K0[p∞] as Zp-  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' However, we will see that one can make the implication much more explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let eΓ′ = |Γ′|−1 �  γ∈Γ′ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have that eΓ′ is a (not necessarily central) idempotent of  Zp[Γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For any Zp[Γ]-module V , we have that eΓ′V = V Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let o := eΓ′SeΓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For any S-  module V , we have that V Γ′ is naturally an o-module, allowing the action to take place in  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let eΓ|Γ′ be the central idempotent that is the sum of all the idempotents ei corresponding  to non-trivial projective Zp[Γ]-modules that have non-trivial Γ′-invariants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' the ei such  24  that eΓ′ei ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let M be the set of the corresponding i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The Vi for i ∈ M are exactly the  non-trivial modules contained in IndΓ  Γ′ Zp [WW21, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For any S-module V , we  have that V Γ′ = (eΓ|Γ′V )Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have eΓ′eΓ|Γ′ = eΓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The ring o is a maximal order in a semi-simple Qp algebra with primitive central idem-  potents eΓ′ei = eieΓ′ for i ∈ M (over Q, this can be found in [WW21, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3,  Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='10], and the argument over Qp is the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  There is a Morita equivalence  between eΓ|Γ′S-modules and o-modules taking V �→ V Γ′ (see [WW21, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The  inverse functor takes U to I(U) := eΓ|Γ′SeΓ′ ⊗o U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If U = ⊕i∈MeΓ′eiU is an o-module, and  V is a eΓ|Γ′S-module with V Γ′ = U (so V ≃ I(U)), then V = ⊕i∈MeiV , with  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1)  |eiV | = |eΓ′eiU|ni/mi  by [CM90, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The finite o-modules are thus indexed by tuples of partitions  (λi)i∈M, given as (Vλ)Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If λ is a tuple of partitions indexed by elements of M, by abuse of  notation, we also use λ to denote the extension by zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' the tuple (λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , λm), where  λi = 0 for i ̸∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The following is an immediate consequence of the Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The measure ν on S-modules from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3, pushed forward to a  measure on finite o-modules via the map V �→ V Γ′ from S-modules to o-modules, gives a  measure νΓ′ on o-modules as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let k ∈ NM and let λ be a tuple of partitions indexed  by the elements of M such that λi  ki+1 = 0 for all i ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then  νΓ′({U | I(U)≤k ≃ Vλ′}) = ν({V | V ≤k ≃ Vλ′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3)  Also, the U-moment of νΓ′ is |(∧2  ZpV )Γ[pr]||V |−u for V = I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The right-hand side of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3) is given by an explicit formula in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 (this is just  the special case of that formula when ki = 0 for k ̸∈ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' It then follows that the probability  of an individual module νΓ′({(Vλ′)Γ′}) is given by a formula that is the same as that given  in Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4, but with the product restricted to i ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Note that the terms  |(∧2  ZpVλ′)Γ[pr]|  |Vλ′|u| Aut(Vλ′)|  can all be given by explicit formulas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' using Lemmas 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Also, by the categorical  equivalence, we have | Aut(Vλ′)| = | Auto((Vλ′)Γ′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  This shows that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 gives explicit conjectures for the distribution of Sylow  p-subgroups of class groups of non-Galois fields in the presence of roots of unity, for p not  dividing the order of the Galois closure, in particular that they are given by the measure νΓ′  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Experimental evidence  Malle [Mal08, Mal10] has done extensive computations on class groups over extensions of  a base field with roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Most of his tables of computed class group data strongly  support his conjectures, showing remarkably good convergence of the probabilities of various  Sylow p-subgroups to the conjectured amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since we see in Section 12 that his conjectures  are (in all cases we shall mention in this section) special cases of our conjecture, these all  provide good evidence for our conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The cases for which this provides data are as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  25  (1) Γ = C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' p = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' u = 1 (K0 = Q(√−3)): [Mal08,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Table 9] and [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Table 1]  on 3-ranks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Table 2] on 3-Sylow  (2) Γ = C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' p = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' u = 1 (K0 = Q(ζ5)): [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Table 3] on 5-ranks  (3) Γ = C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' p = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' u = 1 (K0 = Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Q(√−3)): [Mal08,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Tables 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4] on 2-ranks and  [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Tables 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7] and [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Tables 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='9] on 2-Sylow  (4) Γ = C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' p = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' u = 2 (K0 = Q(  √  5)): [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Table 10] on 2-ranks and [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Table 11] on 2-Sylow  (5) Γ = C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' p = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' u = 1 (K0 = Q(i)): [Mal10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Table 12] on 2-ranks  (6) Γ = C5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' p = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' u = 1 (K0 = Q): [Mal08,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Table 8] on 2-ranks  Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Malle provides a chart of data for Sylow 2-subgroups of class groups of C3-  extensions of Q(i) (so r = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' u = 1) for which he makes no conjecture to explain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' only  remarking they do not seem to fit well with any known formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Our conjecture provides  an excellent match for this data, which we present in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ):  (0)  (1)  (1,1)  (2)  (1,1,1)  (2,1)  (3)  (1,1,1,1)  (2,1,1)  D  1  22  42  24  82  42 × 22  26  162  82 × 22  ≤ 1015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='864  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='25E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='12E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='50E-3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6E-4  ≥ 1016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='859  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='30E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='13E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='42E-3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2E-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4E-4  ≥ 1024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='854  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='123  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='40E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='11E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='64E-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3E-4  ≥ 1032  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='853  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='40E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='10E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='62E-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4E-4  Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='853  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='124  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='0166  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='38E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='10E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='60E-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='38E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='66E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='37E-4  [Mal10, (6-2)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='853  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='133  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='0083  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='44E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+page_content='33E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='21E-4  Cohen-Martinet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='918  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='076  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='0048  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='03E-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='30E-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='25E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='79E-7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='19E-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='16E-4  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Malle’s data on 2-Sylow for C3 extensions K/Q(i) from [Mal10,  Table 13] compared to various conjectures  If K/Q(i) is a C3-extension, then ClK|Q(i)[2∞] is a module for Z2[ζ3], where ζ3 is a primitive  3rd root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Table 1, the columns correspond to the first few finite Z2[ζ3]-modules,  with the top row labeling them in our notation assuming the module is Vλ′, and the second  row labeling them with the notation of [Mal10] which reflects their underlying structure as  abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Malle computed all C3 extensions of Q(i) with discriminant at most 1015,  and for some other values of D the first S such extensions with discriminant at least D (it  appears S is 5 · 105 or 4 · 105 in each case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The numbers in the middle part of the table  show, for each collection of extensions K/Q(i), the proportion with ClK|Q(i)[2∞] which is the  given Z2[ζ3]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In the bottom part of the chart, one sees the proportions predicted  by Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, Malle’s formula [Mal10, (6-2)], and the original conjecture of Cohen and  Martinet [CM90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We see quite good agreement with Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Malle recognized  that the formula [Mal10, (6-2)] was not a good match for these values, but he included this  comparison so we do as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  It would be very interesting to have more experimental evidence for Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The  case of Γ = C7, with p = 2, and K0 = Q is particularly interesting, as it is one of the smallest  cases (being about fields of absolute degree 7) in which we have no current experimental  evidence, and also the first in which there are dual pairs of representations, and so our  26  conjecture predicts a lack of independence between representations (a new phenomenon  worthy of backing up with experimental evidence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Also Γ = C7 ⋊C3 has a transitive degree  7 permutation action, and letting Γ′ be the stabilizer of a point, one could consider the  non-Galois K that are the Γ′ fixed fields of Γ-extensions, with p = 2, and K0 = Q, to check  a non-Galois implication of our conjecture for degree 7 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Other relatively small degree  cases are degree 4 abelian or D4 extensions with p = 3 and K = Q(ζ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The case of Γ = C11,  with p = 2, and K0 = Q, is also of interest, because as we will see below in Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3, our  conjecture differs from a suggestion of Malle’s in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Other interesting cases include  K0 = Q(ζ5) with Γ = C4 and p = 5, or K0 = Q(ζ3) with Γ = C8 and p = 3, as they have  dual irreducible representations and p odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Comparison to previous conjectures in special cases  Malle [Mal08, Mal10] and Adam and Malle [AM15] have made conjectures about class  group distributions in the presence of roots of unity in particular situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this section,  we explain how these conjectures relate to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let Γ be a finite group and Γ′ be a subgroup with �  γ∈Γ γΓ′γ−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let p be a prime  p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Throughout this section, we will consider the distribution of ClKΓ′/K0[p∞], as K  ranges over Γ extensions of K0 for some fixed number field K0 containing prth roots of unity  but not pr+1th roots of unity, and with u infinite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (So we can take Γ′ = 1 to get  the case of relative class groups of Galois extensions, or other Γ′ to consider non-Galois  extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=') We say (Γ, Γ′) is an absolutely irreducible pair if the representation IndΓ  Γ′ C is  the sum of one trivial representation and an irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The conjectures of Malle and Adam are mostly in the case when (Γ, Γ′) is an absolutely  irreducible pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have that (Sn, Sn−1) is an absolutely irreducible pair, so this covers the  case of degree n extensions with Galois closure with Galois group the symmetric group Sn,  including quadratic extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' However, such pairs are rare, with only 14 examples with  |Γ| ≤ 120, and the only example giving Galois fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' with Γ′ = 1) being Γ = S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  When (Γ, Γ′) is an absolutely irreducible pair and r = 1, Malle [Mal10, Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1]  gives a conjecture for the distribution of ClKΓ′/K0[p∞] and we show below that in this case, our  conjecture specializes to [Mal10, Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' 2 When (Γ, Γ′) is an absolutely irreducible  pair and r ≥ 1, Adam and Malle make a conjecture [AM15, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1] that does not  give formulas for their conjectured distribution of ClKΓ′/K0[p∞], but instead conjectures that  the distribution of ClKΓ′/K0[p∞] agrees with a limit of certain distributions from finite matrix  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, we show that the limit of these distributions from matrix groups  exists and, indeed, agrees with our Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 and the explicit formulas we give.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Malle also makes some conjectures in the cases Γ = C3, C5 and p = 2 ([Mal08, (1),(2)],[Mal10,  (6-1),(6-2),(6-3),(6-4)]), and we check in Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 agrees with all of  these conjectures in these specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  However, Malle makes a speculation for the situation in which IndΓ  Γ′ Q is the sum of one  trivial representation and an irreducible representation (over Q) [Mal10, Section 2], and we  show in Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 that in general his speculation does not agree with Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We  show below that the only Galois cases in which Malle’s speculation agrees with Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1  are when Γ is C3 or C5 and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Luckily, these seem to be precisely the cases in which Malle  2Malle also makes some conjectures when p | |Γ|, but we don’t discuss those here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  27  has computational evidence for his speculation (for p ∤ |Γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In many cases of this situation,  our conjecture is qualitatively different from [Mal10, Section 2] because we conjecture (and  prove in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1) that certain modules occur with probability 0 while [Mal10, Section  2] assigns positive probability to each module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Absolutely irreducible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let (Γ, Γ′) be an absolutely irreducible pair and p a  prime p ∤ |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In the notation of Section 10, we then have |M| = 1 (without loss of generality  we say M = {2}) and d2 = 1, and m2 = 1, and q2 = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have that o = Zp ([WW21,  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The argument in [WW21, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='16] shows that o = Zp if and  only if (Γ, Γ′) is an absolutely irreducible pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This highlights the appeal of this condition  in this context, for when it fails, the groups ClKΓ′/K0[p∞] have an o-module structure, for  some o larger than Zp, not just abelian group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Thus in the absolutely irreducible pair situation, we are looking simply for a distribution  of finite Zp-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  For a partition λ, which we extend by 0 to an (m − 1)-tuple of  partitions, we have the o-module (Vλ)Γ′ = �  i Z/pλiZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We also have |Vλ| = | �  i Z/pλiZ|n2 by  Equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1) (and n2 = [Γ : Γ′] − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let (Γ, Γ′) be an absolutely irreducible pair and p a prime p ∤ |Γ|, then Γ has  even order, σ(2) = 2, and ǫ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For a permutation representation, the standard pairing is an invariant symmetric  perfect pairing (since permutation matrices are orthogonal), and thus IndΓ  Γ′ C is self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If IndΓ  Γ′ C = C × W for an irreducible representation W, since W is non-trivial, it must be  self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Odd order groups have no non-trivial self-dual irreducible representations, which  implies |Γ| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let Wp be the mod p representation corresponding to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have that  (Sym2 IndΓ  Γ′ Fp)Γ is at least two dimensional, since it includes the pairing on IndΓ  Γ′ Fp that  first takes the quotient to the trivial representation, as well as the perfect pairing mentioned  above, which implies that (Sym2 Wp)Γ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since p is odd, this implies Wp is self-dual (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  σ(2) = 2) and ǫ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' First,we assume r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We then see that in this case (absolutely irreducible  pair and r = 1) the formula for the measure νΓ′ of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 simplifies significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We have, for a partition λ, and group U = �  i Z/pλ′  iZ, our conjecture for the distribution of  ClKΓ′/K0[p∞] is given by  νΓ′({U}) =  p(λ1  2 )  |U|un2| Aut(U)|  �  ℓ≥0  (1 + p−un2−1−ℓ)−1  λ1  �  ℓ=1  (1 − p−un2−ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2)  Now we compare this to Malle’s [Mal10, Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (We assume that [Mal10, Con-  jecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1] has a typo and the product in the numerator should start at i = u + 1 since  this modification is what is shown in the paper of Adam and Malle [Mal10, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3],  where it is said to be exactly [Mal10, Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=') We need to compute the u in Malle’s  formula (which is not quite our u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since we have that K/K0 is split at all infinite places by  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4, we have that Malle’s χE is the character that is the sum of a copy of the char-  acter of the regular representation for each infinite place of K0 minus one trivial character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  It follows that Malle’s u is our un2 (in the absolutely irreducible pair case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then we see  that Equation (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2) exactly agrees with [Mal10, Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1] (with the typo corrected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  28  Even though both these conjectures imply a conjecture on p-ranks, as a double check, we  compare our conjectured formula for the distribution of p-ranks to Malle’s [Mal10, Propo-  sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' To obtain our formula, we apply Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 with k = 1 (leading us to  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 with k = (1, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  νΓ′({U|rkpU = λ}) =  p(λ  2)  |(Z/pZ)λ|un2| Aut((Z/pZ)λ)|  �  ℓ≥0  (1 + p−un2−1−ℓ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3)  This agrees with the distribution of [Mal10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Now we consider a general r ≥ 1 (still with an absolutely irreducible pair  (Γ, Γ′) and p ∤ |Γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this case, Adam and Malle [AM15, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1] conjecture that  ClKΓ′/K0[p∞] is distributed as the limit of certain distributions coming from matrix groups,  and we describe those distributions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Given a prime p, and positive integers r ≤ k, g, let  Sp(r)  2g (Z/pkZ) be the subgroup of matrices of GL2g(Z/pkZ), that when reduced mod pr are  in Sp2g(Z/prZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In other words, these are matrices that are symplectic mod pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Adam and  Malle define a distribution νr  AM of finite abelian p-groups such that  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4)  νr  AM({G}) := lim  g→∞  |{A ∈ Sp(r)  2g (Z/pkZ) | ker(A − I2g) ≃ G|  |Sp(r)  2g (Z/pkZ)|  ,  where I2g is the 2g × 2g identity matrix, and we take the kernel of A − I2g as a map  (Z/pkZ)2g → (Z/pkZ)2g and k is any positive integer such that pk−1G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' As far as we can  tell, they do not show that these limits exist or that they give a probability distribution, but  we will show that here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For r ≥ 1, the limits defining νr  AM exist and define a probability measure on  the set of (isomorphism classes of) finite abelian p-groups such that  �  G | Sur(G, H)|dνr  AM =  | ∧2 H[pr]| for every finite abelian p-group H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' First, we note that by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 there is a probability measure νr on the set  of finite abelian p-groups with these moments, and the moments are well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For B a  2g×2g matrix over Z/pkZ, we have ker(B) ≃ cok(B), and so we can replace ker(A−Ig) with  coker(A−Ig).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' When pk−1G = 0, the cokernel of a Zp linear map B : Z2g  p → Z2g  p is isomorphic  to G if and only if the same is true for the reduction of ¯B : (Z/pkZ)2g → (Z/pkZ)2g mod pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  So we replace Z/pkZ with Zp (using the analogous definition for Sp(r)  2g (Zp)), and counting  measure on Sp(r)  2g (Z/pkZ) with Haar measure on Sp(r)  2g (Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Let νg be the distribution of coker(A−Ig) for a Haar random A ∈ Sp(r)  2g (Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By Theorem  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2, if we show that  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6)  lim  g→∞  �  G  | Sur(G, H)|dνg = | ∧2 H[pr]|,  for every finite abelian p-group H, then the νg weakly converge to νr, and since the delta  function on a single group is continuous, the limits defining νr  AM({G}) exist and we have  νr  AM({G}) = νr({G}), which would complete the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  To show (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='6), we have that Sp(r)  2g (Zp) acts on Sur(Z2g  p , H), and the fixed points of A ∈  Sp(r)  2g (Zp) are precisely the elements of Sur(coker(A−Ig), H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By Burnside’s Lemma, we then  29  have that the average of Sur(coker(A − Ig), H) over A ∈ Sur(Z2g  p , H) is the number of orbits  of Sp(r)  2g (Zp) on Sur(Z2g  p , H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By [Mic06, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='14], we have that the orbits of Sp2g(Zp)  on Sur(Z2g  p , H), for g sufficiently large given H, are precisely the fibers of a map  F : Sur(Z2g  p , H) → ∧2H  that we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , x2g be a standard basis of Z2g  p , and then F(φ) = �g  i=1 φ(xi)∧  φ(xi+g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The same statement for H/prH, tells us that  ¯F : Sur(Z2g  p , H) → ∧2(H/prH)  is invariant on Sp(r)  2g (Zp) orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Since |∧2 (H/prH)| = |∧2 (H)[pr]|, we will be done if we can show that each fiber of ¯F is a  single Sp(r)  2g (Zp) orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Consider two elements a, b in the same fiber of ¯F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We pick a minimal  generating set h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' , hs of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We write F(a) = �  i<j ai,jhi ∧ hj and F(b) = �  i<j bi,jhi ∧ hj  for ai,j, bi,j ∈ Zp such that ai,j ≡ bi,j (mod pr) for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We chose a bijection u : {i | 1 ≤  i ≤  �s  2  �  } → {(i, j) | 1 ≤ i < j ≤ s} and write u1(i), u2(i) for the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For g large  enough, given H (as above and so g ≥ s +  �s  2  �  ), we can choose a surjection φa ∈ Sur(Z2g  p , H)  such that  φa(xi) = hi for 1 ≤ i ≤ s,  φa(xs+i) = au(i)hu1(i) for 1 ≤ i ≤  �s  2  �  ,  φa(xi) = 0 for s +  �s  2  �  < i ≤ g,  φa(xg+i) = 0 for 1 ≤ i ≤ s and s +  �s  2  �  < i ≤ g, and  φa(xg+s+i) = hu2(i) for 1 ≤ i ≤  �s  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We have F(φa) = F(a), and so φa and a are in the same Sp2g(Zp) orbit and hence the  same Sp(r)  2g (Zp) orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We choose φb similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' There is a Zp-module map sending xs+i �→  xs+i + (bu(i) − au(i))xu1(i) for 1 ≤ i ≤  �s  2  �  and mapping all other xi �→ xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This map is the  identity mod pr and takes φa to φb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Thus we conclude φa and φb (and hence a and b) are in  the same Sp(r)  2g (Zp) orbit, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  □  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 gives explicit formulas for νr  AM, and since we apply the proposition in the  case that ǫ = s = 0, we even have nice product formulas for this measure using Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2,  for any r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Adam and Malle noted it was possible to calculate νr  AM(G) explicitly for any  given r and G [AM15, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5], but did not have a closed formula for r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Finally, we are ready to state Adam and Malle’s conjecture [AM15, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' For  an absolutely irreducible pair (Γ, Γ′), Adam and Malle [AM15, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1] conjecture  that ClKΓ′/K0[p∞] is distributed like the quotient of a random group from νr  AM(G) by un2  uniform, independent random elements (as above Malle’s u is our un2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Such a random  group has H-moment | ∧2 H[pr]||H|−un2 for every H, and by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 there is a  unique distribution with these moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, we see that  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 predicts the distribution with the same moments, and so our conjectures  agree in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  In the special case when Γ = Z/2Z, there is also a conjecture [LST20, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2]  which gives a distribution for the class group together with extra invariants ω, ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Ignoring  the extra invariants, [LST20, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2] implies a conjecture simply for the distribution  of the class group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This agrees with Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' To check this, it suffices to check that  30  the moments of the distribution Qtµ of [LST20, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2] agree with the moments  described in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, since we have seen that these moments uniquely determine  a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The (G, ωG, ψG)-moment of Qtµ is calculated in [LST20, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='12 and  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3] as  1  |G|t| Sym2(G)[ℓn]| for each possible value of the invariants ωG, ψG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' It is straight-  forward to check by writing G as a product of cyclic groups that there are | Sym2(G)[ℓn]|  possible values of ψG for each ωG and | ∧2 (G)[ℓn]| possible values of ωG, so summing over  these possibilities, the G-moment is |∧2(G)[ℓn|  |G|t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' This agrees with Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 since t and u  are both defined to equal the number of infinite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' C3, C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Now let Γ be Cℓ for ℓ = 3, 5 (and Γ′ = 1) and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In this case, there is  only one non-trivial indecomposable module V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have n2 = 1 and d2 = ℓ − 1 and ǫ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  When r = 1, our conjecture for the distribution of ClKΓ′/K0[p∞] (from Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, using  Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 for the formulas) is  ν({Vλ′}) =  2  ℓ−1  2 λ2  1  |Vλ′|u| Aut(Vλ′)|  �  i≥0  (1 + (2ℓ−1)−u− 1  2−i)−1  λ1  �  i=1  (1 − (2ℓ−1)−u−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Note that  �  i≥0  (1+(x2)− 1  2 −i) =  �  i≥0  (1+x−2i−1) =  �  i≥0(1 − (x2)−2i−1)  �  i≥0(1 − x−2i−1)  =  �  i≥0(1 − (x2)−i)  �  i≥0(1 − (x2)−2i)·  �  i≥0(1 − x−2i)  �  i≥0(1 − x−i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  From this one can check that Malle’s conjectures [Mal10, (6-2),(6-4)] in the cases u = 1, 2  agree with our conjecture for C3 and r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' It then formally follows that Malle’s conjectures  [Mal10, (6-1),(6-3)] for the 2-ranks agree with ours (and it is also easily verified using Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' One can also check that Malle’s conjecture [Mal08, Equation (2)] agrees with our  conjecture for C5 and u = r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Malle [Mal10, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='4] suggests that the 2-rank pre-  diction for u = 1 [Mal10, (6-1)] should still hold for r > 1, which agrees with our conjecture  (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Irreducible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We say (Γ, Γ′) is an irreducible pair if the representation IndΓ  Γ′ Q  is the sum of one trivial representation and an irreducible representation W (over Q), and  moreover that W has Schur index 1 (so EndΓ(W) is a field and not just a division algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Irreducible over Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let (Γ, Γ′) be an irreducible pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We first consider the case  when Wp := W ⊗Q Qp is irreducible over Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In the notation of Section 10, we then have  |M| = 1 (without loss of generality we say M = {2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let d = d2 and q = q2 = pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have  m2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' As in the proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, we have that Wp is self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' By [WW21, Section  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3], we have that o is the maximal order in the field EndΓ Wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Then our conjecture for the  distribution of ClKΓ′/K0[p∞], in the case r = 1, is given by, for U = V Γ′  λ′ ,  νΓ′({U}) =  |(∧2  ZpVλ′)Γ[p]|  |Vλ′|u| Auto(U)|  �  ℓ≥0  (1 + q−un2− ǫ2+1  2  −ℓ)−1  λ1  �  ℓ=1  (1 − q−un2−ℓ)  =  q(λ1  2 )+(1−ǫ2) λ1  2  |U|n2u| Auto(U)|  �  ℓ≥0  (1 + q−un2− ǫ2+1  2  −ℓ)−1  λ1  �  ℓ=1  (1 − q−un2−ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7)  31  using Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2 and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3 for the first line, and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1 and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1) for the  second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' In [Mal10, Section 2], for r = 1 Malle suggests that the probability for U should  be  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='8)  c′dλ1  q(λ1  2 )  |U|n2u| Auto(U)|  λ1  �  ℓ=1  (1 − q−un2−ℓ),  for some unspecified c′ not depending on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' (We have absorbed some other constant parts  of the formula into the unspecified constant from [Mal10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Also, one can check that given  that Wp is irreducible, our o is O ⊗Z Zp for Malle’s O, and so our d agrees with Malle’s d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=')  In order for (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='7) and (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='8) to agree, we must have  p  1−ǫ2  2  d = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Since d is a positive integer, and p is a prime, the above equality holds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' our conjectures  agree) exactly when  ǫ2 = 1 and d = 1 (so Wp is absolutely irreducible over Qp and hence Q and we have  an absolutely irreducible pair, as discussed above), or  ǫ2 = 0 and p = 2 and d = 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If we are interested in the Galois case (so Γ′ = 1), then d = |Γ|−1, and the only possibilities  where d = 2, 4 are the Γ = C3, C5 examples discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Reducible over Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let (Γ, Γ′) be an irreducible pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' If Wp is reducible over Qp,  then that means p splits in the field End(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let O be the maximal order End(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Malle  [Mal10, Section 2] gives a suggestion in terms of the “O-rank” of the p-group H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Since O/p  is not a field, it is not clear to the authors what this rank means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Let us consider Γ = C7 and  Γ′ = 1 and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have O = o = Z[ζ7], but O ⊗Z Z2 = R1 × R2 for two extensions Ri of  degree 3 over Z2 (both isomorphic to the same unramified extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We have q = q2 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  In the case r = 1, our conjecture now gives probabilities  ν({Vλ′}) =  qλ2  1λ3  1  |Vλ′|u| Aut(Vλ′)|  ηq(∞)ηq(2u)  ηq(u − λ2  1 + λ3  1)ηq(u + λ2  1 − λ3  1)  λ2  1  �  ℓ=1  (1 − q−u−ℓ)  λ3  1  �  ℓ=1  (1 − q−u−ℓ),  if |λ2  1 − λ3  1| ≤ u and ν({Vλ′}) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' We consider 4 2-torsion O-modules: 1, R1/2,  R2/2, R1/2×R2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' These correspond to the tuples of partitions (0, 0), ((1), 0)(0, (1)), ((1), (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  If we look at the quantities ν({V })|V|u| Aut(V)| for our four modules, they are in ratios  1 : 1 − 2−3u : 1 − 2−3u : 23(1 − 23(−u−1))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  while in [Mal10, Section 2], the corresponding ratio between a rank 0 group and a rank 1  group is  1 : 6(1 − 26(−u−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  We do not see a way to interpret “O-rank” as to make our conjecture align with Malle’s  suggestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  More qualitatively, our conjecture assigns probability 0 to any Vλ′ with |λ2  1 − λ3  1| > u, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  if u = 1, we assign probability 0 to any group which is rank 0 for R1 and rank 2 for R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  In contrast, [Mal10, Section 2] suggests that every 2-group O-module occurs with positive  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' So in this setting our Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, and Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='1, do not agree with Malle’s  suggestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  32  References  [Ach06]  Jeffrey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Achter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The distribution of class groups of function fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Journal of Pure and Applied  Algebra, 204(2):316–333, February 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  [AM15]  Michael Adam and Gunter Malle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' A class group heuristic based on the distribution of 1-  eigenspaces in matrix groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Journal of Number Theory, 149:225–235, April 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+page_content=' Lenstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' On class groups of random number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Proceedings of  the London Mathematical Society, 121(4):927–953, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+page_content=' On the mean number of 2-torsion elements in the class groups,  narrow class groups, and ideal groups of cubic orders and fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+page_content='  [BV16]  Manjul Bhargava and Ila Varma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The mean number of 3-torsion elements in the class groups and  ideal groups of quadratic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Proceedings of the London Mathematical Society, 112(2):235–266,  March 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+page_content=' On unit signatures  and narrow class groups of odd degree abelian number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+page_content=' Homological stability for Hurwitz  spaces and the Cohen-Lenstra conjecture over function fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+page_content='  33  [Smi22b]  Alexander  Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  The  distribution  of  fixed  point  Selmer  groups  in  twist  families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='05143, July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  [SW22a]  Will Sawin and Melanie Matchett Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Finite quotients of 3-manifold groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='01140,  March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  [SW22b]  Will Sawin and Melanie Matchett Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' The moment problem for random objects in a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='06279, October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  [Woo18]  Melanie Matchett Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Cohen-Lenstra heuristics and local conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Research in Number  Theory, 4(4):41, September 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  [WW21]  Weitong Wang and Melanie Matchett Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Moments and interpretations of the Co-  hen–Lenstra–Martinet heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content=' Commentarii Mathematici Helvetici, 96(2):339–387, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='  Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027  USA  Email address: sawin@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='edu  Department of Mathematics, Harvard University, Science Center Room 325, 1 Oxford  Street, Cambridge, MA 02138 USA  Email address: mmwood@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
+page_content='edu  34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQf4vpJ/content/2301.00791v1.pdf'}
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+Prepared for submission to JINST
+Improving primary-vertex reconstruction with a
+minimum-cost lifted multicut graph partitioning algorithm
+V. Kostyukhin,1 M. Keuper,2,3 I. Ibragimov,1 N. Owtscharenko,1 and M. Cristinziani1
+1Center for Particle Physics Siegen, Department Physik, Universität Siegen
+2Visual Computing, Department Elektrotechnik und Informatik, Universität Siegen
+3Max Planck Institute for Informatics, Saarland Informatics Campus
+Abstract: Particle physics experiments often require the simultaneous reconstruction of many
+interaction vertices. Usually, this problem is solved by ad hoc heuristic algorithms. We propose
+a universal approach to address the multiple vertex finding through a principled formulation as
+a minimum-cost lifted multicut problem.
+The suggested algorithm is tested in a typical LHC
+environment with multiple proton–proton interaction vertices. Reconstruction errors caused by the
+particle detectors complicate the solution and require the introduction of special metrics to assess
+the vertex-finding performance. We demonstrate that the minimum-cost lifted multicut approach
+outperforms heuristic algorithms and works well up to the highest vertex multiplicity expected at
+the LHC.
+Keywords: Vertexing algorithms; Pattern recognition, cluster finding, calibration and fitting
+methods
+arXiv:2301.05548v1  [hep-ex]  13 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Minimum-cost multicuts and lifted multicut algorithm for cluster finding
+3
+3
+Data simulation
+5
+4
+Features of simulated data
+6
+5
+Edge weights and constraints
+6
+6
+Performance metrics
+10
+7
+Results
+12
+8
+Conclusion
+19
+A Non-clustered tracks and total reconstructed clusters
+21
+1
+Introduction
+In particle physics experiments, many problems require a precise reconstruction of vertices — points
+in 3D space where particle interactions occur. Knowledge of the positions and features of such
+vertices provides valuable information about the underlying physics of these interactions. There are
+numerous examples: 𝐵 physics, heavy-flavour jet identification, primary event vertex reconstruction,
+search for exotic particles as new physics manifestations, etc. Except for a few specifically designed
+detectors (emulsions, Wilson chamber, etc.), rarely used in modern experiments, the vertices are
+not directly detectable. The presence of vertices is usually inferred from 3D traces of charged stable
+particles produced in the interaction. Various tracking detectors measure the curved trajectories of
+these particles (tracks) in space. The trajectories can be extrapolated to a single 3D point, which
+represents the interaction vertex position [1].
+Despite the simplicity of the vertex reconstruction idea, its real-life exploitation encounters
+problems. For example, at the Large Hadron Collider (LHC) at the end of Run 2, a typical recorded
+event consisted of ∼80 primary proton–proton interactions, and numerous produced charged par-
+ticles underwent further interactions leading to additional vertices, distributed in significant 3D
+volumes. The expected number of proton–proton interactions in a single event at the LHC after
+the planned high-luminosity upgrade (HL-LHC) may reach 200–300, resulting in a few thousand
+reconstructed tracks. Therefore, prior to determining the vertex positions, one needs to determine
+how many vertices are present in a given event and assign the reconstructed tracks to these assumed
+vertices. The track measurement uncertainties, which may differ by a factor of 10 for different tracks
+– 1 –
+
+and often are comparable with the vertex–vertex distances, cause additional complications. These
+uncertainties make an exact crossing of track pairs in 3D space impossible: even if two charged
+particles are produced in the same interaction point, their reconstructed trajectories will only be
+close to the true vertex position and to each other, up to the corresponding uncertainties.
+The explicit reconstruction of multiple vertices in an event can be addressed in a graph-based
+approach. In fact, all space trajectories of the particles produced in a single vertex should be pairwise
+compatible, i.e. every pair of tracks should be close to each other in some volume around the true
+vertex position. Therefore, a compatibility graph can be constructed where every node represents
+a track. Two nodes are connected by an edge if and only if the distance of the corresponding
+trajectories is very small. In the ideal case, every vertex is represented by a fully connected, isolated
+subgraph in such a graph. In a realistic scenario, track measurement errors shuffle tracks among
+different vertices, resulting in a large number of fake edges in the compatibility graph. Yet, it can
+be tried to split the full graph into non-overlapping clusters by minimising the track–track distances
+for all track pairs in a cluster. The obtained set of isolated clusters should be a good approximation
+of the true vertices.
+The present paper focuses on finding primary proton–proton interaction vertices at the LHC.
+Subsequent decays of the particles produced in the detector volume and their interactions with
+the detector material will not be considered here. To illustrate the primary-vertex reconstruction
+problem, Figure 1 shows two zoomed-in regions of a typical LHC event with several pileup
+interactions. The upper plot presents a region where a pileup interaction vertex is identified, which
+has the largest sum of track transverse momenta.
+The bottom plot presents a region where a
+hard-scatter vertex, i.e. the point of interaction of interest, is identified. In both plots, the true
+positions of interaction vertices are shown, together with charged particle trajectories displaced
+due to reconstruction uncertainties. Several true interaction vertices in these plots do not have
+associated tracks because all emanated particles in this interaction are outside of the tracking
+detector’s sensitive volume, see Section 3 for the details. The overlap of the red (from hard-scatter
+vertex), blue and grey (from nearby pileup vertices) tracks in the centre of the bottom plot on
+Figure 1 is clearly visible.
+Experiments at the LHC use heuristic algorithms [3–5] to reconstruct multiple proton–proton
+interaction vertices. Several other approaches can be found in the literature, including medical
+imaging-inspired algorithms [6] and the RAVE package [7] implementing the deterministic anneal-
+ing algorithm [8].
+This article presents an implementation of the Lifted Multicut Graph Partitioning algorithm
+(LMC), which solves the inclusive vertex reconstruction problem described above.
+Section 2
+describes the LMC algorithm and details of its implementation for the vertex finding application.
+Section 3 describes the simulated samples which are used to test the algorithm performance. In
+Section 4, features of the simulated samples are discussed. Section 5 introduces edge cost functions
+used in the graph partitioning. In Section 6, the metrics are introduced to estimate the algorithm
+performance and to compare it with other existing approaches. Section 7 presents the performance
+of the LMC approach in simulation. In Section 8, conclusions are made.
+– 2 –
+
+61
+−
+60
+−
+59
+−
+58
+−
+57
+−
+56
+−
+55
+−
+54
+−
+53
+−
+52
+−
+z [mm]
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+r [mm]
+Reco z = -56.71 mm
+Truth z = -2.59 mm
+2
+ = 470.3 GeV
+T
+2
+ p
+Σ
+w
+T
+ p
+Σ
+PU Vertex chosen by 
+Truth
+Reco
+HS tracks
+PU tracks
+HS jets
+PU jets
+truth PU vertex
+truth HS vertex
+ cut
+0
+ z
+T
+-p
+η
+Simulation Preliminary
+ATLAS 
+7
+−
+6
+−
+5
+−
+4
+−
+3
+−
+2
+−
+1
+−
+0
+1
+2
+z [mm]
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+r [mm]
+Reco z = -2.60 mm
+Truth z = -2.59 mm
+2
+ = 121.2 GeV
+T
+2
+ p
+Σ
+Reconstructed Hard-Scatter Primary Vertex
+Truth
+Reco
+HS tracks
+PU tracks
+HS jets
+PU jets
+truth PU vertex
+truth HS vertex
+ cut
+0
+ z
+T
+-p
+η
+Simulation Preliminary
+ATLAS 
+Figure 1: Two regions of a typical LHC event in the ATLAS detector with many pileup interac-
+tions [2]. True positions of the proton–proton interactions are shown, as well as the reconstructed
+trajectories (tracks) of the produced particles scattered due to reconstruction uncertainties. Some
+truth interaction vertices do not have associated tracks because all emanated particles are outside
+of the sensitive detector phase space and not reconstructed. These pictures illustrate typical track
+densities and overlap of the tracks produced in nearby interaction vertices. Both, tracks associated
+with the hard-scattering (HS) and pileup (PU) are shown.
+2
+Minimum-cost multicuts and lifted multicut algorithm for cluster finding
+We formulate the primary-vertex reconstruction problem as a minimum-cost lifted multicut problem.
+This problem was originally proposed in Reference [9] in the context of image segmentation and
+mesh decomposition. It is a generalization of the better-known minimum cost multicut problem,
+also referred to as the weighted correlation clustering problem [10, 11]. The minimum cost multicut
+problem is a grouping problem defined for a graph 𝐺 = (𝑉, 𝐸) and a cost function 𝑐 : 𝐸 → R
+which assigns to all edges 𝑒 ∈ 𝐸 a real-valued cost or reward for being cut. Then, the minimum
+– 3 –
+
+cost multicut problem is to find a binary edge labelling 𝑦 according to
+min
+𝑦∈{0,1}𝐸
+∑︁
+𝑒∈𝐸
+𝑐𝑒𝑦𝑒
+(2.1)
+subject to
+∀𝐶 ∈ cycles(𝐺)
+∀𝑒 ∈ 𝐶 : 𝑦𝑒 ≤
+∑︁
+𝑒∈𝐶\{𝑒}
+𝑦𝑒 .
+(2.2)
+The constraints on the feasible set of labellings 𝑦 given in Equation (2.2) ensure that the solution
+of the multicut problem relates one-to-one to the decompositions of graph 𝐺, by ensuring for every
+cycle in 𝐺 that if an edge is cut within the cycle (𝑦𝑒 = 1), so needs to be at least one other. Trivial
+optimal solutions are avoided by assigning positive (attractive) costs 𝑐𝑒 to edges between nodes
+𝑣, 𝑤 ∈ 𝑉 that likely belong to the same component, while negative (repulsive) costs are assigned to
+edges that likely belong to different components.
+The minimum cost lifted multicut problem (LMC) generalizes over the problem defined in
+Equation (2.1)–Equation (2.2) by adding a second set of edges that defines additional, potentially
+long-range costs without altering the set of feasible solutions. It thus defines a second set of edges
+𝐹 between the nodes 𝑉 of 𝐺, resulting in a lifted graph 𝐺′ = (𝑉, 𝐸 ∪ 𝐹), on which we can define
+a cost function 𝑐′ : 𝐸 ∪ 𝐹 → R. Then, Equation (2.1) and Equation (2.2) are optimized over all
+edges in 𝐸 ∪ 𝐹 and two additional sets of constraints are defined according to [9]
+∀𝑣, 𝑤 ∈ 𝐹
+∀𝑃 ∈ 𝑣, 𝑤 − paths(𝐺) : 𝑦𝑣𝑤 ≤
+∑︁
+𝑒∈𝑃
+𝑦𝑒
+(2.3)
+∀𝑣, 𝑤 ∈ 𝐹
+∀𝐶 ∈ 𝑣, 𝑤 − cuts(𝐺) : 1 − 𝑦𝑣𝑤 ≤
+∑︁
+𝑒∈𝐶
+(1 − 𝑦𝑒)
+(2.4)
+to ensure that the feasible solutions to the LMC problem still relate one-to-one to the decompositions
+of the original graph 𝐺.
+For the vertex reconstruction problem, this formulation allows encoding Euclidean distance
+constraints in the structure of graph 𝐺 (e.g. point observations that are spatially distant can not
+originate from the same vertex), while the cost function can be naturally defined in the distance
+significance space to take into account the measurement errors. The Euclidean distance and its
+significance can be very different in case of significant reconstruction errors, the lifted multicut
+formulation encodes both metrics in the same graph.
+The minimum cost multicut problem is 𝑛𝑝-hard, and so is the minimum cost LMC problem
+[12]. Yet, efficient heuristic solvers provide practically good solutions [9, 13–16]. Here, we resolve
+to use the primal feasible heuristic KLj that has been proposed in Reference [9] and published in
+an open-source library1. KLj is an iterative approach that produces a sequence of feasible solutions
+whose cost decreases monotonically. It takes as input an initial edge labelling (for example, all
+edge labels are initially set to 0), a lifted graph and costs defined on all edges. In every step, it
+either moves nodes between two neighbouring components, moves nodes from one component into
+a new component or joins two components such as to decrease the cost of the multicut maximally
+according to Equation (2.1).
+1https://github.com/bjoern-andres/graph
+– 4 –
+
+3
+Data simulation
+To estimate the clustering performance, we simulated data using DELPHES [17]. The framework
+allows to perform a fast and realistic simulation of a general-purpose collider detector composed of
+an inner tracker, electromagnetic and hadron calorimeters, and a muon system. For this study, we
+added a detailed parameterisation of the ATLAS detector tracking resolution to the framework.
+To simulate the pileup vertices and hard-scattering events, a sufficiently large amount of
+minimum-bias interaction events was prepared, consisting of single, double, and non-diffractive
+processes. These events have been generated using the Pythia 8 [18] event generator. As the main
+source of hard-scattering interactions, 𝑡¯𝑡 events are used, also generated with Pythia 8. To simulate
+an LHC collision event with full pileup, a single 𝑡¯𝑡 event is mixed with a number of minimum-
+bias events, distributed according to a Poisson distribution with a mean corresponding to a chosen
+luminosity. The interaction vertices are then distributed along the LHC beam trajectory inside the
+detector, according to typical interaction region parameters for ATLAS, i.e. according to a Gaussian
+with 𝜎𝑧 = 42 mm.
+The acceptance of the ATLAS detector allows for reconstructing charged particle trajectories in
+a limited phase space of 𝑝⊥ > 500 MeV and |𝜂| < 2.5. Some minimum-bias proton–proton interac-
+tions produce only particles outside the sensitive phase space of the ATLAS detector, which makes
+them unreconstructable. Positions of interactions with a single track in the ATLAS acceptance can
+be reconstructed, but this vertex category is contaminated by tracks that are strongly displaced by
+measurement errors. In the following, a reconstructable truth vertex refers to the true position of a
+proton–proton interaction producing at least two tracks within the ATLAS detector acceptance.
+All tracks produced in an event and falling into the sensitive ATLAS detector phase space
+are smeared according to the parameterised ATLAS detector resolution.
+Tracks with smeared
+parameters are referred to as reconstructed tracks in the following. The set of reconstructed tracks
+corresponding to a full pileup event is used as input for the performance estimation of the clustering
+algorithms. DELPHES samples used in this paper have been prepared with different energies and
+different pileup conditions (Table 1).
+Energy
+�
+𝜇
+�
+Interaction region 𝜎𝑧
+�
+𝑁event
+trk
+�
+�
+𝑁vrt
+trk =0
+�
+�
+𝑁vrt
+trk =1
+�
+�
+𝑁vrt
+trk >1
+�
+13 TeV
+63
+35 mm
+718
+9
+4
+50
+14 TeV 150
+42 mm
+1674
+22
+9
+119
+14 TeV 200
+42 mm
+2227
+28
+12
+160
+14 TeV 250
+42 mm
+2771
+35
+16
+199
+Table 1: The DELPHES samples used to estimate the LMC performance. Column 𝑁event
+trk
+reports
+the total number of reconstructed tracks in simulated events. The last three columns show the
+numbers of true vertices with 𝑁vrt
+trk = 0, 1, > 1, correspondingly.
+– 5 –
+
+4
+Features of simulated data
+The number of truth tracks in the detector acceptance in the simulated vertices and the position
+measurement errors of these tracks are shown in Figure 2. As can be seen in Figure 2a, in 14% of
+the cases, the simulated vertices do not have tracks in the detector acceptance, and in 6.5% of the
+cases, they have only one track. The number of tracks for all other vertices is widely spread up to
+80.
+0
+20
+40
+60
+80
+Number of tracks in a truth vertex
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+Fraction
+a)
+0
+0.5
+1
+1.5
+2
+Track measurement error
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Fraction
+b)
+Figure 2: a) Number of tracks per simulated vertex and b) Track measurement errors of the
+simulated tracks.
+Track measurement errors are shown in Figure 2b. From the sizes of the luminous regions and
+the number of vertices in Table 1 we can conclude that the track measurement errors are comparable
+or larger than a typical vertex–vertex distance in the simulated data. Smearing of the track positions
+due to measurement errors results in a significant overlap of the tracks from different truth vertices.
+The fraction of cases when a track from one vertex is entirely surrounded by tracks from other
+vertices for different pileup scenarios is shown in Table 2. An example of the track overlap can be
+seen in the bottom panel of Figure 1. Another example is shown in Figure 3.
+�
+𝜇
+�
+63
+150
+200
+250
+Track overlap fraction 20% 41% 53% 66%
+Table 2: Fraction of tracks, positioned in between tracks from other truth vertices due to measure-
+ment errors, as a function of the pileup.
+A priori, well-measured tracks with small errors should be easy to cluster according to the
+truth, while poorly measured tracks with large errors can easily migrate from one cluster to another,
+independently of their true origin. This random migration can be interpreted as noise, and thus, the
+overall problem may be considered as clustering in the presence of significant noise.
+5
+Edge weights and constraints
+To formulate the vertex finding problem in the presence of pileup as a minimum cost lifted multicut
+(LMC) problem, a track-pair compatibility graph needs to be constructed. A node in this graph
+– 6 –
+
+4
+6
+8
+10
+12
+Track and true vertex positions (mm)
+1.5
+−
+1
+−
+0.5
+−
+0
+0.5
+1
+1.5
+Track error (mm)
+Track positions linked to true vertices
+True vertices
+Figure 3: Example display of overlapping tracks from different vertices caused by measurement
+errors (zoom of a simulated DELPHES event with 𝜇 = 150). The crosses at the ordinate value
+of 0 represent the track positions, and the vertical error bars represent the corresponding position
+measurement errors. Squares at ordinate values of 1.3 represent the truth vertex positions. The
+connecting lines show the origin vertex for every track.
+represents a track, and two nodes are connected by an edge if and only if they are close in space and
+can be produced in the same vertex. The degree of track closeness, or equivalently the probability of
+originating in the same vertex, is estimated during the graph construction and is expressed as a weight
+assigned to the edge. The edge weights determine the efficiency of the clustering. Therefore, they
+should incorporate enough information, and the weight assignment procedure should be carefully
+designed. The following approaches are used in our study:
+1. Probability density function (PDF) ratio of the track–track geometrical distance significance
+based on measured uncertainties, 𝑆 =
+√︃
+(𝑧𝑖 − 𝑧 𝑗)2/(𝜎2
+𝑖 + 𝜎2
+𝑗 );
+2. Multivariate binary classification with Boosted Decision Trees (BDT);
+3. Logistic regression based on 𝑆.
+The LMC formulation assumes that the correct edges (two tracks from the same vertex) receive
+positive weights, while random (fake) edges receive negative weights. This can be achieved by
+using a logarithm of the ratio of the probability density functions for the correct and fake edges
+as the cost function of the problem log 𝑝true
+𝑝fake . According to the Neyman–Pearson lemma, this is
+the most efficient test statistic for the true/fake edge classification. An example of the track–track
+distance significance distributions and their ratio are shown in Figure 4. As the PDF of the fake
+edges is independent of the track–track distance significance, its overall normalisation depends on
+the significance range used for the parameterisation. Thus, the exact values of the PDF ratio can
+be scaled by the choice of the parametrisation range, which in principle, should not affect the LMC
+– 7 –
+
+clustering performance if the range is sufficiently large. Such a behaviour can be mimicked by a
+global multiplier of the PDF ratio function. The influence of this multiplier on the clustering will
+be studied in Section 7.3.
+0
+1
+2
+3
+4
+5
+6
+Track-track distance significance
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+0.014
+0.016
+0.018
+0.02
+0.022
+0.024
+True edges
+0
+1
+2
+3
+4
+5
+6
+Track-track distance significance
+0
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+Fake edges
+0
+1
+2
+3
+4
+5
+6
+Track-track distance significance
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Ratio True/Fake pdf's
+Figure 4: Example track–track distance significance for true and fake edges and their ratio. The
+significance distributions are normalized to one.
+A better clustering performance could be achieved by encoding more information in the edge
+weight calculation. To test this approach, we use a BDT classifier combining seven features, listed
+in Table 3, to distinguish true edges from fake ones. The GradientBoost implementation (BDTG)
+from the TMVA [19] package is used to train the classifier. An example of the trained classifier
+response2 is shown in Figure 5. The output is negative for fake edges and positive for true ones,
+exactly as required by the KLj algorithm, and therefore can be used directly as the edge weight.
+n. Description
+1
+Squared significance 𝑆2 (or 𝜒2) of track–track distance along beamline
+2
+Average position of the track pair along beamline
+3
+Position uncertainty of track 1
+4
+Position uncertainty of track 2
+5
+Pseudorapidity 𝜂 of track 1
+6
+Pseudorapidity 𝜂 of track 2
+7
+Number of other tracks crossing the beamline between tracks 1 and 2
+Table 3: Input features for the edge classification BDT.
+Edge weights can also be assigned by using the logistic regression 𝑝 = 𝑒𝑧/(𝑒𝑧 + 1), where
+𝑧 = 𝛽0 + �𝑛
+𝑖=1 𝛽𝑖𝑥𝑖 and 𝑥𝑖 are explanatory variables. The negative inverse of the logistic function,
+logit(𝑝) = log[𝑝/(1 − 𝑝)], provides the necessary edge weight behaviour. Edges that need to be
+removed receive negative weights, and those that need to be preserved receive positive weights.
+The intercept value 𝛽0 is defined by the ratio between the amount of true and fake edges used
+for training, which can be linked to a prior probability of a given edge being true or fake. In the
+2TMVA GradientBoost uses the binomial log-likelihood loss 𝐿(𝐹, 𝑦) = ln[1 + exp(−2𝐹(𝑥)𝑦)] with Gini Index
+separation. We use the following training settings NTree=800, MaxDepth=10, MinNodeSize=1.5%, Shrinkage=0.07.
+– 8 –
+
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+BDTG response
+0
+1
+2
+3
+4
+5
+6
+dx
+ /
+ 
+(1/N) dN
+Signal
+Background
+U/O-flow (S,B): (0.0, 0.0)% / (0.0, 0.0)%
+TMVA response for classifier: BDTG
+Figure 5: Example BDTG classification weight distributions for true and fake edges.
+current problem, the prior probability depends on the true vertex density and cannot be defined
+unambiguously, e.g. it depends on the range of the track–track distance significance 𝑆, see above.
+Therefore, the value of the intercept 𝛽0 in this approach can be modified in some range to achieve
+an over- or undersegmentation in order to validate its optimality. This will be further discussed in
+Section 7.3. A one-dimensional regression is tested in this paper, using variable (1) from Table 3.
+The logistic regression for the edge weight calculation is illustrated in Figure 6.
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+S2
+True
+Figure 6: Example one-variable logistic regression for true and fake edges using the squared
+track–track distance significance 𝑆2.
+The usage of the track–track distance significance for partitioning does not guarantee the com-
+pactness of the obtained cluster in Cartesian space, which may be beneficial when the vertex density
+is large. The compactness requirement can be imposed using the LMC constraint mechanism.
+Some edges in the connectivity graph can be additionally labelled as “have to be cut”, based on a
+priori information, different from the edge probability itself. To make clusters more compact, we
+– 9 –
+
+can constrain the edges to be cut if the corresponding Cartesian track–track distance is larger than
+some scale. In the following, a rather weak requirement of |𝑧𝑖 − 𝑧 𝑗| < 1 mm will be used, which
+removes tracks with very large errors, see Figure 2b. In addition to improving the quality of the
+solution, the constraint limits the phase space of possible solutions, and this leads to a significant
+algorithm speedup.
+6
+Performance metrics
+For a quantitative assessment of the performance of the vertex-finding algorithm, one or several
+metrics are to be established. To compare the performance of the clustering algorithms in, e.g.,
+image segmentation problems, metrics are usually employed, which are based on the assignment
+of graph nodes to clusters. One example of such a metric is the Variation of Information (VI)
+proposed in Reference [20]. The VI metric calculates the degree of compatibility of a clustering 𝐶
+with another clustering 𝐶′ as
+𝑉𝐼(𝐶, 𝐶′) = 𝐻(𝐶) + 𝐻(𝐶′) − 2 · 𝐼(𝐶, 𝐶′)
+(6.1)
+with
+𝐻(𝐶) = −
+𝐾
+∑︁
+𝑘=1
+𝑃(𝑘) · log(𝑃(𝑘)) and 𝐼(𝐶, 𝐶′) =
+𝐾
+∑︁
+𝑘=1
+𝐾 ′
+∑︁
+𝑘′=1
+𝑃(𝑘, 𝑘′) · log
+� 𝑃(𝑘, 𝑘′)
+𝑃(𝑘)𝑃(𝑘′)
+�
+.
+(6.2)
+Here 𝑃(𝑘) = 𝑛𝑘/𝑁, 𝑃(𝑘, 𝑘′) = |𝐶𝑘 ∩ 𝐶′
+𝑘′|/𝑁, 𝑛𝑘 is the number of nodes in the cluster 𝐶𝑘, 𝑁 is
+the total number of nodes in the graph, and 𝐾 and 𝐾′ are the number of elements in 𝐶 and 𝐶′,
+respectively. In our case, the VI metric can be used to compare the truth track-to-vertex assignment
+with the obtained clustering solution. When the obtained set of clusters and the track-to-cluster
+assignment reproduce the truth exactly, 𝑉𝐼 vanishes. Consequently, smaller VI values correspond
+to more truth-like (and therefore better) clustering solutions.
+Another track-to-cluster-based metric, which is investigated in the following, is the Silhou-
+ette [21] score
+𝑠(𝑖) =
+𝑏(𝑖) − 𝑎(𝑖)
+max{𝑎(𝑖), 𝑏(𝑖)}
+(6.3)
+with
+𝑎(𝑖) =
+1
+𝑛𝑘 − 1
+𝐶𝑘
+∑︁
+𝑗, 𝑖≠𝑗
+𝑑(𝑖, 𝑗) and 𝑏(𝑖) =
+min
+𝐶𝑘′≠𝐶𝑘
+1
+𝑛 𝑗
+𝐶𝑘′
+∑︁
+𝑗
+𝑑(𝑖, 𝑗)
+(6.4)
+for node 𝑖 in cluster 𝐶𝑘. Here 𝑑(𝑖, 𝑗) is a distance between nodes 𝑖 and 𝑗. In this study, we use the
+Cartesian distance between tracks and average over all tracks silhouette value
+�
+𝑠(𝑖)
+�
+as a quality
+estimator of the clustering solution. The silhouette value is limited −1 < 𝑠(𝑖) < 1, larger values
+corresponding to more compact clusters, better separated from each other.
+Several other track-to-cluster-based metrics can be found in Reference [20]. These metrics are
+expected to encounter problems in the present case due to the overlap of truth clusters, as explained
+in Section 4. Tracks are assigned most probably to the wrong cluster by any partitioning algorithm
+if placed in between tracks from other clusters by mismeasurement. This phenomenon inevitably
+reduces the accuracy of any track-to-cluster-based metrics. Nevertheless, at least the clustering of
+– 10 –
+
+the well-measured tracks should reproduce the truth closely, which the track-to-cluster metrics can
+still be sensitive to.
+As the metric accuracy is compromised by the presence of tracks with large measurement
+errors, it might be useful to downscale the contribution of such tracks to the metric. For the VI
+metric this can be achieved by weighting every track with 𝜎−2 in the metric calculations, namely
+𝑛𝑘 = �𝑘
+𝑖=1
+1
+𝜎2
+𝑖 , 𝑁 = �𝑁
+𝑖=1
+1
+𝜎2
+𝑖 , etc. For the Silhouette metric the Cartesian distance between two
+tracks can be replaced by its significance 𝑑(𝑖, 𝑗) = 𝑆𝑖 𝑗. The weighted versions of the VI and
+Silhouette metric will be used in the following, along with the original versions.
+The number of reconstructed clusters and the weighted average positions of these clusters,
+dominated by the well-measured tracks, are mostly decoupled from the details of the track-to-
+cluster assignment. The number of clusters can be directly used as a metric (up to the possible
+presence of fake clusters), but a Cartesian distance-based metric is not straightforward. One may
+try to introduce such a metric exploiting the cluster–cluster resolution 𝑅𝑐𝑐, i.e. the minimal distance
+between two reconstructed clusters, see Figure 7. The good, merged, bad cluster categories could
+be defined based on whether the cluster–truth vertex distance is smaller or larger than 𝑅𝑐𝑐. Such
+cluster categories could be used to compare various clustering solutions. But this categorisation
+explicitly depends on 𝑅𝑐𝑐, which itself depends on the clustering algorithm. To avoid such circular
+dependence, a scale-independent Cartesian distance-based metric is needed.
+4
+−
+3
+−
+2
+−
+1
+−
+0
+1
+2
+3
+4
+ z[mm] 
+∆
+ 
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Events / 0.1mm
+=0.35mm
+CC
+R
+Figure 7: Example of a fit to the cluster–cluster distance to determine the resolution. The used
+fitting function is 𝑎/{1 + exp[𝑏 · (𝑅𝑐𝑐 − |𝑥|)]} + 𝑐 where a, b, c are free fitting parameters and 𝑅𝑐𝑐
+is the cluster–cluster resolution, defined as the half-width at the half-depth of the dip in the centre
+of the cluster–cluster weighted centre distances, averaged over all clusters.
+To construct such a metric, we propose the following procedure. Every reconstructable truth
+vertex is linked to the closest reconstructed cluster in the Cartesian space that has 2 or more assigned
+tracks. Thus, a list of linked reconstructed clusters is obtained. Then, every reconstructed cluster is
+classified depending on how many times it enters into this list. If a cluster enters this list only once,
+there is just a single truth vertex referencing this cluster. Therefore it can be called unique, which
+means that a truth vertex is unambiguously reconstructed as a cluster. If a cluster enters several
+times into the list, it is referenced by several truth vertices, and therefore it combines tracks from
+– 11 –
+
+these vertices: this cluster can be called merged. Also, some clusters may not appear in this list
+at all: such clusters are not referenced by any truth vertex and are thus fake. The total number of
+obtained clusters and their classification as unique, merged, fake are scale-independent and can be
+used as a metric to compare various clustering options.
+7
+Results
+7.1
+LHC Run-2 13 TeV data
+First, the LMC clustering algorithm is tested with simulated DELPHES data at a collision energy of
+13 TeV, with pileup
+�
+𝜇
+�
+= 63 and 𝜎𝑧 = 35 mm. These parameters are chosen to provide simulated
+data close to the actual data collected by the ATLAS detector in Run 2. Edge-weight distributions
+for various edge-labelling approaches on these data are shown in Figure 8. The performance of
+the LMC algorithm on these data is shown in Table 4. The rows labelled “cnst” in these tables
+provide performance estimation with the applied constraints |𝑧𝑖 − 𝑧 𝑗| < 1 mm, while the “base”
+rows describe the baseline algorithm performance without constraints.
+15
+−
+10
+−
+5
+−
+0
+5
+Edge weight
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+Density (arbitrary units)
+PDF ratio
+20
+−
+15
+−
+10
+−
+5
+−
+0
+5
+Edge weight
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Density (arbitrary units)
+Logistic regression 1var
+1
+−
+0
+1
+2
+Edge weight
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+Density (arbitrary units)
+BDT
+Figure 8: Typical edge weight distributions for various edge labelling options.
+The column 𝑁wrong
+trk
+in Table 4 is the number of tracks assigned to one cluster but entirely
+surrounded by tracks from other clusters. This number is an estimator for the degree of cluster
+overlap in the obtained solution. The relevant truth data overlap for comparison can be found
+in Table 1.
+In addition, Table 9 in the Appendix gives the number of isolated nodes (tracks)
+reported by the LMC clustering algorithm. These non-assigned tracks do not represent the one-
+track truth vertices, considered non-reconstructable without a priori information, but rather reflect
+the clustering problems.
+The PDF ratio and the regression-based edge weight assignment result in approximately equal
+clustering performance. The BDT-based edge weight assignment leads to a significantly worse
+Silhouette metric value, a smaller value of the cluster overlap and a larger amount of fake clusters.
+As expected, the weighted versions of the VI and Silhouette metrics have significantly better values
+than the standard ones due to downscaling of the noise. Using constraints uniformly improves all
+quality estimators and provides ∼ 30% CPU reduction.
+In total, 70% of the reconstructable truth vertices are reconstructed as unique clusters, while
+the remaining 30% (i.e. 15) truth vertices are squeezed into 7.5 merged vertices. The amount of
+– 12 –
+
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 0.839
+0.407
+0.615
+0.646
+33.3
+8.2
+2.4
+15%
+0.25s
+cnst
+0.782
+0.362
+0.649
+0.660
+33.9
+7.9
+2.3
+8%
+0.18s
+Regression
+base 0.860
+0.416
+0.589
+0.623
+34.7
+7.6
+4.1
+14%
+0.27s
+cnst
+0.829
+0.387
+0.614
+0.633
+35.0
+7.5
+3.9
+8%
+0.18s
+BDT
+base 0.945
+0.399
+0.478
+0.230
+35.0
+7.5
+7.1
+5%
+0.23s
+cnst
+0.937
+0.377
+0.487
+0.234
+35.2
+7.4
+7.0
+4%
+0.14s
+Table 4: LMC performance for the collision energy 13 TeV, pileup 63 and interaction region width
+𝜎𝑧 = 35 mm. These simulation parameters are chosen to match the full ATLAS simulation for Run
+2 results used for comparison. The column 𝑁wrong
+trk
+shows the fraction of tracks wrongly associated
+by the clustering algorithm, which shall be compared to the truth fraction of 20% (Table 2).
+fake clusters is in the range of 5–15%. The number of tracks in the different cluster categories is
+presented in Figure 9. The number of tracks in the unique clusters is close to the track amount in
+the truth vertices, see Figure 2, while the merged clusters contain much more tracks. Finally, fake
+clusters have a very small number of tracks.
+0
+10 20 30 40 50 60 70 80 90 100
+Number of tracks in cluster
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+Density (arbitrary units)
+Unique vertices
+0
+10 20 30 40 50 60 70 80 90 100
+Number of tracks in cluster
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+Density (arbitrary units)
+Merged vertices
+0
+10 20 30 40 50 60 70 80 90 100
+Number of tracks in cluster
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Density (arbitrary units)
+Fake vertices
+Figure 9: Number of tracks in a cluster for the unique, merged and fake cluster categories. The
+distributions are obtained for pileup
+�
+𝜇
+�
+= 63 data using a one-variable logistic regression for the
+edge weight assignment.
+7.2
+High-Luminosity LHC 14 TeV data
+The High Luminosity LHC (HL-LHC) project foresees a significant increase in interaction rates
+to collect significantly more data and thus increase the sensitivity for new physics. The exact
+parameters of the upgraded HL-LHC are not yet final; pileup values of 150, 200, and 250, and an
+interaction region width of 𝜎𝑧 = 42 mm are considered the most probable options. These options
+result in an increase in the density of pileup interaction vertices up to a factor of 4, as compared
+to the current LHC parameters. The degree of truth cluster overlap rises from 20% to 66%, see
+– 13 –
+
+Table 1. It is interesting to check the performance of the LMC problem formulation in such extreme
+conditions.
+For this test, the same PDF ratio and logistic regression function are used for the edge weight
+calculation, while the BDT classification is retrained using 𝜇 = 150, 200, 250 data. Results for
+nominal PDF ratio and logistic regression-based edge weight calculation functions are shown in
+Tables 5, 6, and 7.
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 1.318
+0.690
+0.535
+0.577
+57.7
+27.4
+4.8
+28%
+1.1s
+cnst
+1.211
+0.612
+0.581
+0.609
+59.4
+26.9
+4.1
+14%
+0.42s
+Regression
+base 1.316
+0.682
+0.514
+0.559
+63.0
+25.6
+8.8
+26%
+0.73s
+cnst
+1.259
+0.634
+0.546
+0.582
+63.6
+25.4
+8.2
+14%
+0.50s
+BDT
+base 1.303
+0.658
+0.394
+0.146
+61.8
+25.9
+13
+9%
+0.96s
+cnst
+1.275
+0.616
+0.409
+0.155
+62.8
+25.6
+12
+7%
+0.43s
+Table 5: LMC performance for pileup 𝜇 = 150 in an HL-LHC environment with collision energy
+14 TeV and interaction region size 𝜎𝑧 = 42 mm. The column 𝑁wrong
+trk
+shows the fraction of the
+tracks, wrongly associated by the clustering algorithm, which can be compared to the true fraction
+41% (Table 2).
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 1.574
+0.852
+0.500
+0.546
+64.3
+40.3
+5.7
+36%
+2.3s
+cnst
+1.441
+0.756
+0.552
+0.586
+66.6
+39.8
+4.8
+18%
+0.69s
+Regression
+base 1.546
+0.825
+0.492
+0.539
+70.3
+38.6
+9.0
+32%
+2.4s
+cnst
+1.470
+0.765
+0.529
+0.568
+71.0
+38.4
+8.1
+18%
+0.69s
+BDT
+base 1.512
+0.805
+0.312
+0.040
+69.9
+38.6
+15.6
+13%
+1.8s
+cnst
+1.479
+0.755
+0.332
+0.051
+71.3
+38.2
+15.0
+7%
+0.66s
+Table 6: LMC performance for pileup 𝜇 = 200 in an HL-LHC environment with collision energy
+14 TeV and interaction region size 𝜎𝑧 = 42 mm. The column 𝑁wrong
+trk
+shows the fraction of the
+tracks, wrongly associated by the clustering algorithm, which can be compared to the true fraction
+53% (Table 2).
+Similarly to the 𝜇 = 63 results, the BDT-based edge weight assignment leads to a significantly
+worse Silhouette metric value, a much smaller value of the cluster overlap and a larger number of
+fake clusters, while the PDF ratio and regression-based edge weight calculation approaches provide
+similar performances. The weighted versions of the VI and Silhouette metrics have significantly
+better values than the standard ones due to downscaling of the noise.
+The use of constraints
+– 14 –
+
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 1.782
+0.990
+0.477
+0.526
+68.7
+53.2
+6.4
+42%
+3.0s
+cnst
+1.638
+0.887
+0.531
+0.569
+71.0
+52.7
+5.3
+21%
+1.7s
+Regression
+base 1.753
+0.961
+0.467
+0.517
+77.1
+51.2
+11.
+38%
+3.2s
+cnst
+1.672
+0.895
+0.505
+0.547
+77.8
+51.1
+9.9
+21%
+1.7s
+BDT
+base 1.691
+0.941
+0.307
+0.040
+72.8
+52.4
+15.
+12%
+3.0s
+cnst
+1.651
+0.882
+0.330
+0.055
+74.5
+52.0
+14.
+9%
+1.2s
+Table 7: LMC performance for pileup 𝜇 = 250 in an HL-LHC environment with collision energy
+14 TeV and interaction region size 𝜎𝑧 = 42 mm. The column 𝑁wrong
+trk
+shows the fraction of the
+tracks, wrongly associated by the clustering algorithm, which can be compared to the true fraction
+66% (Table 2).
+significantly improves all quality estimators and provides ∼ 30% CPU reduction.
+The number of unambiguously reconstructed unique clusters is 53% (44%, 37%) out of the
+total amount of the reconstructable truth vertices for the pileup 𝜇 = 150 (200, 250). The remaining
+56 (90, 125) reconstructable truth vertices are clustered into 25 (40, 52) merged clusters. The
+correctness of representation of the initial truth vertices by merged clusters is not granted. Truth
+vertices with a large number of tracks might “absorb” vertices with a small number of tracks.
+7.3
+LMC performance adjustment
+As can be seen from Tables 4–7, different edge weight assignment approaches lead to non-coinciding
+clustering results. For a practical application of the LMC approach for primary vertex finding in
+the LHC experiments, it is important to verify whether a unique optimal clustering solution exists
+in this problem and, if so, whether the different LMC cost functions can be tuned to provide the
+same clustering. As explained in Section 5, parameters of the PDF ratio and regression function for
+the edge weights can be modified to enforce under- or over-segmentation.The PDF ratio function
+can be scaled up and down. In the logistic regression function, the intercept term can be shifted by
+a constant. The cost function modifications are tried on the 𝜇 = 150 data. The obtained clustering
+results are shown in Figure 10 and Figure 11.
+In the performed test, the exploited metrics change monotonically depending on the scale factor
+for the PDF ratio and the intercept shift for the linear regression function. It doesn’t seem possible
+to adjust the PDF ratio and logistic regression parameters so that both approaches provide exactly
+the same clustering performances in all used metrics. In addition, the BDTG-based Silhouette and
+Silhouette weighted metrics results (see Table 5) are not reproducible by any modification of the
+PDF ratio and logistic regression cost functions. However, the overall variations of the clustering
+results remain limited, which means that the LMC approach performance stays close to optimal in
+the full scanned parameter range.
+To conclude, the cost function modification test doesn’t demonstrate the presence of an evident
+unique globally optimal clustering solution for the problem in consideration. Three used edge
+– 15 –
+
+0.8
+0.9
+1
+1.1
+PDF ratio scale
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.8
+0.9
+1
+1.1
+PDF ratio scale
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Resolution
+0
+0.2
+0.4
+Resolution (mm)
+Figure 10: PDF ratio cost-based clustering results as a function of the applied scaling.
+0.2
+−
+0
+0.2
+0.4
+Regression intercept shift
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.2
+−
+0
+0.2
+0.4
+Regression intercept shift
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Resolution
+0
+0.2
+0.4
+Resolution (mm)
+Figure 11: Logistic regression cost-based clustering results as a function of the logistic regression
+intercept term shift.
+weight assignment strategies provide different clustering results, which can be additionally changed
+by simple modification of the cost functions. Therefore, for a practical application as a primary
+vertex finder, an exact LMC formulation should be chosen based on desired physics requirements,
+e.g. minimal amount of fake vertices or best vertex–vertex resolution, disregarding the clustering
+metrics.
+7.4
+Influence of tracks with large measurement errors
+As the truth cluster overlap is caused by the track position mismeasurement, the overlap degree
+can be reduced by removing the badly measured tracks by cutting on the track measurement error
+shown in Figure 2b. A moderate decrease in the total amount of tracks due to this rejection should
+not significantly affect the overall clustering efficiency as the total amount of tracks per truth vertex
+is big enough, see Figure 2a. Reduction of the amount of the selected tracks and the degree of the
+truth cluster overlap due to strongly mismeasured track removal is shown in Table 8. The results
+– 16 –
+
+Track error cut
+𝑁trk
+Truth overlap
+-
+1674
+41%
+0.8
+1540
+31%
+0.6
+1444
+27%
+0.4
+1283
+22%
+Table 8: Number of selected tracks and the truth degree of overlap as a function of the track error
+cut for 𝜇 = 150 data.
+of the clustering are shown in Figure 12 for the PDF ratio cost function and in Figure 13 for the
+nominal logistic regression cost function.
+0.5
+1
+1.5
+Track error cut(mm)
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.5
+1
+1.5
+Track error cut (mm)
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Figure 12: PDF ratio cost-based clustering results as a function of the applied track error cut for
+the 𝜇 = 150 data.
+0.5
+1
+1.5
+Track error cut(mm)
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.5
+1
+1.5
+Track error cut (mm)
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Figure 13: Logistic regression cost-based clustering results as a function of the applied track error
+cut for the 𝜇 = 150 data.
+– 17 –
+
+The distance-based metric demonstrates very small changes in the clustering results in a
+wide range of the badly measured track admixture and, correspondingly, the initial degree of the
+vertex overlap. One may conclude that the amount of clusters identified by the LMC algorithm is
+largely defined by the tracks with small measurement errors and, therefore, is stable with respect
+to significant track noise admixture. Redistribution of the tracks with big errors over the obtained
+clusters doesn’t change their amount but evidently strongly affects all track counting-based clustering
+metrics. The track weighting does mitigate this effect for the VI metric, its weighted version is
+practically independent of the track noise admixture. Surprisingly, the Silhouette metric is only
+weakly sensitive to this noise.
+7.5
+Comparison with the existing approaches
+6
+−
+4
+−
+2
+−
+0
+2
+4
+6
+z [mm]
+∆
+arbitrary units
+          
+t
+AMVF, t
+           
+t
+IVF, t
+Preliminary
+ Simulation
+ATLAS
+ = 60
+〉
+µ
+〈
+ = 13 TeV, 
+s
+0
+10
+20
+30
+40
+50
+60
+70
+80
+ interactions per bunch crossing
+pp
+Number of 
+0
+10
+20
+30
+40
+50
+60
+Average number of reconstructed vertices
+100% interaction reconstruction efficiency
+Reconstruction acceptance
+t
+AMVF, t
+t
+IVF, t
+AMVF - MATCHED
+AMVF - MERGED
+AMVF - SPLIT
+AMVF - FAKE
+ATLAS Simulation Preliminary
+ = 13 TeV
+s
+6
+−
+4
+−
+2
+−
+0
+2
+4
+6
+ z[mm] 
+∆
+ 
+0
+200
+400
+600
+800
+1000
+1200
+1400
+Events / 0.02mm
+DELPHES simulation LMC Cluster-Cluster distance 
+=0.37mm
+CC
+R
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Number of pp interactions per bunch crossing
+0
+10
+20
+30
+40
+50
+60
+Average number of reconstructed clusters
+LMC All
+LMC Unique
+LMC Merged
+100% interaction reconstruction efficiency
+Reconstruction acceptance
+Figure 14: The vertex–vertex resolution and the number of reconstructed vertices as a function
+of the number of 𝑝𝑝 interactions for typical ATLAS data. The upper plots are obtained with the
+the ATLAS baseline AMVF [4] and IVF [3] algorithms. The bottom plots are obtained using the
+LMC algorithm with the PDF ratio-based edge weight assignment on DELPHES 𝜇 = 63 data.
+The DELPHES 𝜇 = 63 simulation is specially tuned to match the ATLAS data used in [4]. The
+cluster–cluster resolution for the LMC algorithm on the bottom left picture is obtained as described
+in Section 6.
+The ATLAS Collaboration used the IVF algorithm [3] to reconstruct the 𝑝𝑝 collision vertices
+in Run 1 and the AMVF algorithm [4] in Run 2 and Run 3. Essential characteristics of a primary-
+vertex reconstruction algorithm are the vertex–vertex resolution and the number of reconstructed
+– 18 –
+
+vertices as a function of the number of 𝑝𝑝 interactions. The upper plots in Figure 14 present the
+corresponding distributions for typical ATLAS data for the AMVF and IVF algorithms. The bottom
+plots show the same distributions provided by the LMC algorithm using DELPHES data tuned to
+the same pileup conditions.
+Figure 14 clearly demonstrates that the LMC algorithm outperforms the ATLAS heuristic
+algorithms. It provides significantly better vertex–vertex resolution. This naturally leads to a larger
+amount of Unique/Matched vertices reconstructed by LMC, while the amount of Merged vertices
+remains practically the same. Routine application of the LMC for the primary vertex reconstruction
+can provide a significant gain in performance for LHC and future collider experiments.
+8
+Conclusion
+In this work, we have addressed a typical particle physics problem of reconstructing multiple
+interaction positions in a dense environment, where each interaction is represented by a cluster
+of tracks. Significant track reconstruction errors lead to a large overlap of truth track clusters,
+which makes their identification challenging. Heuristic algorithms are usually used to address this
+problem. In contrast, we propose to address this problem through a principled formulation as a
+minimum-cost lifted multicut problem. We construct several cost functions for the LMC from
+track–track distances and their significance. We study the performance of the LMC algorithm
+for different vertex densities, cost functions, constraint usage and varying degree of overlap. To
+address potential performance problems of existing track counting clustering metrics for strongly
+overlapped clusters, dedicated metrics are introduced.
+We demonstrate that the LMC approach outperforms the heuristic algorithms in the problem
+of vertex reconstruction in dense environments in terms of vertex–vertex resolution and vertex
+reconstruction efficiency. It works up to the highest vertex density expected at the HL-HLC project
+in spite of the strong truth cluster overlap reaching ∼ 60%. Variations of the LMC algorithm
+parameters and cost functions studied in this work resulted in relatively small variations of the
+obtained clustering solutions.
+Acknowledgments
+This work is supported by the German Science Foundation (DFG) through a research grant and a
+Heisenberg professorship under contracts CR-312/4-1 and CR-312/5-1.
+References
+[1] R. Frühwirth and A. Strandlie, Pattern Recognition, Tracking and Vertex Reconstruction in Particle
+Detectors. Springer, 2021, 10.1007/978-3-030-65771-0.
+[2] ATLAS Collaboration, “Primary Vertex Selection in VBF Higgs to Invisibles at 𝜇 = 200 with the
+ATLAS Experiment.” IDTR-2019-004, 2019.
+[3] ATLAS Collaboration, Reconstruction of primary vertices at the ATLAS experiment in Run 1
+proton–proton collisions at the LHC, Eur. Phys. J. C 77 (2017) 332.
+– 19 –
+
+[4] ATLAS Collaboration, “Development of ATLAS primary vertex reconstruction for LHC Run 3.”
+ATL-PHYS-PUB-2019-015, 2019.
+[5] CMS Collaboration, Description and performance of track and primary-vertex reconstruction with
+the cms tracker, JINST 9 (2014) P10009.
+[6] S. Hageböck and E. von Toerne, Medical imaging inspired vertex reconstruction at LHC, Journal of
+Physics: Conference Series 396 (2012) 022021.
+[7] W. Waltenberger et al., Rave—a detector-independent vertex reconstruction toolkit, Nuclear
+Instruments and Methods in Physics Research A (2007) 549.
+[8] K. Rose, Deterministic annealing for clustering, compression, classification, regression and related
+optimisation problems, Proceedings of the IEEE 86 (1998) 2210.
+[9] M. Keuper, E. Levinkov, N. Bonneel, G. Lavoue, T. Brox and B. Andres, Efficient decomposition of
+image and mesh graphs by lifted multicuts, Proceedings of the IEEE International Conference on
+Computer Vision, ICCV (2015) .
+[10] E. D. Demaine, D. Emanuel, A. Fiat and N. Immorlica, Correlation clustering in general weighted
+graphs, Theoretical Computer Science 361 (2006) 172.
+[11] S. Chopra and M. R. Rao, The partition problem, Mathematical Programming 59 (1993) 87.
+[12] A. Horňáková, J.-H. Lange and B. Andres, Analysis and optimization of graph decompositions by
+lifted multicuts, Proceedings of the International Conference on Machine Learning, ICML (2017)
+[1503.03791].
+[13] T. Beier, T. Kroeger, J. Kappes, U. Köthe and F. Hamprecht, Cut, glue & cut: A fast, approximate
+solver for multicut partitioning, Proceedings of the IEEE Conference on Computer Vision and Pattern
+Recognition, CVPR (2014) .
+[14] J. H. Kappes, B. Andres, F. A. Hamprecht, C. Schnörr, S. Nowozin, D. Batra et al., A Comparative
+Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems,
+International Journal of Computer Vision 115 (2015) 155 [1404.0533].
+[15] T. Beier, B. Andres, U. Köthe and F. A. Hamprecht, An efficient fusion move algorithm for the
+minimum cost lifted multicut problem, Computer Vision – ECCV 2016. Lecture Notes in Computer
+Science (2016) .
+[16] A. Kardoost and M. Keuper, Solving minimum cost lifted multicut problems by node agglomeration,
+Computer Vision – ACCV 2018 (2019) .
+[17] DELPHES 3 collaboration, DELPHES 3: a modular framework for fast simulation of a generic
+collider experiment, JHEP 02 (2014) 057 [1307.6346].
+[18] C. Bierlich et al., A comprehensive guide to the physics and usage of PYTHIA 8.3, 2203.11601.
+[19] A. Hoecker et al., TMVA - Toolkit for Multivariate Data Analysis, physics/0703039.
+[20] M. Meilă, Comparing clusterings—an information based distance, Journal of Multivariate Analysis
+98 (2007) 873.
+[21] P. J. Rousseeuw, Silhouettes: A graphical aid to the interpretation and validation of cluster analysis,
+Journal of Computational and Applied Mathematics 20 (1987) 53.
+– 20 –
+
+A
+Non-clustered tracks and total reconstructed clusters
+In this study, we use four simulated event samples representing realistic proton–proton interactions at
+the LHC with different energies and luminosities. The total amounts of interaction vertices with one
+reconstructed track and two and more tracks are shown in Table 9. Due to the track measurement
+errors, the one-track vertices are difficult to reconstruct correctly without a priori information.
+Finding two and more track vertices becomes problematic if the vertex–vertex distance is less than
+the typical track measurement error. Both problems are illustrated in Table 9, where the amounts
+of the one-track and multi-track clusters are given for every cost function and event sample.
+13 TeV
+14 TeV
+�
+𝜇
+�
+= 63
+�
+𝜇
+�
+= 150
+�
+𝜇
+�
+= 200
+�
+𝜇
+�
+= 250
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+Truth
+4
+50
+9
+119
+12
+160
+16
+199
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+PDF ratio
+11
+44
+19
+90
+23
+110
+25
+128
+Regression
+13
+46
+25
+97
+27
+118
+31
+139
+BDTG
+43
+50
+77
+101
+102
+124
+104
+140
+Table 9: Average numbers of non-clustered tracks and reconstructed clusters obtained by the
+LMC algorithm with different cost functions as compared to the truth numbers of single-track and
+multi-track vertices. Results are shown for all collision energies and pileup densities.
+The number of one-track clusters in each case is significantly larger than the truth amount of
+one-track interaction vertices, especially in the BDTG case. They should be thought of as non-
+clustered tracks, not as reconstructed one-track vertices. The fraction of multi-track clusters found
+decreases with the interaction vertex density, as expected.
+– 21 –
+
diff --git a/E9E5T4oBgHgl3EQfVA-E/content/tmp_files/load_file.txt b/E9E5T4oBgHgl3EQfVA-E/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fe0c39eecde12a5de0365839855a6e2e61e23616
--- /dev/null
+++ b/E9E5T4oBgHgl3EQfVA-E/content/tmp_files/load_file.txt
@@ -0,0 +1,929 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf,len=928
+page_content='Prepared for submission to JINST  Improving primary-vertex reconstruction with a  minimum-cost lifted multicut graph partitioning algorithm  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Kostyukhin,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Keuper,2,3 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Ibragimov,1 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Owtscharenko,1 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Cristinziani1  1Center for Particle Physics Siegen, Department Physik, Universität Siegen  2Visual Computing, Department Elektrotechnik und Informatik, Universität Siegen  3Max Planck Institute for Informatics, Saarland Informatics Campus  Abstract: Particle physics experiments often require the simultaneous reconstruction of many  interaction vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Usually, this problem is solved by ad hoc heuristic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' We propose  a universal approach to address the multiple vertex finding through a principled formulation as  a minimum-cost lifted multicut problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The suggested algorithm is tested in a typical LHC  environment with multiple proton–proton interaction vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Reconstruction errors caused by the  particle detectors complicate the solution and require the introduction of special metrics to assess  the vertex-finding performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' We demonstrate that the minimum-cost lifted multicut approach  outperforms heuristic algorithms and works well up to the highest vertex multiplicity expected at  the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Keywords: Vertexing algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Pattern recognition, cluster finding, calibration and fitting  methods  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='05548v1  [hep-ex]  13 Jan 2023  Contents  1  Introduction  1  2  Minimum-cost multicuts and lifted multicut algorithm for cluster finding  3  3  Data simulation  5  4  Features of simulated data  6  5  Edge weights and constraints  6  6  Performance metrics  10  7  Results  12  8  Conclusion  19  A Non-clustered tracks and total reconstructed clusters  21  1  Introduction  In particle physics experiments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' many problems require a precise reconstruction of vertices — points  in 3D space where particle interactions occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Knowledge of the positions and features of such  vertices provides valuable information about the underlying physics of these interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' There are  numerous examples: 𝐵 physics, heavy-flavour jet identification, primary event vertex reconstruction,  search for exotic particles as new physics manifestations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Except for a few specifically designed  detectors (emulsions, Wilson chamber, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' ), rarely used in modern experiments, the vertices are  not directly detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The presence of vertices is usually inferred from 3D traces of charged stable  particles produced in the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Various tracking detectors measure the curved trajectories of  these particles (tracks) in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The trajectories can be extrapolated to a single 3D point, which  represents the interaction vertex position [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Despite the simplicity of the vertex reconstruction idea, its real-life exploitation encounters  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' For example, at the Large Hadron Collider (LHC) at the end of Run 2, a typical recorded  event consisted of ∼80 primary proton–proton interactions, and numerous produced charged par-  ticles underwent further interactions leading to additional vertices, distributed in significant 3D  volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The expected number of proton–proton interactions in a single event at the LHC after  the planned high-luminosity upgrade (HL-LHC) may reach 200–300, resulting in a few thousand  reconstructed tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Therefore, prior to determining the vertex positions, one needs to determine  how many vertices are present in a given event and assign the reconstructed tracks to these assumed  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The track measurement uncertainties, which may differ by a factor of 10 for different tracks  – 1 –  and often are comparable with the vertex–vertex distances, cause additional complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' These  uncertainties make an exact crossing of track pairs in 3D space impossible: even if two charged  particles are produced in the same interaction point, their reconstructed trajectories will only be  close to the true vertex position and to each other, up to the corresponding uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The explicit reconstruction of multiple vertices in an event can be addressed in a graph-based  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In fact, all space trajectories of the particles produced in a single vertex should be pairwise  compatible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' every pair of tracks should be close to each other in some volume around the true  vertex position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Therefore, a compatibility graph can be constructed where every node represents  a track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Two nodes are connected by an edge if and only if the distance of the corresponding  trajectories is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In the ideal case, every vertex is represented by a fully connected, isolated  subgraph in such a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In a realistic scenario, track measurement errors shuffle tracks among  different vertices, resulting in a large number of fake edges in the compatibility graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Yet, it can  be tried to split the full graph into non-overlapping clusters by minimising the track–track distances  for all track pairs in a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The obtained set of isolated clusters should be a good approximation  of the true vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The present paper focuses on finding primary proton–proton interaction vertices at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Subsequent decays of the particles produced in the detector volume and their interactions with  the detector material will not be considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' To illustrate the primary-vertex reconstruction  problem, Figure 1 shows two zoomed-in regions of a typical LHC event with several pileup  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The upper plot presents a region where a pileup interaction vertex is identified, which  has the largest sum of track transverse momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The bottom plot presents a region where a  hard-scatter vertex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' the point of interaction of interest, is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In both plots, the true  positions of interaction vertices are shown, together with charged particle trajectories displaced  due to reconstruction uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Several true interaction vertices in these plots do not have  associated tracks because all emanated particles in this interaction are outside of the tracking  detector’s sensitive volume, see Section 3 for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The overlap of the red (from hard-scatter  vertex), blue and grey (from nearby pileup vertices) tracks in the centre of the bottom plot on  Figure 1 is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Experiments at the LHC use heuristic algorithms [3–5] to reconstruct multiple proton–proton  interaction vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Several other approaches can be found in the literature, including medical  imaging-inspired algorithms [6] and the RAVE package [7] implementing the deterministic anneal-  ing algorithm [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  This article presents an implementation of the Lifted Multicut Graph Partitioning algorithm  (LMC), which solves the inclusive vertex reconstruction problem described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Section 2  describes the LMC algorithm and details of its implementation for the vertex finding application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Section 3 describes the simulated samples which are used to test the algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In  Section 4, features of the simulated samples are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Section 5 introduces edge cost functions  used in the graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In Section 6, the metrics are introduced to estimate the algorithm  performance and to compare it with other existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Section 7 presents the performance  of the LMC approach in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In Section 8, conclusions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  – 2 –  61  −  60  −  59  −  58  −  57  −  56  −  55  −  54  −  53  −  52  −  z [mm]  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  1  r [mm]  Reco z = -56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='71 mm  Truth z = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='59 mm  2  = 470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3 GeV  T  2  p  Σ  w  T  p  Σ  PU Vertex chosen by  Truth  Reco  HS tracks  PU tracks  HS jets  PU jets  truth PU vertex  truth HS vertex  cut  0  z  T  p  η  Simulation Preliminary  ATLAS  7  −  6  −  5  −  4  −  3  −  2  −  1  −  0  1  2  z [mm]  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  1  r [mm]  Reco z = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='60 mm  Truth z = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='59 mm  2  = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2 GeV  T  2  p  Σ  Reconstructed Hard-Scatter Primary Vertex  Truth  Reco  HS tracks  PU tracks  HS jets  PU jets  truth PU vertex  truth HS vertex  cut  0  z  T  p  η  Simulation Preliminary  ATLAS  Figure 1: Two regions of a typical LHC event in the ATLAS detector with many pileup interac-  tions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' True positions of the proton–proton interactions are shown, as well as the reconstructed  trajectories (tracks) of the produced particles scattered due to reconstruction uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Some  truth interaction vertices do not have associated tracks because all emanated particles are outside  of the sensitive detector phase space and not reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' These pictures illustrate typical track  densities and overlap of the tracks produced in nearby interaction vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Both, tracks associated  with the hard-scattering (HS) and pileup (PU) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  2  Minimum-cost multicuts and lifted multicut algorithm for cluster finding  We formulate the primary-vertex reconstruction problem as a minimum-cost lifted multicut problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  This problem was originally proposed in Reference [9] in the context of image segmentation and  mesh decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' It is a generalization of the better-known minimum cost multicut problem,  also referred to as the weighted correlation clustering problem [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The minimum cost multicut  problem is a grouping problem defined for a graph 𝐺 = (𝑉, 𝐸) and a cost function 𝑐 : 𝐸 → R  which assigns to all edges 𝑒 ∈ 𝐸 a real-valued cost or reward for being cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Then, the minimum  – 3 –  cost multicut problem is to find a binary edge labelling 𝑦 according to  min  𝑦∈{0,1}𝐸  ∑︁  𝑒∈𝐸  𝑐𝑒𝑦𝑒  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1)  subject to  ∀𝐶 ∈ cycles(𝐺)  ∀𝑒 ∈ 𝐶 : 𝑦𝑒 ≤  ∑︁  𝑒∈𝐶\\{𝑒}  𝑦𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2)  The constraints on the feasible set of labellings 𝑦 given in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2) ensure that the solution  of the multicut problem relates one-to-one to the decompositions of graph 𝐺, by ensuring for every  cycle in 𝐺 that if an edge is cut within the cycle (𝑦𝑒 = 1), so needs to be at least one other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Trivial  optimal solutions are avoided by assigning positive (attractive) costs 𝑐𝑒 to edges between nodes  𝑣, 𝑤 ∈ 𝑉 that likely belong to the same component, while negative (repulsive) costs are assigned to  edges that likely belong to different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The minimum cost lifted multicut problem (LMC) generalizes over the problem defined in  Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1)–Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2) by adding a second set of edges that defines additional, potentially  long-range costs without altering the set of feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' It thus defines a second set of edges  𝐹 between the nodes 𝑉 of 𝐺, resulting in a lifted graph 𝐺′ = (𝑉, 𝐸 ∪ 𝐹), on which we can define  a cost function 𝑐′ : 𝐸 ∪ 𝐹 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Then, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1) and Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2) are optimized over all  edges in 𝐸 ∪ 𝐹 and two additional sets of constraints are defined according to [9]  ∀𝑣, 𝑤 ∈ 𝐹  ∀𝑃 ∈ 𝑣, 𝑤 − paths(𝐺) : 𝑦𝑣𝑤 ≤  ∑︁  𝑒∈𝑃  𝑦𝑒  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3)  ∀𝑣, 𝑤 ∈ 𝐹  ∀𝐶 ∈ 𝑣, 𝑤 − cuts(𝐺) : 1 − 𝑦𝑣𝑤 ≤  ∑︁  𝑒∈𝐶  (1 − 𝑦𝑒)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4)  to ensure that the feasible solutions to the LMC problem still relate one-to-one to the decompositions  of the original graph 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  For the vertex reconstruction problem, this formulation allows encoding Euclidean distance  constraints in the structure of graph 𝐺 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' point observations that are spatially distant can not  originate from the same vertex), while the cost function can be naturally defined in the distance  significance space to take into account the measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The Euclidean distance and its  significance can be very different in case of significant reconstruction errors, the lifted multicut  formulation encodes both metrics in the same graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The minimum cost multicut problem is 𝑛𝑝-hard, and so is the minimum cost LMC problem  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Yet, efficient heuristic solvers provide practically good solutions [9, 13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Here, we resolve  to use the primal feasible heuristic KLj that has been proposed in Reference [9] and published in  an open-source library1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' KLj is an iterative approach that produces a sequence of feasible solutions  whose cost decreases monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' It takes as input an initial edge labelling (for example, all  edge labels are initially set to 0), a lifted graph and costs defined on all edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In every step, it  either moves nodes between two neighbouring components, moves nodes from one component into  a new component or joins two components such as to decrease the cost of the multicut maximally  according to Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='com/bjoern-andres/graph  – 4 –  3  Data simulation  To estimate the clustering performance, we simulated data using DELPHES [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The framework  allows to perform a fast and realistic simulation of a general-purpose collider detector composed of  an inner tracker, electromagnetic and hadron calorimeters, and a muon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' For this study, we  added a detailed parameterisation of the ATLAS detector tracking resolution to the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  To simulate the pileup vertices and hard-scattering events, a sufficiently large amount of  minimum-bias interaction events was prepared, consisting of single, double, and non-diffractive  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' These events have been generated using the Pythia 8 [18] event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' As the main  source of hard-scattering interactions, 𝑡¯𝑡 events are used, also generated with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' To simulate  an LHC collision event with full pileup, a single 𝑡¯𝑡 event is mixed with a number of minimum-  bias events, distributed according to a Poisson distribution with a mean corresponding to a chosen  luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The interaction vertices are then distributed along the LHC beam trajectory inside the  detector, according to typical interaction region parameters for ATLAS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' according to a Gaussian  with 𝜎𝑧 = 42 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The acceptance of the ATLAS detector allows for reconstructing charged particle trajectories in  a limited phase space of 𝑝⊥ > 500 MeV and |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Some minimum-bias proton–proton interac-  tions produce only particles outside the sensitive phase space of the ATLAS detector, which makes  them unreconstructable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Positions of interactions with a single track in the ATLAS acceptance can  be reconstructed, but this vertex category is contaminated by tracks that are strongly displaced by  measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In the following, a reconstructable truth vertex refers to the true position of a  proton–proton interaction producing at least two tracks within the ATLAS detector acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  All tracks produced in an event and falling into the sensitive ATLAS detector phase space  are smeared according to the parameterised ATLAS detector resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Tracks with smeared  parameters are referred to as reconstructed tracks in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The set of reconstructed tracks  corresponding to a full pileup event is used as input for the performance estimation of the clustering  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' DELPHES samples used in this paper have been prepared with different energies and  different pileup conditions (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Energy  �  𝜇  �  Interaction region 𝜎𝑧  �  𝑁event  trk  �  �  𝑁vrt  trk =0  �  �  𝑁vrt  trk =1  �  �  𝑁vrt  trk >1  �  13 TeV  63  35 mm  718  9  4  50  14 TeV 150  42 mm  1674  22  9  119  14 TeV 200  42 mm  2227  28  12  160  14 TeV 250  42 mm  2771  35  16  199  Table 1: The DELPHES samples used to estimate the LMC performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Column 𝑁event  trk  reports  the total number of reconstructed tracks in simulated events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The last three columns show the  numbers of true vertices with 𝑁vrt  trk = 0, 1, > 1, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  – 5 –  4  Features of simulated data  The number of truth tracks in the detector acceptance in the simulated vertices and the position  measurement errors of these tracks are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' As can be seen in Figure 2a, in 14% of  the cases, the simulated vertices do not have tracks in the detector acceptance, and in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5% of the  cases, they have only one track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The number of tracks for all other vertices is widely spread up to  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  0  20  40  60  80  Number of tracks in a truth vertex  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='14  Fraction  a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  2  Track measurement error  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  Fraction  b)  Figure 2: a) Number of tracks per simulated vertex and b) Track measurement errors of the  simulated tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Track measurement errors are shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' From the sizes of the luminous regions and  the number of vertices in Table 1 we can conclude that the track measurement errors are comparable  or larger than a typical vertex–vertex distance in the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Smearing of the track positions  due to measurement errors results in a significant overlap of the tracks from different truth vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The fraction of cases when a track from one vertex is entirely surrounded by tracks from other  vertices for different pileup scenarios is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' An example of the track overlap can be  seen in the bottom panel of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Another example is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  �  𝜇  �  63  150  200  250  Track overlap fraction 20% 41% 53% 66%  Table 2: Fraction of tracks, positioned in between tracks from other truth vertices due to measure-  ment errors, as a function of the pileup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  A priori, well-measured tracks with small errors should be easy to cluster according to the  truth, while poorly measured tracks with large errors can easily migrate from one cluster to another,  independently of their true origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' This random migration can be interpreted as noise, and thus, the  overall problem may be considered as clustering in the presence of significant noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  5  Edge weights and constraints  To formulate the vertex finding problem in the presence of pileup as a minimum cost lifted multicut  (LMC) problem, a track-pair compatibility graph needs to be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' A node in this graph  – 6 –  4  6  8  10  12  Track and true vertex positions (mm)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  −  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  Track error (mm)  Track positions linked to true vertices  True vertices  Figure 3: Example display of overlapping tracks from different vertices caused by measurement  errors (zoom of a simulated DELPHES event with 𝜇 = 150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The crosses at the ordinate value  of 0 represent the track positions, and the vertical error bars represent the corresponding position  measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Squares at ordinate values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3 represent the truth vertex positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The  connecting lines show the origin vertex for every track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  represents a track, and two nodes are connected by an edge if and only if they are close in space and  can be produced in the same vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The degree of track closeness, or equivalently the probability of  originating in the same vertex, is estimated during the graph construction and is expressed as a weight  assigned to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The edge weights determine the efficiency of the clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Therefore, they  should incorporate enough information, and the weight assignment procedure should be carefully  designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The following approaches are used in our study:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Probability density function (PDF) ratio of the track–track geometrical distance significance  based on measured uncertainties, 𝑆 =  √︃  (𝑧𝑖 − 𝑧 𝑗)2/(𝜎2  𝑖 + 𝜎2  𝑗 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Multivariate binary classification with Boosted Decision Trees (BDT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Logistic regression based on 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The LMC formulation assumes that the correct edges (two tracks from the same vertex) receive  positive weights, while random (fake) edges receive negative weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' This can be achieved by  using a logarithm of the ratio of the probability density functions for the correct and fake edges  as the cost function of the problem log 𝑝true  𝑝fake .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' According to the Neyman–Pearson lemma, this is  the most efficient test statistic for the true/fake edge classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' An example of the track–track  distance significance distributions and their ratio are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' As the PDF of the fake  edges is independent of the track–track distance significance, its overall normalisation depends on  the significance range used for the parameterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Thus, the exact values of the PDF ratio can  be scaled by the choice of the parametrisation range, which in principle, should not affect the LMC  – 7 –  clustering performance if the range is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Such a behaviour can be mimicked by a  global multiplier of the PDF ratio function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The influence of this multiplier on the clustering will  be studied in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='024  True edges  0  1  2  3  4  5  6  Track-track distance significance  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='006  Fake edges  0  1  2  3  4  5  6  Track-track distance significance  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content="5  5  Ratio True/Fake pdf's  Figure 4: Example track–track distance significance for true and fake edges and their ratio." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The  significance distributions are normalized to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  A better clustering performance could be achieved by encoding more information in the edge  weight calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' To test this approach, we use a BDT classifier combining seven features, listed  in Table 3, to distinguish true edges from fake ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The GradientBoost implementation (BDTG)  from the TMVA [19] package is used to train the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' An example of the trained classifier  response2 is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The output is negative for fake edges and positive for true ones,  exactly as required by the KLj algorithm, and therefore can be used directly as the edge weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Description  1  Squared significance 𝑆2 (or 𝜒2) of track–track distance along beamline  2  Average position of the track pair along beamline  3  Position uncertainty of track 1  4  Position uncertainty of track 2  5  Pseudorapidity 𝜂 of track 1  6  Pseudorapidity 𝜂 of track 2  7  Number of other tracks crossing the beamline between tracks 1 and 2  Table 3: Input features for the edge classification BDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Edge weights can also be assigned by using the logistic regression 𝑝 = 𝑒𝑧/(𝑒𝑧 + 1), where  𝑧 = 𝛽0 + �𝑛  𝑖=1 𝛽𝑖𝑥𝑖 and 𝑥𝑖 are explanatory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The negative inverse of the logistic function,  logit(𝑝) = log[𝑝/(1 − 𝑝)], provides the necessary edge weight behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Edges that need to be  removed receive negative weights, and those that need to be preserved receive positive weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The intercept value 𝛽0 is defined by the ratio between the amount of true and fake edges used  for training, which can be linked to a prior probability of a given edge being true or fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In the  2TMVA GradientBoost uses the binomial log-likelihood loss 𝐿(𝐹, 𝑦) = ln[1 + exp(−2𝐹(𝑥)𝑦)] with Gini Index  separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' We use the following training settings NTree=800, MaxDepth=10, MinNodeSize=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5%, Shrinkage=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  – 8 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  BDTG response  0  1  2  3  4  5  6  dx  /  (1/N) dN  Signal  Background  U/O-flow (S,B): (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='0)% / (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='0)%  TMVA response for classifier: BDTG  Figure 5: Example BDTG classification weight distributions for true and fake edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  current problem, the prior probability depends on the true vertex density and cannot be defined  unambiguously, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' it depends on the range of the track–track distance significance 𝑆, see above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Therefore, the value of the intercept 𝛽0 in this approach can be modified in some range to achieve  an over- or undersegmentation in order to validate its optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' This will be further discussed in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' A one-dimensional regression is tested in this paper, using variable (1) from Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The logistic regression for the edge weight calculation is illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  0  5  10  15  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='0  S2  True  Figure 6: Example one-variable logistic regression for true and fake edges using the squared  track–track distance significance 𝑆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The usage of the track–track distance significance for partitioning does not guarantee the com-  pactness of the obtained cluster in Cartesian space, which may be beneficial when the vertex density  is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The compactness requirement can be imposed using the LMC constraint mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Some edges in the connectivity graph can be additionally labelled as “have to be cut”, based on a  priori information, different from the edge probability itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' To make clusters more compact, we  – 9 –  can constrain the edges to be cut if the corresponding Cartesian track–track distance is larger than  some scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In the following, a rather weak requirement of |𝑧𝑖 − 𝑧 𝑗| < 1 mm will be used, which  removes tracks with very large errors, see Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In addition to improving the quality of the  solution, the constraint limits the phase space of possible solutions, and this leads to a significant  algorithm speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  6  Performance metrics  For a quantitative assessment of the performance of the vertex-finding algorithm, one or several  metrics are to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' To compare the performance of the clustering algorithms in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=',  image segmentation problems, metrics are usually employed, which are based on the assignment  of graph nodes to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' One example of such a metric is the Variation of Information (VI)  proposed in Reference [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The VI metric calculates the degree of compatibility of a clustering 𝐶  with another clustering 𝐶′ as  𝑉𝐼(𝐶, 𝐶′) = 𝐻(𝐶) + 𝐻(𝐶′) − 2 · 𝐼(𝐶, 𝐶′)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1)  with  𝐻(𝐶) = −  𝐾  ∑︁  𝑘=1  𝑃(𝑘) · log(𝑃(𝑘)) and 𝐼(𝐶, 𝐶′) =  𝐾  ∑︁  𝑘=1  𝐾 ′  ∑︁  𝑘′=1  𝑃(𝑘, 𝑘′) · log  � 𝑃(𝑘, 𝑘′)  𝑃(𝑘)𝑃(𝑘′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2)  Here 𝑃(𝑘) = 𝑛𝑘/𝑁, 𝑃(𝑘, 𝑘′) = |𝐶𝑘 ∩ 𝐶′  𝑘′|/𝑁, 𝑛𝑘 is the number of nodes in the cluster 𝐶𝑘, 𝑁 is  the total number of nodes in the graph, and 𝐾 and 𝐾′ are the number of elements in 𝐶 and 𝐶′,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In our case, the VI metric can be used to compare the truth track-to-vertex assignment  with the obtained clustering solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' When the obtained set of clusters and the track-to-cluster  assignment reproduce the truth exactly, 𝑉𝐼 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Consequently, smaller VI values correspond  to more truth-like (and therefore better) clustering solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Another track-to-cluster-based metric, which is investigated in the following, is the Silhou-  ette [21] score  𝑠(𝑖) =  𝑏(𝑖) − 𝑎(𝑖)  max{𝑎(𝑖), 𝑏(𝑖)}  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3)  with  𝑎(𝑖) =  1  𝑛𝑘 − 1  𝐶𝑘  ∑︁  𝑗, 𝑖≠𝑗  𝑑(𝑖, 𝑗) and 𝑏(𝑖) =  min  𝐶𝑘′≠𝐶𝑘  1  𝑛 𝑗  𝐶𝑘′  ∑︁  𝑗  𝑑(𝑖, 𝑗)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4)  for node 𝑖 in cluster 𝐶𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Here 𝑑(𝑖, 𝑗) is a distance between nodes 𝑖 and 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In this study, we use the  Cartesian distance between tracks and average over all tracks silhouette value  �  𝑠(𝑖)  �  as a quality  estimator of the clustering solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The silhouette value is limited −1 < 𝑠(𝑖) < 1, larger values  corresponding to more compact clusters, better separated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Several other track-to-cluster-based metrics can be found in Reference [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' These metrics are  expected to encounter problems in the present case due to the overlap of truth clusters, as explained  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Tracks are assigned most probably to the wrong cluster by any partitioning algorithm  if placed in between tracks from other clusters by mismeasurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' This phenomenon inevitably  reduces the accuracy of any track-to-cluster-based metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Nevertheless, at least the clustering of  – 10 –  the well-measured tracks should reproduce the truth closely, which the track-to-cluster metrics can  still be sensitive to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  As the metric accuracy is compromised by the presence of tracks with large measurement  errors, it might be useful to downscale the contribution of such tracks to the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' For the VI  metric this can be achieved by weighting every track with 𝜎−2 in the metric calculations, namely  𝑛𝑘 = �𝑘  𝑖=1  1  𝜎2  𝑖 , 𝑁 = �𝑁  𝑖=1  1  𝜎2  𝑖 , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' For the Silhouette metric the Cartesian distance between two  tracks can be replaced by its significance 𝑑(𝑖, 𝑗) = 𝑆𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The weighted versions of the VI and  Silhouette metric will be used in the following, along with the original versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The number of reconstructed clusters and the weighted average positions of these clusters,  dominated by the well-measured tracks, are mostly decoupled from the details of the track-to-  cluster assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The number of clusters can be directly used as a metric (up to the possible  presence of fake clusters), but a Cartesian distance-based metric is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' One may  try to introduce such a metric exploiting the cluster–cluster resolution 𝑅𝑐𝑐, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' the minimal distance  between two reconstructed clusters, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The good, merged, bad cluster categories could  be defined based on whether the cluster–truth vertex distance is smaller or larger than 𝑅𝑐𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Such  cluster categories could be used to compare various clustering solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' But this categorisation  explicitly depends on 𝑅𝑐𝑐, which itself depends on the clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' To avoid such circular  dependence, a scale-independent Cartesian distance-based metric is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  4  −  3  −  2  −  1  −  0  1  2  3  4  z[mm]  ∆  0  20  40  60  80  100  120  140  160  180  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1mm  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='35mm  CC  R  Figure 7: Example of a fit to the cluster–cluster distance to determine the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The used  fitting function is 𝑎/{1 + exp[𝑏 · (𝑅𝑐𝑐 − |𝑥|)]} + 𝑐 where a, b, c are free fitting parameters and 𝑅𝑐𝑐  is the cluster–cluster resolution, defined as the half-width at the half-depth of the dip in the centre  of the cluster–cluster weighted centre distances, averaged over all clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  To construct such a metric, we propose the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Every reconstructable truth  vertex is linked to the closest reconstructed cluster in the Cartesian space that has 2 or more assigned  tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Thus, a list of linked reconstructed clusters is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Then, every reconstructed cluster is  classified depending on how many times it enters into this list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' If a cluster enters this list only once,  there is just a single truth vertex referencing this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Therefore it can be called unique, which  means that a truth vertex is unambiguously reconstructed as a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' If a cluster enters several  times into the list, it is referenced by several truth vertices, and therefore it combines tracks from  – 11 –  these vertices: this cluster can be called merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Also, some clusters may not appear in this list  at all: such clusters are not referenced by any truth vertex and are thus fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The total number of  obtained clusters and their classification as unique, merged, fake are scale-independent and can be  used as a metric to compare various clustering options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  7  Results  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1  LHC Run-2 13 TeV data  First, the LMC clustering algorithm is tested with simulated DELPHES data at a collision energy of  13 TeV, with pileup  �  𝜇  �  = 63 and 𝜎𝑧 = 35 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' These parameters are chosen to provide simulated  data close to the actual data collected by the ATLAS detector in Run 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Edge-weight distributions  for various edge-labelling approaches on these data are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The performance of  the LMC algorithm on these data is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The rows labelled “cnst” in these tables  provide performance estimation with the applied constraints |𝑧𝑖 − 𝑧 𝑗| < 1 mm, while the “base”  rows describe the baseline algorithm performance without constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  15  −  10  −  5  −  0  5  Edge weight  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='24  Density (arbitrary units)  PDF ratio  20  −  15  −  10  −  5  −  0  5  Edge weight  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='2  Density (arbitrary units)  Logistic regression 1var  1  −  0  1  2  Edge weight  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='14  Density (arbitrary units)  BDT  Figure 8: Typical edge weight distributions for various edge labelling options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The column 𝑁wrong  trk  in Table 4 is the number of tracks assigned to one cluster but entirely  surrounded by tracks from other clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' This number is an estimator for the degree of cluster  overlap in the obtained solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The relevant truth data overlap for comparison can be found  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  In addition, Table 9 in the Appendix gives the number of isolated nodes (tracks)  reported by the LMC clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' These non-assigned tracks do not represent the one-  track truth vertices, considered non-reconstructable without a priori information, but rather reflect  the clustering problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The PDF ratio and the regression-based edge weight assignment result in approximately equal  clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The BDT-based edge weight assignment leads to a significantly worse  Silhouette metric value, a smaller value of the cluster overlap and a larger amount of fake clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  As expected, the weighted versions of the VI and Silhouette metrics have significantly better values  than the standard ones due to downscaling of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Using constraints uniformly improves all  quality estimators and provides ∼ 30% CPU reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  In total, 70% of the reconstructable truth vertices are reconstructed as unique clusters, while  the remaining 30% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='5 merged vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The amount of  – 12 –  Edge weight  VI  VI  Silhouette  Silhouette  Unique  Merged  Fake 𝑁wrong  trk  CPU  weighted  weighted  PDF ratio  base 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='14s  Table 4: LMC performance for the collision energy 13 TeV, pileup 63 and interaction region width  𝜎𝑧 = 35 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' These simulation parameters are chosen to match the full ATLAS simulation for Run  2 results used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The column 𝑁wrong  trk  shows the fraction of tracks wrongly associated  by the clustering algorithm, which shall be compared to the truth fraction of 20% (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  fake clusters is in the range of 5–15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The number of tracks in the different cluster categories is  presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The number of tracks in the unique clusters is close to the track amount in  the truth vertices, see Figure 2, while the merged clusters contain much more tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Finally, fake  clusters have a very small number of tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='1  Density (arbitrary units)  Unique vertices  0  10 20 30 40 50 60 70 80 90 100  Number of tracks in cluster  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='7  Density (arbitrary units)  Fake vertices  Figure 9: Number of tracks in a cluster for the unique, merged and fake cluster categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The  distributions are obtained for pileup  �  𝜇  �  = 63 data using a one-variable logistic regression for the  edge weight assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='2  High-Luminosity LHC 14 TeV data  The High Luminosity LHC (HL-LHC) project foresees a significant increase in interaction rates  to collect significantly more data and thus increase the sensitivity for new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The exact  parameters of the upgraded HL-LHC are not yet final;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' pileup values of 150, 200, and 250, and an  interaction region width of 𝜎𝑧 = 42 mm are considered the most probable options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content=' It is interesting to check the performance of the LMC problem formulation in such extreme  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  For this test, the same PDF ratio and logistic regression function are used for the edge weight  calculation, while the BDT classification is retrained using 𝜇 = 150, 200, 250 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content=' The column 𝑁wrong  trk  shows the fraction of the  tracks, wrongly associated by the clustering algorithm, which can be compared to the true fraction  66% (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  significantly improves all quality estimators and provides ∼ 30% CPU reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The number of unambiguously reconstructed unique clusters is 53% (44%, 37%) out of the  total amount of the reconstructable truth vertices for the pileup 𝜇 = 150 (200, 250).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The remaining  56 (90, 125) reconstructable truth vertices are clustered into 25 (40, 52) merged clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The  correctness of representation of the initial truth vertices by merged clusters is not granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Truth  vertices with a large number of tracks might “absorb” vertices with a small number of tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3  LMC performance adjustment  As can be seen from Tables 4–7, different edge weight assignment approaches lead to non-coinciding  clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' For a practical application of the LMC approach for primary vertex finding in  the LHC experiments, it is important to verify whether a unique optimal clustering solution exists  in this problem and, if so, whether the different LMC cost functions can be tuned to provide the  same clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' As explained in Section 5, parameters of the PDF ratio and regression function for  the edge weights can be modified to enforce under- or over-segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='The PDF ratio function  can be scaled up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In the logistic regression function, the intercept term can be shifted by  a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The cost function modifications are tried on the 𝜇 = 150 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The obtained clustering  results are shown in Figure 10 and Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  In the performed test, the exploited metrics change monotonically depending on the scale factor  for the PDF ratio and the intercept shift for the linear regression function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' It doesn’t seem possible  to adjust the PDF ratio and logistic regression parameters so that both approaches provide exactly  the same clustering performances in all used metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In addition, the BDTG-based Silhouette and  Silhouette weighted metrics results (see Table 5) are not reproducible by any modification of the  PDF ratio and logistic regression cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' However, the overall variations of the clustering  results remain limited, which means that the LMC approach performance stays close to optimal in  the full scanned parameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  To conclude, the cost function modification test doesn’t demonstrate the presence of an evident  unique globally optimal clustering solution for the problem in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Three used edge  – 15 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Metrics  VI  VI weighted  Silhouette  Silhouette weighted  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1  PDF ratio scale  0  20  40  60  80  100  Clusters  All  Unique  Merged  Fake  Resolution  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Resolution (mm)  Figure 10: PDF ratio cost-based clustering results as a function of the applied scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='4  Regression intercept shift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Metrics  VI  VI weighted  Silhouette  Silhouette weighted  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Regression intercept shift  0  20  40  60  80  100  Clusters  All  Unique  Merged  Fake  Resolution  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Resolution (mm)  Figure 11: Logistic regression cost-based clustering results as a function of the logistic regression  intercept term shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  weight assignment strategies provide different clustering results, which can be additionally changed  by simple modification of the cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Therefore, for a practical application as a primary  vertex finder, an exact LMC formulation should be chosen based on desired physics requirements,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' minimal amount of fake vertices or best vertex–vertex resolution, disregarding the clustering  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Influence of tracks with large measurement errors  As the truth cluster overlap is caused by the track position mismeasurement, the overlap degree  can be reduced by removing the badly measured tracks by cutting on the track measurement error  shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' A moderate decrease in the total amount of tracks due to this rejection should  not significantly affect the overall clustering efficiency as the total amount of tracks per truth vertex  is big enough, see Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Reduction of the amount of the selected tracks and the degree of the  truth cluster overlap due to strongly mismeasured track removal is shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The results  – 16 –  Track error cut  𝑁trk  Truth overlap 1674  41%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  1540  31%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  1444  27%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  1283  22%  Table 8: Number of selected tracks and the truth degree of overlap as a function of the track error  cut for 𝜇 = 150 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  of the clustering are shown in Figure 12 for the PDF ratio cost function and in Figure 13 for the  nominal logistic regression cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  Track error cut(mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Metrics  VI  VI weighted  Silhouette  Silhouette weighted  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  Track error cut (mm)  0  20  40  60  80  100  Clusters  All  Unique  Merged  Fake  Figure 12: PDF ratio cost-based clustering results as a function of the applied track error cut for  the 𝜇 = 150 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  Track error cut(mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='4  Metrics  VI  VI weighted  Silhouette  Silhouette weighted  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  Track error cut (mm)  0  20  40  60  80  100  Clusters  All  Unique  Merged  Fake  Figure 13: Logistic regression cost-based clustering results as a function of the applied track error  cut for the 𝜇 = 150 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  – 17 –  The distance-based metric demonstrates very small changes in the clustering results in a  wide range of the badly measured track admixture and, correspondingly, the initial degree of the  vertex overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' One may conclude that the amount of clusters identified by the LMC algorithm is  largely defined by the tracks with small measurement errors and, therefore, is stable with respect  to significant track noise admixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Redistribution of the tracks with big errors over the obtained  clusters doesn’t change their amount but evidently strongly affects all track counting-based clustering  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The track weighting does mitigate this effect for the VI metric, its weighted version is  practically independent of the track noise admixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Surprisingly, the Silhouette metric is only  weakly sensitive to this noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='5  Comparison with the existing approaches  6  −  4  −  2  −  0  2  4  6  z [mm]  ∆  arbitrary units  t  AMVF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' t  t  IVF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' t  Preliminary  Simulation  ATLAS  = 60  〉  µ  〈  = 13 TeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  s  0  10  20  30  40  50  60  70  80  interactions per bunch crossing  pp  Number of  0  10  20  30  40  50  60  Average number of reconstructed vertices  100% interaction reconstruction efficiency  Reconstruction acceptance  t  AMVF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' t  t  IVF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' t  AMVF - MATCHED  AMVF - MERGED  AMVF - SPLIT  AMVF - FAKE  ATLAS Simulation Preliminary  = 13 TeV  s  6  −  4  −  2  −  0  2  4  6  z[mm]  ∆  0  200  400  600  800  1000  1200  1400  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='02mm  DELPHES simulation LMC Cluster-Cluster distance  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='37mm  CC  R  0  10  20  30  40  50  60  70  80  Number of pp interactions per bunch crossing  0  10  20  30  40  50  60  Average number of reconstructed clusters  LMC All  LMC Unique  LMC Merged  100% interaction reconstruction efficiency  Reconstruction acceptance  Figure 14: The vertex–vertex resolution and the number of reconstructed vertices as a function  of the number of 𝑝𝑝 interactions for typical ATLAS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The upper plots are obtained with the  the ATLAS baseline AMVF [4] and IVF [3] algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The bottom plots are obtained using the  LMC algorithm with the PDF ratio-based edge weight assignment on DELPHES 𝜇 = 63 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The DELPHES 𝜇 = 63 simulation is specially tuned to match the ATLAS data used in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The  cluster–cluster resolution for the LMC algorithm on the bottom left picture is obtained as described  in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The ATLAS Collaboration used the IVF algorithm [3] to reconstruct the 𝑝𝑝 collision vertices  in Run 1 and the AMVF algorithm [4] in Run 2 and Run 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Essential characteristics of a primary-  vertex reconstruction algorithm are the vertex–vertex resolution and the number of reconstructed  – 18 –  vertices as a function of the number of 𝑝𝑝 interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The upper plots in Figure 14 present the  corresponding distributions for typical ATLAS data for the AMVF and IVF algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The bottom  plots show the same distributions provided by the LMC algorithm using DELPHES data tuned to  the same pileup conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Figure 14 clearly demonstrates that the LMC algorithm outperforms the ATLAS heuristic  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' It provides significantly better vertex–vertex resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' This naturally leads to a larger  amount of Unique/Matched vertices reconstructed by LMC, while the amount of Merged vertices  remains practically the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Routine application of the LMC for the primary vertex reconstruction  can provide a significant gain in performance for LHC and future collider experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  8  Conclusion  In this work, we have addressed a typical particle physics problem of reconstructing multiple  interaction positions in a dense environment, where each interaction is represented by a cluster  of tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Significant track reconstruction errors lead to a large overlap of truth track clusters,  which makes their identification challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Heuristic algorithms are usually used to address this  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' In contrast, we propose to address this problem through a principled formulation as a  minimum-cost lifted multicut problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' We construct several cost functions for the LMC from  track–track distances and their significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' We study the performance of the LMC algorithm  for different vertex densities, cost functions, constraint usage and varying degree of overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' To  address potential performance problems of existing track counting clustering metrics for strongly  overlapped clusters, dedicated metrics are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  We demonstrate that the LMC approach outperforms the heuristic algorithms in the problem  of vertex reconstruction in dense environments in terms of vertex–vertex resolution and vertex  reconstruction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' It works up to the highest vertex density expected at the HL-HLC project  in spite of the strong truth cluster overlap reaching ∼ 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Variations of the LMC algorithm  parameters and cost functions studied in this work resulted in relatively small variations of the  obtained clustering solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Acknowledgments  This work is supported by the German Science Foundation (DFG) through a research grant and a  Heisenberg professorship under contracts CR-312/4-1 and CR-312/5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content=' The total amounts of interaction vertices with one  reconstructed track and two and more tracks are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Due to the track measurement  errors, the one-track vertices are difficult to reconstruct correctly without a priori information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  Finding two and more track vertices becomes problematic if the vertex–vertex distance is less than  the typical track measurement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Both problems are illustrated in Table 9, where the amounts  of the one-track and multi-track clusters are given for every cost function and event sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='13 TeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='14 TeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='𝜇  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='= 63  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='𝜇  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
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+page_content='104  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='Table 9: Average numbers of non-clustered tracks and reconstructed clusters obtained by the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='LMC algorithm with different cost functions as compared to the truth numbers of single-track and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='multi-track vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' Results are shown for all collision energies and pileup densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  The number of one-track clusters in each case is significantly larger than the truth amount of  one-track interaction vertices, especially in the BDTG case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' They should be thought of as non-  clustered tracks, not as reconstructed one-track vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content=' The fraction of multi-track clusters found  decreases with the interaction vertex density, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
+page_content='  – 21 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E5T4oBgHgl3EQfVA-E/content/2301.05548v1.pdf'}
diff --git a/F9AyT4oBgHgl3EQfSveK/content/tmp_files/2301.00092v1.pdf.txt b/F9AyT4oBgHgl3EQfSveK/content/tmp_files/2301.00092v1.pdf.txt
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+arXiv:2301.00092v1  [stat.ML]  31 Dec 2022
+Inference on Time Series Nonparametric Conditional
+Moment Restrictions Using General Sieves
+Xiaohong Chen∗
+Yuan Liao†
+Weichen Wang‡
+First draft: September 2020, revised January 3, 2023
+Abstract
+General nonlinear sieve learnings are classes of nonlinear sieves that can approxi-
+mate nonlinear functions of high dimensional variables much more flexibly than various
+linear sieves (or series). This paper considers general nonlinear sieve quasi-likelihood ra-
+tio (GN-QLR) based inference on expectation functionals of time series data, where the
+functionals of interest are based on some nonparametric function that satisfy conditional
+moment restrictions and are learned using multilayer neural networks. While the asymp-
+totic normality of the estimated functionals depends on some unknown Riesz representer
+of the functional space, we show that the optimally weighted GN-QLR statistic is asymp-
+totically Chi-square distributed, regardless whether the expectation functional is regular
+(root-n estimable) or not. This holds when the data are weakly dependent beta-mixing
+condition. We apply our method to the off-policy evaluation in reinforcement learning,
+by formulating the Bellman equation into the conditional moment restriction frame-
+work, so that we can make inference about the state-specific value functional using the
+proposed GN-QLR method with time series data. In addition, estimating the averaged
+partial means and averaged partial derivatives of nonparametric instrumental variables
+and quantile IV models are also presented as leading examples. Finally, a Monte Carlo
+study shows the finite sample performance of the procedure
+∗Cowles Foundation for Research in Economics,
+Yale University,
+New Haven,
+CT 06520, USA.
+xiaohong.chen@yale.edu.
+†Department of Economics, Rutgers University, New Brunswick, NJ 08901, USA. yuan.liao@rutgers.edu
+‡Faculty of Business and Economics, The University of Hong Kong, Pokfulam Road, Hong Kong
+weichenw@hku.hk.
+1
+
+1
+Introduction
+Consider a conditional moment restriction model
+E[ρ(Yt+1, α0)|σt(X )] = 0 ,
+(1.1)
+where ρ is a scalar residual function; α0 = (θ0, h0) contains a finite dimensional parameter θ0
+and an infinite dimensional parameter h0, which may depend on some endogenous variables Wt.
+The conditioning filtration σt(X ) is the sigma-algebra generated by variables {Xs : s ≤ t},
+where Xs is a vector of multivariate (finite dimensional) exogenous variables, including all
+relevant lagged variables of Yt and other instrumental variables. The model therefore allows
+for endogenous variables and weakly dependent data.
+This paper considers optimal estimation and inference for linear functionals φ(α0) of the
+infinite dimension. The functional may be either known or not. When it is unknown, it is
+assumed to take the form
+φ(α0) = El(h0(Wt)) ,
+where l is a known linear function and h0(Wt) is the nonparametric function on endogenous
+variable. We use general nonlinear sieve learning spaces, whose complexity grows with the
+sample size, to estimate the infinite dimensional parameter, such as multi-layer neural networks
+and Gaussian radial basis. The motivation of using general nonlinear sieve learning space,
+besides being adaptive to high dimensional covariates, is that they allow unbounded supports
+of the covariates. This is particularly desirable for models of dependent time series data, such
+as nonlinear autoregressive models.
+We formally establish inferential theories of these functionals learned using the general
+nonlinear sieve learning space, and conduct inference using quasi-likelihood ratio (QLR) statis-
+tics based on the optimally weighted minimum distance. Of particular interest is the estima-
+tion of an expectation functional, such as averaged partial means, weighted average derivatives
+and averaged squared partial derivatives, of a nonparametric conditional moment restriction
+via nonlinear sieve learning sieves. An important insight from our main theory is that the
+asymptotic distribution does not depend on the actual choice of the learning space, but is only
+determined by the functional and the loss function. Therefore, estimators produced by either
+deep neural networks, Gaussian radial basis, or other nonlinear sieve learning basis, have the
+same asymptotic distribution.
+In general, machine learning inference often relies on sample splitting/ cross-fitting, which
+does not work well in the time series setting. We propose a new time series efficient inference
+2
+
+based on the optimal quasi-likelihood ratio test, without requiring cross-fitting. It is shown
+that the optimally weighted QLR statistic, based on the general nonlinear sieve learning of
+h0(), is asymptotically chi-square distributed regardless of whether the information bound for
+the expectation functional is singular or not, which can be used to construct confidence sets
+without the need to compute standard errors. We present a Monte Carlo study to illustrate
+finite sample performance of our inference procedure.
+Depending on the specific applications, our model may involve Fredholm integral equation
+of either the first kind (NPIV and NPQIV) or the second kind (Bellman equations). In the for-
+mer case, it is well known that estimating h0 is an ill-posed problem and the rate of convergence
+might be slow. In the latter case, the problem can be well-posed. As one of the leading ex-
+amples of the Fredholm integral equation of second kind, we show that our framework implies
+a natural neural network-based inference in the context of Reinforcement Learning (RL), a
+popular learning device behind many successful applications of artificial intelligence such as Al-
+phaGo, video games, robotics, and autonomous driving (Sutton and Barto, 2018; Silver et al.,
+2016; Vinyals et al., 2019; Shalev-Shwartz et al., 2016). Due to the dynamics of the RL model,
+theoretical analysis of reinforcement learning naturally requires to explicitly allow time series
+dependency among the observed data. Earlier theoretical studies focused on the settings where
+the value function are approximated by linear functions. More recent developments on non-
+linear learning space include Farahmand et al. (2016); Geist et al. (2019); Fan et al. (2020);
+Duan et al. (2021); Long et al. (2021); Chen and Qi (2022); Shi et al. (2020), among others.
+Our innovation lies in making inference about the functionals (such as the value functional
+for specific states) of the Q-function using general nonlinear sieve learning spaces. While the
+reinforcement learning is based on the well known Bellman equation, it can be formulated as
+the conditional moment restriction model with time series data. Therefore, one can apply
+the GN-QLR inference to estimating the state-specific value function in the setting of the
+off-policy evaluation. These applications are potentially useful for dynamic causal inference.
+In the i.i.d. case, existing theoretical works on neural networks have focused on deriv-
+ing approximation theories and optimal rates of convergence for estimations. Theoretically,
+deep learning has been shown to be able to approximate a broad class of highly nonlinear
+functions, see, e.g.
+Mhaskar et al. (2016); Rolnick and Tegmark (2017); Lin et al. (2017);
+Shen et al. (2021); Hsu et al. (2021); Schmidt-Hieber (2020). Yang and Barron (1999) ob-
+tained the minimax L2- rate of convergence for neural network models. Recently, Chen, Chen
+and Tamer (2021) considered NN efficient estimation of the (weighted) average derivatives in
+a NPIV model for i.i.d. data, and presented consistent variance estimation. In contrast, using
+3
+
+a general theory of Riesz representations, we derive the asymptotic distribution of the finite
+dimensional parameter θ0 and functionals of the infinite dimensional parameter h0 that is
+learned from the general learning space. The uncertainty of the general nonlinear sieve learn-
+ing estimator plays an essential role in the asymptotic distributions.
+Chernozhukov et al.
+(2018a,b,c) proposed double machine learning and debias methods to achieve valid inference;
+Dikkala et al. (2020) studied a minimax criterion function to study the unknown functional
+approximated by neural networks for NPIV models. In addition, the Riesz representation is
+playing a central role in our inferential theory. See Newey (1994); Shen (1997); Chen and Shen
+(1998); Chernozhukov et al. (2020) for related approaches.
+In the time series setting, the neural networks have been applied to economic demand
+estimations as in Chen and Ludvigson (2009), and is widely applicable in financial asset pric-
+ing such as Guijarro-Ordonez et al. (2021); Gu et al. (2020); Bali et al. (2021). These papers
+approximate unknown functions by neural networks, but without rigorous theoretical justifi-
+cations. All these models can be formulated as an inference problem for conditional moments.
+The rest of the paper is organized as follows. Section 2 first introduces the model, the
+NN sieve space, the estimation and inference procedures. Section 3 establishes the converence
+rate of the NN sieve estimator for the unknown function satisfying the conditional moment
+restrictions with weakly dependent data. Section 4 provides the limiting distribution of the
+estimator for functionals that can be regular or irregular. Section 5 reveals that the NN sieve
+QLR statistics is asymptotically Chi-square distributed for both the regular and irregular
+functionals for time series. In Section 6 we apply our approach to the estimation of the value
+function of RL and the weighted average derivative of NPIV and NPQIV as leading examples.
+Section 7 contains simulation studies and Section 8 briefly concludes.
+2
+The model
+2.1
+The General sieve learning space
+This paper studies inference with the general nonlinear sieve learning space. The unknown
+function is estimated on a learning space, denoted by Hn, is a general approximation space
+that consists of either linear or nonlinear sieves, provided that the function of interest can be
+approximated well by the learning space.
+The popular feedforward neural network (NN) is one of the leading examples that fits
+into this context. Many theoretical studies have shown that NN can well approximate a broad
+class of functions and achieves nice statistical properties.
+The multilayer feedforward NN
+4
+
+composites functions taking the form:
+h(x) = θJ+1hJ(x),
+· · ·
+hj(x) = σ(θjhj−1(x)),
+· · ·
+, h0(x) = x
+where the parameters θ = (θ1, · · · , θJ) with θj ∈ Rdj×dj−1 , hj(x) ∈ Rdj, and σ : Rdj → Rdj is
+a elementwise nonlinear activation function, usually the same across components and layers.
+One of the popularly used activation functions is known as ReLU, defined as σ(x) = max(0, x).
+The number of neurons being used in layer j, denoted by dj, is called the width of that layer.
+We could also use other nonlinear approximation learning spaces, which uses nonlinear
+combinations of inputs and neurons. One such example is the space spanned by Gaussian
+radial bases, which is a multilayer compositions of functions of the form:
+h(x) = α0 +
+J
+�
+j=1
+αjG(σ−1
+j ∥x − γj∥),
+α0, αj, γj ∈ R, σj > 0,
+where G is the standard normal density function.
+A key feature is that here inputs and
+neurons (e.g., a vector of x) are “nonlinearly combined” as ∥x − γj∥, while they are linearly
+combined as indices θjx in the ordinary neural networks. Additional examples of nonlinear
+sieves include spline and wavelet sieves. They are very flexible and enjoy better approximation
+properties than linear sieves.
+One of the key motivations of using general nonlinear sieve learning space, besides being
+adaptive to high dimensional covariates, is that it allows unbounded supports of input covari-
+ates. This is particularly desirable for time series models dependent data, such as nonlinear
+autoregressive models.
+2.2
+Semiparametric learning
+We shall assume a finite-order Markov property: for some known and fixed integer r ≥ 1, let
+Xt := (Xt, ..., Xt−r) for all t = 1, ..., n. define
+m(Xt, α)
+=
+E[ρ(Yt+1, α)|σt(X )],
+Σ(Xt)
+=
+Var(ρ(Yt+1, α0)|σt(X )),
+where we assume that E[ρ(Yt+1, α)|σt(X )] and Var(ρ(Yt+1, α0)|σt(X )) only depend on (Xt, ..., Xt−r)
+for all α. The model is then equivalent to Q(α0) = 0 where
+Q(α) = Em(Xt, α)2Σ(Xt)−1.
+5
+
+Here we use the optimal weighting function Σ(Xt).
+Suppose there are nonparametric es-
+timators �m(X, α) and �Σ(Xt) for m(Xt., α) and Σ(Xt), we then define the sample criterion
+function
+Qn(α) = 1
+n
+n
+�
+t=1
+�m(Xt, α)2�Σ(Xt)−1.
+The estimated optimal weighting matrix is needed for the quasi-likelihood inference. In prac-
+tice, one can start with the identity weighting function to obtain an initial estimator for α0,
+use it to estimate Σ(Xt), then update the estimator using the estimated optimal weighting
+matrix.
+We focus on the general nonlinear sieve learning approximation to the true nonparametric
+function, and restrict to the following estimation space:
+An := Θ × Hn.
+Here Θ is a compact set as the parameter space for θ0 but not necessarily for Hn. In addition,
+let Pen(h) denote some functional penalty for the infinite dimensional parameter. We then
+define the estimator �α = (�θ,�h) ∈ An as an approximate minimizer of the penalized loss
+function restricted to the general nonlinear sieve learning space:
+Qn(�α) + λnPen(�h) ≤ inf
+α∈An Qn(α) + λnPen(h) + oP(n−1).
+The tuning parameter λn is chosen to decay relatively fast, so that the penalization Pen(·)
+does not have a first-order impact on the asymptotic theory. Nevertheless, the functional
+penalization is imposed to overcome undesirable properties associated with estimates based
+on a large parameter space. Essentially, it plays a role of forcing the optimization to be carried
+out within a weakly compact set (Shen, 1997).
+The functions (x, α) �→ �m(x, α) and x �→ �Σ(x) are nonparametric estimators of (x, α) �→
+m(x, α) and x �→ Σ(x) (a positive definite weighting matrix) respectively. The projection
+m(Xt, α) can be also estimated using linear sieves:
+�m(·, α) = min
+m∈Dn
+T
+�
+t=1
+[ρ(Yt+1, α) − m(Xt)]2
+6
+
+where we consider linear sieve space: let {Ψj : j = 1, . . . , kn} denote a set of sieve bases,
+Dn :=
+�
+g(x) =
+kn
+�
+j=1
+πjΨj(x) : ∥g∥∞,ω < ∞, πj ∈ R
+�
+.
+So we use the general nonlinear sieve learning space Hn to approximate the function space
+for h0, and a linear sieve space Dn to approximate the instrumental space, which is easier
+to implement computationally than using nonlinear sieve approximations to the instrumental
+space. A more important motivation of using linear sieve space to estimate the conditional
+mean function E[ρ(Yt+1, α)|σt(X )] is that the sample loss function Qn(α) can be shown to
+have a local quadratic approximation (LQA): for some Bn = OP(1) and Zn →d N(0, 1),
+Qn(α + xun) − Qn(α) = Bnx2 + 2x[n−1/2Zn + ⟨un, α − α0⟩] + oP(n−1)
+(2.1)
+uniformly for all α in a shrinking neighborhood of α0 and |x| ≤ Cn−1/2; here ⟨un, α − α0⟩ is
+some inner product between α − α0 and some function un, to be defined explicitly later. This
+LQA plays a fundamental role for the inferential theory of semiparametric inference using
+general nonlinear sieve learning methods.
+2.3
+Semiparametric efficient estimations
+Let the parameter space of the true function be H0 and let A0 = Θ × H0. We are interested
+in the inference of φ(α0), where φ : A0 → R can be a known functional of α0. We also study
+the inference problem of unknown functionals, taking the form
+φ(α0) = El(h0(Wt)) ,
+where l(·) is a known function. While the naive plug-in estimator
+1
+n
+�n
+t=1 l(�h(Wt)) is also
+asymptotically normal, when the model contains endogenous variables, it is not semipara-
+metrically efficient.
+An important example of φ(α0) is the weighted average derivative of
+nonparametric instrumental variable regression (NPIV), defined as
+φ(α0) = E[Ω(Wt)′∇h0(Wt)] ,
+where Ω(·) is a known positive weight function and ∇h0 denotes the gradient of the non-
+parametric regression function h0. As documented by Ai and Chen (2012), the simple plug-in
+estimator is not an efficient estimator. To obtain a more efficient estimator, on the popu-
+lation level consider conditional (given Xt) projection of l(h0(Wt)) onto ρ(Yt+1, α0), and the
+7
+
+corresponding functional of interest also can be represented as φ(α0) with the functional:
+φ(α)
+=
+E [l(h(Wt)) − Γ0(Xt)ρ(Yt+1, α)] ,
+(2.2)
+where Γ0(Xt) = E[l(h0(Wt))ρ(Yt+1, α0)|σt(X )]Σ(Xt)−1 is the projection coefficient. We shall
+obtain efficient estimator of φ(α0) based on this expectation expression. It is worthy to know
+that the added term Γ0(Xt)ρ(Yt+1,α0) is in effect only for endogenous regressors.
+In pure
+exogeneous models where Wt = Xt, we have Γ0(Xt) = 0. In this case the moment condition
+(2.2) reduces to the original one φ(α) = El(h(Wt)).
+Let
+�φ(α) = 1
+n
+n
+�
+t=1
+[l(h(Wt)) − �Γtρ(Yt+1, α)]
+(2.3)
+for some estimator �Γt to be defined later. Then we estimate the functional by �φ(�α). Asymp-
+totically, we shall show that
+�φ(�α) − φ(α0) = [φ(�α) − φ(α0)] + 1
+n
+n
+�
+t=1
+[Wt − EWt] + oP(σn−1/2),
+(2.4)
+where Wt = l(h0(Wt)) − Γ0(Xt)ρ(Yt+1, α0) and σ2 is the asymptotic variance. It is clear that
+the asymptotic distribution arises from two sources of uncertainties, and importantly, the
+nonparametric learning error φ(�α) − φ(α0) plays a first-order role.
+We shall show that in both known and unknown functional case, estimated φ(α0) is
+asymptotically normal. We then provide quasi-likelihood inference to construct confidence
+intervals for φ(α0).
+3
+Rates of Convergence for Semi-parametric Neural Network
+3.1
+Weighted function space and sieve learning space
+Since the supports of the endogenous variable Wt could be unbounded, we use a weighted
+sup-norm metric defined as
+∥h∥∞,ω = sup
+s |h(s)|(1 + |s|2)−ω/2,
+for some ω > 0.
+(3.1)
+This is known as “admissible weight” which is often used for h0(Wt) when Wt has fat tailed dis-
+tribution (Remark 2.6 of Haroske and Skrzypczak (2020)). Smooth functions with unbounded
+support might still be well approximated under the weighted sup-norm. The L2(W)-norm can
+8
+
+be bounded by the weighted sup-norm as: for any function h(w):
+∥h∥2
+L2(W ) =
+�
+h(s)2fW(s)ds ≤ ∥h∥2
+∞,ω
+�
+(1 + |s|2)ωfW(s)ds,
+provided the distribution of the endogenous variable W has as density fW such that fW(s)(1+
+|s|2)ω is integrable.
+We do not consider the overparametrized regime, but impose restrictions on the complex-
+ity of the general nonlinear sieve learning space Hn, measured by the “number of parameters”
+of the space, denoted by p(Hn). More specifically, we impose the following condition.
+Assumption 3.1 (function and learning space). (i) The function space: The unknown func-
+tion h0 ∈ H0, which is a weighted H¨older ball: for some γ > 0, g ≥ 0,
+H0 = {h : ∥h(·)(1 + | · |2)−g/2∥Λγ ≤ c}
+where
+∥f∥Λγ = sup
+w |f(w)| + max
+|a|=d sup
+w1̸=w2
+|∇af(w1) − ∇af(w2)|
+∥w1 − w2∥γ−d
+.
+Also, we require g < ω for ω defined in (3.1).
+(ii) Approximation rate under the ∥∥∞,ω norm:
+inf
+h∈H0 ∥h0 − h∥∞,ω ≤ cp(Hn)−m
+for some m > 0, and some sequence p(Hn) → ∞, p(Hn) log n = o(n).
+(iii) Complexity: Let N (δ, Hn, ∥.∥∞,ω) denote the minimal covering number, that is, the
+minimal number of closed balls of radius δ with respect to ∥.∥∞,ω needed to cover Hn. We
+assume, there is a constant C > 0, so that for any δ > 0,
+N (δ, Hn, ∥.∥∞,ω) ≤
+�Cn
+δ
+�p(Hn)
+.
+We need to assume that h(w) is smooth in some sense with respect to h(w). Condition
+(i) is a standard weighted smoothness condition for functions with unbounded support. Here
+two weighted norms are being defined, the weighted sup norm ∥.∥∞,ω with a weight parameter
+ω in (3.1).
+The weighted sup norm intead of the usual sup norm is being considered, as
+discussed above, for the purpose of allowing the nonparametric function h(·) to have possibly
+9
+
+unbounded support, which is the typical case for autoregressive models. The other norm is
+∥.∥Λγ for the H¨older ball with a weight parameter g. Here we require g < ω so that the
+closure of the function space H0 with respec to the norm ∥.∥∞,ω is compact, following from
+Gallant and Nychka (1987).
+In Condition (ii), p(Hn) → ∞ measures the dimension of of the learning space. For mul-
+tilayer neural networks with ReLU activation functions, Anthony and Bartlett (2009) showed
+that the bound holds with p(Hn) being the pseudo-dimension of the space and is bounded by
+CJ2K2 log(JK2), where J and K respectively denote the width and depth of the network.
+For finite-dimensional linear sieve, the inequality also holds with p(Hn) being bounded by the
+number of sieve bases.
+When the function h has bounded support, Condition (ii) has been verified for numerous
+learning spaces. For instance, for feed forward multilayer neural networks, Bauer and Kohler
+(2019) showed that the approximation rate is n−c, for c =
+p
+2p+d∗ and p = a + γ, with properly
+chosen depth and width of layers. Importantly, d∗ ≤ dim(Wt) is the “intrinsic dimension”
+of the true function.
+For instance if h0 has a hierarchical interaction structure or multi-
+index structure, d∗ is the number of index. When the function h has unbounded support,
+it is known that for linear sieves such as B-splines and wavelets the approximation rate is
+m = p(Hn)−γ/ dim(Wt) where p(Hn) is the number of basis. The approximation rate is however
+still an open question for feed forward neural networks in this case.
+3.2
+Ill-posedness
+In this section we present the rate of convergence. For simplicity throughout the rest of the
+paper, we focus on the case dim(ρ(Yt+1, α)) = 1. By the identification condition, Q(α) = 0 if
+and only if α = α0. So the usual risk consistency refers to Q(�α) = oP(1). In the presence of
+endogenous variables, the risk consistency however, is not sufficient to guarantee the estimation
+consistency. The latter is often defined under a strong norm:
+∥α1 − α2∥∞,ω := ∥θ1 − θ2∥ + ∥h1 − h2∥∞,ω.
+We first introduce a pseudometric on An that is weaker than ∥.∥∞,ω. To do so, recall the
+general Gateaux derivative. Given generic α = (θ, h) and v = (vθ, vh), let F(x, α) = F(x, θ, h)
+be a function that is assumed to be differentiable with respect to θ. Define
+dF(x, α)
+dα
+[v]
+=
+∂F(x, α)
+∂θ
+′
+vθ + dF(x, θ, h + τvh)
+dτ
+����
+τ=0
+,
+10
+
+where we implicitly assume dF (x,θ,h+τvh)
+dτ
+exists at τ = 0. Then the weak norm is defined to be
+∥v∥2 := E
+�dm(Xt, α0)
+dα
+[v]
+�2
+Σ(Xt)−1.
+Define πnα0 ∈ An be such that
+∥πnα0 − α0∥∞,ω = min
+α∈An ∥α − α0∥∞,ω.
+The following assumption imposes conditions on the local curvature of the criterion function.
+Assumption 3.2 (criterion function). There are c1, c2 > 0 so that
+(i) ∥α − α0∥2 ≤ c1Em(Xt, α)2Σ(Xt)−1 for all α ∈ An.
+(ii) Em(Xt, πnα0)2Σ(Xt)−1 ≤ c2∥α0 − πnα0∥2.
+We now discuss the ill-posedness which reflects the relation between the risk consistency
+and estimation consistency. Let the sieve modulus of continuity be
+ωn(δ) :=
+sup
+α∈An:∥α−πnα0∥≤δ
+∥α − πnα0∥∞,ω.
+We say that the problem is ill-posed if δ = o(ωn(δ)) as δ → 0. The growth of ωn(δ)δ−1
+reflects the difficulty of recovering α0 through minimizing the criterion function.
+3.3
+Rates of convergence
+Below we present regularity conditions to achieve the rates of convergence. We allow weakly
+dependent time series data satisfying β-mixing conditions. Define the mixing coefficient
+β(j) := sup
+t E sup{|P(B|F t
+−∞) − P(B)| : B ∈ F ∞
+t+j}
+where F t
+s denotes the σ-field generated by (Ys+1, Xs), ..., (Yt+1, Xt).
+Assumption 3.3 (Dependences). (i) {(Yt+1, Xt)}n
+t=1 is a strictly stationary and β-mixing
+sequence with β(j) ≤ β0 exp(−cj) for some β0, c > 0.
+(ii) There is a known and finite integer r ≥ 1 so that for each α ∈ An and t = 1, ..., n, The
+conditional expectation E[f(St, α)|σt(X )] depend on σt(X ) only through Xt := (Xt, ..., Xt−r),
+for St = (Yt+1, Wt) and
+f(St, α) ∈ {ρ(Yt+1, α),
+ρ(Yt+1, α0)2,
+l(h0(Wt))ρ(Yt+1, α0)}.
+11
+
+Assumption 3.4. Q(α) = 0 if and only if α = α0. In addition, Q(α) is lower semicontinuous.
+The lower semicontinuity of the criteria function is satisfied by the risk function of many
+interesting models. This condition ensures that it has a minimum on any compact set.
+Assumption 3.5 (Penalty). (i) There is M0 > 0, Pen(h) ≤ M0 for all h ∈ Hn ∪ {h0}.
+(ii) Pen is lower semicompact on (An, ∥.∥∞,ω), i.e. {h : Pen(h) ≤ M} is compact for any
+M > 0.
+(iii) kn
+n + Q(πnα0) = O(λn) where recall kn is the number of linear sieve bases in Dn.
+Define
+ǫ(St, α) := ρ(Yt+1, α) − m(Xt, α).
+One of the major technical steps is to establish the stochastic equicontinuity for the function
+class Ψj(Xt)ǫ(St, α) for β-mixing observations, where α belongs to the class of deep neural
+networks. More specifically, we shall derive the bound for, with Ψ(Xt) := (Ψj(Xt) : j ≤ kn):
+sup
+α∈An:Em(Xt,α)≤r2n
+�����
+1
+√n
+n
+�
+t=1
+Ψ(Xt)[ǫ(St, α) − ǫ(St, α0)]
+�����
+for a given convergence sequence rn → 0. This is achieved under the following Assumption.
+Assumption 3.6. There is C > 0,
+(i) There are κ > 0 and C > 0 so that for all δ > 0 and all α1, α2 ∈ An,
+max
+j≤kn E[Ψj(Xt)2 + 1]
+sup
+∥α1−α2∥∞,ω<δ
+|ǫ(St, α1) − ǫ(St, α2)|2 ≤ Cδ2κ.
+(ii) E maxj≤kn Ψj(Xt)2 supα∈An ρ(Yt+1, α)2 ≤ C.
+(iii) There is a ∥.∥∞,ω- neighborhood of α0 on which m(·, α) is continuously pathwise
+differentiable with respect to α, and there is a constant C > 0 such that ∥α − α0∥ ≤ C∥α −
+α0∥∞,ω.
+Next we present regularity conditions on the linear sieve space Dn used to approximate
+the conditional mean function m(X, α).
+Assumption 3.7 (Linear sieve space). (i) There is ϕn → 0 so that uniformly for α ∈ An,
+there is kn × 1 vector bα,
+E[g(Xt, α) − Ψ(Xt)′bα]2 = O(ϕ2
+n),
+for all g(Xt, α) ∈ {m(Xt, α), E[l(h0(Wt))ρ(Yt+1, α0)|Xt], dm(Xt,α)
+dα
+[un], dm(Xt,α)
+dα
+[un]Σ(Xt)−1}.
+12
+
+(ii) Let Ψn be the n × kn matrix of the linear sieve bases: Ψn = (Ψ(Xt) : t = 1...n)n×kn:
+and let A := 1
+nEΨ′
+nΨn. The linear sieve satisfies: λmin(A) > c and ∥ 1
+nΨ′
+nΨn − A∥ = oP(1).
+Finally, we apply the pseudo dimension to quantify the complexity of the neural network
+class.
+Assumption 3.8. (i) supx[Σ(x)−1 + Σ(x)] < C. Also, supx |�Σ(x) − Σ(x)| = oP(1).
+(ii) The distribution of the endogenous variable Wt has a density function fW, which
+satisfies
+�
+w(x)−2fW(x)dx < ∞.
+Recall that kn denotes the number of sieve bases being used to estimate the expectation
+function m(X, α); ϕn is the approximation rate in Assumption 3.7. Let
+dn
+:=
+�
+p(Hn) log2 n
+n
+,
+¯δn
+:=
+∥πnα0 − α0∥ +
+�
+λn +
+�
+kndn + ϕn,
+δn
+:=
+∥πnα0 − α0∥∞,ω + ωn(¯δn).
+(3.2)
+Theorem 3.1 (Rate of convergence). Under Assumptions 3.2-3.8, for any ǫ > 0,
+∥�α − α0∥∞,ω = OP(δn),
+Q(�α) = OP(¯δ2
+n).
+The derived rate of convergence is comparable with that of Chen and Pouzo (2012). In
+¯δn, the term ∥πnα0 − α0∥ is the approximation error on the general nonlinear sieve learning
+space; √λn is the effect of penalization. In addition, ϕn and √kndn respectively arise from the
+bias and variance of estimating m(X, α). In particular, the variance term √kndn depends on
+the complexity of the general nonlinear sieve learning space, which arises from the stochastic
+equicontinuity. In addition, ωn(¯δn) connects the convergence under the weak norm OP(¯δn) to
+the convergence under the strong norm via the sieve modulus of continuity. When there are
+no endogeneity, ¯δn and ωn(¯δn) are of the same order. General nonlinear sieve spaces with more
+complicated structures (with larger “dimension” p(Hn)) have increased covering numbers on
+the learning space, and thus lead to slower decays of these two terms.
+4
+Asymptotic Distributions for NN Functionals
+We now study estimating linear functionals of α0. We establish the asymptotically normality
+of the estimated functionals formed via pluging-in the general learning estimators.
+13
+
+4.1
+Riesz representation
+A key ingredient of our analysis, as in Chen and Pouzo (2015), relies on representing the
+estimation error φ(�α) − φ(α0) using a linear inner product induced from the loss function via
+the Riesz representation theorem. We define an inner product space as follows.
+For any space H, let span{H} denote the closed linear span of H.
+For any v1, v2 in
+span(An ∪ {α0}), the linear span of An ∪ {α0}, define the inner product:
+⟨v1, v2⟩ = EΣ(Xt)−1
+�dm(Xt, α0)
+dα
+[v1]
+� �dm(Xt, α0)
+dα
+[v2]
+�
+.
+Let α0,n ∈ span(An) be such that
+∥α0,n − α0∥ =
+min
+α∈span(An) ∥α − α0∥.
+We note that it is likely α0,n ̸= πnα0 because πnα0 ∈ An, which is not the same as span(An),
+when An is a nonlinear NN space.
+Given Theorem 3.1, we can focus on shrinking neighborhoods
+Aosn
+:=
+{α ∈ An : ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤ C¯δ2
+n}
+Cn
+:=
+{α + xun : α ∈ Aosn, |x| ≤ Cn−1/2},
+un := v∗
+n/∥v∗
+n∥,
+¯Vn
+:=
+span(Aosn − {α0,n}) ⊂ span(An).
+(4.1)
+for a generic constant C > 0, where v∗
+n is the Riesz representer to be defined below.
+Because both Aosn and α0,n are functions inside the general nonlinear sieve learning space,
+( ¯Vn, ⟨.⟩) is a finite dimensional Hilbert space under the weak-norm ∥v∥ =
+�
+⟨v, v⟩. Suppose
+dφ(α0)
+dα [v] is a linear functional. As any linear functional on a finite dimensional Hilbert space
+is bounded, by the Riesz representation Theorem, there is v∗
+n ∈ ¯Vn so that
+dφ(α0)
+dα
+[v] = ⟨v∗
+n, v⟩,
+∀v ∈ ¯Vn.
+To appreciate the role of Riesz representation in the semiparametric inference, note that
+�α − α0,n ∈ ¯Vn, and we have,
+φ(�α) − φ(α0)
+=
+dφ(α0)
+dα
+[�α − α0]
+=
+dφ(α0)
+dα
+[�α − α0,n] + dφ(α0)
+dα
+[α0,n − α0]
+14
+
+=
+⟨v∗
+n, �α − α0,n⟩ + dφ(α0)
+dα
+[α0,n − α0]
+�
+��
+�
+negligible
+.
+where the first equality follows from the smoothness condition (Assumption 4.1 below) of the
+functional; the second equality is to the linearity of the functional pathwise derivative. In
+addition, suppose dφ(α0)
+dα [α0,n − α0] is negligible, a claim we shall discuss in Remark 4.1 later,
+we can then apply the Riesz representation theorem to reach the last line of the expansion.
+In addition, one of the key technical steps in the proof, by locally expanding the risk
+function, is to prove:
+√n⟨v∗
+n, �α − α0,n⟩ = √n⟨v∗
+n, �α − α0⟩ = − 1
+√n
+�
+t
+Zt + oP(∥v∗
+n∥)
+where Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt,α0)
+dα
+[v∗
+n], and ∥v∗
+n∥2 = Var( 1
+√n
+�
+t Zt). Then together we
+have
+√n(φ(�α) − φ(α0))
+∥v∗n∥
+→d N (0, 1).
+Importantly, our inference procedure does not require estimating the Riesz representer
+v∗
+n or ∥v∗
+n∥. Instead, we propose a quasi-likelihood ratio (QLR) inference. We shall provide
+regularity conditions in the next section to formalize the above derivations, and subsequently
+address estimating the known and unknown functionals.
+4.2
+Asymptotic distributions for known functionals
+We have the following assumptions.
+Assumption 4.1 (smoothness). (i) The functional φ is linear in the sense that the functional
+φ is linear in the sense that φ(α) − φ(α0) = dφ(α0)
+dα [α − α0].
+(ii) √ndφ(α0)
+dα [α0,n − α0] = oP(∥v∗
+n∥).
+Remark 4.1. Assumption 4.1 (iii) requires that the neural network bias term dφ(α0)
+dα [α0,n −
+α0] should be negligible. Here we present a sufficient condition following the discussion of
+Chen and Pouzo (2015). First, since α0,n is the projection of α0 on to span(An) and v∗
+n ∈ ¯Vn ⊂
+span(An), we have ⟨v∗
+n, α0,n−α0⟩ = 0. In addition, define an infinite dimensional Hilbert space
+¯V as the closure of the linear span of A − {α0}. Suppose dφ(α0)
+dα [·] is bounded, then there is a
+unique Riesz representer v∗ ∈ ¯V so that
+dφ(α0)
+dα
+[v] = ⟨v∗, v⟩,
+∀v ∈ ¯V .
+15
+
+As α0,n − α0 ∈ ¯V , we have
+����
+√ndφ(α0)
+dα
+[α0,n − α0]
+���� =
+��√n⟨v∗ − v∗
+n, α0,n − α0⟩
+�� ≤ √n∥v∗ − v∗
+n∥∥α0,n − α0∥.
+So condition (iii) holds as long as √n∥v∗ − v∗
+n∥∥α0,n − α0∥ = oP(∥v∗
+n∥).
+To allow quantile applications that involve nonsmooth loss functions, we need to show
+that the sample criterion function Qn(α) can be replaced with a smoothed criterion �Qn(α) :=
+1
+n
+�
+t ℓ(Xt, α)2�Σ(Xt)−1, where Ψn = (Ψ(Xt) : t = 1...n)n×kn:
+ℓ(x, α) := �m(x, α) + �m(x, α0),
+�m(x, α) := Ψ(x)′(Ψ′
+nΨn)−1Ψ′
+nmn(α),
+and mn(α) denotes the n × 1 vector of m(Xt, α). The replacement error is negligible:
+sup
+α∈Aosn
+sup
+|x|≤Cn−1/2 |Qn(α + xun) − �Qn(α + xun)| = oP(n−1).
+Therefore, theoretical analysis of Qn(α) is asymptotically equivalent to that of �Qn(α), while
+the latter is second-order pathwise differentiable, and admits a local quadratic approximation.
+Formalizing this argument would require the following conditions.
+Assumption 4.2. m(x, t) is twice differentiable with respect to t, and there is C > 0, so that,
+recall that un = v∗
+n/∥v∗
+n∥ being the “normalized Riesz representer”:
+(i) E|ρ(Yt+1, α0)|2+ζ ���dm(Xt,α0)
+dα
+[un]
+���
+2+ζ
++ E|ρ(Yt+1, α0)|2+ζ < C for some ζ > 0;
+(ii) E supα∈Cn sup|τ|≤Cn−1/2 1
+n
+�
+t
+�
+d2
+dτ 2m(Xt, α + τun)|
+�2
+< C;
+(iii) supτ∈(0,1) supα∈Cn E
+�
+d2
+dτ 2m(Xt, α0 + τ(α − α0))
+�2
+= o(n−1);
+(iv) kn supα∈Cn
+1
+n
+�
+t[ dm(Xt,α)
+dα
+[un] − dm(Xt,α0)
+dα
+[un]]2 = oP(1);
+(v) E
+�
+[maxj≤kn Ψj(Xt)2+1] supα∈Cn(ρ(Yt+1, α)−ρ(Yt+1, α0))2�
+< Cδ2η
+n for some κ, η > 0.
+Finally, we need to strengthen conditions on the penalty and some rates of convergence
+as follows.
+Assumption 4.3. (i) Let Ch := {h : (θ, h) ∈ Cn for some θ ∈ Θ}, which is the local neighbor-
+hood for the estimated h(·). We assume
+λn sup
+h∈Ch
+|Pen(h) − Pen(h0)| + λn sup
+h∈Ch
+|Pen(πnh) − Pen(h0)| = o(n−1).
+16
+
+(ii) √n¯δn∥�Σn − Σn∥ = o(1), where �Σn and Σn be the diagonal matrix of �Σ(Xt) and Σ(Xt) for
+all t, and furthermore ϕ2
+n¯δ2
+n + knd2
+nδ2η
+n + √kndnδη
+n¯δn = o(n−1).
+The following condition is similar to Condition C in Shen (1997), which is used to control
+the approximation error of the NN space for locally perturbed elements.
+Assumption 4.4. There is µn → 0 so that µn¯δn = o(n−1), we have
+sup
+α∈Cn
+1
+n
+n
+�
+t=1
+[m(Xt, πnα) − m(Xt, α)]2 = OP(µ2
+n).
+Theorem 4.1 (Limiting distribution). Under Assumptions 3.3-4.4,
+√nφ(�α) − φ(α0)
+∥v∗
+n∥
+→d N (0, 1).
+An important insight from this theorem is that the asymptotic distribution does not
+depend on the actual choice of the learning space. The asymptotic variance
+∥v∗
+n∥2 = EΣ(Xt)−1
+�dm(Xt, α0)
+dα
+[v∗
+n]
+�2
+is only determined by the functional forms φ and m(X, α), and more generally, the loss
+function. So whether the multilayer neural network, B-spline, Gaussian radial basis, etc, are
+being used to estimate α0, the asymptotic distribution is the same. What really matters is
+the loss function.
+4.3
+Estimation for unknown functionals
+We now consider estimating unknown (probably not √n-estimable) functionals, taking the
+form
+γ0 := El(h0(Wt)),
+where l(·) is a known function. Ai and Chen (2012) used the following moment condition (4.2)
+to construct the optimal criterion function:
+γ0 = EWt,
+Wt = l(h0(Wt)) − Γ(Xt)ρ(Yt+1, α0),
+(4.2)
+where Γ(Xt) = E[l(h0(Wt))ρ(Yt+1, α0)|σt(X )]Σ(Xt)−1. They showed that estimating γ0 based
+on this moment condition leads to more efficient estimator than based on the naive plug-in
+method 1
+n
+�
+i l(�h(Wt)), whenever Wt is endogenous. Because the naive plug-in estimator does
+17
+
+not take into account the potential correlations between the moment functions m(Xt, α) and
+l(h(Wt)).
+Using the more efficient moment condition of γ0, and letting
+φ(α) := El(h(Wt)) − EΓ(Xt)ρ(Yt+1, α),
+we note that φ(α0) = γ0. Suppose the functional φ(·) were known, and Assumption 4.1
+continues to hold for φ(α), then we can show
+√n(φ(�α) − φ(α0)) ≈ − 1
+√n
+n
+�
+t=1
+Zt,
+Zt := ρ(Yt+1, α0)Σ(Xt)−1dm(Xt, α0)
+dα
+[v∗
+n],
+where v∗
+n is the Riesz representer.
+But we in fact are facing a problem of estimating an
+unknown functional φ(·). To do so, we first estimate Γ(Xt) by
+�Γt :=
+n
+�
+s=1
+l(�h(Ws))ρ(Ys+1, �α)φ(Xs)′(Ψ′
+nΨn)−1Ψ(Xt)�Σ(Xt)−1.
+Then define the final estimator:
+�γ := �φ(�α), where �φ(α) = 1
+n
+n
+�
+t=1
+[l(h(Wt)) − �Γtρ(Yt+1, α)].
+(4.3)
+The following asymptotic expansion holds for the estimated functional:
+�γ − γ0
+=
+[φ(�α) − φ(α0)] + 1
+n
+n
+�
+t=1
+[Wt − EWt] + oP(σn−1/2)
+=
+1
+n
+n
+�
+t=1
+[−Zt + Wt − EWt] + oP(σn−1/2)
+where Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt,α0)
+dα
+[v∗
+n]. This explicitly presents two leading sources for
+the asymptotic distribution, where the asymptotic variance is given by
+σ2 := 1
+n Var
+� n
+�
+t=1
+(Wt − Zt)
+�
+= 1
+n Var
+� n
+�
+t=1
+Wt
+�
++ ∥v∗
+n∥2.
+(4.4)
+where Wt and Zt are uncorrelated.
+We impose the following conditions
+18
+
+Assumption 4.5. (i) supx |Γ(x)|2 + supw suph∈Hn l(h(w))2 < C.
+(ii) l(h) is linear in h.
+(iii) E supα∈Cn |l(h(Wt)) − l(h0(Wt))|2 ≤ Cδ2η
+n , where for simplicity we assume the same
+η as in Assumption 4.2 (v).
+Assumption 4.5 regulates the approximation quality of the instrumental space using linear
+sieves, which is not stringent since E(l(h(Wt))ρ(Yt+1, α)|σt(X )) is a function of the instrumen-
+tal variable.
+The next assumption imposes a condition on the accuracy of estimating the optimal
+weighting function Σ(Xt). For the NPQIV model this assumption is trivially satisfied since
+�Σ(Xt) = Σ(Xt) = ̟(1 − ̟) is known (see Section 6.3 for the definition of ̟). We shall verify
+it for the NPIV model in Section 6.2.
+Assumption 4.6. There is a sequence pn so that pn¯δnσ = o(n−1) and
+1
+n
+�
+t
+Γ(Xt)Σ(Xt)(�Σ(Xt)−1 − Σ(Xt)−1)ρ(Yt+1, α0) = OP(pn).
+The asymptotic normality requires some rate restrictions, which we impose below.
+Assumption 4.7. (i) There is c0 > 0 so that σ2 > c0.
+(ii) Let νn := δη
+n supx |�Σ(x) − Σ(x)| + √kndnδη
+n + ϕ2
+n. Then νn¯δnσ = o(n−1).
+Theorem 4.2. Suppose Assumptions 3.3-4.4 hold for φ(α) = El(h(Wt)) − EΓ(Xt)ρ(Yt+1, α).
+In addition, Assumptions 4.5-4.7 hold. Then
+√nσ−1(�γ − γ0) →d N (0, 1).
+5
+Quasi-Likelihood Ratio Inference for Functionals
+As shown by Theorems 4.1 and 4.2, computing the asymptotic variance requires estimating
+Riesz representer. While Chen and Pouzo (2015) and Chernozhukov et al. (2018c) proposed
+framework of estimating the Riesz representer, the task is in general quite challenging when
+its does not have closed-form approximations. In this section we propose to make inference
+directly using the optimally weighted quas-likelihood ratio statistic (QLR).
+19
+
+5.1
+QLR Inference for known functionals
+Consider testing
+H0 : φ(α0) = φ0
+for some known φ0 ∈ R. Consider the restricted null space AR
+n := {α ∈ An : φ(α) = φ0}. The
+GN-QLR statistic is defined as
+Sn(φ0) = n
+�
+Qn(�αR) − Qn(�α)
+�
+where �αR ∈ AR
+n approximately minimizes the penalized loss function over the general nonlinear
+sieve learning restricted on the null space:
+Qn(�αR) + λnPen(�hR) ≤ inf
+α∈AR
+n
+Qn(α) + λnPen(α) + oP(n−1).
+Define
+πR
+n α = arg
+min
+b∈An,φ(b)=φ0 ∥b − α∥∞,ω.
+Assumption 5.1. (i) Recall µn as defined in Assumption 4.4. It also satisfies:
+sup
+α∈Aosn,φ(α)=φ0
+1
+n
+n
+�
+t=1
+[m(Xt, πR
+n (α + xun)) − m(Xt, α + xun)]2 = OP(µ2
+n)
+(ii) (1 + ∥v∗
+n∥) supα∈Cn |φ(πnα) − φ(α)| = o(n−1/2).
+The following theorem shows the asymptotic null distribution of Sn(φ0).
+Theorem 5.1. Suppose conditions of Theorem 4.1 and Assumption 5.1 hold. Then under
+H0 : φ(α0) = φ0,
+Sn(φ0) →d χ2
+1.
+5.2
+QLR inference for unknown functionals
+We now move on to the inference for the unknown functional γ0 := El(h0(Wt)), which is
+estimated by �γ as defined in (4.3). Consider testing
+H0 : El(h0(Wt)) = φ0
+20
+
+for some known φ0. Define
+Ln(α, γ) := Qn(α) + (�φ(α) − γ)2�Σ−1
+2 ,
+where �Σ2 consistently estimates the long-run variance (e.g. Newey and West (1987)):
+Σ2 := Var
+�
+1
+√n
+n
+�
+t=1
+Wt
+�
+= 1
+n
+n
+�
+t=1
+Var(Wt) + 1
+n
+�
+t̸=s
+cov(Wt, Ws).
+We recall that Wt = l(h0(Wt)) − Γ(Xt)ρ(Yt+1, α0).
+Note that (�α, �γ) is numerically equivalent to the solution to the following problem:
+Ln(�α, �γ) + λnPen(�h) ≤ inf
+α∈An min
+γ
+Ln(α, γ) + λnPen(h) + oP(n−1).
+We define the GN-QLR statistic as
+�Sn(φ0) = n
+�
+Ln(�αR, φ0) − Ln(�α, �γ)
+�
+,
+where �αR ∈ AR
+n approximately minimizes the penalized loss function in the learning space
+Hn, but fixing γ = φ0:
+Ln(�αR, φ0) + λnPen(�hR) ≤ inf
+α∈An Ln(α, φ0) + λnPen(α) + oP(n−1).
+The asymptotic analysis of �Sn(φ0) is rather sophisticated, which requires additional rate
+constraints stated as follows.
+Theorem 5.2. Suppose �Σ2 − Σ2 = oP(1)Σ2 and conditions of Theorem 4.2 hold. Then under
+H0 : γ0 = φ0
+�Sn(φ0) →d χ2
+1.
+6
+Examples
+In this section, we illustrate our main results using three important models: Reinforcement
+learning, NPIV and NPQIV. We impose premitive conditions to verify the high level Assump-
+tions 3.2, 3.6 and 4.2 respectively in the two models.
+21
+
+6.1
+Reinforcement learning
+Reinforcement learning (RL) has been an important learning device behind many successes in
+applications of artificial intelligence. Theories of RL have been developed in the literature of
+statistical learning and computer science. Most of the existing theoretical works formulate the
+problem as a least-square regression and approximate the value function by a linear function,
+such as Bradtke and Barto (1996), etc. Nonlinear approximations using kernel methods or
+deep learning appeared in the more recent literature, for example Farahmand et al. (2016);
+Geist et al. (2019); Fan et al. (2020); Duan et al. (2021); Long et al. (2021); Chen and Qi
+(2022). Shi et al. (2020) also conducted inference for the optimal policy using linear sieve
+representations.
+We proceed learning using neural networks, and study the inference for a given policy.
+We follow the recent literature on the off-policy evaluation problem, and formulate the rein-
+forcement learning problem as a conditional moment restriction model. Assume the observed
+data trajectory {(St, At, Rt)}t≥0 is obtained from an unknown behavior policy probability
+πb(a|s), where (St, At, Rt) denote the state, action and observed reward at time t respectively
+and πb(a|s) is the distribution to take action a at state s. We denote the space of states and
+actions as S and A. It is assumed that the reward Rt is jointly determined by (St, At, St+1).
+Standing at state St at period t, one takes action At and receives reward Rt. The state then
+transits to St+1 at the next period.
+The value of a given policy π is measured by the so-called Q-function. Specifically, for
+any given π and any state-action pair (s, a), Q-function is defined as the expected discounted
+reward:
+Qπ(s, a) =
+∞
+�
+t=0
+γtEπ(Rt|S0 = s, A0 = a) ,
+where Eπ or in short E is the expectation when we take actions according to π, 0 ≤ γ < 1 is
+the discount factor and we consider the discounted infinite-horizon sum of expected rewards.
+To estimate Qπ, a classical approach is to solve the Bellman equation below:
+Qπ(s, a) = E
+�
+Rt + γ
+�
+x∈A
+π(x|St+1)Qπ(St+1, x)dx
+����St = s, At = a
+�
+.
+The goal is to recover Qπ of a given target policy π.
+In practice, multiple trajectories
+{(Si,t, Ai,t, Ri,t, Si,t+1)}0≤t≤T,1≤i≤N may be observed to help estimate the Q-function.
+But
+for simplicity we assume N = 1 and T = n. The more general case can be cast by merging
+the N time series into a single series of size n = TN.
+22
+
+The Bellman equation can be formulated as a conditional moment restriction with respect
+to Qπ for weakly dependent time series:
+E[ρ(Yt+1, Qπ)|St, At] = 0,
+Yt+1 = (Rt, St, At, St+1),
+Xt = (St, At),
+where
+ρ(Yt+1, h) = Rt − h(St, At) + γ
+�
+x∈A
+π(x|St+1)h(St+1, x)dx.
+In this framework, the estimation of the function Qπ(s, a) can be conducted on the neural
+network space, and we assume that computationally the integration in the ρ-function can be
+well approximately by the Monte Carlo method. For off-policy evaluations, the following value
+function is of the major interest in this section: given state s ∈ S,
+φs(Qπ) =
+�
+a∈A
+π(a|s)Qπ(s, a)da,
+(6.1)
+which is a known functional φs(·) for a single state s.
+The Bellman equation also admits a Fredholm integral equation of the second kind (Kress,
+1989), which is a well-posed problem. Therefore, estimating the Q-function may achieve fast-
+rate of convergence. That is, the sieve modulus of continuity satisfies:
+ωn(δ) :=
+sup
+α∈An:∥α−πnα0∥≤δ
+∥α − πnα0∥s ≍ δ
+Recently Chen and Qi (2022) showed this result for ∥.∥s to be either the sup-norm or the
+ℓ2-norm. The inner product is defined, in this case, as ⟨v1, v2⟩ = EΣ(Xt)−1 � dm
+dh [v1]
+� �dm
+dh [v2]
+�
+,
+where
+dm
+dh [v] = γ
+�
+x∈A
+E [π(x|St+1)v(St+1, x)|St, At] dx − v(St, At),
+(6.2)
+and induced a Riesz representer v∗ whose closed form is unavailable. Meanwhile, it follows
+from the Bellman equation that m(Xt, h) = dm
+dh [h − Qπ] for all h ∈ Hn. Therefore, the weak
+norm ∥.∥ can be expressed as:
+∥h − Qπ∥2 = Em(Xt, h)2Σ(Xt)−1,
+which shows that the employed minimum distance criterion function is directly estimating the
+squared weak norm.
+Let �Qπ be the estimated Qπ using the general nonlinear learning space, and the functional
+23
+
+is naturally estimated using
+φs( �Qπ) =
+�
+a∈A
+π(a|s) �Qπ(s, a)da
+As the moment restriction function E[ρ(Yt+1, h)|St, At] is linear in h in this case, it is straight-
+forward to verify the high-level conditions as follows.
+Assumption 6.1. (i) For some ζ > 4, the Riesz representer satisfies
+E
+�
+π(x|St+1)|v∗
+n(St+1, x)|ζdx + E|v∗
+n(St, At)|ζ ≤ ∥v∗
+n∥ζ.
+(ii) ER4
+t < ∞, E maxj≤kn Ψj(Xt)4 < ∞, E(1+|St|2 +|At|2)2ω < ∞ and EM(St+1)4 < ∞,
+where M(St+1) :=
+�
+π(x|St+1)(1 + x2 + S2
+t+1)ω/2dx, and ω is the degree of the weighted-sup
+metric ∥.∥∞,ω.
+Proposition 6.1. For the Reinforcement Learning model considered here, Assumption 6.1
+implies Assumptions 3.2, 3.6 and 4.2.
+It then follows from Theorem 4.1 that
+∥v∗
+n∥−1√n
+�
+φs( �Qπ) − φs(Qπ)
+�
+→d N (0, 1)
+Inference about φs(Qπ) based on pivotal statistics can be conducted using the GN-QLR test.
+6.2
+The NPIV model
+In the nonparametric instrumental variable model (NPIV), consider
+yt+1 = h0(Wt) + Ut+1,
+E(Ut+1|σt(X )) = 0.
+where σt(X ) is the filtration generated from instrumental variables Xt. Then m(Xt, α) =
+E[(yt+1−h(Wt))|σt(X )] and the Gateaux derivative is defined as dm(Xt,α)
+dh
+[v] = E(v(Wt)|σt(X )),
+implying
+⟨un, h − h0⟩ = E
+�
+E(un(Wt)|σt(X ))E(h − h0|σt(X ))Σ(Xt)−1�
+.
+We estimate the conditional variance Σ(Xt) by �Σt = �A′
+nΨn(Ψ′
+nΨn)−1Ψ(Xt) where �An is a n×1
+vector of ρ(Yt+1, �α)2. Recall that for δn and ¯δn defined in (3.2),
+∥�h − h∥∞,ω = OP(δn),
+∥�h − h∥ = OP(¯δn).
+We impose the following low-level conditions to verify Assumptions 3.6 and 4.2.
+24
+
+Assumption 6.2. (i) δ2
+n¯δnσ = o(n−1), E maxj≤kn |Ψj(Xt)|2(U2
+t +1) < C, and E(U2
+t |σt(X )) <
+C almost surely. Also, E(1 + |Wt|2)ω < C and E maxj≤kn Ψj(Xt)2(1 + |Wt|2)ω < C.
+(ii) The Riesz representer v∗
+n satisfies: there are C, ζ > 0,
+E(maxj≤kn Ψj(Xt)2 + 1)v∗
+n(Wt)2 < CEK2
+t and E|Ut|2+ζ|Kt|2+ζ ≤ C(EK2
+t )1+ζ/2, where
+Kt := E(v∗
+n(Wt)|σt(X )).
+Proposition 6.2. For the NPIV model,
+(i) Assumption 6.2 implies Assumptions 3.2, 3.6, 4.2 and 4.6.
+(ii) For the known functional φ(·), in addition Assumptions 3.3, 3.4, 3.5, 3.7, 3.8, 4.1,
+4.3, 4.4 hold. Then
+√n(φ(�α) − φ(α0))
+σn
+→d N (0, 1),
+where σ2
+n := Var (E(v∗
+n(Wt)|σt(X ))Σ(Xt)−1Ut) .
+(iii) For the unknown functional γ0 = El(h0(Wt)), if additionally Assumptions 4.5,4.7
+hold, then
+√nv−1(�γ − γ0) →d N (0, 1),
+where v2 := 1
+n Var(�
+t Wt−Zt) with Wt = l(h0(Wt))−Γ0(Xt)Ut+1 and Zt = Ut+1Σ(Xt)−1E[v∗
+n(Wt)|σt(X )].
+6.3
+The NPQIV model
+Consider the nonparametric quantile instrumental variable (NPQIV) model
+E[1{yt+1 ≤ h0(Wt)}|σt(X )] = ̟ ∈ (0, 1).
+Then m(Xt, α) = P(Ut+1 < h−h0|σt(X ))−̟ where Ut+1 = yt+1 −h0(Wt) and α = h. Within
+this framework, we now verify the high-level assumptions presented in the previous sections.
+Suppose the conditional distribution of Ut given (Xt, Wt) is absolutely continuous with
+density function fUt|σt(X),Wt(u). In this context, Σ(Xt) is known, given by
+Σ(Xt) = Var(1{yt+1 ≤ h0(Wt)}|σt(X )) = ̟ − ̟2.
+Then the Gateaux derivative is defined as
+dm(Xt, α)
+dh
+[v] = E(fUt|σt(X),Wt(h(Wt) − h0(Wt))v(Wt)|σt(X )),
+25
+
+implying, for g1 = fUt|σt(X),Wt(0)un(Wt) and g2 = fUt|σt(X),Wt(0)(h(Wt) − h0(Wt)),
+⟨un, h − h0⟩ = E [E(g1|σt(X ))E(g2|σt(X ))] (̟ − ̟2)−1.
+Also, ∥v∗
+n∥2 = (̟ − ̟2)−1Eg(Xt)2 where g(Xt) = E[fUt|σt(X),Wt(0)v∗
+n(Wt)|σt(X )].
+We impose the following low-level conditions to verify Assumptions 3.6 and 4.2. Let
+At(v)
+:=
+� 1
+0
+fUt|σt(X),Wt (x(v(Wt) − h0(Wt))) dx
+Bt(v, h)
+:=
+E {At(v)[h(Wt) − h0(Wt)]|Xt} .
+Assumption 6.3. (i) There are c1, c2, ǫ0 > 0 so that for all ∥h − h0∥∞,ω < ǫ0,
+c2EBt(h, h)2Σ(Xt)−1 ≤ EBt(h0, h)2Σ(Xt)−1 ≤ c1EBt(h, h)2Σ(Xt)−1.
+(ii) Almost surely, supu f
+′
+Ut|σt(X),Wt(u) < C and supu,x,w fUt|σt(X),Wt(u) < C. Also and
+there is L > 0, for all u, almost surely, supx,w |fUt|σt(X),Wt(u) − fUt|σt(X),Wt(0)| ≤ L|u|.
+(iii) E[maxj≤kn Ψj(Xt)2 + At(h0)2](1 + |Wt|2)ω < C and E[un(Wt)4|σt(X )] < C.
+(iv) δ2
+nkn = o(1) and δ4
+n = o(n−1).
+The following proposition, proved in the appendix, is the main result in this subsection,
+which verifies the high-level conditions in the NPQIV context.
+Proposition 6.3. For the NPQIV model,
+(i) Assumption 6.3 implies Assumptions 3.2, 3.6, 4.2 and 4.6.
+(ii) For the known functional φ(·), in addition Assumptions 3.3, 3.4, 3.5, 3.7, 3.8, 4.1,
+4.3, 4.4 hold. Then
+√n(φ(�α) − φ(α0))
+σn
+→d N (0, 1),
+where σ2
+n := (̟ − ̟2)−1E
+�
+[EfUt|σt(X),Wt(0)v∗
+n(Wt)|σt(X )]2�
+.
+(iii) For the unknown functional γ0 = El(h0(Wt)), if additionally Assumptions 4.5,4.7
+hold, then
+√nv−1(�γ − γ0) →d N (0, 1),
+where v2 := 1
+n Var(�
+t Wt − Zt) with Wt = l(h0(Wt)) − Γ0(Xt)Ut+1 and
+26
+
+Zt = (̟ − ̟2)−11{Ut+1 ≤ 0}EfUt|σt(X),Wt(0)v∗
+n(Wt)|σt(X ).
+7
+Simulation Studies
+In this section, we set up nonparametric endogenous models to illustrate the performance
+of our proposed estimators and testing statistics using some synthetic data. Consider the
+following data generating process
+Yt = h(Zt, Yt−1, . . . , Yt−L) + et ,
+where
+h(Zt, Yt−1, . . . , Yt−L) = Ztϑ0 + f(
+L
+�
+l=1
+blYt−l) ,
+and φ(α) = E[∂h/∂Zt] = ϑ0 = 1 is the quantity to be estimated. We choose L = 3, bl = 0.4l
+and consider the nonlinear mapping f(x) = 1−exp(−x)
+1+exp(−x). The endogenous Zt is generated using
+the following auto-regressive model:
+Zt = 0.3Zt−1 + ut,
+(ut, εt) ∼iid N(0, Σ),
+Σ =
+�
+1
+ρ
+ρ
+1
+�
+.
+And et is generated with the following ARCH model using εt as the innovation:
+et = σtεt,
+σ2
+t = 0.5 + 0.5(1 − 0.32)Z2
+t−1 .
+We set ρ = 0.5 to make Zt endogenous. We also make et heterogeneous. Note that E[e2
+t] =
+E[σ2
+t ] = 1. The endogenous variable is Wt = Zt. The instruments are Xt = (Zt−1, Yt−1, ..., Yt−L).
+We chose to generate n = 5000 samples (some burning period has been thrown away to make
+sure data are stationary). Note that the model can be used for both NPIV and NPQIV with
+̟ = 0.5.
+We applied a fully-connected J-layer ReLU-activated NN with hidden layer width of
+K. The optimization of the unconstrained NPIV or NPQIV objective used vanilla gradient
+descent. We did not apply mini-batch in gradient descent training as using mini-batches may
+hurt performance due to insufficient smoothing. The training epoch was as large as 10000 with
+learning rate 0.01 for NPIV and 0.1 for NPQIV. Furthermore we did not apply any penalty
+term for this example since the problem is relatively easy and the NN under consideration is
+of a small scale. The linear sieve bases (Ψ1, ..., Ψkn) for the instrumental variable space were
+˜kn cubic B-splines for X and each of the three Y lags concatenated together. For simplicity,
+no interaction terms between X and Y lags were included. Thus in total, we have kn = 4˜kn −3
+27
+
+Table 1: Estimation and hypothesis testing under NPIV and NPQIV with synthetic data.
+Here (J, K, kn) respectively denote the number of layers, width of the neural nets and the
+number of sieve bases for estimating the instrumental space. The true value for ϑ0 = 1. 95%
+qtl refers to the empirical 95% quantile, where the theoretical quantile for the chi square
+distribution is 3.84.
+Layer
+Width
+Basis
+Estimator of ϑ0
+Testing Statistic for ϑ0
+Problem
+J
+K
+kn
+mean
+std
+mean
+std
+95% qtl
+size
+NPIV
+3
+10
+17
+0.968
+0.116
+0.999
+1.432
+3.814
+5.0%
+NPIV
+3
+10
+13
+0.957
+0.115
+0.874
+1.236
+3.727
+4.8%
+NPIV
+1
+40
+13
+0.984
+0.108
+1.032
+1.418
+4.215
+6.0%
+NPQIV
+3
+10
+49
+0.997
+0.129
+1.086
+1.565
+4.280
+6.4%
+NPQIV
+3
+10
+45
+0.994
+0.130
+1.002
+1.409
+3.955
+5.6%
+NPQIV
+1
+40
+29
+0.977
+0.126
+1.050
+1.421
+3.678
+4.9%
+bases (since all B-spline bases sum up to 1, we remove the last basis for each dimension and
+finally add the intercept term as another basis). In our simulations, we find that NPQIV
+requires more number of sieve basis kn for estimating the instrumental space.
+For the NPIV problem, we first optimize the equal weighted quadratic loss to obtain �h,
+which is used to estimate Σ(Xt) and Γ(Xt) consistently. In the second step, we optimize the
+optimally weighted quadratic loss with the weighting matrix �Σ(Xt)−1 and apply the forward
+filter to estimate our expectation functional, which in this example is the constant ϑ0 = 1.
+Finally, we carry out the hypothesis testing for H0 : φ(h) = E[∂h/∂Zt] = φ0 = 1 to check
+the size of the testing statistic. Specifically, we estimated the forward filtered residuals as
+�
+Wt = ∂�h(Wt)/∂Wt−�Γt(Yt−�h(Wt)) and estimated Σ2 = Var(Wt) by the Newey-West estimator
+given �
+Wt, then solved the constrained optimization of Ln(h, φ0) and finally constructed the
+testing statistic. For NPQIV problem, since the optimal weighting is proportional to equal
+weighting, we do not need the initial step to estimate Σ(Xt). So we directly optimized the
+optimally weighted quadratic loss and estimated Γ(Xt) using the results and then used the
+forward filter to correct the estimation of the average partial derivative. Finally, similar to
+NPIV, we conduct the hypothesis testing for H0 : φ(α) = 1 under NPQIV.
+As for the computational practice, we find that for NPQIV models, it is helpful to apply
+truncations to the learned gradients in each step of training the network. Specifically, we
+smooth the loss function of the NPQIV model and truncate the updated gradient:
+θk+1 = θk − lr ∗ min{|∇Ln,k|, 0.001} ∗ sgn(∇Ln,k)
+28
+
+where lr is the learning rate, fixed to be 0.1 for NPQIV; ∇Ln,k is the gradient of the NN
+at the current step; θk+1 is the updated neural network coefficients at the current step. The
+truncation prevents the network from having very large gradients during iterations, helping
+stabilize the training process empirically.
+We repeat each setting for 1000 times. For the efficient estimation, we report the mean
+and standard deviation of the forward filtered average gradient for the optimal weighting
+optimizaiton in Table 1. For hypothesis testing, we also report in Table 1 the mean, standard
+deviation and 95% quantile of the empirical testing statistic.
+In addition, if we use the
+theoretical critical value corresponding to 5% significance level, which is 3.84 for χ2
+1, the
+p-value is also reported.
+As we can see from Table 1, for NPIV, optimal weighting estimates φ(h) accurately in
+the sense that the mean insignificantly differs from the true value ϑ0 = 1. NPQIV is less
+efficient with a larger standard deviation, and thus requires more samples to be estimated to
+the same accuracy. Note that the instrumental space with a step function can be harder to
+approximate with the cubic B-spline linear sieve bases. In terms of the performance of QLR
+testing statistic, the p-values are all close to the nominal 5% level for the NPIV and NPQIV
+models.
+Admittedly through our experiments the results can be sensitive to some tuning
+parameters, which is typically the case when applying deep learning for statistical inference:
+at the moment we still heavily rely on ad-hoc tuning in many problems. In comparison, the
+estimation of φ(h) is more stable with respect to different J and K values. Here we only mean
+to present some results without heavily tuning the parameters. Methods using NN for real
+applications require more extensive tuning in practice and some rough sense on the model
+complexity would be useful to determine the balance between the dimensions of the NN sieve
+and the linear IV sieve.
+8
+Conclusion
+In this paper we establish neural network estimation and inference on functionals of unknown
+function satisfies a general time series conditional moment restrictions containing endogenous
+variables. We consider quasi-likelihood ratio (GN-QLR) based inference, where nonparametric
+functions are learned using multilayer neural networks. While the asymptotic normality of the
+estimated functionals depends on some unknown Riesz representer of the functional space, we
+show that the GN-QLR statistic is asymptotically Chi-square distributed, regardless whether
+the expectation functional is regular (root-n estimable) or not. This holds when the data are
+weakly dependent and satisfy the beta-mixing condition.
+In addition to estimating partial derivatives in nonparametric endogenous problems as
+29
+
+examples, our study is well motivated by the setting of reinforcement learning where data are
+time series in nature. We apply our method to the off-policy evaluation, by formulating the
+Bellman equation into the conditional moment restriction framework, so that we can make
+inference about the state-specific value functional using the proposed GN-QLR method with
+time series data.
+30
+
+A
+Stochastic equicontinuity on the NN space for β-mixing obser-
+vations
+A key technical result is the stochastic equicontinuity of the residual function on the general nonlinear
+sieve learning space, which is established in the following proposition in this section.
+Let St =
+(Yt+1, Xt) and
+ǫt(α) ≡ ǫ(St, α) := ρ(Yt+1, α) − m(Xt, α).
+We derive bounds that require the pseudo dimension of the deep neural network class. Recall
+δn := ∥πnα0 − α0∥∞,ω + ωn(¯δn),
+¯δ2
+n := ∥πnα0 − α0∥2 + λn + knd2
+n + ϕ2
+n
+where dn :=
+�
+p(Hn) log2 n
+n
+.
+Proposition A.1. Let Cn = {α + xun : α ∈ An, ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤ C¯δ2
+n, |x| ≤ Cn−1/2}.
+Suppose :
+(a) E maxj≤kn Ψj(Xt)2 supα∈Cn(ρ(Yt+1, α) − ρ(Yt+1, α0))2 = Cδ2η
+n for some η, C > 0.
+(b) For some κ, C > 0 , EΨj(Xt)2 sup∥α1−α∥∞,ω<δ |ǫt(α1) − ǫt(α)|2 ≤ Cδ2κ for all δ > 0 and
+α, α1 ∈ cl{a + xb : a, b ∈ An, x ∈ R}.
+Then
+max
+j≤kn
+sup
+|x|≤Cn−1/2 sup
+α∈Cn
+| 1
+n
+�
+t
+Ψj(Xt)(ǫ(St, α + xun) − ǫ(St, α0))| ≤ OP (dnδη
+n).
+Proof. Let E := {(ǫ(, α + xun) − ǫ(, α0))Ψj : α ∈ Cn, j ≤ kn, |x| ≤ Cn−1/2} and let St = (Yt+1, Xt).
+We divide the proof into several steps.
+Step 1: construct blocks. Consider the following independent blocks: for any integer pair
+(an, bn), with bn = [n/(2an)], divide {St : t ≤ n} into 2bn blocks with length an and the remaining
+block of length n − 2anbn:
+H1,l
+=
+{i : 2(l − 1)an + 1 ≤ i ≤ (2l − 1)an}
+H2,l
+=
+{i : (2l − 1)an + 1 ≤ i ≤ 2lan}
+where l = 1, ..., bn. Let Υ = {i : 2anbn + 1 ≤ i ≤ n}. Now let {�S1, ..., �Sn} be a random sequence that
+is independent of {S1, .., Sn} and has independent blocks such that each block has the same joint
+distribution as the corresponding block of the St-sequence. Because the St-sequence is β-mixing, by
+Lemma 2 of Eberlein (1984), for any measurable set A, with the mixing coefficient β(),
+|P
+�
+{�St : t ∈ H1,l, l = 1, ..., bn} ∈ A
+�
+− P ({St : t ∈ H1,l, l = 1, ..., bn} ∈ A) | ≤ (bn − 1)β(an). (A.1)
+31
+
+The same inequality holds when H1,l is replaced with H2,l. In addition, for any function f, define
+U1,f(�Sl) = 1
+an
+�
+t∈H1,l
+f(�St),
+U2,f(�Sl) = 1
+an
+�
+t∈H2,l
+f(�St),
+where �Sl = {�St : t ∈ H1,l}.
+By construction, U1,f(�Sl) and U2,f(�Sl) are independent across l.
+Similarly, let Sl = {St : t ∈ H1,l}. Then
+1
+n
+�
+t
+f(St) − Ef(St)
+=
+1
+n
+�
+t∈Υ
+f(St) − Ef(St) + 1
+bn
+�
+l≤bn
+anbnn−1[U1,f(Sl) − EU1,f(Sl)]
++ 1
+bn
+�
+l≤bn
+anbnn−1[U2,f(Sl) − EU2,f(Sl)].
+(A.2)
+Next, we shall bound each term on the right hand side uniformly for f ∈ E. We replace U1,f(Sl)
+with U1,f(�Sl); the latter is easier to bound because blocks �Sl are independent. We then show that
+the effect of such replacements is negligible due to (A.1) by properly chosen (an, bn).
+Step 2: the envelop function for U1,f. Note that Ef = 0 for f ∈ E and that �St and St are
+identically distributed within each block H1,l. By Cauchy-Schwarz,
+E sup
+f∈E
+U1,f(�Sl)2
+≤
+E sup
+f∈E
+
+ 1
+an
+�
+t∈H1,l
+f(�St)
+
+
+2
+≤ 1
+an
+�
+t∈H1,l
+E sup
+f∈E
+f(St)2
+≤
+2E max
+j≤kn Ψj(Xt)2
+sup
+|x|≤Cn−1/2 sup
+α∈Cn
+(ρ(Yt+1, α + xun) − ρ(Yt+1, α0))2 ≤ Cδ2η
+n .
+Now take some p > η. Let F = {U1,f : f ∈ E} and let F := max{n−p, supf∈E |U1,f|}. Then both
+supf∈E |U1,f| and F are envelope functions of F, and
+n−p ≤ G := ∥F∥L2(St) ≤ Cn−p + Cδη
+n ≤ 2Cδη
+n.
+Step 3: the bracketing number. We aim to apply Theorem 2.14.2 of van der Vaart and Wellner
+(1996) to bound
+1
+bn
+�
+l≤bn anbnn−1U1,f(�Sl), which requires bounding the bracketing number of F.
+To do so, suppose h1, ..., hN is a δ-cover of Hn under the norm ∥h∥∞,ω and N := N(δ, Hn, ∥.∥∞,ω);
+θ1, ..., θR is a δ-cover of Θ and R := N(δ, Θ, ∥.∥) (the Euclidean norm in Θ). Here N(δ, A, .) de-
+notes the covering number for space A. Also let x1...xMn be a δ-cover of [−Cn−1/2, Cn−1/2], with
+Mn ≤ 4Cn−1/2/δ.
+Then for any f = (ǫ(, α + xun) − ǫ(, α0))Ψj ∈ E, there are Ψj, xq and αik = (θk, hi) so that
+32
+
+∥α−αik∥∞,ω ≤ ∥h−hi∥∞,ω +∥θ−θk∥ ≤ 2δ and |x−xq| < δ. Let fijkq = (ǫ(, αik +xqun)−ǫ(, α0))Ψj.
+sup
+f=(ǫ(,α+xun)−ǫ(,α0))Ψj:∥α−αik∥∞,ω<2δ,|x−xq|<δ
+|U1,f(�Sl) − U1,fijkq(�Sl)|
+≤
+sup
+f=(ǫ(,α+xun)−ǫ(,α0))Ψj:∥α−αik∥∞,ω<2δ,|x−xq|<δ
+| 1
+an
+�
+t∈H1,l
+f(�St) − fijkq(�St)|
+≤
+1
+an
+�
+t∈H1,l
+|Ψj( �
+Xt)|
+sup
+∥α−αik∥∞,ω<2δ
+sup
+|x−xq|<δ
+|ǫ(�St, α + xun) − ǫ(�St, αik + xqun)| := bijkq(�Sl, δ).
+Then U1,f ∈ [lijkq, uijkq], where lijkq := U1,fijkq −bijkq(, δ) and uijkq = U1,fijkq +bijkq(, δ). In addition,
+E[uijkq − lijkq]2
+≤
+4Ebijkq(�Sl, δ)2
+≤
+CE
+
+ 1
+an
+�
+t∈H1,l
+|Ψj( �
+Xt)|
+sup
+|x−xq|<δ
+sup
+∥α−αik∥∞,ω<2δ
+|ǫ(�St, α + xun) − ǫ(�St, αik + xqun)|
+
+
+2
+≤
+CEΨj( �
+Xt)2
+sup
+∥α−αik∥∞,ω<2δ
+sup
+|x−xq|<δ
+|ǫ(�St, α + xun) − ǫ(�St, αik + xqun)|2 ≤ Cδ2κ.
+Hence {[lijkq, uijkq] : i ≤ N, j ≤ kn, k ≤ R} is a Cδκ bracket of F, whose bracketing number satisfies
+N[](Cδκ, F, ∥.∥L2(�St)) ≤ N(δ, Hn, ∥.∥∞,ω)
+�
+��
+�
+N
+(C/δ)d
+� �� �
+R
+(n−1/2/δ)
+�
+��
+�
+Mn
+kn,
+where we used R ≤ (C/δ)d for d = dim(θ0) since θ0 ∈ Θ is compact. Then for a generic constant
+C > 0,
+N[](Gx, F, ∥.∥L2(�St)) ≤ CN(x1/κ(G/C)1/κ, Hn, ∥.∥∞,ω)G−(d+1)/κx−(d+1)/κkn,
+∀x > 0.
+Step 4: bound independent blocks. Note that U1,f(�Sl) are independent across l and is
+mean-zero. For the envelop G defined in step 2 and some constant ¯
+M > 0,
+E sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(�Sl)
+������
+≤ 1
+2E sup
+g∈F
+������
+1
+bn
+�
+l≤bn
+g(�Sl)
+������
+≤(i)
+b−1/2
+n
+G
+� 1
+0
+�
+1 + log N[](Gx, F, ∥.∥L2(�St))dx
+≤
+C
+√bn
+δη
+n
+� 1
+0
+�
+1 + log N(x1/κ(G/C)1/κ, Hn, ∥.∥∞) + log
+C
+n1/2G(d+1)/κx(d+1)/κ + log kndx
+≤(ii)
+Cδη
+n
+√bn
+� 1
+0
+�
+1 + p(Hn) log
+Cn
+x1/κG1/κ + (d + 1) log
+C
+G1/κx1/κ + log kndx
+≤(iii)
+Cδη
+n
+√bn
+� 1
+0
+�
+2 log kn + 2p(Hn) log
+Cn
+x1/κG1/κ dx
+33
+
+≤(iv)
+δη
+n
+�
+Cp(Hn) log n
+bn
+,
+where (i) follows from Theorem 2.14.2 of van der Vaart and Wellner (1996); (ii) follows from As-
+sumption 3.1; (iii) is due to p(Hn) → ∞ .
+We now prove the inequality (iv), which is to show
+� 1
+0
+�
+2 log kn + g(x)dx ≤ C
+�
+p(Hn) log n
+where g(x) = 2p(Hn) log
+Cn
+x1/κG1/κ .
+Let A := 2 log kn + 2p(Hn) log
+Cn
+G1/κ − 2κ−1p(Hn).
+We have
+log Cn
+G → ∞, hence 2κ−1p(Hn) ≤ A. Note that log(y) ≤ y − 1 for all y > 0. Hence
+2 log kn + g(x)
+=
+2 log kn + 2p(Hn) log Cn
+G1/κ + 2κ−1p(Hn) log 1
+x
+≤
+2 log kn + 2p(Hn) log Cn
+G1/κ + 2κ−1p(Hn)(1
+x − 1)
+=
+A + 2κ−1p(Hn)x−1 ≤ A + Ax−1 ≤ 2Ax−1.
+The last inequality holds for x < 1. Thus with n−10 ≤ G, and kn = O(bn),
+� 1
+0
+�
+2 log kn + g(x)dx
+≤
+√
+2A
+� 1
+0
+x−1/2dx ≤ 4
+�
+2 log kn + 2p(Hn)[log(Cn) + log G−1/κ]
+≤
+4
+�
+2 log kn + 2p(Hn)[log(Cn) + log n10/κ] ≤ C
+�
+p(Hn) log n.
+Therefore by the Markov inequality, for any ε > 0, with probability at least 1 − ε/4,
+sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(�Sl)
+������
+≤ cn
+ε ,
+cn = δη
+n
+�
+Cp(Hn) log n
+bn
+.
+Step 5: completion. By (A.1) and step 4,
+P
+
+sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(Sl)
+������
+> cn
+ε
+
+ ≤ P
+
+sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(�Sl)
+������
+> cn
+ε
+
++(bn−1)β(an).
+We now take an = M log n/2 with M > 0 and bn = [n/(M log n)]. Then (bn − 1)β(an) → 0 for
+sufficiently large M. Also, the requirement in step 4 that p(Hn) = o(bn) holds as long as p(Hn) log n =
+o(n). Hence with this choice of bn,
+sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(Sl)
+������
+= OP
+
+δη
+n
+�
+p(Hn) log2 n
+n
+
+ .
+The same rate applies when U1,f is replaced with U2,f following from the same proof of steps 2,3,4.
+34
+
+In addition, |Υ|0 ≤ 2an. Hence
+E sup
+f∈E
+�����
+1
+n
+�
+t∈Υ
+f(St) − Ef(St)
+�����
+≤
+E 1
+n
+�
+t∈Υ
+sup
+f∈E
+|f(St)| ≤ Can
+n E max
+j≤kn |Ψj(Xt)| sup
+α∈Cn
+|ǫt(α) − ǫt(α0)|
+≤
+Cδη
+n log n
+n
+.
+Together, by (A.2) maxj≤kn supα∈Cn | 1
+n
+�
+t Ψj(Xt)(ǫt(St, α) − ǫ(St, α0))| = OP (δη
+ndn) .
+B
+Proof of Theorem 3.1
+B.1
+Consistency
+Lemma B.1 (Consistency). Suppose kn
+n + Q(πnα0) = O(λ). Also suppose Pen(h) is lower semi-
+compact on (Hn, ∥.∥∞,ω) and Q(α) is lower semicontinuous. Then ∥�α − α0∥∞,ω = oP (1).
+Proof. The proof of this lemma does not depend on Assumption 3.2. First we show Pen(�h) = OP (1).
+Let ρn(α), mn(α) be the n × 1 vectors of ρ(Yt+1, α) and m(Xt, α). Let �Σ−1
+n
+be the diagonal matrix
+of �Σ(Xt)−1 for all t. By steps 1, 3 of the proof of Theorem 3.1 below,
+λPen(�h)
+≤
+Qn(πnα0) + λPen(πnh0) + oP (n−1)
+≤
+2
+n
+�
+t
+[ �m(Xt, πnα0) − �m(Xt, πnα0)]2�Σ(Xt)−1 + CE �m(Xt, πnα0)2 + λPen(πnh0) + oP(n−1)
+≤
+2[ρn(πnα0) − mn(πnα0)]′Pn�Σ−1
+n Pn[ρn(πnα0) − mn(πnα0)]
++CEm(Xt, πnα0)2 + λPen(πnh0) + oP (n−1)
+≤
+OP (kn
+n + Q(πnα0) + λ) = OP (λ)
+with the condition that kn
+n +Q(πnα0) = O(λ). So let M0 > 0 be a large constant so that Pen(�h) ≤ M0
+with probability arbitrarily close to one.
+Now take an arbitrary ǫ > 0, let Bǫ = {α = (θ, h) ∈ An : ∥α − α0∥∞,ω ≥ ǫ, Pen(h) ≤ M0}. Be-
+cause Pen(h) is lower semicompact on (H0, ∥.∥∞,ω) and Q(α) is lower semicontinuous, minα∈Bǫ Q(α)
+exists, that is, there is α∗ ∈ Bǫ so that infα∈Bǫ Q(α) = Q(α∗) > c0. If ∥�α − α0∥∞,ω > ǫ, then
+Q(�α) ≥ infα∈Bǫ Q(α) > c0. Meanwhile, by (B.1) (to be proved below),
+c0 ≤ Q(�α) ≤ Q(πnα0) + λn|Pen(πnh0) − Pen(�h)| + OP (knd2
+n + ϕ2
+n).
+But the right hand side is oP (1). Hence we must have ∥�α − α0∥∞,ω = oP (1).
+35
+
+B.2
+Proof of Theorem 3.1
+The proof depends on some important technical lemmas, one of which is the stochastic equicontinuity
+of ǫ(St, α) = ρ(Yt+1, α) − m(Xt, α), given by Proposition A.1.
+Proof. We divide the proof in the following steps.
+Let Dn be the sieve space used to estimate
+m(X, α), and
+�m(X, α) = arg min
+�m∈Dn
+n
+�
+t=1
+(m(Xt, α) − �m(Xt))2.
+We show the following steps:
+step 1.
+Show that for c, C > 0, uniformly in α ∈ An,
+cE �m(Xt, α)2 ≤ 1
+n
+n
+�
+t=1
+�m(Xt, α)2 ≤ CE �m(Xt, α)2.
+To prove it, we shall apply an empirical identifiability result that first proved by Huang (1998)
+for the i.i.d. case and then extended by Chen and Christensen (2015) to more general setting with
+a much simpler proof. We note that �m(·, α) ∈ Dn := {g(x) = �kn
+j=1 πjΨj(x) : ∥g∥∞,ω < ∞}. Let
+Ψn be the n × kn matrix of the linear sieve bases, and let A := 1
+nEΨ′
+nΨn. Suppose the linear sieve
+satisfies: λmin(A) > c and ∥ 1
+nΨ′
+nΨn − A∥ = oP (1). Then ∥A−1/2 1
+nΨ′
+nΨnA−1/2 − I∥ = oP(1), so the
+conditions of Lemma 4.1 of Chen and Christensen (2015) are satisfied. We then apply this lemma
+to reach that
+sup
+α∈An
+| 1
+n
+�
+t �m(Xt, α)2 − E �m(Xt, α)2|
+E �m(Xt, α)2
+≤ sup
+g∈Dn
+| 1
+n
+�
+t g(Xt)2 − Eg(Xt)2|
+Eg(Xt)2
+= oP (1).
+This then leads to the desired result.
+step 2.
+Show that
+sup
+α∈An
+1
+n
+n
+�
+t=1
+[ �m(Xt, α) − �m(Xt, α)]2 = OP (knd2
+n),
+d2
+n := p(Hn) log2 n
+n
+.
+Let ǫ(St, α) = ρ(Yt+1, α) − m(Xt, α). Also let Pn = Ψn(Ψ′
+nΨn)−1Ψ′
+n and ¯ǫn(α) be the n × 1
+vector of ǫ(St, α). We then have
+sup
+α∈An
+1
+n
+n
+�
+t=1
+[ �m(Xt, α) − �m(Xt, α)]2 = sup
+α∈An
+1
+n¯ǫn(α)′Pn¯ǫn(α) = OP (1) sup
+α
+∥ 1
+nΨ′
+n¯ǫn(α)∥2
+≤
+OP (kn) sup
+α max
+j≤kn | 1
+n
+n
+�
+t=1
+Ψj(Xt)ǫ(St, α)|2 = OP (knd2
+n).
+36
+
+The last bound is given by Lemma B.2.
+step 3.
+Show that supα∈An E[ �m(Xt, α) − m(Xt, α)]2 = O(ϕ2
+n).
+Let �mn(α) and mn(α) respectively be the n × 1 vectors of �m(Xt, α) and m(Xt, α). Also let
+mn(α) = Ψnbα + rα where rα is the sieve approximation error and bα is the sieve coefficient to
+approximate mn(X, α). Then �mn(α) = Pnmn(α) and
+sup
+α∈An
+E[ �m(Xt, α) − m(Xt, α)]2 = 1
+n sup
+α∈An
+Emn(α)′(I − Pn)mn(α) = 1
+n sup
+α∈An
+Er′
+α(I − Pn)rα
+≤
+1
+n sup
+α E∥rα∥2 = OP (ϕ2
+n).
+After achieving the above three steps, then we have (since �Σ(Xt)−1 and Σ(Xt)−1 are bounded
+away from zero)
+Qn(�α)
+≥
+c
+n
+�
+t
+�m(Xt, �α)2 ≥ 0.5c
+n
+�
+t
+�m(Xt, �α)2 − c
+n
+�
+t
+[ �m(Xt, �α) − �m(Xt, �α)]2
+≥(i)
+cE �m(Xt, �α)2 − OP (knd2
+n) ≥(ii) cEm(Xt, �α)2 − OP (knd2
+n + ϕ2
+n) ≥ Q(�α) − OP (knd2
+n)
+Qn(πnα0)
+≤
+C
+n
+�
+t
+�m(Xt, πnα0)2 ≤ 2C
+n
+�
+t
+�m(Xt, πnα0)2 + 2C
+n
+�
+t
+[ �m(Xt, πnα0) − �m(Xt, πnα0)]2
+≤(iii)
+CE �m(Xt, πnα0)2 + OP (knd2
+n) ≤(iv) CEm(Xt, πnα0)2 + OP (knd2
+n + ϕ2
+n)
+≤
+Q(πnα0) + OP (knd2
+n + ϕ2
+n)
+where (i) (iii) follow from steps 1,2; (ii) (iv) follow from step 3.
+Hence Qn(�α) + λnPen(�h) ≤ Qn(πnα0) + λnPen(πnh0) + oP (n−1) implies
+Q(�α) ≤ Q(πnα0) + λn|Pen(πnh0) − Pen(�h)| + OP (knd2
+n + ϕ2
+n).
+(B.1)
+Now by Assumption 3.2,
+∥�α − α0∥2 ≤ C∥πnα0 − α0∥2 + OP (λn + knd2
+n + ϕ2
+n).
+Hence ∥�α − πnα0∥ ≤ ∥�α − α0∥ + ∥πnα0 − α0∥ ≤ C∥πnα0 − α0∥ + OP (√λn + √kndn + ϕn). Thus
+∥�α − α0∥∞,ω
+≤
+∥�α − πnα0∥∞,ω + ∥πnα0 − α0∥∞,ω
+≤
+OP (∥πnα0 − α0∥∞,ω + ωn(∥πnα0 − α0∥ +
+�
+λn +
+�
+kndn + ϕn)).
+Lemma B.2. Suppose
+(a) E maxj≤kn Ψj(Xt)2 supα∈An ρ(Yt+1, α)2 ≤ C2
+37
+
+(b) There are κ > 0 and C > 0 so that EΨj(Xt)2 sup∥α1−α2∥∞,ω<δ |ǫ(St, α1) − ǫ(St, α2)|2 ≤ Cδ2κ
+holds for any δ > 0.
+(c) p(Hn) → ∞ and p(Hn) log n = o(n).
+Then supα maxj≤kn | 1
+n
+�n
+t=1 Ψj(Xt)ǫ(St, α)| = OP (
+�
+p(Hn) log2 n
+n
+).
+Proof. Let E := {ǫ(·, α)Ψj : α ∈ An, j ≤ kn}. We divide the proof into several steps.
+Step 1: construct blocks. This step is the same as that of the proof of Proposition A.1.
+Step 2: the envelop function for U1,f. Note that Ef = 0 for f ∈ E and that �St and St are
+identically distributed within each block H1,l. By Cauchy-Schwarz,
+E sup
+f∈E
+U1,f(�Sl)2
+≤
+E sup
+f∈E
+
+ 1
+an
+�
+t∈H1,l
+f(�St)
+
+
+2
+≤ 1
+an
+�
+t∈H1,l
+E sup
+f∈E
+f(St)2
+≤
+2E max
+j≤kn Ψj(Xt)2 sup
+α∈An
+ρ(Yt+1, α)2 ≤ C2.
+Let F = {U1,f : f ∈ E} and let F := max{n−10, supf∈E |U1,f|}. Then both supf∈E |U1,f| and F are
+envelope functions of F, and
+n−10 ≤ G := ∥F∥L2(St) ≤ C.
+Step 3: the bracketing number. We aim to apply Theorem 2.14.2 of van der Vaart and Wellner
+(1996) to bound
+1
+bn
+�
+l≤bn anbnn−1U1,f(�Sl), which requires bounding the bracketing number of F.
+To do so, suppose h1, ..., hN is a δ-cover of Hn under the norm ∥h∥∞,ω and N := N(δ, Hn, ∥.∥∞,ω);
+θ1, ..., θR is a δ-cover of Θ and R := N(δ, Θ, ∥.∥) (the Euclidean norm in Θ). Here N(δ, A, .) denotes
+the covering number for space A.
+Then for any f = ǫ(, α)Ψj ∈ E, there are Ψj and αik = (θk, hi) so that ∥α − αik∥∞,ω ≤
+∥h − hi∥∞,ω + ∥θ − θk∥ ≤ 2δ. Let fijk = ǫ(·, αik)Ψj. We have
+sup
+f=ǫ(·,α)Ψj:∥α−αik∥∞,ω<2δ
+|U1,f(�Sl) − U1,fijk(�Sl)| ≤
+sup
+f=ǫ(·,α)Ψj:∥α−αik∥∞,ω<2δ
+| 1
+an
+�
+t∈H1,l
+f(�St) − fijk(�St)|
+≤
+1
+an
+�
+t∈H1,l
+|Ψj( �
+Xt)|
+sup
+∥α−αik∥∞,ω<2δ
+|ǫt(�St, α) − ǫt(�St, αik)| := bijk(�Sl, δ).
+Then U1,f ∈ [lijk, uijk], where lijk := U1,fijk − bijk(, δ) and uijk = U1,fijk + bijk(, δ). In addition,
+E[uijk − lijk]2
+≤
+4Ebijk(�Sl, δ)2
+≤
+CE
+
+ 1
+an
+�
+t∈H1,l
+|Ψj( �
+Xt)|
+sup
+∥α−αik∥∞,ω<2δ
+|ǫt(�St, α) − ǫt(�St, αik)|
+
+
+2
+38
+
+≤
+CEΨj( �
+Xt)2
+sup
+∥α−αik∥∞,ω<2δ
+|ǫ(�St, α) − ǫ(�St, αik)|2 ≤ Cδ2κ.
+Hence {[lijk, uijk] : i ≤ N, j ≤ kn, k ≤ R} is a Cδκ bracket of F, whose bracketing number satisfies
+N[](Cδκ, F, ∥.∥L2(�St)) ≤ N(δ, Hn, ∥.∥∞,ω)
+�
+��
+�
+N
+(C/δ)d
+� �� �
+R
+kn,
+where we used R ≤ (C/δ)d for d = dim(θ0) since θ0 ∈ Θ is compact. Then for a generic constant
+C > 0,
+N[](Gx, F, ∥.∥L2(�St)) ≤ CN(x1/κ(G/C)1/κ, Hn, ∥.∥∞,ω)G−d/κx−d/κkn,
+∀x > 0.
+Step 4: bound independent blocks. Note that U1,f(�Sl) are independent across l and is
+mean-zero. For the envelop G defined in step 2 and some constant ¯
+M > 0,
+E sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(�Sl)
+������
+≤ 1
+2E sup
+g∈F
+������
+1
+bn
+�
+l≤bn
+g(�Sl)
+������
+≤(i)
+b−1/2
+n
+G
+� 1
+0
+�
+1 + log N[](Gx, F, ∥.∥L2(�St))dx
+≤
+C
+√bn
+� 1
+0
+�
+1 + log N(x1/κ(G/C)1/κ, Hn, ∥.∥∞) + log
+C
+Gd/κxd/κ + log kndx
+≤(ii)
+C
+√bn
+� 1
+0
+�
+1 + p(Hn) log
+Cn
+x1/κG1/κ + d log
+C
+G1/κx1/κ + log kndx
+≤(iii)
+C
+√bn
+� 1
+0
+�
+2 log kn + 2p(Hn) log
+Cn
+x1/κG1/κ dx ≤(iv)
+�
+Cp(Hn) log n
+bn
+,
+where (i) follows from Theorem 2.14.2 of van der Vaart and Wellner (1996); (ii) follows from As-
+sumption 3.1; (iii) is due to p(Hn) → ∞ . (iv) follows from the same proof as that of Proposition
+A.1.
+Step 5: completion. By an inequality similar to (A.1) and step 4,
+P
+
+sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(Sl)
+������
+> cn
+ε
+
+ ≤ P
+
+sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(�Sl)
+������
+> cn
+ε
+
++(bn−1)β(an).
+We now take an = M log n/2 with M > 0 and bn = [n/(M log n)]. Then (bn − 1)β(an) → 0 for
+sufficiently large M. Also, the requirement in step 4 that p(Hn) = o(bn) holds as long as p(Hn) log n =
+39
+
+o(n). Hence with this choice of bn,
+sup
+f∈E
+������
+1
+bn
+�
+l≤bn
+anbnn−1U1,f(Sl)
+������
+= OP
+
+
+�
+p(Hn) log2 n
+n
+
+ .
+The same rate applies when U1,f is replaced with U2,f following from the same proof of steps 2,3,4.
+In addition, |Υ|0 ≤ 2an. Hence
+E sup
+f∈E
+�����
+1
+n
+�
+t∈Υ
+f(St) − Ef(St)
+�����
+≤
+2E 1
+n
+�
+t∈Υ
+sup
+f∈E
+|f(St)| ≤ Can
+n E max
+j≤kn |Ψj(Xt)| sup
+α
+|ǫ(St, α)| ≤ C log n
+n
+.
+Together, maxj≤kn supα∈An | 1
+n
+�
+t Ψj(Xt)ǫ(St, α)| = OP
+��
+p(Hn) log2 n
+n
+�
+.
+C
+Proofs for Section 4
+C.1
+Local quadratic approximation
+Proposition C.1 (LQA). Let Cn = {α + xun : |x| < Cn−1/2, α ∈ An, ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤
+C¯δ2
+n}. Suppose for un = v∗
+n/∥v∗
+n∥, there are C > 0, so that
+(a) √n¯δn∥�Σn − Σn∥ = o(1), ϕ2
+n¯δ2
+n + knd2
+nδ2η
+n + √kndnδη
+n¯δn = o(n−1).
+(b)
+1
+√n∥(I − Pn)Σ−1
+n
+dmn(α)
+dα
+[un]∥ +
+1
+√n∥(I − Pn)dmn(α)
+dα
+[un]∥ = OP (ϕn).
+(c) kn supα∈Cn
+1
+n
+�
+t[dm(Xt,α)
+dα
+[un] − dm(Xt,α0)
+dα
+[un]]2 = oP (1).
+(d) conditions of Proposition A.1 hold.
+(e) supτ∈(0,1) supα∈Cn E
+�
+d2m(Xt,α0+τ(α−α0))
+dτ 2
+|
+�2
+= o(n−1) and
+(f) E supα∈Cn sup|τ|≤Cn−1/2 1
+n
+�
+t
+�
+d2
+dτ 2 m(Xt, α + τun)|
+�2
+= O(1).
+Then
+sup
+α∈Aosn
+sup
+|x|≤Cn−1/2 |Qn(α + xun) − Qn(α) − An(α(x))| = oP(n−1),
+where
+(a1) An(α(x)) := 2x[n−1/2Zn + ⟨un, α − α0⟩] + Bnx2
+(a2) Bn = 1
+n
+dmn(α0)
+dα
+[un]′Σ−1
+n
+dmn(α0)
+dα
+[un] →P 1, and
+(a3) Zn →d N(0, 1).
+40
+
+Proof. Let �Qn(α) = 1
+n
+�
+t ℓ(Xt, α)2�Σ(Xt)−1, and
+ℓ(x, α) := �m(x, α) + �m(x, α0),
+�m(x, α) := Ψ(x)′(Ψ′
+nΨn)−1Ψ′
+nmn(α).
+Step 1: expansions. By assumption, �Qn(α) is differentiable. So we shall prove the LQA for
+�Qn(α) via the mean value theorem, and show that �Qn(α)−Qn(α) is “small” locally. Indeed, Lemma
+C.1 shows that supα∈Cn |Qn(α) − �Qn(α)| = oP (n−1).
+We write f(s) := �Qn(α+sxun) and by the second order mean value theorem, for some s ∈ (0, 1),
+�Qn(α + xun) − �Qn(α) = f ′(0) + 1
+2f ′′(s) = 2xG(α) + x2Bx + x2Dx,
+G(α)
+:=
+1
+n
+�
+t
+ℓ(Xt, α)�Σ(Xt)−1 d �m(Xt, α)
+dα
+[un]
+Bx
+:=
+1
+n
+�
+t
+�d �m(α + sxun)
+dα
+[un]
+�2
+�Σ(Xt)−1
+Dx
+:=
+1
+n
+�
+t
+ℓ(α + sxun)�Σ(Xt)−1 d2
+dτ 2 �m(α + τxun)|τ=s.
+Lemma C.2 shows that uniformly Dx = oP (1). Hence sup|x|≤Cn−1/2 x2|Dx| = oP(n−1).
+Step 2: convergence of Bx. Let dmn(α)
+dα
+[un] and ρn be the n × 1 vectors of dm(Xt,α)
+dα
+[un] and
+ρ(Yt+1, α0). Also let ∥v∥2
+Σ := v′Σ−1
+n v. Write Bn := 1
+n∥dmn(α0)
+dα
+[un]∥2
+Σ = OP (1). Uniformly for α(x), s,
+Bx − Bn
+≤
+1
+n∥d �mn(α + sxun)
+dα
+[un]∥2
+�Σn − 1
+n∥d �mn(α0)
+dα
+[un]∥2
+�Σn
++ 1
+n∥d �mn(α0)
+dα
+[un]∥2
+�Σn − 1
+n∥dmn(α0)
+dα
+[un]∥2
+�Σn
++ 1
+n∥dmn(α0)
+dα
+[un]∥2
+�Σn − 1
+n∥dmn(α0)
+dα
+[un]∥2
+Σn = oP(1).
+Hence sup|x|≤Cn−1/2 |Bx − Bn|x2 = oP (n−1). To show that Bn →P 1, we have
+Bn = ⟨un, un⟩ +
+� 1
+n
+dmn(α0)
+dα
+[un]′Σ−1
+n
+dmn(α0)
+dα
+[un] − ⟨un, un⟩
+�
+= ⟨un, un⟩ + oP (1).
+Let Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt,α0)
+dα
+[v]. Then for each v,
+∥v∥2
+=
+Var( 1
+√n
+�
+t
+Zt) = Var(Zt) + 2
+n
+�
+s>t
+EZtE(Zs|σs(X)) = Var(Zt) = ⟨v, v⟩.
+Hence ⟨un, un⟩ = ⟨v∗
+n, v∗
+n⟩∥v∗
+n∥−2 = 1. Hence Bn = 1 + oP (1).
+Step 3: expansion of G(α). We have supα∈Cn
+1
+√n∥mn(α)′Pn∥+supα∈Cn
+1
+√n∥mn(α)∥ = OP (¯δn)
+41
+
+and (√n¯δn + √kn)∥�Σn − Σn∥ = o(1).
+G(α)
+=
+1
+nmn(α)′Pn�Σ−1
+n
+d �mn(α)
+dα
+[un] + 1
+nρ′
+nPn�Σ−1
+n
+d �mn(α)
+dα
+[un]
+=
+1
+nmn(α)′PnΣ−1
+n
+d �mn(α)
+dα
+[un] + 1
+nρ′
+nPnΣ−1
+n
+d �mn(α)
+dα
+[un] + oP (n−1/2)
+=
+1
+nmn(α)′PnΣ−1
+n
+dmn(α)
+dα
+[un] + 1
+nρ′
+nPnΣ−1
+n
+dmn(α)
+dα
+[un]
++ 1
+nmn(α)′PnΣ−1
+n (Pn − I)dmn(α)
+dα
+[un] + 1
+nρ′
+nPnΣ−1
+n (Pn − I)dmn(α)
+dα
+[un] + oP (n−1/2)
+=
+1
+nmn(α)′PnΣ−1
+n
+dmn(α)
+dα
+[un] + 1
+nρ′
+nPnΣ−1
+n
+dmn(α)
+dα
+[un] + OP (ϕn¯δn) + oP (n−1/2)
+=
+1
+nmn(α)′Σ−1
+n
+dmn(α)
+dα
+[un] + 1
+nρ′
+nPnΣ−1
+n
+dmn(α)
+dα
+[un] + oP (n−1/2)
+=(a)
+⟨un, α − α0⟩ + 1
+nρ′
+nPnΣ−1
+n
+dmn(α)
+dα
+[un] + oP(n−1/2)
+=(b)
+⟨un, α − α0⟩ + 1
+nρn(α0)′Σ−1
+n
+dmn(α0)
+dα
+[un]
+�
+��
+�
+1
+√nZn
++oP (n−1/2),
+where (a) follows from Lemma C.2; (b) is due to
+�
+Eρn(α0)′Pnρn(α0)
+�
+sup
+α∈Cn
+1
+n
+�
+t
+[dm(Xt, α)
+dα
+[un] − dm(Xt, α0)
+dα
+[un]]2
+≤
+�
+EtrPnΣ(Xt)−1
+�
+sup
+α∈Cn
+1
+n
+�
+t
+[dm(Xt, α)
+dα
+[un] − dm(Xt, α0)
+dα
+[un]]2 = oP (1).
+Step 4: weak convergence of Zn. It then remains to show Zn →d N(0, 1). Note that
+Zn =
+1
+√n
+�
+t
+Zt∥v∗
+n∥−1,
+Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt, α0)
+dα
+[v∗
+n],
+where un = v∗
+n/∥v∗
+n∥. When s > t, we have Zt ∈ σs(X). Hence E(ZtZs|σs(X)) = ZtE(Zs|σs(X)) = 0.
+Thus
+Var( 1
+√n
+�
+t
+Zt)
+=
+Var(Zt) + 2 1
+n
+�
+s>t
+EE(ZtZs|σs(X)) = Var(Zt)
+=
+E Var
+�
+ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt, α0)
+dα
+[v∗
+n]
+����σt(X)
+�
+=
+EΣ(Xt)−2
+�dm(Xt, α0)
+dα
+[v∗
+n]
+�2
+Var(ρ(Yt+1, α0)|σs(X))
+=
+⟨v∗
+n, v∗
+n⟩ = ∥v∗
+n∥2
+where we used Var(ρ(Yt+1, α0)|σs(X)) = Σ(Xt).
+42
+
+Next, it is assumed that there is some ζ > 0,
+E|Zt∥v∗
+n∥−1|2+ζ ≤ CE|ρ(Yt+1, α0)|2+ζ
+����
+dm(Xt, α0)
+dα
+[un]
+����
+2+ζ
+< ∞.
+In addition, Zt is strictly stationary, satisfying the β-mixing condition (Assumption 3.3). Let α(n)
+denote the α-mixing coefficient (the strong mixing coefficient). We have that, by Assumption 3.3,
+α(n) ≤ 1
+2β(n) ≤ C exp(−cn) for some c, C > 0. Hence
+∞
+�
+n=1
+α(n)ζ/(2+ζ) ≤ C
+∞
+�
+n=1
+exp(−cζn/(2 + ζ)) < ∞.
+Then by Theorem 1.7 of Ibragimov (1962), Zn →d N(0, 1).
+Lemma C.1. Let �Qn(α) = 1
+n
+�
+t ℓ(Xt, α)2�Σ(Xt)−1, and
+ℓ(x, α) := �m(x, α) + �m(x, α0),
+�m(x, α) := Ψ(Xt)′(Ψ′
+nΨn)−1Ψ′
+nmn(α).
+Suppose knd2
+nδ2η
+n + √kndnδη
+n¯δn = o(n−1) and
+1
+√n∥mn(πnα) − mn(α)∥¯δn ≤ o(n−1).
+Then for Cn = {α + xun : α ∈ An : ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤ C¯δ2
+n, |x| ≤ Cn−1/2},
+(i)
+sup
+α∈Cn
+|Qn(α) − �Qn(α)| = oP (n−1).
+(ii)
+sup
+α∈Cn
+|Qn(α) − Qn(πnα)| = oP(n−1).
+Proof. (i) Recall that ǫt(α) = ρ(Yt+1, α) − m(Xt, α) and mn(α), ¯ǫn(α) and ρn(α) are n × 1 vectors
+of m(Xt, α), ǫt(α) and ρ(Yt+1, α). Also write α(x) := α + xun.
+Qn(α + xun) − �Qn(α + xun) = 1
+n
+�
+t
+[ �m(Xt, α(x))2 − ℓ(Xt, α(x))2]�Σ(Xt)−1
+=
+1
+n[¯ǫn(α + xun) − ¯ǫn(α0)]′Pn�Σ−1
+n Pn[¯ǫn(α + xun) − ¯ǫn(α0) + 2mn(α + xun) + 2ρn(α0)]
+≤
+OP (1) 1
+n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥2 + OP (1) 1
+n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥∥Pnmn(α + xun)∥
++OP(1) 1
+n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥∥Pnρn(α0)∥
+≤
+OP (d2
+1 + d1 × d2 + d1 × d3)
+d1
+:=
+1
+√n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥
+d2
+:=
+1
+√n∥Pnmn(α + xun)∥,
+d3 :=
+1
+√n∥Pnρn(α0)∥.
+43
+
+We shall respectively calculate d1 ∼ d3. By Proposition A.1, d1 = OP (√kndnδη
+n) uniformly in α(x).
+As for d2, by steps 1 and 3 in the proof of Theorem 3.1, uniformly in α(x),
+d2
+2 ≤ 1
+n
+�
+t
+�m(Xt, α(x))2 ≤ CE �m(Xt, α(x))2 ≤ C(ϕ2
+n + Em(Xt, α(x))2) ≤ C¯δ2
+n.
+Finally, d2
+3 = OP (kn
+n ). Together, Qn(α+xun)− �Qn(α+xun) ≤ OP (knd2
+nδ2η
+n +√kndnδη
+n¯δn) = oP (n−1).
+(ii) Let mn(α) and �mn(α) respectively be the n × 1 vectors of m(Xt, α) and �m(Xt, α). First,
+1
+√n∥ �mn(πnα) − �mn(α)∥ ≤
+1
+√n∥mn(πnα) − mn(α)∥ ≤ OP (µn). Second, 1
+n∥ �mn(α)∥2 ≤ OP (¯δ2
+n). Third,
+1
+√n∥ �mn(α0)∥ = OP (1)
+�
+1
+nρn(α0)′Pnρn(α0) = OP (
+�
+kn
+n ).
+Hence for �Qn(α) = 1
+n[ �mn(α) + �mn(α0)]′�Σ−1
+n [ �mn(α) + �mn(α0)], we have
+�Qn(α) − �Qn(πnα)
+≤
+OP (1) 1
+√n∥ �mn(πnα) − �mn(α)∥
+� 1
+√n∥ �mn(πnα) − �mn(α)∥ +
+1
+√n∥ �mn(α)∥ + 1
+√n∥ �mn(α0)∥
+�
+≤
+OP (µn¯δn) = o(n−1).
+Finally, by part (i) | �Qn(α) − Qn(α)| = oP(n−1) uniformly in α.
+Lemma C.2. Suppose supτ∈(0,1) supα∈Cn E
+�
+d2m(Xt,α0+τ(α−α0))
+dτ 2
+|
+�2
+= o(n−1) and
+E supα∈Cn sup|τ|≤Cn−1/2 1
+n
+�
+t
+�
+d2
+dτ 2 m(Xt, α + τun)|
+�2
+= O(1). Then uniformly for α ∈ Cn,
+(i) supα∈Cn sup|s|≤1,|x|≤Cn−1/2
+��� 1
+n
+�
+t ℓ(Xt, α + sxun)�Σ(Xt)−1 d2
+dτ 2 �m(Xt, α + τxun)|τ=s
+��� = oP (1).
+(ii) supα∈Cn
+√n| 1
+nmn(α)′Σ−1
+n
+dmn(α0)
+dα
+[un] − ⟨un, α − α0⟩| = oP (1).
+Proof. (i) We have that
+��� 1
+n
+�
+t ℓ(Xt, α + sxun)�Σ(Xt)−1 d2
+dτ 2 �m(Xt, α + τxun)|τ=s
+���
+2
+≤ OP (1)AB where
+A := 1
+n
+�
+t
+ℓ(Xt, α + sxun)2,
+B := 1
+n
+�
+t
+d2
+dτ 2 �m(Xt, α + τxun)|2
+τ=s.
+Let mn and ρn denote the n × 1 vectors of m(Xt, ·) and ρ(Yt+1, α0). Uniformly for α ∈ Cn,
+A ≤ 2
+n∥Pnmn(α + sxun)∥2 + 2
+n∥Pnρn∥2 = oP (1).
+We have B ≤ OP (1)E supα∈Cn sup|τ|≤Cn−1/2 | d2
+dτ 2 m(Xt, α + τun)|2 = OP (1).
+44
+
+(ii) By the second order mean value theorem, for some ξ ∈ (0, 1),
+1
+nmn(α)′Σ−1
+n
+dmn(α0)
+dα
+[un] = 1
+n
+�
+t
+[m(Xt, α) − m(Xt, α0)]Σ(Xt)−1 dm(Xt, α0)
+dα
+[un]
+=
+1
+n
+�
+t
+f(Xt) − Ef(Xt) + E[m(Xt, α) − m(Xt, α0)]Σ(Xt)−1 dm(Xt, α0)
+dα
+[un]
+=
+1
+n
+�
+t
+f(Xt) − Ef(Xt) + Edm(Xt, α0)
+dα
+[α − α0]Σ(Xt)−1 dm(Xt, α0)
+dα
+[un]
++1
+2Ed2m(Xt, α0 + τ(α − α0))
+dτ 2
+|τ=ξΣ(Xt)−1 dm(Xt, α0)
+dα
+[un]
+=
+1
+n
+�
+t
+f(Xt) − Ef(Xt) + ⟨un, α − α0⟩ + o(n−1/2) = ⟨un, α − α0⟩ + oP (n−1/2)
+where f(Xt) = [m(Xt, α) − m(Xt, α0)]Σ(Xt)−1 dm(Xt,α0)
+dα
+[un] and the last equality follows from
+sup
+f∈En
+| 1
+√n
+�
+t
+(f(Xt) − Ef(Xt))| = oP (1)
+(C.1)
+with En := {m(Xt, α)Σ(Xt)−1 dm(Xt,α0)
+dα
+[un] : α ∈ Cn} and that m(Xt, α0) = 0.
+C.2
+Proof of Theorem 4.1
+Proof. By the Riesz representation Theorem, there is v∗
+n ∈ cl{An − α0}
+dφ(α0)
+dα
+[�α − α0] = ⟨v∗
+n, �α − α0⟩.
+Next, we show √n⟨un, �α − α0⟩ →d N(0, 1), or more precisely, for Zn →d N(0, 1),
+Zn + √n⟨un, �α − α0⟩ = oP (1).
+(C.2)
+The proof of Theorem 3.1 implies that for any ǫ > 0, there is C > 0 so that with probability at least
+1 − ǫ, �α ∈ Aosn := {α ∈ An : Q(α) ≤ C¯δ2
+n, ∥α − α0∥∞,ω ≤ Cδn}. We now condition on this event.
+By Proposition C.1,
+sup
+α∈Aosn
+sup
+|x|≤Cn−1/2 |Qn(α + xun) − Qn(α) − An(α(x))| = oP (n−1),
+(C.3)
+where An(α(x)) := 2x[n−1/2Zn + ⟨un, α − α0⟩] + Bnx2 with Bn = OP (1), Zn →d N(0, 1). Write
+un = (uγ, uh). Now let ∆n be such that sup|x|≤Cn−1/2 |Pen(πn(�h + xuh)) − Pen(�h)| = OP (∆n). Then
+En := λnPen(πn(�h + xuh)) − λnPen(�h) = OP (λn∆n).
+45
+
+Now by definition, πn(�α + xun) ∈ An, hence
+0
+≤
+Qn(πn(�α + xun)) − Qn(�α) + En
+≤
+Qn(�α + xun) − Qn(�α) + En + |Qn(�α + xun) − Qn(πn(�α + xun))|
+≤
+Qn(�α + xun) − Qn(�α) + En + oP (n−1)
+≤
+2x[n−1/2Zn + ⟨un, �α − α0⟩] + Bnx2 + En + oP (n−1),
+where the third inequality follows from Lemma C.1 and the last inequality follows from (C.3).
+By the assumption λn∆n = oP (n−1). Hence there is ηn = o(n−1), so that
+0 ≤ x[n−1/2Zn + ⟨un, �α − α0⟩] + Bnx2 + OP (ηn).
+From n1/2ηn = o(n−1/2), we can find ǫn → 0+ so that n1/2ηn ≪ ǫn ≪ n−1/2. Set x ∈ {ǫn, −ǫn}.
+Multiply by (2ǫn)−1n1/2 on both sides,
+−1
+2
+√nBnǫn ≤ Zn + √n⟨un, �α − α0⟩ + OP (ηnǫ−1
+n n1/2) ≤ 1
+2
+√nBnǫn.
+We have ηnǫ−1
+n n1/2 + √nBnǫn = oP (1). Therefore we reach Zn + √n⟨un, �α − α0⟩ = oP(1), which
+implies √n⟨un, �α − α0⟩ = −Zn + oP (1) →d N(0, 1).
+Finally, let ζn = ∥v∗
+n∥n−1/2. Apply Assumption 4.1 with α = �α and un = v∗
+n/∥v∗
+n∥,
+ζ−1
+n (φ(�α) − φ(α0))
+=
+ζ−1
+n
+dφ(α0)
+dα
+[�α − α0] + oP (1)
+=
+ζ−1
+n
+dφ(α0)
+dα
+[�α − α0,n] + ζ−1
+n
+dφ(α0)
+dα
+[α0,n − α0] + oP (1)
+=
+√n⟨un, �α − α0,n⟩ + oP (1)
+=
+√n⟨un, �α − α0⟩ + oP(1) →d N(0, 1).
+where in the last equality we used √n⟨un, α0,n − α0⟩ = 0 because α0,n is the projection (under ∥.∥)
+of α0 onto span{An} and un ∈span{An}.
+C.3
+Proof of Theorem 4.2
+Proof. We divide the proof in the following steps.
+Step 1: decompose �γ. Write σ2 := Var
+�
+1
+√n
+�
+t Wt
+�
++ ∥v∗
+n∥2, which will be shown to be
+the asymptotic variance. Also write bn(α) and ¯bn(α) respectively as the n × 1 vectors of b(St, α) :=
+46
+
+l(h(Wt))ρ(Yt+1, α) and ¯b(Xt, α) := E(l(h(Wt))ρ(Yt+1, α)|σt(X)). Then
+�γ − γ
+=
+[φn(α0) − φ(α0)] + [φ(�α) − φ(α0)] + 1
+n
+n
+�
+t=1
+(Γ(Xt) − �Γt)ρ(Yt+1, �α) + a1,
+a1
+:=
+φn(�α) − φ(�α) − [φn(α0) − φ(α0)]
+φn(α)
+=
+1
+n
+�
+t
+l(h(Wt)) − Γ(Xt)ρ(Yt+1, α)
+φ(α)
+=
+Eφn(α).
+Bounding a1 is based on the stochastic equicontinuity of φn − φ, established in Lemma C.3, which
+yields a1 = OP (dnδη
+n) = oP (σn−1/2) by the assumption that dnδη
+n = o(σn−1/2).
+Step 2: decompose �Γ(Xt).
+We have Γ(Xt) = ¯b(Xt, α0)Σ(Xt)−1. Let �α ∈ Cn denote the
+estimated α0 used in defining �Γt. Then
+�Γt = Ψ(Xt)′(Ψ′
+nΨn)−1Ψ′
+nbn(�α)�Σ(Xt)−1.
+We then achieve the following decomposition:
+1
+n
+�n
+t=1(�Γt − Γ(Xt))ρ(Yt+1, �α) = 1
+nbn(�α)′Pn�Σ−1
+n ρn(�α) − 1
+n¯bn(α0)′Σ−1
+n ρn(�α) = a2 + ... + a8 where
+a2
+:=
+1
+n[bn(�α) − ¯bn(�α)]′Pn�Σ−1
+n ρn(�α)
+a3
+:=
+1
+n[¯bn(�α) − ¯bn(α0)]′Pn�Σ−1
+n ρn(�α)
+a4
+:=
+1
+n
+¯bn(α0)′(Pn − I)(�Σ−1
+n − Σ−1
+n )ρn(�α)
+a5
+:=
+1
+n
+¯bn(α0)′(Pn − I)Σ−1
+n (ρn(�α) − mn(�α))
+a6
+:=
+1
+n
+¯bn(α0)′(Pn − I)Σ−1
+n mn(�α)
+a7
+:=
+1
+n
+¯bn(α0)′(�Σ−1
+n − Σ−1
+n )(ρn(�α) − ρn(α0))
+a8
+:=
+1
+n
+¯bn(α0)′(�Σ−1
+n − Σ−1
+n )ρn(α0)
+(C.4)
+Lemma C.3 shows a2 + ... + a7 = OP (δη
+n supx |�Σ(x) − Σ(x)| + √kndnδη
+n + ϕ2
+n), which is oP(σn−1/2).
+The bound for a8 = oP (σn−1/2) is from Assumption 4.6. Hence
+1
+n
+n
+�
+t=1
+(�Γt − Γ(Xt))ρ(Yt+1, �α) = oP (σn−1/2).
+Step 3: Complete proofs. By the same proof of that of Theorem 4.1,
+φ(�α) − φ(α0)
+=
+∥v∗
+n∥⟨un, �α − α0⟩ + oP(∥v∗
+n∥n−1/2)
+=
+−∥v∗
+n∥n−1/2Zn + oP (∥v∗
+n∥n−1/2)
+47
+
+=
+− 1
+n
+�
+t
+Zt + oP(∥v∗
+n∥n−1/2)
+Zt
+:=
+ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt, α0)
+dα
+[v∗
+n]
+φn(α0) − φ(α0)
+=
+1
+n
+n
+�
+t=1
+Wt − EWt,
+Wt := l(h0(Wt)) − Γ(Xt)ρ(Yt+1, α0).
+Putting together, �γ − γ = [φn(α0) − φ(α0)] + [φ(�α) − φ(α0)] + oP (σn−1/2), and
+[φn(α0) − φ(α0)] + [φ(�α) − φ(α0)] = 1
+n
+n
+�
+t=1
+Wt − EWt − Zt + oP (σn−1/2).
+Next, Wt − EWt − Zt is strictly stationary, satisfying the strong mixing condition (Assumption
+3.3) with �∞
+n=1 α(n)ζ/(2+ζ) ≤ C �∞
+n=1 exp(−cζn/(2 + ζ)) < ∞ for any constant ζ > 0. In addition,
+E
+��(Wt − EWt − Zt)σ−1��2+ζ ≤ CE
+��Wtσ−1��2+ζ + CE
+��Zt∥v∗
+n∥−1��2+ζ
+≤
+CE
+����ρ(Yt+1, α0)dm(Xt, α0)
+dα
+[un]
+����
+2+ζ
++ CE |ρ(Yt+1, α0)|2+ζ < C.
+Then by Theorem 1.7 of Ibragimov (1962),
+√nσ−1 [φn(α0) − φ(α0) + φ(�α) − φ(α0)] → N(0, 1).
+(C.5)
+This implies the asymptotic normality of �γ − γ.
+Lemma C.3 (for Theorems 4.2, 5.2). Recall that bn(α) and ¯bn(α) are the n × 1 vectors of
+l(h(Wt))ρ(Yt+1, α) and E(l(h(Wt))ρ(Yt+1, α)|σt(X)). Suppose
+(a) supx |Γ(x)|2 + supw supHn l(h(w))2 < C.
+(b) |l(h1(w)) − l(h2(w))| ≤ C|h1(w) − h2(w)| uniformly for all h1, h2 ∈ Hn,and w.
+(c) E supα∈Cn |l(h(Wt)) − l(h0(Wt))|2 ≤ Cδ2η
+n .
+(d) E supα∈Cn(ρ(Yt+1, α) − ρ(Yt+1, α0))2 = Cδ2η
+n for some η, C > 0.
+(e) For some κ, C > 0 , E sup∥α1−α∥∞,ω<δ |ǫt(α1) − ǫt(α)|2 ≤ Cδ2κ for all δ > 0 and α, α1 ∈
+cl{a + xb : a, b ∈ An, x ∈ R}.
+(f)
+1
+√n∥¯bn(α0)′(Pn − I)∥ = OP (ϕn).
+Then for ¯ǫn(α) as the n × 1 vector of ρ(Yt+1, α) − m(Xt, α),
+48
+
+(i) supα1,α2∈Cn∪{α0} |φn(α1) − φ(α1) − [φn(α2) − φ(α2)]| = OP (dnδη
+n).
+(ii) supCn | 1
+n¯bn(α0)′(�Σ−1
+n − Σ−1
+n )[ρn(α) − ρn(α0)]| = OP (δη
+n) supx |�Σ(x) − Σ(x)|.
+(iii) supCn
+1
+√n∥Pn�Σ−1
+n ρn(α)∥ = OP (supx ∥�Σ(x) − Σ(x)∥ + dnδη
+n + ¯δn).
+(iv) supCn
+1
+n¯bn(α0)′(Pn − I)Σ−1
+n ¯ǫn(α) = OP (√kndnδη
+n).
+(v) supCn
+1
+n[bn(α) − ¯bn(α)]′Pn�Σ−1
+n ρn(α) + supCn
+1
+n[¯bn(α) − ¯bn(α0)]′Pn�Σ−1
+n ρn(α)
+= OP (δη
+n supx ∥�Σ(x) − Σ(x)∥ + dnδ2η
+n + √kndnδη
+n¯δn +
+�
+kn/n¯δn).
+(vi) supCn
+1
+n¯bn(α0)′(Pn − I)(�Σ−1
+n − Σ−1
+n )ρn(α) = OP (ϕn supx ∥�Σ(x) − Σ(x)∥)
+(vii) supCn
+1
+n¯bn(α0)′(Pn − I)Σ−1
+n mn(α) = OP (ϕ2
+n).
+Proof. (i) First recall ǫ(St, α) = ρ(Yt+1, α) − m(Xt, α).
+a
+:=
+sup
+α1,α2∈Cn∪{α0}
+| 1
+n
+n
+�
+t=1
+Γ(Xt)[ρ(Yt+1, α1) − ρ(Yt+1, α2)] − EΓ(Xt)[ρ(Yt+1, α1) − ρ(Yt+1, α2)]|
+=
+sup
+α1,α2∈Cn∪{α0}
+| 1
+n
+n
+�
+t=1
+Γ(Xt)[ǫ(St, α1) − ǫ(St, α2)]|
+≤
+2
+sup
+α∈Cn∪{α0}
+| 1
+n
+n
+�
+t=1
+Γ(Xt)[ǫ(St, α) − ǫ(St, α0)]|.
+b
+:=
+sup
+α1,α2∈Cn∪{α0}
+| 1
+n
+n
+�
+t=1
+l(h1(Wt)) − l(h2(Wt)) − E[l(h1(Wt)) − l(h2(Wt))]|
+≤
+2
+sup
+α∈Cn∪{α0}
+| 1
+n
+n
+�
+t=1
+l(h(Wt)) − l(h0(Wt)) − E[l(h(Wt)) − l(h0(Wt))]|.
+Note E supα∈Cn Γ(Xt)2[ǫ(St, α) − ǫ(St, α0)]2 ≤ CE supα∈Cn[ǫ(St, α) − ǫ(St, α0)]2 ≤ Cδ2η
+n , η ≤ 1.
+Then the convergence of a and b follow from the same argument of that of Proposition A.1 with
+Ψj(Xt) replaced with Γ(Xt). Term b follows from the same proof of this Proposition. We reach
+a + b = OP (dnδη
+n).
+Therefore supα1,α2∈Cn∪{α0} |φn(α1) − φ(α1) − [φn(α2) − φ(α2)]| ≤ a + b = OP (dnδη
+n).
+(ii) First, E supα∈Cn[ρ(Yt+1, α) − ρ(Yt+1, α0)]2 ≤ O(δ2η
+n ). This implies
+1
+√n∥ρn(α) − ρn(α0)∥ =
+OP (δη
+n). The target of interest is then bounded by
+∥�Σn − Σn∥ 1
+√n∥ρn(α) − ρn(α0)∥ = OP (δη
+n) sup
+x ∥�Σ(x) − Σ(x)∥.
+(iii) First, write �mΣ(Xt, α) := Ψ(Xt)′(Ψ′
+nΨn)−1Ψ′
+nΣ−1
+n mn(α).
+Then step 1 of the proof of
+49
+
+Theorem 3.1 carries over, leading to
+sup
+Cn
+1
+n∥PnΣ−1
+n mn(α)∥2 = sup
+Cn
+1
+n
+�
+t
+�mΣ(Xt, α)2 ≤ C sup
+Cn
+E �mΣ(Xt, α)2
+≤
+C sup
+Cn
+E[ �mΣ(Xt, α) − m(Xt, α)Σ(Xt)−1]2 + C sup
+Cn
+Em(Xt, α)2 = OP (¯δ2
+n).
+Also, for ¯ǫn(α) := ρn(α)−mn(α), the first inequality below follows from the same proof of Proposition
+A.1,
+sup
+Cn
+1
+√n∥PnΣ−1
+n (¯ǫn(α) − ¯ǫn(α0))∥ ≤ OP (dnδη
+n)
+sup
+Cn
+1
+√n∥Pn(�Σ−1
+n − Σ−1
+n )(ρn(α) − ρn(α0))∥ ≤ OP (δη
+n) sup
+x ∥�Σ(x) − Σ(x)∥
+sup
+Cn
+1
+√n∥PnΣ−1
+n (ρn(α) − ρn(α0))∥ ≤ sup
+Cn
+1
+√n∥PnΣ−1
+n (¯ǫn(α) − ¯ǫn(α0))∥ + sup
+Cn
+1
+√n∥PnΣ−1
+n mn(α)∥
+≤
+OP (dnδη
+n + ¯δn)
+sup
+Cn
+1
+√n∥Pn�Σ−1
+n (ρn(α) − ρn(α0))∥
+≤
+sup
+Cn
+1
+√n∥Pn(�Σ−1
+n − Σ−1
+n )(ρn(α) − ρn(α0))∥ + sup
+Cn
+1
+√n∥PnΣ−1
+n (ρn(α) − ρn(α0))∥
+≤
+OP (δη
+n) sup
+x ∥�Σ(x) − Σ(x)∥ + OP (dnδη
+n + ¯δn)
+1
+√n∥Pn�Σ−1
+n ρn(α0)∥ = OP (1) sup
+x ∥�Σ(x) − Σ(x)∥ + OP (
+�
+kn/n).
+Together, supCn
+1
+√n∥Pn�Σ−1
+n ρn(α)∥ ≤ supCn
+1
+√n∥Pn�Σ−1
+n (ρn(α) − ρn(α0))∥ +
+1
+√n∥Pn�Σ−1
+n ρn(α0)∥ whose
+final rate is OP (supx ∥�Σ(x) − Σ(x)∥ + dnδη
+n + ¯δn).
+(iv) First, 1
+n¯bn(α0)′(Pn −I)Σ−1
+n ¯ǫn(α0) = OP (ϕnn−1/2). Next, from the proof of Proposition A.1,
+sup
+Cn
+1
+n
+¯bn(α0)′(Pn − I)Σ−1
+n [¯ǫn(α) − ¯ǫn(α0)]
+≤
+sup
+Cn
+1
+n∥Ψ′
+nΣ−1
+n [¯ǫn(α) − ¯ǫn(α0)]∥ + sup
+Cn
+1
+n
+¯bn(α0)′Σ−1
+n [¯ǫn(α) − ¯ǫn(α0)] = OP (
+�
+kndnδη
+n).
+So supCn
+1
+n¯bn(α0)′(Pn − I)Σ−1
+n ¯ǫn(α) = OP (√kndnδη
+n + ϕnn−1/2) = OP (√kndnδη
+n).
+(v) The same proof of Proposition A.1 carries over to here. So
+sup
+α∈Cn
+1
+n∥Ψ′
+n(¯bn(α) − bn(α)) − Ψ′
+n(¯bn(α0) − bn(α0))∥ = OP (
+�
+kndnδη
+n).
+In addition, 1
+n∥Ψ′
+n(¯bn(α0) − bn(α0))∥ = OP (√knn−1/2). This implies
+sup
+Cn
+1
+√n[bn(α) − ¯bn(α)]′Pn ≤ OP (1) sup
+α∈Cn
+1
+n∥Ψ′
+n(¯bn(α) − bn(α))∥ = OP (dnδη
+n + n−1/2)
+�
+kn.
+50
+
+Also,
+1
+n∥¯bn(α) − ¯bn(α0)∥2
+≤
+OP (1)E sup
+Cn
+[E|ρ(Yt+1, α) − ρ(Yt+1, α0)||σt(X)]2 + OP (1) sup
+Cn
+E|l(h) − l(h0)|2
+≤
+OP (δ2η
+n ).
+Hence supCn
+1
+n[bn(α)−¯bn(α)]′Pn�Σ−1
+n ρn(α) = OP (√kndnδη
+n+
+�
+kn/n)(supx ∥�Σ(x)−Σ(x)∥+dnδη
+n +¯δn)
+and supCn
+1
+n[¯bn(α) − ¯bn(α0)]′Pn�Σ−1
+n ρn(α) = OP (supx ∥�Σ(x) − Σ(x)∥δη
+n + dnδ2η
+n + ¯δnδη
+n). So the
+final rate of the sum of the two is δη
+n supx ∥�Σ(x) − Σ(x)∥ + dnδ2η
+n + √kndnδη
+n¯δn +
+�
+kn/n¯δn).
+(vi) (vii) The proof is straightforward.
+D
+Proofs for Section 5
+D.1
+Proof of Theorem 5.1
+Proof. Proposition C.1 shows the following LQA:
+sup
+α∈Cn
+sup
+|x|≤Cn−1/2 |Qn(α + xun) − Qn(α) − An(α(x))| = oP (n−1)
+(D.1)
+where An(α(x)) := 2x[n−1/2Zn + ⟨un, α − α0⟩] + Bnx2 with Bn = 1 + oP(1), Zn →d N(0, 1). We
+respectively provide lower and upper bounds for Qn(�αR) − Qn(�α).
+Step 1: lower bound. To apply the LQA, we need to first show that �αR ∈ Cn with a high
+probability. In fact, there is πR
+n α0 ∈ AR
+n so that
+Qn(�αR) + λnPen(�hR) ≤ Qn(πR
+n α0) + λnPen(πR
+n h0).
+Given the above inequalities, the proof of Theorem 3.1 carries over, establishing that �αR ∈ Cn with
+a high probability. We now condition on this event. Hence by (D.1), uniformly for all |x| ≤ Cn−1/2,
+Qn(�αR + xun) − Qn(�αR)
+=
+2x[n−1/2Zn + ⟨un, �αR − α0⟩] + Bnx2 + oP(n−1)
+=
+2xn−1/2Zn + Bnx2 + oP (n−1),
+(D.2)
+where the second equality follows from Lemma D.1. Next, we note one technical difficulty that the
+inequality Qn(�α) + λnPen(�α) ≤ Qn(α) + λnPen(α) may not hold for α = �αR + xun, as An is a
+nonlinear space so �αR + xun is not necessarily in An. Nevertheless, we can apply this inequality for
+α = πn(�αR + xun), and show that |Qn(πn(�αR + xun)) − Qn(�αR + xun)| is negligible. Specifically, by
+51
+
+Lemma D.1 and Assumption 4.3,
+Qn(�α) − Qn(�αR + xun)
+≤
+λnPen(πn(�αR + xun)) − λnPen(�α) + Qn(πn(�αR + xun)) − Qn(�αR + xun)
+≤
+oP(n−1).
+(D.3)
+Thus (D.2) and (D.3) imply Qn(�αR)−Qn(�α) ≥ −2xn−1/2Zn−Bnx2−oP (n−1). Take x = −ZnB−1
+n n−1/2
+which maximizes −2xn−1/2Zn − Bnx2, then
+Qn(�αR) − Qn(�α) ≥ Z2
+nB−1
+n n−1 − oP(n−1).
+Step 2: upper bound. Fix x∗ determined as in Lemma D.1, this lemma shows that x∗ =
+n−1/2ZnB−1
+n
++ oP (n−1/2) and that |Qn(πR
+n (�α + x∗un)) − Qn(�α + x∗un)| = oP (n−1). Hence by (D.1)
+again,
+Qn(�αR) − Qn(�α)
+≤
+Qn(πR
+n (�α + x∗un)) − Qn(�α) + λn(Pen(πR
+n (�α + x∗un)) − Pen(�αR))
+=
+Qn(�α + x∗un) − Qn(�α) + oP (n−1)
+=
+2x∗n−1/2[Zn + n1/2⟨un, �α − α0⟩] + Bnx∗2 + oP (n−1)
+=
+Bnx∗2 + oP (n−1) = Z2
+nB−1
+n n−1 + oP (n−1),
+where the third equality is due to the proof of Theorem 4.1 that Zn + √n⟨un, �α − α0⟩ = oP (1).
+Step 3: matching bounds.
+Together, we have
+Sn(φ0) = n(Qn(�αR) − Qn(�α)) = B−1
+n Z2
+n + oP (1) →d χ2
+1
+given that Bn →P 1 proved in Proposition C.1.
+Lemma D.1 (for Theorem 5.1). Suppose
+(a) supα∈Cn
+1
+n
+�n
+t=1[m(Xt, πnα) − m(Xt, α)]2 = OP (µ2
+n) and
+supα∈Cn,φ(α)=φ0
+1
+n
+�n
+t=1[m(Xt, πR
+n (α + xun)) − m(Xt, α + xun)]2 = OP (µ2
+n).
+(b) µn¯δn = o(n−1).
+(c) t → φ(α + tun) is continuous.
+Then
+(i) ⟨un, �αR − α0⟩ = oP (n−1/2).
+(ii) sup|x|≤Cn−1/2 |Qn(πn(�αR + xun)) − Qn(�αR + xun)| = oP (n−1).
+52
+
+(iii) there is x∗ so that φ(�α + x∗un) = φ0 and |Qn(πR
+n (�α + x∗un)) − Qn(�α + x∗un)| = oP(n−1)
+and x∗ = n−1/2ZnB−1
+n
++ oP(n−1/2).
+Proof. (i) Note that φ(�αR) − φ(α0) = 0. By Assumption 4.1,
+���dφ(α0)
+dα
+[�αR − α0]
+��� = o(∥v∗
+n∥n−1/2). By
+the Riesz representation Theorem,
+dφ(α0)
+dα
+[�αR − α0] = dφ(α0)
+dα
+[�αR − α0,n] + dφ(α0)
+dα
+[α0,n − α0] = ∥v∗
+n∥⟨un, �αR − α0⟩ + o(∥v∗
+n∥n−1/2)
+with the definition un = v∗
+n/∥v∗
+n∥. This finishes the proof.
+(ii) The proof is the same as that of Lemma C.1.
+(iii) First, we prove there is x∗ so that φ(�α+x∗un) = φ0. Define F(x) := ⟨v∗
+n, α−α0⟩+x∥v∗
+n∥. Also
+define R(x) := φ(α+xun)−φ(α0). By Assumption 4.1, there is a positive sequence bn = o(∥v∗
+n∥n−1/2),
+uniformly for all α ∈ Cn, for all x ≤ Cn−1/2,
+|R(x) − F(x)| ≤ bn
+Now fix some r such that |r−⟨v∗
+n, α−α0⟩| ≤ C∥v∗
+n∥n−1/2 and define x1 = (r−⟨v∗
+n, α−α0⟩−2bn)∥v∗
+n∥−1
+and x2 = (r − ⟨v∗
+n, α − α0⟩ + 2bn)∥v∗
+n∥−1. This ensures that F(x1) + 2bn = F(x2) − 2bn = r and
+|x1| + |x2| ≤ Cn−1/2. Therefore,
+R(x1) ≤ F(x1) + bn < r,
+R(x2) ≥ F(x2) − bn > r.
+Hence there is x∗ between x1, x2 so that R(x∗) = r.
+In the above proof, suppose r = 0 and
+α = �α are admitted, then φ(�α + x∗un) = φ(α0). To show the admissibility, we note (C.2) that
+n−1/2Zn + ∥v∗
+n∥−1⟨v∗
+n, �α − α0⟩ = oP (n−1/2). Hence indeed, for any ǫ > 0, there is C > 0,
+|⟨v∗
+n, α − α0⟩| = ∥v∗
+n∥n−1/2|Zn| + oP (∥v∗
+n∥n−1/2) ≤ C∥v∗
+n∥n−1/2
+with probability at least 1 − ǫ.
+Now |x∗ − n−1/2Zn| ≤ |x1 − n−1/2Zn| + |x2 − n−1/2Zn| ≤ 2|bn|
+∥v∗n∥ + oP (n−1/2) = oP (n−1/2).
+Finally, the proof of |Qn(πR
+n (�α + x∗un)) − Qn(�α + x∗un)| = oP (n−1) is the same as part (ii).
+53
+
+D.2
+Proof of Theorem 5.2
+Proof. As in the proof of Theorem 5.1, we respectively provide lower and upper bounds for 1
+n �Sn(φ0) =
+Ln(�αR, φ0) − Ln(�α, �γ). Note that Ln(�α, �γ) = Qn(�α). Let
+g1
+=
+(�φ(�αR) − γ0)2�Σ−1
+2
+g2
+=
+φ(�αR) − φ(α0)
+g3
+=
+[φn(α0) − φ(α0)] + [φ(�α) − γ0]
+g4
+=
+[φn(α0) − γ0 + g2]2�Σ−1
+2
+g5
+=
+φn(α0) − γ0 − ∥v∗
+n∥n−1/2Zn
+g6
+=
+n−1/2Zn + ∥v∗
+n∥−1g2
+Also note that �αR ∈ Cn with a high probability, by Lemma D.2. We now condition on this event.
+Step 1: lower bound. Due to λnPen(�αR + xun) − λnPen(�α) = oP (n−1) and �αR ∈ An, so
+uniformly for all |x| ≤ Cn−1/2,
+Ln(�α, �γ) − Ln(�αR, φ0) = Qn(�α) − Qn(�αR) − g1 + oP (n−1)
+≤(a)
+Qn(πn(�αR + xun)) − Qn(�αR + xun) + Qn(�αR + xun) − Qn(�αR) − g1 + oP (n−1)
+=(b)
+Qn(�αR + xun) − Qn(�αR) − g1 + oP (n−1)
+=(c)
+2x[n−1/2Zn + ⟨un, �αR − α0⟩] + x2 − g1 + oP (n−1),
+=(d)
+2x[n−1/2Zn + ∥v∗
+n∥−1 dφ(α0)
+dα
+[�αR − α0]] + x2 − g1 + oP (n−1),
+=(e)
+2x[n−1/2Zn + ∥v∗
+n∥−1g2] + x2 − g1 + oP (n−1)
+=(f)
+2x[n−1/2Zn + ∥v∗
+n∥−1g2] + x2 − g4
+�
+��
+�
+F (x)
++oP(n−1),
+where in (a) we used Qn(�α) ≤ Qn(πn(�αR + xun)); (b) follows from |Qn(πn(�αR + xun)) − Qn(�αR +
+xun)| ≤ oP (n−1) following the same proof of that of Lemma D.1(ii); (c) is from (D.1); (d) is from
+the Riesz representation: (⟨v∗
+n, α0 − α0,n⟩ = 0)
+dφ(α0)
+dα
+[�αR − α0] = dφ(α0)
+dα
+[�αR − α0,n] + dφ(α0)
+dα
+[α0,n − α0] = ⟨v∗
+n, �αR − α0,n⟩ + oP (n−1/2∥v∗
+n∥);
+(e) is from Assumption 4.1; (f) is from Lemma D.3.
+We choose x = x∗ to minimize F(x) on the right hand side, leading to the choice x∗ =
+−[n−1/2Zn + ∥v∗
+n∥−1g2] = −g6. We shall verify that |x∗| = OP (n−1/2) in Step 3 below. Suppose for
+now this is true, then we have obtained the lower bound: 1
+n �Sn(φ0) ≥ −F(x∗) − oP(n−1), where
+−F(x∗) = [n−1/2Zn + ∥v∗
+n∥−1g2]2 + g4 = g2
+6 + g4.
+54
+
+Step 2: upper bound. Uniformly for all |x| ≤ Cn−1/2,
+Ln(�αR, φ0) − Ln(�α, �γ)
+≤
+Ln(πn(�α + xun), φ0) − Ln(�α, �γ) + λnPen(πn(�α + xun)) − λnPen(�αR) + oP (n−1)
+≤(g)
+Ln(�α + xun, φ0) − Ln(�α, �γ) + oP(n−1)
+=
+Qn(�α + xun) − Qn(�α) + (�φ(�α + xun) − φ0)2�Σ−1
+2
++ oP(n−1)
+=(h)
+x2 + 2x[n−1/2Zn + ⟨�α − α0, un⟩] + (�φ(�α + xun) − φ0)2�Σ−1
+2
++ oP(n−1)
+=(i)
+x2 + (�φ(�α + xun) − γ0)2�Σ−1
+2
++ oP(n−1)
+=(j)
+x2 + [x∥v∗
+n∥ + g3]2�Σ−1
+2
+�
+��
+�
+G(x)
++oP (n−1)
+where (g) follows from Lemma D.2 and that λnPen(πn(�α + xun)) − λnPen(�αR) = oP (n−1); (h) is
+from (D.1); (i) is from (C.2); (j) is from Lemma D.3. We choose x = τ ∗ to minimize G(x), leading
+to the choice τ ∗ = −g3∥v∗
+n∥(∥v∗
+n∥2 + �Σ2)−1. It is easy to see that |τ ∗| = OP (n−1/2), following this
+argument: from the proof of Theorem 4.2, g3 = OP (σn−1/2), and σ2 = Σ2 + ∥v∗
+n∥2. So provided that
+�Σ2 − Σ2 = oP(1)Σ2,
+|τ ∗| = OP (n−1/2)
+�
+Σ2 + ∥v∗n∥2∥v∗
+n∥
+∥v∗n∥2 + �Σ2
+= OP (n−1/2).
+Thus τ ∗ is admitted. Then we have obtained the upper bound: 1
+n �Sn(φ0) ≤ G(τ ∗) + oP(n−1), where
+G(τ ∗) =
+g2
+3
+∥v∗n∥2 + �Σ2
+.
+Step 3: matching bounds.
+We now show that the lower and upper bounds match, that is, −F(x∗) = G(τ ∗)+oP(n−1), which
+requires analyzing g2 = φ(�αR) − φ(α0) and g6. First, Lemma D.3 yields, uniformly in |x| ≤ Cn−1/2,
+(�φ(�αR + xun) − γ0)2�Σ−1
+2
+− (�φ(�αR) − γ0)2�Σ−1
+2
+= H(x) + oP (n−1)
+(D.4)
+where H(x) = �Σ−1
+2 x2∥v∗
+n∥2 + 2x�Σ−1
+2 ∥v∗
+n∥[�φ(α0) − γ0 + g2]. Next, the basic inequality yields
+Ln(�αR, φ0) ≤ Ln(πn(�αR + xun), γ0) + oP (n−1) ≤ Ln(�αR + xun, γ0) + oP(n−1)
+(D.5)
+where the first inequality follows from with the assumption that λnPen(�αR + xun) − λnPen(�αR) =
+oP (n−1); the second inequality follows from Lemma D.2. Uniformly for |x| ≤ Cn−1/2,
+Qn(�αR + xun) − Qn(�αR)
+=
+oP (n−1) + x2 + 2x[n−1/2Zn + ⟨un, �αR − α0⟩]
+=
+oP (n−1) + x2 + 2x[n−1/2Zn + g2∥v∗
+n∥−1],
+55
+
+where ⟨un, �αR − α0⟩ = ∥v∗
+n∥−1 dφ(α0)
+dα
+[�αR − α0] + oP (n−1/2) = ∥v∗
+n∥−1g2 + oP(n−1/2). This along with
+(D.4) (D.5) give rise to,
+0
+≤
+x2 + 2x[n−1/2Zn + g2∥v∗
+n∥−1] + H(x) + oP(n−1)
+=
+(1 + ∥v∗
+n∥2)x2 + 2x[n−1/2Zn + g2∥v∗
+n∥−1 + ∥v∗
+n∥(�φ(α0) − γ0 + g2)] + oP (n−1)
+=
+x2(1 + �Σ−1
+2 ∥v∗
+n∥2) + 2x[n−1/2Zn + g2∥v∗
+n∥−1 + �Σ−1
+2 ∥v∗
+n∥(�φ(α0) − γ0 + g2)] + oP (n−1)
+=
+x2(1 + �Σ−1
+2 ∥v∗
+n∥2) + 2x[g6 + �Σ−1
+2 ∥v∗
+n∥(φn(α0) − γ0 + g2)] + oP (n−1),
+(D.6)
+where in the last equality, �Σ−1
+2 ∥v∗
+n∥(�φ(α0) − φn(α0)) = oP (n−1/2), from Lemma D.2:
+|�Σ−1
+2 ∥v∗
+n∥(�φ(α0) − φn(α0))| ≤ |�Σ−1
+2 ∥v∗
+n∥ 1
+n
+n
+�
+t=1
+(Γ(Xt) − �Γt)ρ(Yt+1, α0)| = oP (n−1/2).
+Hence (D.6) implies there is some ¯ηn = oP (n−1) so that
+x2(1 + �Σ−1
+2 ∥v∗
+n∥2) + 2x[g6 + �Σ−1
+2 ∥v∗
+n∥(φn(α0) − γ0 + g2)] + ¯ηn ≥ 0.
+(D.7)
+We now derive some important intermediate results from (D.7). First, let
+Cn := min{∥v∗
+n∥Σ−1/2
+2
+, ∥v∗
+n∥2
+∗Σ−1
+2 , ∥v∗
+n∥σΣ−1
+2 }.
+It is known that (Cn+C2
+n)/(1+�Σ−1
+2 ∥v∗
+n∥2) ≤ 2 because ∥v∗
+n∥ ≤ σ. So ¯ηn = oP (n−1)C2
+n/(1+�Σ−1
+2 ∥v∗
+n∥2),
+implying ¯ηnn1/2C−1
+n
+≪ n−1/2Cn/(1+ �Σ−1
+2 ∥v∗
+n∥2). Hence there is a positive sequence ǫn = OP (n−1/2)
+so that ¯ηnn1/2C−1
+n
+≪ ǫn ≪ n−1/2Cn/(1+�Σ−1
+2 ∥v∗
+n∥2). Hence ¯ηn/ǫn+ǫn(1+�Σ−1
+2 ∥v∗
+n∥2) = oP (n−1/2Cn).
+Take x = ±ǫn and divide by 2ǫn on (D.7). We reach four intermediate results:
+g6 + �Σ−1
+2 ∥v∗
+n∥(φn(α0) − γ0 + g2) = OP (¯ηn/ǫn + ǫn(1 + �Σ−1
+2 ∥v∗
+n∥2)) = oP (n−1/2Cn) (D.8)
+(1 + �Σ−1
+2 ∥v∗
+n∥2)g6 + �Σ−1
+2 ∥v∗
+n∥g5 = oP (n−1/2Cn)
+(D.9)
+g4 = (oP (n−1/2Cn) − g6)2�Σ2∥v∗
+n∥−2
+(D.10)
+g6 = OP (n−1/2)
+(D.11)
+where (D.8) follows from (D.7) with x = ±ǫn; the left hand sides of (D.8) and (D.9) are equal;
+(D.10) is from (D.8) and the definition of g4; (D.11) is from (D.9), g5 = OP (σn−1/2) and that
+oP (n−1/2Cn) = σ2Σ−1
+2 OP (n−1/2). Also, the proof of (D.11) does not rely on the conclusion of Step
+1, so it verifies that |x∗| = OP (n−1/2), a claim used in step 1.
+We are now ready to match the bounds. From (D.10) and (D.11),
+−F(x∗)
+=
+g2
+6 + g4 = g2
+6 + (oP (n−1/2Cn) − g6)2�Σ2∥v∗
+n∥−2 = g2
+6(1 + �Σ2∥v∗
+n∥−2) + oP (n−1)
+56
+
+=(k)
+[oP (n−1/2Cn) − �Σ−1
+2 ∥v∗
+n∥g5]2
+(1 + �Σ−1
+2 ∥v∗n∥2)2
+(1 + �Σ2∥v∗
+n∥−2) + oP (n−1)
+=
+g2
+5
+�Σ2 + ∥v∗n∥2 + oP(n−1) =(l)
+g2
+3
+�Σ2 + ∥v∗n∥2 + oP (n−1) = G(τ ∗) + oP (n−1),
+where (k) is from (D.9); (l) is from the fact that (due to (C.2))
+|g2
+3 − g2
+5|
+≤
+|φ(�α) − φ(α0) + ∥v∗
+n∥n−1/2Zn|OP (n−1/2σ)
+≤
+����
+dφ(α0)
+dα
+[�α − α0] + ∥v∗
+n∥n−1/2Zn
+���� OP (n−1/2σ) + oP(σ2n−1)
+=
+���⟨�α − α0, un⟩ + n−1/2Zn + oP (n−1/2)
+��� OP (∥v∗
+n∥n−1/2σ) + oP(σ2n−1) = oP (σ2n−1).
+Thus we have proved that the upper and lower bounds match up to oP (n−1), implying
+�Sn(φ0) = nG(τ ∗) + oP (1) = ng2
+3
+�σ2 + oP (1) →d χ2
+1
+where the convergence in distribution follows from (C.5).
+Lemma D.2 (for Theorem 5.2). Suppose (δη
+n supx |�Σ(x)−Σ(x)|+√kndnδη
+n+ϕ2
+n) = oP (n−1/2 min{1, σ−1}).
+In addition, suppose (1 + ∥v∗
+n∥) supα∈Cn |φ(πnα) − φ(α)| = o(n−1/2).
+Write �φ(α) := 1
+n
+�n
+t=1[l(h(Wt)) − �Γtρ(Yt+1, α)]. Then
+(i) ∥�αR − α∥∞,ω = OP (δn), Q(�αR) ≤ OP (¯δ2
+n).
+(ii) supα1,α2∈Cn[�φ(α1) − �φ(α2)] − [φ(α1) − φ(α2)] = OP (δη
+n supx |�Σ(x) − Σ(x)| + √kndnδη
+n + ϕ2
+n).
+(iii) |Ln(πn(�α + xun), γ0) − Ln(�α + xun, γ0)| = oP (n−1)
+(iv) |Ln(πn(�αR + xun), γ0) − Ln(�αR + xun, γ0)| = oP(n−1).
+Proof. (i) The inequality Ln(�αR, φ0) + λnPen(�hR) ≤ Ln(πnα0, γ0) + λnPen(πnh0) implies
+Qn(�αR) ≤ Qn(πnα0) + Fn(πnα0) + OP (λ) = O(¯δ2
+n) + Fn(πnα0)
+where Fn(α) := (�φ(α) − γ0)′�Σ−1
+2 (�φ(α) − γ0). We now bound Fn(πnα0). Note that
+�φ(πnα0) − γ0
+=
+1
+n
+n
+�
+t=1
+l(πnh0(Wt)) − El(h0(Wt)) − 1
+n
+n
+�
+t=1
+(�Γt − Γ(Xt))ρ(Yt+1, πnα0)
+− 1
+n
+n
+�
+t=1
+[Γ(Xt)ρ(Yt+1, πnα0) − EΓtρ(Yt+1, α0)] − EΓt[m(Xt, α0) − m(Xt, πnα0)].
+57
+
+The first term is bounded by OP (n−1/2)+E[l(πnh0(Wt))−l(h0(Wt))]; the second term is bounded by
+OP (¯δn), following from the same argument as those for (C.4); the third and fourth terms are bounded
+by OP (n−1/2)+EΓ(Xt)[m(Xt, πnα0)−m(Xt, α0)] ≤ OP (
+�
+Q(πnα0)). Hence �φ(πnα0)−γ0 = OP (¯δn).
+This implies Fn(πnα0) = OP (¯δ2
+n). This yields Qn(�αR) = OP (¯δ2
+n). Then from the proof of Theorem
+3.1,
+Q(�αR) ≤ CE �m(Xt, �αR)2 + OP (¯δ2
+n) ≤ C 1
+n
+�
+t
+�m(Xt, �αR)2 + OP (¯δ2
+n) ≤ Qn(�αR) + OP (¯δ2
+n) = OP (¯δ2
+n).
+It also implies ∥�αR − πnα0∥ ≤ ∥�αR − α0∥ + ∥πnα0 − α0∥ = OP (¯δn), and hence ∥�αR − α0∥∞,ω ≤
+∥πnα0 − α0∥∞,ω + ∥�αR − πnα0∥∞,ω = ∥πnα0 − α0∥∞,ω + OP (ωn(¯δn)) = OP (δn).
+(ii) Let a1 = [(φn(α1) − φ(α1)) − (φn(α2) − φ(α2))]. We have
+[�φ(α1) − �φ(α2)] − [φ(α1) − φ(α2)] = a1 + 1
+n
+n
+�
+t=1
+(Γ(Xt) − �Γt)ρ(Yt+1, α1) + 1
+n
+n
+�
+t=1
+(�Γt − Γ(Xt))ρ(Yt+1, α2)
+≤
+OP (δη
+n sup
+x |�Σ(x) − Σ(x)| +
+�
+kndnδη
+n + ϕ2
+n),
+where the first inequality follows from bounds for (C.4) and Lemma C.3.
+(iii) Let α = �α+xun. By the same proof of Lemma D.1(ii), Qn(πnα)−Qn(α) = oP (n−1). Next,
+�φ(α) − γ0
+=
+a1(α) + a3 + a4(α) − 1
+n
+�
+t
+(�Γt − Γ(Xt))ρ(Yt+1, α),
+a1(α)
+:=
+φn(α) − φ(α) − [φn(α0) − φ(α0)]
+a3
+:=
+φn(α0) − φ(α0)
+a4(α)
+:=
+φ(α) − φ(α0).
+By Lemma C.3 and the same proof for bounding (C.4),
+a1− 1
+n
+�
+t
+(�Γt−Γ(Xt))ρ(Yt+1, α) = OP (δη
+n sup
+x |�Σ(x)−Σ(x)|+
+�
+kndnδη
+n+ϕ2
+n) = oP (n−1/2 min{1, σ−1}),
+where the last equality follows from the assumption (δη
+n supx |�Σ(x) − Σ(x)| + √kndnδη
+n + ϕ2
+n) =
+oP (n−1/2 min{1, σ−1}). The same bound holds when α is replaced with πnα. Meanwhile, by the
+proof of Theorem 4.2, a3 = OP (σn−1/2).
+To bound a4(α), first note that
+∥πn(�α + xun) − α0∥2 ≤ CEm(Xt, πn(�α + xun))2 ≤ CE[m(Xt, πn(�α + xun)) − m(Xt, �α + xun)]2
++CE[m(Xt, �α + xun) − m(Xt, �α)]2 + CQ(�α) ≤ OP (µ2
+n + ¯δ2
+n)
+∥(�α + xun) − α0∥2 ≤ OP (¯δ2
+n).
+58
+
+So by Assumption 4.1, and that ⟨v∗
+n, α0 − α0,n⟩ = 0,
+a4(�α + xun)
+≤
+|φ(�α + xun) − φ(α0)| ≤
+����
+dφ(α0)
+dα
+[�α + xun − α0]
+����
+≤
+����
+dφ(α0)
+dα
+[�α + xun − α0,n]
+���� +
+����
+dφ(α0)
+dα
+[α0,n − α0]
+����
+=
+oP (∥v∗
+n∥)n−1/2 + |⟨�α − α0, v∗
+n⟩ + x⟨un, v∗
+n⟩| = OP (∥v∗
+n∥n−1/2).
+a4(πn(�α + xun))
+≤
+|φ(πn(�α + xun)) − φ(α0)| ≤
+����
+dφ(α0)
+dα
+[πn(�α + xun) − α0]
+����
+≤
+����
+dφ(α0)
+dα
+[πn(�α + xun) − α0,n]
+���� +
+����
+dφ(α0)
+dα
+[α0,n − α0]
+����
+=
+oP (∥v∗
+n∥)n−1/2 + |⟨πn(�α + xun) − α0, v∗
+n⟩|
+≤
+oP (∥v∗
+n∥)n−1/2 + ∥πn(�α + xun) − α0∥∥v∗
+n∥
+≤
+OP (¯δn)∥v∗
+n∥.
+Also, by the proof of Lemma D.1(ii), Qn(πn(�α + xun)) − Qn(�α + xun) = oP(n−1). Together, with
+α = �α + xun, |�φ(α) − γ0| ≤ oP(n−1/2 min{1, σ−1}) + OP (∥v∗
+n∥n−1/2), and cn := |φ(πnα) − φ(α)|,
+Ln(πn(�α + xun), γ0) − Ln(�α + xun, γ0) = Qn(πnα) − Qn(α)
++(�φ(πnα) − γ0)′�Σ−1
+2 (�φ(πnα) − γ0) − (�φ(α) − γ0)′�Σ−1
+2 (�φ(α) − γ0)
+=
+oP (n−1) + (�φ(πnα) − �φ(α))′ �Σ−1
+2 (�φ(πnα) − �φ(α)) + 2(�φ(πnα) − �φ(α))′�Σ−1
+2 (�φ(α) − γ0)
+=
+oP (n−1) + OP (1)|φ(πnα) − φ(α)|2 + OP (1)|φ(πnα) − φ(α)||�φ(α) − γ0|
+≤
+oP (n−1) + OP (c2
+n) + oP (n−1/2 min{1, σ−1})cn + OP (∥v∗
+n∥n−1/2cn) = oP (n−1).
+(iv) The proof is the same for part (iii).
+Lemma D.3. Write νn := δη
+n supx |�Σ(x) − Σ(x)| + √kndnδη
+n + ϕ2
+n. Suppose
+νn = oP (n−1/2σ−1), (pn + νn)¯δn∥v∗
+n∥ = o(n−1).
+Then Uniformly for |x| ≤ Cn−1/2, for b(x) := �φ(α0) − γ0 + φ(�αR) − φ(α0),
+(i) (�φ(�α + xun) − γ0)2�Σ−1
+2
+= [x∥v∗
+n∥ + g3]2�Σ−1
+2
++ oP (n−1)
+(ii) (�φ(�αR + xun) − γ0)2�Σ−1
+2
+− (�φ(�αR) − γ0)2�Σ−1
+2
+= x2∥v∗
+n∥2�Σ−1
+2
++ 2x∥v∗
+n∥b(x)�Σ−1
+2
++ oP (n−1).
+(iii) (�φ(�αR) − γ0)2�Σ−1
+2
+= [φn(α0) − γ0 + φ(�αR) − φ(α0)]2�Σ−1
+2
++ oP(n−1).
+Proof. (i) Let g3 = [φn(α0) − φ(α0)] + [φ(�α) − φ(α0)].
+�φ(�α + xun) − γ0
+=
+φ(�α + xun) − φ(�α) + g3 + b1 + b2
+b1
+=
+[�φ(�α + xun) − �φ(�α)] − [φ(�α + xun) − φ(�α)]
+59
+
+b2
+=
+�γ − γ0 − g3.
+We now work with φ(�α + xun) − φ(�α). By Assumption 4.1 and the Riesz representation,
+φ(�α + xun) − φ(�α)
+=
+φ(�α + xun) − φ(α0) − [φ(�α) − φ(α0)]
+=
+dφ(α0)
+dα
+[�α − α0 + xun] − dφ(α0)
+dα
+[�α − α0]
+=
+⟨v∗
+n, �α − α0 + xun⟩ − ⟨v∗
+n, �α − α0⟩ + oP (∥v∗
+n∥)n−1/2
+=
+⟨v∗
+n, un⟩x = x∥v∗
+n∥.
+(D.12)
+Hence �φ(�α + xun) − γ0 = x∥v∗
+n∥ + g3 + b1 + b2. Also note that g3 = OP (σn−1/2). Together
+(�φ(�α + xun) − γ0)2�Σ−1
+2
+is bounded by
+[x∥v∗
+n∥ + g3]2�Σ−1
+2
++ oP(n−1) + OP (b2
+1 + b2
+2) + OP (b1 + b2)(∥v∗
+n∥ + σ)n−1/2.
+By Lemma D.2, b1 = OP (νn). By the proof of Theorem 4.2, b2 = OP (νn) = oP (n−1/2σ−1). So the
+above is bounded by (σ ≥ ∥v∗
+n∥),
+[x∥v∗
+n∥ + g3]2�Σ−1
+2
++ oP (n−1) + OP (νnn−1/2)(∥v∗
+n∥ + σ) = [x∥v∗
+n∥ + g3]2�Σ−1
+2
++ oP (n−1).
+(ii) (�φ(�αR + xun) − γ0)2�Σ−1
+2
+− (�φ(�αR) − γ0)2�Σ−1
+2
+equals
+∆1 := (�φ(�αR + xun) − �φ(�αR))2�Σ−1
+2
++ 2(�φ(�αR + xun) − �φ(�αR))�Σ−1
+2 (�φ(�αR) − γ0).
+The same argument as in (D.12) yields �φ(�αR + xun) − �φ(�αR) = x∥v∗
+n∥. Meanwhile,
+�φ(�αR) − γ0
+=
+�φ(α0) − γ0 + φ(�αR) − φ(α0) + [�φ(�αR) − �φ(α0)] − [φ(�αR) − φ(α0)]
+�
+��
+�
+=OP (νn) by Lemma D.2
+.
+(D.13)
+�φ(�αR + xun) − �φ(�αR)
+=
+OP (µn) + φ(�αR + xun) − φ(�αR) = ∥v∗
+n∥x + OP (νn),
+φ(�αR) − φ(α0)
+=
+dφ(α0)
+dα
+[�αR − α0] = ⟨�αR − α0, v∗
+n⟩
+≤
+∥�αR − α0∥∥v∗
+n∥ ≤ C
+�
+Q(�αR)∥v∗
+n∥
+≤
+OP (¯δn∥v∗
+n∥).
+(D.14)
+Hence with �φ(α0) − γ0 = OP (n−1/2σ), and ̟nσ = O(1)
+∆1
+=
+x2∥v∗
+n∥2�Σ−1
+2
++ oP (n−1) + 2[x∥v∗
+n∥][�φ(α0) − γ0 + φ(�αR) − φ(α0) + OP (µn)]�Σ−1
+2
+=
+x2∥v∗
+n∥2�Σ−1
+2
++ 2x∥v∗
+n∥[�φ(α0) − γ0 + φ(�αR) − φ(α0)]�Σ−1
+2
++ oP (n−1)
++OP (µn)n−1/2∥v∗
+n∥
+60
+
+=
+x2∥v∗
+n∥2�Σ−1
+2
++ 2x∥v∗
+n∥[�φ(α0) − γ0 + φ(�αR) − φ(α0)]�Σ−1
+2
++ oP (n−1),
+(iii) Define
+z1
+:=
+(�φ(�αR) − γ0)2�Σ−1
+2
+z2
+:=
+[�φ(α0) − γ0 + φ(�αR) − φ(α0)]2�Σ−1
+2
+z3
+:=
+[φn(α0) − γ0 + φ(�αR) − φ(α0)]2�Σ−1
+2 .
+First the proof for bounding (C.4) can be simplified to yield
+|�φ(α0) − φn(α0)| ≤ 1
+n
+n
+�
+t=1
+(�Γt − Γ(Xt))ρ(Yt+1, α0)
+=
+1
+n
+¯bn(α0)′(�Σ−1
+n − Σ−1
+n )ρn(α0) + OP (δη
+n)[sup
+x ∥�Σ(x) − Σ(x)∥ +
+�
+kn
+n ].
+√z2 + √z3 = OP (n−1/2σ + ¯δn∥v∗
+n∥).
+By (D.13), and with the assumption (νn + pn)¯δn∥v∗
+n∥ = oP (n−1),
+|z1 − z2|
+≤
+OP (ν2
+n) + OP (µn)|�φ(α0) − γ0 + φ(�αR) − φ(α0)|
+=
+oP(n−1) + OP (νn)(n−1/2σ + ¯δn∥v∗
+n∥) = oP(n−1).
+|z2 − z3|
+≤
+OP (1)|�φ(α0) − φn(α0)|(√z2 + √z3) = oP (n−1) + OP (pn¯δn∥v∗
+n∥) = oP(n−1).
+Hence z1 − z3 = oP (n−1).
+E
+Verifying conditions for RL, NPIV and NPQIV in Section 6
+E.1
+Reinforcement learning model: proof of Proposition 6.1
+Proof. Recall that Qπ denotes the true Q-function. Let Xt = (St, At). We have
+m(Xt, h) = E(Rt|Xt) − h(St, At) + γE
+��
+x∈A
+π(x|St+1)h(St+1, x)
+����St, At
+�
+dx
+In addition, for dm
+dh [v] defined in (6.2), ∥v∥2 := E
+� dm
+dh [v]
+�2 Σ(Xt)−1.
+Verifying Assumption 3.2.
+The Bellman equation implies m(Xt, Qπ) = 0 so for all h ∈ Hn,
+m(Xt, h) = m(Xt, h) − m(Xt, Qπ). Hence m(Xt, h) = dm
+dh [h − Qπ].
+∥h − Qπ∥2 = E
+�dm
+dh [h − Qπ]
+�2
+Σ(Xt)−1 = Em(Xt, h)2Σ(Xt)−1.
+61
+
+This shows condition (i). For condition (ii), it is also easy to see:
+Em(Xt, πnα0)2Σ(Xt)−1 = E
+�dm
+dh [πnQπ − Qπ]
+�2
+Σ(Xt)−1 = ∥πnQπ − Qπ∥2.
+Verifying Assumption 3.6.
+For condition (i), let Tt = Ψj(Xt)2 + 1. Also,
+ρ(Yt+1, h) = Rt − h(St, At) + γK(h),
+K(h) =
+�
+x∈A
+π(x|St+1)h(St+1, x)dx.
+and ǫ(St, h1) − ǫ(St, h2) = γK(h1) − γK(h2) − γE[K(h1) − K(h2)|St, At]. Now
+|K(h1) − K(h2)| ≤ ∥h1 − h2∥∞,ωM(St+1),
+M(St+1) :=
+�
+π(x|St+1)(1 + x2 + S2
+t+1)ω/2dx.
+Uniform in j, with E maxj≤kn Ψj(Xt)4 < ∞, and EM(St+1)4 < ∞,
+ETt
+sup
+∥h1−h∥∞,ω<δ
+|ǫ(St, h1) − ǫ(St, h)|2 ≤ 4γ2
+sup
+∥h1−h∥∞,ω<δ
+∥h1 − h∥2
+∞,ωC ≤ Cδ2.
+For (ii), Let T1 := maxj≤kn Ψj(Xt)2. Also ER4
+t < C, ET1R2
+t < C. Since suph∈Hn ∥h∥2
+∞,ω < C,
+ET1 suph∈Hn h(St, At)2 ≤ ET1(1+|St|2+|At|2)ω∥h∥2
+∞,ω < C. Also ET1 suph K(h)2 ≤ EM(St+1)2 suph ∥h∥2
+∞,ω.
+Hence ET1 suph∈Hn ρ(Yt+1, h)2 ≤ C.
+For (iii), the pathwise derivative of m is given by (6.2), for C :=
+�
+E(1 + |St|2 + |At|2)ω + EM(St+1)2�
+,
+∥h − Qπ∥2 ≤ CE[h − Qπ]2 + CE|E(K(h) − K(Qπ)|St, At)|2 ≤ C∥h − Qπ∥∞,ω
+Verifying Assumption 4.2. We note that for any h ∈ Hn ∪ {Qπ},
+dm(Xt, h)
+dh
+[un] = γ
+�
+x∈A
+E [π(x|St+1)un(St+1, x)|St, At] dx − un(St, At),
+which does not depend on h. Also, for any h, τ, v, because of the linearity,
+d2
+dτ 2 m(Xt, h + τv) = 0.
+For condition (i), let r := 2 + ζ, a := |ρ(Yt+1, Qπ)|2+ζ, b :=
+���dm(Xt,Qπ)
+dh
+[un]
+��� then we have
+b ≤ |γEtEπun(St+1, A)|+|un(St, At)| where Et = E(.|St, At) and Eπ is with respect to the distribution
+π(.|St+1) for A. Let dπ := EEπ|un(St+1, A)|2r and d := E|un(St, At)|2r. Then
+Ea2 ≤ C + E|Rt|4+2ζ < C.
+Also, d + dπ ≤ C because EEπ|un(St+1, A)|2r + E|v∗
+n(St, At)|2r ≤ ∥v∗
+n∥2r. Hence
+E|ρ(Yt+1, Qπ)|2+ζ
+����
+dm(Xt, Qπ)
+dh
+[un]
+����
+2+ζ
++ E|ρ(Yt+1, Qπ)|2+ζ ≤ C(Ea2)1/2 �
+d1/2 + d1/2
+π
++ 1
+�
+< C.
+62
+
+Conditions (ii)(iii)(iv) are trivially satisfied because of the linearity.
+For condition (v), let T2 = maxj≤kn Ψj(Xt)2 + 1. Recall that for h ∈ Cn, h = h1 + xun where
+∥h1 − Qπ∥∞,ω < Cδn and |x| ≤ Cn−1/2. Hence
+ET2 sup
+h∈Cn
+(ρ(Yt+1, h) − ρ(Yt+1, Qπ))2 ≤ CET2 sup
+h∈Cn
+|h(Xt) − Qπ(Xt)|2 + ET2 sup
+h∈Cn
+γ2[K(h) − K(Qπ)]2
+≤
+�
+CET2(1 + ∥Xt∥2)ω + γ2ET2M(St+1)2�
+sup
+h∈Cn
+∥h − Qπ∥2
+∞,ω
+≤
+O(δ2
+n + n−1) ≤ Cδ2
+n.
+E.2
+NPIV model: proof of Proposition 6.2
+In this case m(Xt, h) = E(h0(Wt) − h(Wt)|σt(X)) and ǫ(St, α) = Ut + h0(Wt) − h(Wt) − E(h0(Wt) −
+h(Wt)|σt(X)), dm(Xt,α)
+dh
+[v] = E(v(Wt)|σt(X)), and
+d2
+dτ 2 m(Xt, h + τv) = 0 because of the linearity.
+Proof. We sequentially verify conditions in Assumptions 3.2, 3.6, 4.2 and 4.6.
+Verifying Assumption 3.2 This assumption follows immediately from
+∥α1 − α2∥2 = E (E(h1(Wt) − h2(Wt)|σt(X)))2 Σ(Xt)−1 = E[m(Xt, h1) − m(Xt, h2)]2Σ(Xt)−1.
+Verifying Assumption 3.6 (i) Uniformly in j ≤ kn, for Mt := (1 + |Wt|2)ω,
+E[Ψj(Xt)2 + 1]
+sup
+∥α−α1∥∞,ω<δ
+|ǫ(St, α1) − ǫ(St, α)|2
+≤
+E[Ψj(Xt)2 + 1]
+sup
+∥h−h1∥∞,ω<δ
+[h1(Wt) − h(Wt) − E(h1(Wt) − h(Wt)|σt(X))]2
+≤
+4E[Ψj(Xt)2 + 1] [Mt + E(Mt|σt(X))]
+sup
+∥h−h1∥∞,ω<δ
+∥h1 − h∥2
+∞,ω ≤ Cδ2
+given that E[Ψj(Xt)2 + 1]Mt < ∞.
+(ii) Suppose E maxj≤kn Ψj(Xt)2[(1 + |Wt|2)ω + U 2
+t ] < ∞,
+E max
+j≤kn Ψj(Xt)2 sup
+α∈An
+ρ(Yt+1, α)2
+≤
+2E max
+j≤kn Ψj(Xt)2 sup
+α∈An
+[h0(Wt) − h(Wt)]2 + 2E max
+j≤kn Ψj(Xt)2U 2
+t ≤ C.
+(iii) We have ∥α1 − α2∥2 ≤ C∥α1 − α2∥2
+∞,ωE(1 + |Wt|2)ω.
+Verifying Assumption 4.2
+63
+
+For (i) we have
+E|ρ(Yt+1, α0)|2+ζ
+����
+dm(Xt, α0)
+dα
+[un]
+����
+2+ζ
+= E|Ut|2+ζ|E(v∗
+n(Wt)|Xt)|2+ζ∥v∗
+n∥−(2+ζ) < C.
+For (ii)(iii), we have
+d2
+dτ 2 m(Xt, h + τv) = 0 for any h and v inside H0 ∪ Hn because of the
+linearity. For (iv), we have supα∈Cn
+1
+n
+�
+t[dm(Xt,α)
+dα
+[un] − dm(Xt,α0)
+dα
+[un]]2 = 0.
+For (v), let A := maxj≤kn Ψj(Xt)2 +1. For h ∈ Cn, we know there is hn ∈ Hn and |x| ≤ Cn−1/2
+so that
+h = hn + xun,
+∥hn − h0∥∞,ω ≤ Cδn.
+Because EAun(W)2 < C, hence
+EA sup
+α∈Cn
+(ρ(Yt+1, h) − ρ(Yt+1, α0))2 = EA sup
+α∈Cn
+(h(Wt) − h0(Wt))2
+≤
+2EA sup
+Cn
+|hn(Wt) − h0(Wt)|2 + Cn−1EAun(W)2
+≤
+Cδ2
+n + Cn−1 ≤ Cδ2
+n.
+Verifying Assumption 4.6. For notational simplicity, write Γt = Γ(Xt), Σt = Σ(Xt), �Σt = �Σ(Xt)
+and ρt = ρ(Yt+1, α0). Using �Σ−1
+t
+− Σ−1
+t
+= �Σ−1
+t (Σt − �Σt)Σ−1
+t , the triangular inequality yields
+1
+n
+�
+t
+ΓtΣt(�Σ−1
+t
+− Σ−1
+t )ρt ≤ | 1
+n
+�
+t
+ΓtΣt(�Σ−1
+t
+− Σ−1
+t )(�Σt − Σt)Σ−1
+t ρt|
++| 1
+n
+�
+t
+Γt(�Σt − Σt)Σ−1
+t ρt|
+≤
+| 1
+n
+�
+t
+Γt(�Σt − Σt)Σ−1
+t ρt| + OP (1) 1
+n
+�
+t
+|(�Σt − Σt)|2.
+Note �Σt = �A′
+nΨn(Ψ′
+nΨn)−1Ψ(Xt) where �An is a n × 1 vector of �ρ2
+t . Also let (An, E(An|X), Gn, Un)
+respectively be n × 1 vectors of (ρ2
+t , Σt, gt, ut) where gt = ΓtΣ−1
+t ρt and ut = ρ2
+t − E(ρ2
+t|σt(X)). Let Jt
+be the t th element of (I − Pn)E(An|X).
+We have ( 1
+√n∥ �An − An∥)2 ≤ C 1
+n
+�
+t(�ρt − ρt)2ρ2
+t + C 1
+n
+�
+t(�ρt − ρt)4 = OP (δ2
+n). In addition, let
+D be the diagonal matrix of ΓtΣ−1
+t . Then
+1
+√n∥PnGn∥ = OP ( 1
+√n
+�
+ρ′nDPnDρn) = OP (
+�
+kn/n). So
+we have the following decomposition
+1
+n
+�
+t
+Γt(�Σt − Σt)Σ−1
+t ρt = 1
+n[ �A′
+nPn − E(An|X)]Gn = a1 + a2 + a3,
+1
+n
+�
+t
+|(�Σt − Σt)|2 = 1
+n∥Pn �An − E(An|X)∥2 ≤ C(a4 + a5 + a6)
+a1
+=
+1
+nE(An|X)′(Pn − I)Gn = 1
+n
+�
+t
+JtΓtΣ−1
+t ρt = OP ( 1
+√n)
+�
+EJ2
+t Γ2
+t Σ−1
+t
+64
+
+=
+OP (n−1/2
+�
+EJ2
+t ) = OP (
+�
+ϕ2n/n).
+a2
+=
+1
+nU ′
+nPnGn ≤ OP (1)∥ 1
+nU ′
+nΨn∥∥ 1
+n
+�
+t
+Ψ(Xt)ΓtΣ−1
+t ρt∥ = OP (kn
+n )
+a3
+=
+1
+n[ �An − An]′PnGn ≤
+1
+√n∥ �An − An∥ 1
+√n∥PnGn∥ ≤ OP (δ2
+n + kn
+n ) = OP (δ2
+n)
+a4
+=
+1
+n∥(I − Pn)E(An|X)∥2 = OP (ϕ2
+n).
+a5
+=
+1
+n∥PnUn∥2 ≤ OP (1)∥ 1
+nU ′
+nΨn∥2 = OP (kn
+n )
+a6
+=
+1
+n∥Pn( �An − An)∥2 = OP (δ2
+n)
+Putting together, 1
+n
+�
+t ΓtΣt(�Σ−1
+t
+− Σ−1
+t )ρt = OP (pn) where pn = ϕ2
+n + kn
+n + δ2
+n ≤ Cδ2
+n.
+E.3
+NPQIV model: proof of Proposition 6.3
+In this model m(Xt, α) = P(Ut < h−h0|σt(X))−̟ where Ut = Yt−h0(Wt). Suppose the conditional
+distribution of Ut given (Xt, Wt) is absolutely continuous with density function fUt|σt(X),Wt(u). Then
+the derivative is defined as
+dm(Xt, α)
+dh
+[v] = E(fUt|σt(X),Wt(h(Wt) − h0(Wt))v(Wt)|σt(X)).
+Proof. Verifying Assumption 3.2.
+Let
+At(h)
+:=
+� 1
+0
+fUt|σt(X),Wt (x(h(Wt) − h0(Wt))) dx
+Bt(v, h)
+:=
+E {At(v)[h(Wt) − h0(Wt)]|σt(X)} .
+Then m(Xt, h) = Bt(h, h), Em(Xt, h)2Σ(Xt)−1 = EBt(h, h)2Σ(Xt)−1 and ∥α−α0∥2 = EBt(h0, h)2Σ(Xt)−1.
+This assumption then follows from the condition that c2EBt(h, h)2Σ(Xt)−1 ≤ EBt(h0, h)2Σ(Xt)−1 ≤
+c1EBt(h, h)2Σ(Xt)−1 for all ∥h − h0∥ < ǫ0.
+Verifying Assumption 3.6 (i) Let Aj := [Ψj(Xt)2 + 1]. Fix any α = h ∈ An,
+EAj
+sup
+∥α−α1∥∞,ω<δ
+|ǫ(St, α1) − ǫ(St, α)|2
+≤
+2EAj
+sup
+∥α−α1∥∞,ω<δ
+|ρ(Yt+1, α1) − ρ(Yt+1, α)|2 + 2EAj
+sup
+∥α−α1∥∞,ω<δ
+|m(Xt, α1) − m(Xt, α)|2.
+On one hand,
+EAj
+sup
+∥α−α1∥∞,ω<δ
+|m(Xt, α1) − m(t, α)|2
+65
+
+≤
+2EAj
+sup
+∥h−h1∥∞,ω<δ
+P(h(Wt) − h0(Wt) ≤ Ut ≤ h1(Wt) − h0(Wt)|X)21{h1(Wt) > h(Wt)}
++2EAj
+sup
+∥h−h1∥∞,ω<δ
+P(h1(Wt) − h0(Wt) ≤ Ut ≤ h(Wt) − h0(Wt)|X)21{h(Wt) > h1(Wt)}
+≤
+2EAj sup
+u
+fUt|σt(X),Wt(u)2(1 + |Wt|2)ω
+sup
+∥h−h1∥∞,ω<δ
+∥h1 − h2∥2
+∞,ω
+≤
+2EAj sup
+u fUt|σt(X),Wt(u)2(1 + |Wt|2)ωδ2 ≤ Cδ2.
+On the other hand, for notational simplicity, write a = h(Wt) − h0(Wt), and a1 = h1(Wt) − h0(Wt).
+Then ∥h − h1∥∞,ω < δ implies |a − a1| ≤ δ(1 + |Wt|2)ω/2 := gt(δ). So
+EAj
+sup
+∥α−α1∥∞,ω<δ
+|ρ(Yt+1, α1) − ρ(Yt+1, α)|2
+≤
+EAj
+sup
+∥h−h1∥∞,ω<δ
+1{a ≤ Ut ≤ a1}1{a1 > a} + EAj
+sup
+∥h−h1∥∞,ω<δ
+1{a1 ≤ Ut ≤ a}1{a > a1}
+≤
+EAj
+�
+sup
+h1:∥h−h1∥∞,ω<δ
+1{a ≤ Ut ≤ a1}fUt|σt(X),Wt(u)du1{a1 > a}
++EAj
+�
+sup
+h1:∥h−h1∥∞,ω<δ
+1{a1 ≤ Ut ≤ a}fUt|σt(X),Wt(u)du1{a > a1}
+≤
+EAj
+� a+gt(δ)
+a
+fUt|σt(X),Wt(u)du1{a1 > a} + EAj
+� a
+a−gt(δ)
+fUt|σt(X),Wt(u)du1{a > a1}
+≤
+2 sup
+u fUt|σt(X),Wt(u)δEAj(1 + |Wt|2)ω/2 ≤ Cδ.
+(ii) We have
+E max
+j≤kn Ψj(Xt)2 sup
+α∈An
+ρ(Yt+1, α)2 ≤ CE max
+j≤kn Ψj(Xt)2 < C.
+(iii) Because Bt(h0, h)2 ≤ E
+�
+At(h0)2(1 + W 2
+t )ω|σt(X)
+�
+∥h − h0∥2
+∞,ω, we have
+∥h − h0∥2 ≤ EBt(h0, h)2Σ(Xt)−1 ≤ ∥h − h0∥2
+∞,ωEAt(h0)2(1 + W 2
+t )ωΣ(Xt)−1.
+Verifying Assumption 4.2 (i). Trivially |ρ(y, h)| + |m(x, h)| ≤ 4. Also
+dm(Xt, α)
+dh
+[un] = E(fUt|σt(X),Wt(0)un(Wt)|σt(X)) < C.
+So E|ρ(Yt+1, α0)|2+ζ ���dm(Xt,α0)
+dα
+[un]
+���
+2+ζ
++ E|ρ(Yt+1, α0)|2+ζ < C.
+Verifying Assumption 4.2 (ii). Let f ′
+Ut|σt(X),Wt denote the first derivative of fUt|σt(X),Wt. We
+have
+d2
+dτ 2 m(Xt, h + τv) = E[f ′
+Ut|σt(X),Wt(h(Wt) − h0(Wt) + τv(Wt))v(Wt)2|σt(X)]
+66
+
+Hence
+E sup
+α∈Cn
+sup
+|τ|≤Cn−1/2
+1
+n
+�
+t
+� d2
+dτ 2 m(Xt, α + τun)|
+�2
+≤
+E sup
+α∈Cn
+sup
+x
+sup
+|τ|≤Cn−1/2 E
+�
+f
+′2
+Ut|σt(X),Wt(h(Wt) − h0(Wt) + τv(Wt))un(Wt)4|σt(X) = x
+�
+≤
+sup
+u,x,w f
+′2
+Ut,x,w(u)E[un(Wt)4|σt(X)] < C.
+Verifying Assumption 4.2 (iii).
+sup
+τ∈(0,1)
+sup
+α∈Cn
+E
+� d2
+dτ 2 m(Xt, α0 + τ(α − α0))
+�2
+≤
+sup
+τ∈(0,1)
+sup
+α∈Cn
+E
+�
+E[f ′
+Ut|σt(X),Wt(τ(h − h0))(h − h0)2|σt(X)]
+�2
+≤ sup
+α∈Cn
+E
+�
+E(h − h0)2|σt(X)
+�2
+≤
+sup
+h∈Cn
+sup
+w |h(w) − h(w)|4 ≤ O(δ4
+n) = o(n−1).
+Verifying Assumption 4.2 (iv). Let g1 := h(Wt) − h0(Wt).
+kn sup
+α∈Cn
+1
+n
+�
+t
+[dm(Xt, α)
+dα
+[un] − dm(Xt, α0)
+dα
+[un]]2
+≤
+kn sup
+α∈Cn
+1
+n
+�
+t
+[E(fUt|σt(X),Wt(g1) − fUt|σt(X),Wt(0))un(Wt)|σt(X)]2
+≤
+knL sup
+α∈Cn
+1
+n
+�
+t
+E(g2
+1|σt(X))E(un(Wt)2|σt(X))
+≤
+Ckn sup
+α∈Cn
+1
+n
+�
+t
+(E(h(Wt) − h0(Wt))2|σt(X)) = O(knδ2
+n) = oP(1).
+Verifying Assumption 4.2 (v). Let A = maxj≤kn Ψj(Xt)2 + 1.
+EA sup
+h∈Cn
+(ρ(Yt+1, h) − ρ(Yt+1, h0))2
+≤
+EA sup
+h∈Cn
+1{−|h − h0| < Ut < |h − h0|} ≤ EA1{− sup
+h∈Cn
+|h − h0| < Ut < sup
+h∈Cn
+|h − h0|}
+=
+EA
+� suph∈Cn |h−h0|
+− suph∈Cn |h−h0|
+fUt|σt(X),Wt(u)du
+≤
+2EA sup
+u fu|σt(X),Wt(u) sup
+Cn
+|h(Wt) − h0(Wt)| ≤ O(δn)EA(1 + Wt)ω/2.
+Finally, Assumption 4.6 is naturally satisfied in the NPQIV model where �Σ(Xt) = Σ(Xt) =
+̟(1 − ̟).
+67
+
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diff --git a/F9AyT4oBgHgl3EQfSveK/content/tmp_files/load_file.txt b/F9AyT4oBgHgl3EQfSveK/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4c3c69c4b19f6b6c8a5f757d1445141b58d1fc10
--- /dev/null
+++ b/F9AyT4oBgHgl3EQfSveK/content/tmp_files/load_file.txt
@@ -0,0 +1,2010 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf,len=2009
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='00092v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='ML]  31 Dec 2022  Inference on Time Series Nonparametric Conditional  Moment Restrictions Using General Sieves  Xiaohong Chen∗  Yuan Liao†  Weichen Wang‡  First draft: September 2020, revised January 3, 2023  Abstract  General nonlinear sieve learnings are classes of nonlinear sieves that can approxi-  mate nonlinear functions of high dimensional variables much more flexibly than various  linear sieves (or series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This paper considers general nonlinear sieve quasi-likelihood ra-  tio (GN-QLR) based inference on expectation functionals of time series data, where the  functionals of interest are based on some nonparametric function that satisfy conditional  moment restrictions and are learned using multilayer neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' While the asymp-  totic normality of the estimated functionals depends on some unknown Riesz representer  of the functional space, we show that the optimally weighted GN-QLR statistic is asymp-  totically Chi-square distributed, regardless whether the expectation functional is regular  (root-n estimable) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This holds when the data are weakly dependent beta-mixing  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We apply our method to the off-policy evaluation in reinforcement learning,  by formulating the Bellman equation into the conditional moment restriction frame-  work, so that we can make inference about the state-specific value functional using the  proposed GN-QLR method with time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, estimating the averaged  partial means and averaged partial derivatives of nonparametric instrumental variables  and quantile IV models are also presented as leading examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Finally, a Monte Carlo  study shows the finite sample performance of the procedure  ∗Cowles Foundation for Research in Economics,  Yale University,  New Haven,  CT 06520, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  xiaohong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='chen@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  †Department of Economics, Rutgers University, New Brunswick, NJ 08901, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' yuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='liao@rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='edu  ‡Faculty of Business and Economics, The University of Hong Kong, Pokfulam Road, Hong Kong  weichenw@hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  1  1  Introduction  Consider a conditional moment restriction model  E[ρ(Yt+1, α0)|σt(X )] = 0 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  where ρ is a scalar residual function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0 = (θ0, h0) contains a finite dimensional parameter θ0  and an infinite dimensional parameter h0, which may depend on some endogenous variables Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The conditioning filtration σt(X ) is the sigma-algebra generated by variables {Xs : s ≤ t},  where Xs is a vector of multivariate (finite dimensional) exogenous variables, including all  relevant lagged variables of Yt and other instrumental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The model therefore allows  for endogenous variables and weakly dependent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  This paper considers optimal estimation and inference for linear functionals φ(α0) of the  infinite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The functional may be either known or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' When it is unknown, it is  assumed to take the form  φ(α0) = El(h0(Wt)) ,  where l is a known linear function and h0(Wt) is the nonparametric function on endogenous  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We use general nonlinear sieve learning spaces, whose complexity grows with the  sample size, to estimate the infinite dimensional parameter, such as multi-layer neural networks  and Gaussian radial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The motivation of using general nonlinear sieve learning space,  besides being adaptive to high dimensional covariates, is that they allow unbounded supports  of the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This is particularly desirable for models of dependent time series data, such  as nonlinear autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We formally establish inferential theories of these functionals learned using the general  nonlinear sieve learning space, and conduct inference using quasi-likelihood ratio (QLR) statis-  tics based on the optimally weighted minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Of particular interest is the estima-  tion of an expectation functional, such as averaged partial means, weighted average derivatives  and averaged squared partial derivatives, of a nonparametric conditional moment restriction  via nonlinear sieve learning sieves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' An important insight from our main theory is that the  asymptotic distribution does not depend on the actual choice of the learning space, but is only  determined by the functional and the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Therefore, estimators produced by either  deep neural networks, Gaussian radial basis, or other nonlinear sieve learning basis, have the  same asymptotic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In general, machine learning inference often relies on sample splitting/ cross-fitting, which  does not work well in the time series setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We propose a new time series efficient inference  2  based on the optimal quasi-likelihood ratio test, without requiring cross-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' It is shown  that the optimally weighted QLR statistic, based on the general nonlinear sieve learning of  h0(), is asymptotically chi-square distributed regardless of whether the information bound for  the expectation functional is singular or not, which can be used to construct confidence sets  without the need to compute standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We present a Monte Carlo study to illustrate  finite sample performance of our inference procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Depending on the specific applications, our model may involve Fredholm integral equation  of either the first kind (NPIV and NPQIV) or the second kind (Bellman equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In the for-  mer case, it is well known that estimating h0 is an ill-posed problem and the rate of convergence  might be slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In the latter case, the problem can be well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' As one of the leading ex-  amples of the Fredholm integral equation of second kind, we show that our framework implies  a natural neural network-based inference in the context of Reinforcement Learning (RL), a  popular learning device behind many successful applications of artificial intelligence such as Al-  phaGo, video games, robotics, and autonomous driving (Sutton and Barto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=',  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Shalev-Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Due to the dynamics of the RL model,  theoretical analysis of reinforcement learning naturally requires to explicitly allow time series  dependency among the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Earlier theoretical studies focused on the settings where  the value function are approximated by linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' More recent developments on non-  linear learning space include Farahmand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Geist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Chen and Qi (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2020), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Our innovation lies in making inference about the functionals (such as the value functional  for specific states) of the Q-function using general nonlinear sieve learning spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' While the  reinforcement learning is based on the well known Bellman equation, it can be formulated as  the conditional moment restriction model with time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Therefore, one can apply  the GN-QLR inference to estimating the state-specific value function in the setting of the  off-policy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' These applications are potentially useful for dynamic causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' case, existing theoretical works on neural networks have focused on deriv-  ing approximation theories and optimal rates of convergence for estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Theoretically,  deep learning has been shown to be able to approximate a broad class of highly nonlinear  functions, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Mhaskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Rolnick and Tegmark (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Schmidt-Hieber (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Yang and Barron (1999) ob-  tained the minimax L2- rate of convergence for neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Recently, Chen, Chen  and Tamer (2021) considered NN efficient estimation of the (weighted) average derivatives in  a NPIV model for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' data, and presented consistent variance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In contrast, using  3  a general theory of Riesz representations, we derive the asymptotic distribution of the finite  dimensional parameter θ0 and functionals of the infinite dimensional parameter h0 that is  learned from the general learning space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The uncertainty of the general nonlinear sieve learn-  ing estimator plays an essential role in the asymptotic distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (2018a,b,c) proposed double machine learning and debias methods to achieve valid inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Dikkala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2020) studied a minimax criterion function to study the unknown functional  approximated by neural networks for NPIV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, the Riesz representation is  playing a central role in our inferential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' See Newey (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Shen (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Chen and Shen  (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2020) for related approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In the time series setting, the neural networks have been applied to economic demand  estimations as in Chen and Ludvigson (2009), and is widely applicable in financial asset pric-  ing such as Guijarro-Ordonez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Bali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' These papers  approximate unknown functions by neural networks, but without rigorous theoretical justifi-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' All these models can be formulated as an inference problem for conditional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Section 2 first introduces the model, the  NN sieve space, the estimation and inference procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Section 3 establishes the converence  rate of the NN sieve estimator for the unknown function satisfying the conditional moment  restrictions with weakly dependent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Section 4 provides the limiting distribution of the  estimator for functionals that can be regular or irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Section 5 reveals that the NN sieve  QLR statistics is asymptotically Chi-square distributed for both the regular and irregular  functionals for time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In Section 6 we apply our approach to the estimation of the value  function of RL and the weighted average derivative of NPIV and NPQIV as leading examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Section 7 contains simulation studies and Section 8 briefly concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  2  The model  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  The General sieve learning space  This paper studies inference with the general nonlinear sieve learning space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The unknown  function is estimated on a learning space, denoted by Hn, is a general approximation space  that consists of either linear or nonlinear sieves, provided that the function of interest can be  approximated well by the learning space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The popular feedforward neural network (NN) is one of the leading examples that fits  into this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Many theoretical studies have shown that NN can well approximate a broad  class of functions and achieves nice statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The multilayer feedforward NN  4  composites functions taking the form:  h(x) = θJ+1hJ(x),  · ·  hj(x) = σ(θjhj−1(x)),  · ·  , h0(x) = x  where the parameters θ = (θ1, · · · , θJ) with θj ∈ Rdj×dj−1 , hj(x) ∈ Rdj, and σ : Rdj → Rdj is  a elementwise nonlinear activation function, usually the same across components and layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  One of the popularly used activation functions is known as ReLU, defined as σ(x) = max(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The number of neurons being used in layer j, denoted by dj, is called the width of that layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We could also use other nonlinear approximation learning spaces, which uses nonlinear  combinations of inputs and neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' One such example is the space spanned by Gaussian  radial bases, which is a multilayer compositions of functions of the form:  h(x) = α0 +  J  �  j=1  αjG(σ−1  j ∥x − γj∥),  α0, αj, γj ∈ R, σj > 0,  where G is the standard normal density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  A key feature is that here inputs and  neurons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', a vector of x) are “nonlinearly combined” as ∥x − γj∥, while they are linearly  combined as indices θjx in the ordinary neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Additional examples of nonlinear  sieves include spline and wavelet sieves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' They are very flexible and enjoy better approximation  properties than linear sieves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  One of the key motivations of using general nonlinear sieve learning space, besides being  adaptive to high dimensional covariates, is that it allows unbounded supports of input covari-  ates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This is particularly desirable for time series models dependent data, such as nonlinear  autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Semiparametric learning  We shall assume a finite-order Markov property: for some known and fixed integer r ≥ 1, let  Xt := (Xt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Xt−r) for all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' define  m(Xt, α)  =  E[ρ(Yt+1, α)|σt(X )],  Σ(Xt)  =  Var(ρ(Yt+1, α0)|σt(X )),  where we assume that E[ρ(Yt+1, α)|σt(X )] and Var(ρ(Yt+1, α0)|σt(X )) only depend on (Xt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Xt−r)  for all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The model is then equivalent to Q(α0) = 0 where  Q(α) = Em(Xt, α)2Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  5  Here we use the optimal weighting function Σ(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Suppose there are nonparametric es-  timators �m(X, α) and �Σ(Xt) for m(Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', α) and Σ(Xt), we then define the sample criterion  function  Qn(α) = 1  n  n  �  t=1  �m(Xt, α)2�Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The estimated optimal weighting matrix is needed for the quasi-likelihood inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In prac-  tice, one can start with the identity weighting function to obtain an initial estimator for α0,  use it to estimate Σ(Xt), then update the estimator using the estimated optimal weighting  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We focus on the general nonlinear sieve learning approximation to the true nonparametric  function, and restrict to the following estimation space:  An := Θ × Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Here Θ is a compact set as the parameter space for θ0 but not necessarily for Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition,  let Pen(h) denote some functional penalty for the infinite dimensional parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We then  define the estimator �α = (�θ,�h) ∈ An as an approximate minimizer of the penalized loss  function restricted to the general nonlinear sieve learning space:  Qn(�α) + λnPen(�h) ≤ inf  α∈An Qn(α) + λnPen(h) + oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The tuning parameter λn is chosen to decay relatively fast, so that the penalization Pen(·)  does not have a first-order impact on the asymptotic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Nevertheless, the functional  penalization is imposed to overcome undesirable properties associated with estimates based  on a large parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Essentially, it plays a role of forcing the optimization to be carried  out within a weakly compact set (Shen, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The functions (x, α) �→ �m(x, α) and x �→ �Σ(x) are nonparametric estimators of (x, α) �→  m(x, α) and x �→ Σ(x) (a positive definite weighting matrix) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The projection  m(Xt, α) can be also estimated using linear sieves:  �m(·, α) = min  m∈Dn  T  �  t=1  [ρ(Yt+1, α) − m(Xt)]2  6  where we consider linear sieve space: let {Ψj : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' , kn} denote a set of sieve bases,  Dn :=  �  g(x) =  kn  �  j=1  πjΨj(x) : ∥g∥∞,ω < ∞, πj ∈ R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  So we use the general nonlinear sieve learning space Hn to approximate the function space  for h0, and a linear sieve space Dn to approximate the instrumental space, which is easier  to implement computationally than using nonlinear sieve approximations to the instrumental  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' A more important motivation of using linear sieve space to estimate the conditional  mean function E[ρ(Yt+1, α)|σt(X )] is that the sample loss function Qn(α) can be shown to  have a local quadratic approximation (LQA): for some Bn = OP(1) and Zn →d N(0, 1),  Qn(α + xun) − Qn(α) = Bnx2 + 2x[n−1/2Zn + ⟨un, α − α0⟩] + oP(n−1)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  uniformly for all α in a shrinking neighborhood of α0 and |x| ≤ Cn−1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' here ⟨un, α − α0⟩ is  some inner product between α − α0 and some function un, to be defined explicitly later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This  LQA plays a fundamental role for the inferential theory of semiparametric inference using  general nonlinear sieve learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3  Semiparametric efficient estimations  Let the parameter space of the true function be H0 and let A0 = Θ × H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We are interested  in the inference of φ(α0), where φ : A0 → R can be a known functional of α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We also study  the inference problem of unknown functionals, taking the form  φ(α0) = El(h0(Wt)) ,  where l(·) is a known function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' While the naive plug-in estimator  1  n  �n  t=1 l(�h(Wt)) is also  asymptotically normal, when the model contains endogenous variables, it is not semipara-  metrically efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  An important example of φ(α0) is the weighted average derivative of  nonparametric instrumental variable regression (NPIV), defined as  φ(α0) = E[Ω(Wt)′∇h0(Wt)] ,  where Ω(·) is a known positive weight function and ∇h0 denotes the gradient of the non-  parametric regression function h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' As documented by Ai and Chen (2012), the simple plug-in  estimator is not an efficient estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' To obtain a more efficient estimator, on the popu-  lation level consider conditional (given Xt) projection of l(h0(Wt)) onto ρ(Yt+1, α0), and the  7  corresponding functional of interest also can be represented as φ(α0) with the functional:  φ(α)  =  E [l(h(Wt)) − Γ0(Xt)ρ(Yt+1, α)] ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  where Γ0(Xt) = E[l(h0(Wt))ρ(Yt+1, α0)|σt(X )]Σ(Xt)−1 is the projection coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We shall  obtain efficient estimator of φ(α0) based on this expectation expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' It is worthy to know  that the added term Γ0(Xt)ρ(Yt+1,α0) is in effect only for endogenous regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In pure  exogeneous models where Wt = Xt, we have Γ0(Xt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In this case the moment condition  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2) reduces to the original one φ(α) = El(h(Wt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let  �φ(α) = 1  n  n  �  t=1  [l(h(Wt)) − �Γtρ(Yt+1, α)]  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3)  for some estimator �Γt to be defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then we estimate the functional by �φ(�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Asymp-  totically, we shall show that  �φ(�α) − φ(α0) = [φ(�α) − φ(α0)] + 1  n  n  �  t=1  [Wt − EWt] + oP(σn−1/2),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4)  where Wt = l(h0(Wt)) − Γ0(Xt)ρ(Yt+1, α0) and σ2 is the asymptotic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' It is clear that  the asymptotic distribution arises from two sources of uncertainties, and importantly, the  nonparametric learning error φ(�α) − φ(α0) plays a first-order role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We shall show that in both known and unknown functional case, estimated φ(α0) is  asymptotically normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We then provide quasi-likelihood inference to construct confidence  intervals for φ(α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  3  Rates of Convergence for Semi-parametric Neural Network  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Weighted function space and sieve learning space  Since the supports of the endogenous variable Wt could be unbounded, we use a weighted  sup-norm metric defined as  ∥h∥∞,ω = sup  s |h(s)|(1 + |s|2)−ω/2,  for some ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  This is known as “admissible weight” which is often used for h0(Wt) when Wt has fat tailed dis-  tribution (Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 of Haroske and Skrzypczak (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Smooth functions with unbounded  support might still be well approximated under the weighted sup-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The L2(W)-norm can  8  be bounded by the weighted sup-norm as: for any function h(w):  ∥h∥2  L2(W ) =  �  h(s)2fW(s)ds ≤ ∥h∥2  ∞,ω  �  (1 + |s|2)ωfW(s)ds,  provided the distribution of the endogenous variable W has as density fW such that fW(s)(1+  |s|2)ω is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We do not consider the overparametrized regime, but impose restrictions on the complex-  ity of the general nonlinear sieve learning space Hn, measured by the “number of parameters”  of the space, denoted by p(Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' More specifically, we impose the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (function and learning space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) The function space: The unknown func-  tion h0 ∈ H0, which is a weighted H¨older ball: for some γ > 0, g ≥ 0,  H0 = {h : ∥h(·)(1 + | · |2)−g/2∥Λγ ≤ c}  where  ∥f∥Λγ = sup  w |f(w)| + max  |a|=d sup  w1̸=w2  |∇af(w1) − ∇af(w2)|  ∥w1 − w2∥γ−d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Also, we require g < ω for ω defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Approximation rate under the ∥∥∞,ω norm:  inf  h∈H0 ∥h0 − h∥∞,ω ≤ cp(Hn)−m  for some m > 0, and some sequence p(Hn) → ∞, p(Hn) log n = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) Complexity: Let N (δ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω) denote the minimal covering number, that is, the  minimal number of closed balls of radius δ with respect to ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω needed to cover Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We  assume, there is a constant C > 0, so that for any δ > 0,  N (δ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω) ≤  �Cn  δ  �p(Hn)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We need to assume that h(w) is smooth in some sense with respect to h(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Condition  (i) is a standard weighted smoothness condition for functions with unbounded support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Here  two weighted norms are being defined, the weighted sup norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω with a weight parameter  ω in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The weighted sup norm intead of the usual sup norm is being considered, as  discussed above, for the purpose of allowing the nonparametric function h(·) to have possibly  9  unbounded support, which is the typical case for autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The other norm is  ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥Λγ for the H¨older ball with a weight parameter g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Here we require g < ω so that the  closure of the function space H0 with respec to the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω is compact, following from  Gallant and Nychka (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In Condition (ii), p(Hn) → ∞ measures the dimension of of the learning space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For mul-  tilayer neural networks with ReLU activation functions, Anthony and Bartlett (2009) showed  that the bound holds with p(Hn) being the pseudo-dimension of the space and is bounded by  CJ2K2 log(JK2), where J and K respectively denote the width and depth of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For finite-dimensional linear sieve, the inequality also holds with p(Hn) being bounded by the  number of sieve bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  When the function h has bounded support, Condition (ii) has been verified for numerous  learning spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For instance, for feed forward multilayer neural networks, Bauer and Kohler  (2019) showed that the approximation rate is n−c, for c =  p  2p+d∗ and p = a + γ, with properly  chosen depth and width of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Importantly, d∗ ≤ dim(Wt) is the “intrinsic dimension”  of the true function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For instance if h0 has a hierarchical interaction structure or multi-  index structure, d∗ is the number of index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' When the function h has unbounded support,  it is known that for linear sieves such as B-splines and wavelets the approximation rate is  m = p(Hn)−γ/ dim(Wt) where p(Hn) is the number of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The approximation rate is however  still an open question for feed forward neural networks in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Ill-posedness  In this section we present the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For simplicity throughout the rest of the  paper, we focus on the case dim(ρ(Yt+1, α)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By the identification condition, Q(α) = 0 if  and only if α = α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So the usual risk consistency refers to Q(�α) = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In the presence of  endogenous variables, the risk consistency however, is not sufficient to guarantee the estimation  consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The latter is often defined under a strong norm:  ∥α1 − α2∥∞,ω := ∥θ1 − θ2∥ + ∥h1 − h2∥∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We first introduce a pseudometric on An that is weaker than ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' To do so, recall the  general Gateaux derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Given generic α = (θ, h) and v = (vθ, vh), let F(x, α) = F(x, θ, h)  be a function that is assumed to be differentiable with respect to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Define  dF(x, α)  dα  [v]  =  ∂F(x, α)  ∂θ  ′  vθ + dF(x, θ, h + τvh)  dτ  ����  τ=0  ,  10  where we implicitly assume dF (x,θ,h+τvh)  dτ  exists at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then the weak norm is defined to be  ∥v∥2 := E  �dm(Xt, α0)  dα  [v]  �2  Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Define πnα0 ∈ An be such that  ∥πnα0 − α0∥∞,ω = min  α∈An ∥α − α0∥∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The following assumption imposes conditions on the local curvature of the criterion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (criterion function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' There are c1, c2 > 0 so that  (i) ∥α − α0∥2 ≤ c1Em(Xt, α)2Σ(Xt)−1 for all α ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Em(Xt, πnα0)2Σ(Xt)−1 ≤ c2∥α0 − πnα0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We now discuss the ill-posedness which reflects the relation between the risk consistency  and estimation consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let the sieve modulus of continuity be  ωn(δ) :=  sup  α∈An:∥α−πnα0∥≤δ  ∥α − πnα0∥∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We say that the problem is ill-posed if δ = o(ωn(δ)) as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The growth of ωn(δ)δ−1  reflects the difficulty of recovering α0 through minimizing the criterion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3  Rates of convergence  Below we present regularity conditions to achieve the rates of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We allow weakly  dependent time series data satisfying β-mixing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Define the mixing coefficient  β(j) := sup  t E sup{|P(B|F t  −∞) − P(B)| : B ∈ F ∞  t+j}  where F t  s denotes the σ-field generated by (Ys+1, Xs), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', (Yt+1, Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3 (Dependences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) {(Yt+1, Xt)}n  t=1 is a strictly stationary and β-mixing  sequence with β(j) ≤ β0 exp(−cj) for some β0, c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) There is a known and finite integer r ≥ 1 so that for each α ∈ An and t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', n, The  conditional expectation E[f(St, α)|σt(X )] depend on σt(X ) only through Xt := (Xt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Xt−r),  for St = (Yt+1, Wt) and  f(St, α) ∈ {ρ(Yt+1, α),  ρ(Yt+1, α0)2,  l(h0(Wt))ρ(Yt+1, α0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  11  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Q(α) = 0 if and only if α = α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, Q(α) is lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The lower semicontinuity of the criteria function is satisfied by the risk function of many  interesting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This condition ensures that it has a minimum on any compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5 (Penalty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) There is M0 > 0, Pen(h) ≤ M0 for all h ∈ Hn ∪ {h0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Pen is lower semicompact on (An, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' {h : Pen(h) ≤ M} is compact for any  M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) kn  n + Q(πnα0) = O(λn) where recall kn is the number of linear sieve bases in Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Define  ǫ(St, α) := ρ(Yt+1, α) − m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  One of the major technical steps is to establish the stochastic equicontinuity for the function  class Ψj(Xt)ǫ(St, α) for β-mixing observations, where α belongs to the class of deep neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' More specifically, we shall derive the bound for, with Ψ(Xt) := (Ψj(Xt) : j ≤ kn):  sup  α∈An:Em(Xt,α)≤r2n  �����  1  √n  n  �  t=1  Ψ(Xt)[ǫ(St, α) − ǫ(St, α0)]  �����  for a given convergence sequence rn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This is achieved under the following Assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' There is C > 0,  (i) There are κ > 0 and C > 0 so that for all δ > 0 and all α1, α2 ∈ An,  max  j≤kn E[Ψj(Xt)2 + 1]  sup  ∥α1−α2∥∞,ω<δ  |ǫ(St, α1) − ǫ(St, α2)|2 ≤ Cδ2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) E maxj≤kn Ψj(Xt)2 supα∈An ρ(Yt+1, α)2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) There is a ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω- neighborhood of α0 on which m(·, α) is continuously pathwise  differentiable with respect to α, and there is a constant C > 0 such that ∥α − α0∥ ≤ C∥α −  α0∥∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Next we present regularity conditions on the linear sieve space Dn used to approximate  the conditional mean function m(X, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7 (Linear sieve space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) There is ϕn → 0 so that uniformly for α ∈ An,  there is kn × 1 vector bα,  E[g(Xt, α) − Ψ(Xt)′bα]2 = O(ϕ2  n),  for all g(Xt, α) ∈ {m(Xt, α), E[l(h0(Wt))ρ(Yt+1, α0)|Xt], dm(Xt,α)  dα  [un], dm(Xt,α)  dα  [un]Σ(Xt)−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  12  (ii) Let Ψn be the n × kn matrix of the linear sieve bases: Ψn = (Ψ(Xt) : t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n)n×kn:  and let A := 1  nEΨ′  nΨn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The linear sieve satisfies: λmin(A) > c and ∥ 1  nΨ′  nΨn − A∥ = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, we apply the pseudo dimension to quantify the complexity of the neural network  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) supx[Σ(x)−1 + Σ(x)] < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also, supx |�Σ(x) − Σ(x)| = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) The distribution of the endogenous variable Wt has a density function fW, which  satisfies  �  w(x)−2fW(x)dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Recall that kn denotes the number of sieve bases being used to estimate the expectation  function m(X, α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' ϕn is the approximation rate in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let  dn  :=  �  p(Hn) log2 n  n  ,  ¯δn  :=  ∥πnα0 − α0∥ +  �  λn +  �  kndn + ϕn,  δn  :=  ∥πnα0 − α0∥∞,ω + ωn(¯δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (Rate of convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8, for any ǫ > 0,  ∥�α − α0∥∞,ω = OP(δn),  Q(�α) = OP(¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The derived rate of convergence is comparable with that of Chen and Pouzo (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In  ¯δn, the term ∥πnα0 − α0∥ is the approximation error on the general nonlinear sieve learning  space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' √λn is the effect of penalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, ϕn and √kndn respectively arise from the  bias and variance of estimating m(X, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In particular, the variance term √kndn depends on  the complexity of the general nonlinear sieve learning space, which arises from the stochastic  equicontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, ωn(¯δn) connects the convergence under the weak norm OP(¯δn) to  the convergence under the strong norm via the sieve modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' When there are  no endogeneity, ¯δn and ωn(¯δn) are of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' General nonlinear sieve spaces with more  complicated structures (with larger “dimension” p(Hn)) have increased covering numbers on  the learning space, and thus lead to slower decays of these two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  4  Asymptotic Distributions for NN Functionals  We now study estimating linear functionals of α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We establish the asymptotically normality  of the estimated functionals formed via pluging-in the general learning estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Riesz representation  A key ingredient of our analysis, as in Chen and Pouzo (2015), relies on representing the  estimation error φ(�α) − φ(α0) using a linear inner product induced from the loss function via  the Riesz representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We define an inner product space as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For any space H, let span{H} denote the closed linear span of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For any v1, v2 in  span(An ∪ {α0}), the linear span of An ∪ {α0}, define the inner product:  ⟨v1, v2⟩ = EΣ(Xt)−1  �dm(Xt, α0)  dα  [v1]  � �dm(Xt, α0)  dα  [v2]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let α0,n ∈ span(An) be such that  ∥α0,n − α0∥ =  min  α∈span(An) ∥α − α0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We note that it is likely α0,n ̸= πnα0 because πnα0 ∈ An, which is not the same as span(An),  when An is a nonlinear NN space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Given Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1, we can focus on shrinking neighborhoods  Aosn  :=  {α ∈ An : ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤ C¯δ2  n}  Cn  :=  {α + xun : α ∈ Aosn, |x| ≤ Cn−1/2},  un := v∗  n/∥v∗  n∥,  ¯Vn  :=  span(Aosn − {α0,n}) ⊂ span(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  for a generic constant C > 0, where v∗  n is the Riesz representer to be defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Because both Aosn and α0,n are functions inside the general nonlinear sieve learning space,  ( ¯Vn, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='⟩) is a finite dimensional Hilbert space under the weak-norm ∥v∥ =  �  ⟨v, v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose  dφ(α0)  dα [v] is a linear functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' As any linear functional on a finite dimensional Hilbert space  is bounded, by the Riesz representation Theorem, there is v∗  n ∈ ¯Vn so that  dφ(α0)  dα  [v] = ⟨v∗  n, v⟩,  ∀v ∈ ¯Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  To appreciate the role of Riesz representation in the semiparametric inference, note that  �α − α0,n ∈ ¯Vn, and we have,  φ(�α) − φ(α0)  =  dφ(α0)  dα  [�α − α0]  =  dφ(α0)  dα  [�α − α0,n] + dφ(α0)  dα  [α0,n − α0]  14  =  ⟨v∗  n, �α − α0,n⟩ + dφ(α0)  dα  [α0,n − α0]  �  ��  �  negligible  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  where the first equality follows from the smoothness condition (Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 below) of the  functional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' the second equality is to the linearity of the functional pathwise derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In  addition, suppose dφ(α0)  dα [α0,n − α0] is negligible, a claim we shall discuss in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 later,  we can then apply the Riesz representation theorem to reach the last line of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition, one of the key technical steps in the proof, by locally expanding the risk  function, is to prove:  √n⟨v∗  n, �α − α0,n⟩ = √n⟨v∗  n, �α − α0⟩ = − 1  √n  �  t  Zt + oP(∥v∗  n∥)  where Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt,α0)  dα  [v∗  n], and ∥v∗  n∥2 = Var( 1  √n  �  t Zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then together we  have  √n(φ(�α) − φ(α0))  ∥v∗n∥  →d N (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Importantly, our inference procedure does not require estimating the Riesz representer  v∗  n or ∥v∗  n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Instead, we propose a quasi-likelihood ratio (QLR) inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We shall provide  regularity conditions in the next section to formalize the above derivations, and subsequently  address estimating the known and unknown functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Asymptotic distributions for known functionals  We have the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (smoothness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) The functional φ is linear in the sense that the functional  φ is linear in the sense that φ(α) − φ(α0) = dφ(α0)  dα [α − α0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) √ndφ(α0)  dα [α0,n − α0] = oP(∥v∗  n∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (iii) requires that the neural network bias term dφ(α0)  dα [α0,n −  α0] should be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Here we present a sufficient condition following the discussion of  Chen and Pouzo (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' First, since α0,n is the projection of α0 on to span(An) and v∗  n ∈ ¯Vn ⊂  span(An), we have ⟨v∗  n, α0,n−α0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, define an infinite dimensional Hilbert space  ¯V as the closure of the linear span of A − {α0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose dφ(α0)  dα [·] is bounded, then there is a  unique Riesz representer v∗ ∈ ¯V so that  dφ(α0)  dα  [v] = ⟨v∗, v⟩,  ∀v ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  15  As α0,n − α0 ∈ ¯V , we have  ����  √ndφ(α0)  dα  [α0,n − α0]  ���� =  ��√n⟨v∗ − v∗  n, α0,n − α0⟩  �� ≤ √n∥v∗ − v∗  n∥∥α0,n − α0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  So condition (iii) holds as long as √n∥v∗ − v∗  n∥∥α0,n − α0∥ = oP(∥v∗  n∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  To allow quantile applications that involve nonsmooth loss functions, we need to show  that the sample criterion function Qn(α) can be replaced with a smoothed criterion �Qn(α) :=  1  n  �  t ℓ(Xt, α)2�Σ(Xt)−1, where Ψn = (Ψ(Xt) : t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n)n×kn:  ℓ(x, α) := �m(x, α) + �m(x, α0),  �m(x, α) := Ψ(x)′(Ψ′  nΨn)−1Ψ′  nmn(α),  and mn(α) denotes the n × 1 vector of m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The replacement error is negligible:  sup  α∈Aosn  sup  |x|≤Cn−1/2 |Qn(α + xun) − �Qn(α + xun)| = oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Therefore, theoretical analysis of Qn(α) is asymptotically equivalent to that of �Qn(α), while  the latter is second-order pathwise differentiable, and admits a local quadratic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Formalizing this argument would require the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' m(x, t) is twice differentiable with respect to t, and there is C > 0, so that,  recall that un = v∗  n/∥v∗  n∥ being the “normalized Riesz representer”:  (i) E|ρ(Yt+1, α0)|2+ζ ���dm(Xt,α0)  dα  [un]  ���  2+ζ  + E|ρ(Yt+1, α0)|2+ζ < C for some ζ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) E supα∈Cn sup|τ|≤Cn−1/2 1  n  �  t  �  d2  dτ 2m(Xt, α + τun)|  �2  < C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) supτ∈(0,1) supα∈Cn E  �  d2  dτ 2m(Xt, α0 + τ(α − α0))  �2  = o(n−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iv) kn supα∈Cn  1  n  �  t[ dm(Xt,α)  dα  [un] − dm(Xt,α0)  dα  [un]]2 = oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (v) E  �  [maxj≤kn Ψj(Xt)2+1] supα∈Cn(ρ(Yt+1, α)−ρ(Yt+1, α0))2�  < Cδ2η  n for some κ, η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, we need to strengthen conditions on the penalty and some rates of convergence  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) Let Ch := {h : (θ, h) ∈ Cn for some θ ∈ Θ}, which is the local neighbor-  hood for the estimated h(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We assume  λn sup  h∈Ch  |Pen(h) − Pen(h0)| + λn sup  h∈Ch  |Pen(πnh) − Pen(h0)| = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  16  (ii) √n¯δn∥�Σn − Σn∥ = o(1), where �Σn and Σn be the diagonal matrix of �Σ(Xt) and Σ(Xt) for  all t, and furthermore ϕ2  n¯δ2  n + knd2  nδ2η  n + √kndnδη  n¯δn = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The following condition is similar to Condition C in Shen (1997), which is used to control  the approximation error of the NN space for locally perturbed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' There is µn → 0 so that µn¯δn = o(n−1), we have  sup  α∈Cn  1  n  n  �  t=1  [m(Xt, πnα) − m(Xt, α)]2 = OP(µ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (Limiting distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4,  √nφ(�α) − φ(α0)  ∥v∗  n∥  →d N (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  An important insight from this theorem is that the asymptotic distribution does not  depend on the actual choice of the learning space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The asymptotic variance  ∥v∗  n∥2 = EΣ(Xt)−1  �dm(Xt, α0)  dα  [v∗  n]  �2  is only determined by the functional forms φ and m(X, α), and more generally, the loss  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So whether the multilayer neural network, B-spline, Gaussian radial basis, etc, are  being used to estimate α0, the asymptotic distribution is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' What really matters is  the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3  Estimation for unknown functionals  We now consider estimating unknown (probably not √n-estimable) functionals, taking the  form  γ0 := El(h0(Wt)),  where l(·) is a known function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Ai and Chen (2012) used the following moment condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  to construct the optimal criterion function:  γ0 = EWt,  Wt = l(h0(Wt)) − Γ(Xt)ρ(Yt+1, α0),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  where Γ(Xt) = E[l(h0(Wt))ρ(Yt+1, α0)|σt(X )]Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' They showed that estimating γ0 based  on this moment condition leads to more efficient estimator than based on the naive plug-in  method 1  n  �  i l(�h(Wt)), whenever Wt is endogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Because the naive plug-in estimator does  17  not take into account the potential correlations between the moment functions m(Xt, α) and  l(h(Wt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Using the more efficient moment condition of γ0, and letting  φ(α) := El(h(Wt)) − EΓ(Xt)ρ(Yt+1, α),  we note that φ(α0) = γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose the functional φ(·) were known, and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  continues to hold for φ(α), then we can show  √n(φ(�α) − φ(α0)) ≈ − 1  √n  n  �  t=1  Zt,  Zt := ρ(Yt+1, α0)Σ(Xt)−1dm(Xt, α0)  dα  [v∗  n],  where v∗  n is the Riesz representer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  But we in fact are facing a problem of estimating an  unknown functional φ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' To do so, we first estimate Γ(Xt) by  �Γt :=  n  �  s=1  l(�h(Ws))ρ(Ys+1, �α)φ(Xs)′(Ψ′  nΨn)−1Ψ(Xt)�Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then define the final estimator:  �γ := �φ(�α), where �φ(α) = 1  n  n  �  t=1  [l(h(Wt)) − �Γtρ(Yt+1, α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3)  The following asymptotic expansion holds for the estimated functional:  �γ − γ0  =  [φ(�α) − φ(α0)] + 1  n  n  �  t=1  [Wt − EWt] + oP(σn−1/2)  =  1  n  n  �  t=1  [−Zt + Wt − EWt] + oP(σn−1/2)  where Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt,α0)  dα  [v∗  n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This explicitly presents two leading sources for  the asymptotic distribution, where the asymptotic variance is given by  σ2 := 1  n Var  � n  �  t=1  (Wt − Zt)  �  = 1  n Var  � n  �  t=1  Wt  �  + ∥v∗  n∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4)  where Wt and Zt are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We impose the following conditions  18  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) supx |Γ(x)|2 + supw suph∈Hn l(h(w))2 < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) l(h) is linear in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) E supα∈Cn |l(h(Wt)) − l(h0(Wt))|2 ≤ Cδ2η  n , where for simplicity we assume the same  η as in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5 regulates the approximation quality of the instrumental space using linear  sieves, which is not stringent since E(l(h(Wt))ρ(Yt+1, α)|σt(X )) is a function of the instrumen-  tal variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The next assumption imposes a condition on the accuracy of estimating the optimal  weighting function Σ(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For the NPQIV model this assumption is trivially satisfied since  �Σ(Xt) = Σ(Xt) = ̟(1 − ̟) is known (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3 for the definition of ̟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We shall verify  it for the NPIV model in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' There is a sequence pn so that pn¯δnσ = o(n−1) and  1  n  �  t  Γ(Xt)Σ(Xt)(�Σ(Xt)−1 − Σ(Xt)−1)ρ(Yt+1, α0) = OP(pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The asymptotic normality requires some rate restrictions, which we impose below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) There is c0 > 0 so that σ2 > c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Let νn := δη  n supx |�Σ(x) − Σ(x)| + √kndnδη  n + ϕ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then νn¯δnσ = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4 hold for φ(α) = El(h(Wt)) − EΓ(Xt)ρ(Yt+1, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition, Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  √nσ−1(�γ − γ0) →d N (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  5  Quasi-Likelihood Ratio Inference for Functionals  As shown by Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, computing the asymptotic variance requires estimating  Riesz representer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' While Chen and Pouzo (2015) and Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2018c) proposed  framework of estimating the Riesz representer, the task is in general quite challenging when  its does not have closed-form approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In this section we propose to make inference  directly using the optimally weighted quas-likelihood ratio statistic (QLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  QLR Inference for known functionals  Consider testing  H0 : φ(α0) = φ0  for some known φ0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Consider the restricted null space AR  n := {α ∈ An : φ(α) = φ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The  GN-QLR statistic is defined as  Sn(φ0) = n  �  Qn(�αR) − Qn(�α)  �  where �αR ∈ AR  n approximately minimizes the penalized loss function over the general nonlinear  sieve learning restricted on the null space:  Qn(�αR) + λnPen(�hR) ≤ inf  α∈AR  n  Qn(α) + λnPen(α) + oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Define  πR  n α = arg  min  b∈An,φ(b)=φ0 ∥b − α∥∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) Recall µn as defined in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' It also satisfies:  sup  α∈Aosn,φ(α)=φ0  1  n  n  �  t=1  [m(Xt, πR  n (α + xun)) − m(Xt, α + xun)]2 = OP(µ2  n)  (ii) (1 + ∥v∗  n∥) supα∈Cn |φ(πnα) − φ(α)| = o(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The following theorem shows the asymptotic null distribution of Sn(φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 and Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then under  H0 : φ(α0) = φ0,  Sn(φ0) →d χ2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  QLR inference for unknown functionals  We now move on to the inference for the unknown functional γ0 := El(h0(Wt)), which is  estimated by �γ as defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Consider testing  H0 : El(h0(Wt)) = φ0  20  for some known φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Define  Ln(α, γ) := Qn(α) + (�φ(α) − γ)2�Σ−1  2 ,  where �Σ2 consistently estimates the long-run variance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Newey and West (1987)):  Σ2 := Var  �  1  √n  n  �  t=1  Wt  �  = 1  n  n  �  t=1  Var(Wt) + 1  n  �  t̸=s  cov(Wt, Ws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We recall that Wt = l(h0(Wt)) − Γ(Xt)ρ(Yt+1, α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Note that (�α, �γ) is numerically equivalent to the solution to the following problem:  Ln(�α, �γ) + λnPen(�h) ≤ inf  α∈An min  γ  Ln(α, γ) + λnPen(h) + oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We define the GN-QLR statistic as  �Sn(φ0) = n  �  Ln(�αR, φ0) − Ln(�α, �γ)  �  ,  where �αR ∈ AR  n approximately minimizes the penalized loss function in the learning space  Hn, but fixing γ = φ0:  Ln(�αR, φ0) + λnPen(�hR) ≤ inf  α∈An Ln(α, φ0) + λnPen(α) + oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The asymptotic analysis of �Sn(φ0) is rather sophisticated, which requires additional rate  constraints stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose �Σ2 − Σ2 = oP(1)Σ2 and conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then under  H0 : γ0 = φ0  �Sn(φ0) →d χ2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  6  Examples  In this section, we illustrate our main results using three important models: Reinforcement  learning, NPIV and NPQIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We impose premitive conditions to verify the high level Assump-  tions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 respectively in the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  21  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Reinforcement learning  Reinforcement learning (RL) has been an important learning device behind many successes in  applications of artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Theories of RL have been developed in the literature of  statistical learning and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Most of the existing theoretical works formulate the  problem as a least-square regression and approximate the value function by a linear function,  such as Bradtke and Barto (1996), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Nonlinear approximations using kernel methods or  deep learning appeared in the more recent literature, for example Farahmand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Geist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Chen and Qi  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2020) also conducted inference for the optimal policy using linear sieve  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We proceed learning using neural networks, and study the inference for a given policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We follow the recent literature on the off-policy evaluation problem, and formulate the rein-  forcement learning problem as a conditional moment restriction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Assume the observed  data trajectory {(St, At, Rt)}t≥0 is obtained from an unknown behavior policy probability  πb(a|s), where (St, At, Rt) denote the state, action and observed reward at time t respectively  and πb(a|s) is the distribution to take action a at state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We denote the space of states and  actions as S and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' It is assumed that the reward Rt is jointly determined by (St, At, St+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Standing at state St at period t, one takes action At and receives reward Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The state then  transits to St+1 at the next period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The value of a given policy π is measured by the so-called Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Specifically, for  any given π and any state-action pair (s, a), Q-function is defined as the expected discounted  reward:  Qπ(s, a) =  ∞  �  t=0  γtEπ(Rt|S0 = s, A0 = a) ,  where Eπ or in short E is the expectation when we take actions according to π, 0 ≤ γ < 1 is  the discount factor and we consider the discounted infinite-horizon sum of expected rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  To estimate Qπ, a classical approach is to solve the Bellman equation below:  Qπ(s, a) = E  �  Rt + γ  �  x∈A  π(x|St+1)Qπ(St+1, x)dx  ����St = s, At = a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The goal is to recover Qπ of a given target policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In practice, multiple trajectories  {(Si,t, Ai,t, Ri,t, Si,t+1)}0≤t≤T,1≤i≤N may be observed to help estimate the Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  But  for simplicity we assume N = 1 and T = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The more general case can be cast by merging  the N time series into a single series of size n = TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  22  The Bellman equation can be formulated as a conditional moment restriction with respect  to Qπ for weakly dependent time series:  E[ρ(Yt+1, Qπ)|St, At] = 0,  Yt+1 = (Rt, St, At, St+1),  Xt = (St, At),  where  ρ(Yt+1, h) = Rt − h(St, At) + γ  �  x∈A  π(x|St+1)h(St+1, x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In this framework, the estimation of the function Qπ(s, a) can be conducted on the neural  network space, and we assume that computationally the integration in the ρ-function can be  well approximately by the Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For off-policy evaluations, the following value  function is of the major interest in this section: given state s ∈ S,  φs(Qπ) =  �  a∈A  π(a|s)Qπ(s, a)da,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  which is a known functional φs(·) for a single state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The Bellman equation also admits a Fredholm integral equation of the second kind (Kress,  1989), which is a well-posed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Therefore, estimating the Q-function may achieve fast-  rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' That is, the sieve modulus of continuity satisfies:  ωn(δ) :=  sup  α∈An:∥α−πnα0∥≤δ  ∥α − πnα0∥s ≍ δ  Recently Chen and Qi (2022) showed this result for ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥s to be either the sup-norm or the  ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The inner product is defined, in this case, as ⟨v1, v2⟩ = EΣ(Xt)−1 � dm  dh [v1]  � �dm  dh [v2]  �  ,  where  dm  dh [v] = γ  �  x∈A  E [π(x|St+1)v(St+1, x)|St, At] dx − v(St, At),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  and induced a Riesz representer v∗ whose closed form is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Meanwhile, it follows  from the Bellman equation that m(Xt, h) = dm  dh [h − Qπ] for all h ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Therefore, the weak  norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥ can be expressed as:  ∥h − Qπ∥2 = Em(Xt, h)2Σ(Xt)−1,  which shows that the employed minimum distance criterion function is directly estimating the  squared weak norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let �Qπ be the estimated Qπ using the general nonlinear learning space, and the functional  23  is naturally estimated using  φs( �Qπ) =  �  a∈A  π(a|s) �Qπ(s, a)da  As the moment restriction function E[ρ(Yt+1, h)|St, At] is linear in h in this case, it is straight-  forward to verify the high-level conditions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) For some ζ > 4, the Riesz representer satisfies  E  �  π(x|St+1)|v∗  n(St+1, x)|ζdx + E|v∗  n(St, At)|ζ ≤ ∥v∗  n∥ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) ER4  t < ∞, E maxj≤kn Ψj(Xt)4 < ∞, E(1+|St|2 +|At|2)2ω < ∞ and EM(St+1)4 < ∞,  where M(St+1) :=  �  π(x|St+1)(1 + x2 + S2  t+1)ω/2dx, and ω is the degree of the weighted-sup  metric ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For the Reinforcement Learning model considered here, Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  implies Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  It then follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 that  ∥v∗  n∥−1√n  �  φs( �Qπ) − φs(Qπ)  �  →d N (0, 1)  Inference about φs(Qπ) based on pivotal statistics can be conducted using the GN-QLR test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  The NPIV model  In the nonparametric instrumental variable model (NPIV), consider  yt+1 = h0(Wt) + Ut+1,  E(Ut+1|σt(X )) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  where σt(X ) is the filtration generated from instrumental variables Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then m(Xt, α) =  E[(yt+1−h(Wt))|σt(X )] and the Gateaux derivative is defined as dm(Xt,α)  dh  [v] = E(v(Wt)|σt(X )),  implying  ⟨un, h − h0⟩ = E  �  E(un(Wt)|σt(X ))E(h − h0|σt(X ))Σ(Xt)−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We estimate the conditional variance Σ(Xt) by �Σt = �A′  nΨn(Ψ′  nΨn)−1Ψ(Xt) where �An is a n×1  vector of ρ(Yt+1, �α)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Recall that for δn and ¯δn defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2),  ∥�h − h∥∞,ω = OP(δn),  ∥�h − h∥ = OP(¯δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We impose the following low-level conditions to verify Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  24  Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) δ2  n¯δnσ = o(n−1), E maxj≤kn |Ψj(Xt)|2(U2  t +1) < C, and E(U2  t |σt(X )) <  C almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also, E(1 + |Wt|2)ω < C and E maxj≤kn Ψj(Xt)2(1 + |Wt|2)ω < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) The Riesz representer v∗  n satisfies: there are C, ζ > 0,  E(maxj≤kn Ψj(Xt)2 + 1)v∗  n(Wt)2 < CEK2  t and E|Ut|2+ζ|Kt|2+ζ ≤ C(EK2  t )1+ζ/2, where  Kt := E(v∗  n(Wt)|σt(X )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For the NPIV model,  (i) Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 implies Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) For the known functional φ(·), in addition Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  √n(φ(�α) − φ(α0))  σn  →d N (0, 1),  where σ2  n := Var (E(v∗  n(Wt)|σt(X ))Σ(Xt)−1Ut) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) For the unknown functional γ0 = El(h0(Wt)), if additionally Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7  hold, then  √nv−1(�γ − γ0) →d N (0, 1),  where v2 := 1  n Var(�  t Wt−Zt) with Wt = l(h0(Wt))−Γ0(Xt)Ut+1 and Zt = Ut+1Σ(Xt)−1E[v∗  n(Wt)|σt(X )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3  The NPQIV model  Consider the nonparametric quantile instrumental variable (NPQIV) model  E[1{yt+1 ≤ h0(Wt)}|σt(X )] = ̟ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then m(Xt, α) = P(Ut+1 < h−h0|σt(X ))−̟ where Ut+1 = yt+1 −h0(Wt) and α = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Within  this framework, we now verify the high-level assumptions presented in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Suppose the conditional distribution of Ut given (Xt, Wt) is absolutely continuous with  density function fUt|σt(X),Wt(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In this context, Σ(Xt) is known, given by  Σ(Xt) = Var(1{yt+1 ≤ h0(Wt)}|σt(X )) = ̟ − ̟2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then the Gateaux derivative is defined as  dm(Xt, α)  dh  [v] = E(fUt|σt(X),Wt(h(Wt) − h0(Wt))v(Wt)|σt(X )),  25  implying, for g1 = fUt|σt(X),Wt(0)un(Wt) and g2 = fUt|σt(X),Wt(0)(h(Wt) − h0(Wt)),  ⟨un, h − h0⟩ = E [E(g1|σt(X ))E(g2|σt(X ))] (̟ − ̟2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Also, ∥v∗  n∥2 = (̟ − ̟2)−1Eg(Xt)2 where g(Xt) = E[fUt|σt(X),Wt(0)v∗  n(Wt)|σt(X )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We impose the following low-level conditions to verify Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let  At(v)  :=  � 1  0  fUt|σt(X),Wt (x(v(Wt) − h0(Wt))) dx  Bt(v, h)  :=  E {At(v)[h(Wt) − h0(Wt)]|Xt} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) There are c1, c2, ǫ0 > 0 so that for all ∥h − h0∥∞,ω < ǫ0,  c2EBt(h, h)2Σ(Xt)−1 ≤ EBt(h0, h)2Σ(Xt)−1 ≤ c1EBt(h, h)2Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Almost surely, supu f  ′  Ut|σt(X),Wt(u) < C and supu,x,w fUt|σt(X),Wt(u) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also and  there is L > 0, for all u, almost surely, supx,w |fUt|σt(X),Wt(u) − fUt|σt(X),Wt(0)| ≤ L|u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) E[maxj≤kn Ψj(Xt)2 + At(h0)2](1 + |Wt|2)ω < C and E[un(Wt)4|σt(X )] < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iv) δ2  nkn = o(1) and δ4  n = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The following proposition, proved in the appendix, is the main result in this subsection,  which verifies the high-level conditions in the NPQIV context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For the NPQIV model,  (i) Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3 implies Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) For the known functional φ(·), in addition Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  √n(φ(�α) − φ(α0))  σn  →d N (0, 1),  where σ2  n := (̟ − ̟2)−1E  �  [EfUt|σt(X),Wt(0)v∗  n(Wt)|σt(X )]2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) For the unknown functional γ0 = El(h0(Wt)), if additionally Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7  hold, then  √nv−1(�γ − γ0) →d N (0, 1),  where v2 := 1  n Var(�  t Wt − Zt) with Wt = l(h0(Wt)) − Γ0(Xt)Ut+1 and  26  Zt = (̟ − ̟2)−11{Ut+1 ≤ 0}EfUt|σt(X),Wt(0)v∗  n(Wt)|σt(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  7  Simulation Studies  In this section, we set up nonparametric endogenous models to illustrate the performance  of our proposed estimators and testing statistics using some synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Consider the  following data generating process  Yt = h(Zt, Yt−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' , Yt−L) + et ,  where  h(Zt, Yt−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' , Yt−L) = Ztϑ0 + f(  L  �  l=1  blYt−l) ,  and φ(α) = E[∂h/∂Zt] = ϑ0 = 1 is the quantity to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We choose L = 3, bl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4l  and consider the nonlinear mapping f(x) = 1−exp(−x)  1+exp(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The endogenous Zt is generated using  the following auto-regressive model:  Zt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3Zt−1 + ut,  (ut, εt) ∼iid N(0, Σ),  Σ =  �  1  ρ  ρ  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  And et is generated with the following ARCH model using εt as the innovation:  et = σtεt,  σ2  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='32)Z2  t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We set ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5 to make Zt endogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We also make et heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that E[e2  t] =  E[σ2  t ] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The endogenous variable is Wt = Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The instruments are Xt = (Zt−1, Yt−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Yt−L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We chose to generate n = 5000 samples (some burning period has been thrown away to make  sure data are stationary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that the model can be used for both NPIV and NPQIV with  ̟ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We applied a fully-connected J-layer ReLU-activated NN with hidden layer width of  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The optimization of the unconstrained NPIV or NPQIV objective used vanilla gradient  descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We did not apply mini-batch in gradient descent training as using mini-batches may  hurt performance due to insufficient smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The training epoch was as large as 10000 with  learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='01 for NPIV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 for NPQIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Furthermore we did not apply any penalty  term for this example since the problem is relatively easy and the NN under consideration is  of a small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The linear sieve bases (Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Ψkn) for the instrumental variable space were  ˜kn cubic B-splines for X and each of the three Y lags concatenated together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For simplicity,  no interaction terms between X and Y lags were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Thus in total, we have kn = 4˜kn −3  27  Table 1: Estimation and hypothesis testing under NPIV and NPQIV with synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Here (J, K, kn) respectively denote the number of layers, width of the neural nets and the  number of sieve bases for estimating the instrumental space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The true value for ϑ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' 95%  qtl refers to the empirical 95% quantile, where the theoretical quantile for the chi square  distribution is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Layer  Width  Basis  Estimator of ϑ0  Testing Statistic for ϑ0  Problem  J  K  kn  mean  std  mean  std  95% qtl  size  NPIV  3  10  17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='968  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='116  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='999  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='432  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='814  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='0%  NPIV  3  10  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='957  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='874  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='236  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='727  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8%  NPIV  1  40  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='984  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='032  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='418  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='215  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='0%  NPQIV  3  10  49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='997  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='129  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='086  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='565  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='280  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4%  NPQIV  3  10  45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='130  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='409  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='955  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6%  NPQIV  1  40  29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='977  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='126  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='421  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='678  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='9%  bases (since all B-spline bases sum up to 1, we remove the last basis for each dimension and  finally add the intercept term as another basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In our simulations, we find that NPQIV  requires more number of sieve basis kn for estimating the instrumental space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For the NPIV problem, we first optimize the equal weighted quadratic loss to obtain �h,  which is used to estimate Σ(Xt) and Γ(Xt) consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In the second step, we optimize the  optimally weighted quadratic loss with the weighting matrix �Σ(Xt)−1 and apply the forward  filter to estimate our expectation functional, which in this example is the constant ϑ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, we carry out the hypothesis testing for H0 : φ(h) = E[∂h/∂Zt] = φ0 = 1 to check  the size of the testing statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Specifically, we estimated the forward filtered residuals as  �  Wt = ∂�h(Wt)/∂Wt−�Γt(Yt−�h(Wt)) and estimated Σ2 = Var(Wt) by the Newey-West estimator  given �  Wt, then solved the constrained optimization of Ln(h, φ0) and finally constructed the  testing statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For NPQIV problem, since the optimal weighting is proportional to equal  weighting, we do not need the initial step to estimate Σ(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So we directly optimized the  optimally weighted quadratic loss and estimated Γ(Xt) using the results and then used the  forward filter to correct the estimation of the average partial derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Finally, similar to  NPIV, we conduct the hypothesis testing for H0 : φ(α) = 1 under NPQIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  As for the computational practice, we find that for NPQIV models, it is helpful to apply  truncations to the learned gradients in each step of training the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Specifically, we  smooth the loss function of the NPQIV model and truncate the updated gradient:  θk+1 = θk − lr ∗ min{|∇Ln,k|, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='001} ∗ sgn(∇Ln,k)  28  where lr is the learning rate, fixed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 for NPQIV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' ∇Ln,k is the gradient of the NN  at the current step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' θk+1 is the updated neural network coefficients at the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The  truncation prevents the network from having very large gradients during iterations, helping  stabilize the training process empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We repeat each setting for 1000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For the efficient estimation, we report the mean  and standard deviation of the forward filtered average gradient for the optimal weighting  optimizaiton in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For hypothesis testing, we also report in Table 1 the mean, standard  deviation and 95% quantile of the empirical testing statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition, if we use the  theoretical critical value corresponding to 5% significance level, which is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='84 for χ2  1, the  p-value is also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  As we can see from Table 1, for NPIV, optimal weighting estimates φ(h) accurately in  the sense that the mean insignificantly differs from the true value ϑ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' NPQIV is less  efficient with a larger standard deviation, and thus requires more samples to be estimated to  the same accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that the instrumental space with a step function can be harder to  approximate with the cubic B-spline linear sieve bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In terms of the performance of QLR  testing statistic, the p-values are all close to the nominal 5% level for the NPIV and NPQIV  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Admittedly through our experiments the results can be sensitive to some tuning  parameters, which is typically the case when applying deep learning for statistical inference:  at the moment we still heavily rely on ad-hoc tuning in many problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In comparison, the  estimation of φ(h) is more stable with respect to different J and K values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Here we only mean  to present some results without heavily tuning the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Methods using NN for real  applications require more extensive tuning in practice and some rough sense on the model  complexity would be useful to determine the balance between the dimensions of the NN sieve  and the linear IV sieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  8  Conclusion  In this paper we establish neural network estimation and inference on functionals of unknown  function satisfies a general time series conditional moment restrictions containing endogenous  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We consider quasi-likelihood ratio (GN-QLR) based inference, where nonparametric  functions are learned using multilayer neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' While the asymptotic normality of the  estimated functionals depends on some unknown Riesz representer of the functional space, we  show that the GN-QLR statistic is asymptotically Chi-square distributed, regardless whether  the expectation functional is regular (root-n estimable) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This holds when the data are  weakly dependent and satisfy the beta-mixing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition to estimating partial derivatives in nonparametric endogenous problems as  29  examples, our study is well motivated by the setting of reinforcement learning where data are  time series in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We apply our method to the off-policy evaluation, by formulating the  Bellman equation into the conditional moment restriction framework, so that we can make  inference about the state-specific value functional using the proposed GN-QLR method with  time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  30  A  Stochastic equicontinuity on the NN space for β-mixing obser-  vations  A key technical result is the stochastic equicontinuity of the residual function on the general nonlinear  sieve learning space, which is established in the following proposition in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let St =  (Yt+1, Xt) and  ǫt(α) ≡ ǫ(St, α) := ρ(Yt+1, α) − m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We derive bounds that require the pseudo dimension of the deep neural network class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Recall  δn := ∥πnα0 − α0∥∞,ω + ωn(¯δn),  ¯δ2  n := ∥πnα0 − α0∥2 + λn + knd2  n + ϕ2  n  where dn :=  �  p(Hn) log2 n  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let Cn = {α + xun : α ∈ An, ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤ C¯δ2  n, |x| ≤ Cn−1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Suppose :  (a) E maxj≤kn Ψj(Xt)2 supα∈Cn(ρ(Yt+1, α) − ρ(Yt+1, α0))2 = Cδ2η  n for some η, C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (b) For some κ, C > 0 , EΨj(Xt)2 sup∥α1−α∥∞,ω<δ |ǫt(α1) − ǫt(α)|2 ≤ Cδ2κ for all δ > 0 and  α, α1 ∈ cl{a + xb : a, b ∈ An, x ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then  max  j≤kn  sup  |x|≤Cn−1/2 sup  α∈Cn  | 1  n  �  t  Ψj(Xt)(ǫ(St, α + xun) − ǫ(St, α0))| ≤ OP (dnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let E := {(ǫ(, α + xun) − ǫ(, α0))Ψj : α ∈ Cn, j ≤ kn, |x| ≤ Cn−1/2} and let St = (Yt+1, Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 1: construct blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Consider the following independent blocks: for any integer pair  (an, bn), with bn = [n/(2an)], divide {St : t ≤ n} into 2bn blocks with length an and the remaining  block of length n − 2anbn:  H1,l  =  {i : 2(l − 1)an + 1 ≤ i ≤ (2l − 1)an}  H2,l  =  {i : (2l − 1)an + 1 ≤ i ≤ 2lan}  where l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let Υ = {i : 2anbn + 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Now let {�S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', �Sn} be a random sequence that  is independent of {S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='., Sn} and has independent blocks such that each block has the same joint  distribution as the corresponding block of the St-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Because the St-sequence is β-mixing, by  Lemma 2 of Eberlein (1984), for any measurable set A, with the mixing coefficient β(),  |P  �  {�St : t ∈ H1,l, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', bn} ∈ A  �  − P ({St : t ∈ H1,l, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', bn} ∈ A) | ≤ (bn − 1)β(an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  31  The same inequality holds when H1,l is replaced with H2,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, for any function f, define  U1,f(�Sl) = 1  an  �  t∈H1,l  f(�St),  U2,f(�Sl) = 1  an  �  t∈H2,l  f(�St),  where �Sl = {�St : t ∈ H1,l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  By construction, U1,f(�Sl) and U2,f(�Sl) are independent across l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Similarly, let Sl = {St : t ∈ H1,l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  1  n  �  t  f(St) − Ef(St)  =  1  n  �  t∈Υ  f(St) − Ef(St) + 1  bn  �  l≤bn  anbnn−1[U1,f(Sl) − EU1,f(Sl)]  + 1  bn  �  l≤bn  anbnn−1[U2,f(Sl) − EU2,f(Sl)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  Next, we shall bound each term on the right hand side uniformly for f ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We replace U1,f(Sl)  with U1,f(�Sl);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' the latter is easier to bound because blocks �Sl are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We then show that  the effect of such replacements is negligible due to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1) by properly chosen (an, bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 2: the envelop function for U1,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that Ef = 0 for f ∈ E and that �St and St are  identically distributed within each block H1,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By Cauchy-Schwarz,  E sup  f∈E  U1,f(�Sl)2  ≤  E sup  f∈E  \uf8eb  \uf8ed 1  an  �  t∈H1,l  f(�St)  \uf8f6  \uf8f8  2  ≤ 1  an  �  t∈H1,l  E sup  f∈E  f(St)2  ≤  2E max  j≤kn Ψj(Xt)2  sup  |x|≤Cn−1/2 sup  α∈Cn  (ρ(Yt+1, α + xun) − ρ(Yt+1, α0))2 ≤ Cδ2η  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Now take some p > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let F = {U1,f : f ∈ E} and let F := max{n−p, supf∈E |U1,f|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then both  supf∈E |U1,f| and F are envelope functions of F, and  n−p ≤ G := ∥F∥L2(St) ≤ Cn−p + Cδη  n ≤ 2Cδη  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 3: the bracketing number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We aim to apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 of van der Vaart and Wellner  (1996) to bound  1  bn  �  l≤bn anbnn−1U1,f(�Sl), which requires bounding the bracketing number of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  To do so, suppose h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', hN is a δ-cover of Hn under the norm ∥h∥∞,ω and N := N(δ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', θR is a δ-cover of Θ and R := N(δ, Θ, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥) (the Euclidean norm in Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Here N(δ, A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=') de-  notes the covering number for space A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also let x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='xMn be a δ-cover of [−Cn−1/2, Cn−1/2], with  Mn ≤ 4Cn−1/2/δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then for any f = (ǫ(, α + xun) − ǫ(, α0))Ψj ∈ E, there are Ψj, xq and αik = (θk, hi) so that  32  ∥α−αik∥∞,ω ≤ ∥h−hi∥∞,ω +∥θ−θk∥ ≤ 2δ and |x−xq| < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let fijkq = (ǫ(, αik +xqun)−ǫ(, α0))Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  sup  f=(ǫ(,α+xun)−ǫ(,α0))Ψj:∥α−αik∥∞,ω<2δ,|x−xq|<δ  |U1,f(�Sl) − U1,fijkq(�Sl)|  ≤  sup  f=(ǫ(,α+xun)−ǫ(,α0))Ψj:∥α−αik∥∞,ω<2δ,|x−xq|<δ  | 1  an  �  t∈H1,l  f(�St) − fijkq(�St)|  ≤  1  an  �  t∈H1,l  |Ψj( �  Xt)|  sup  ∥α−αik∥∞,ω<2δ  sup  |x−xq|<δ  |ǫ(�St, α + xun) − ǫ(�St, αik + xqun)| := bijkq(�Sl, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then U1,f ∈ [lijkq, uijkq], where lijkq := U1,fijkq −bijkq(, δ) and uijkq = U1,fijkq +bijkq(, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition,  E[uijkq − lijkq]2  ≤  4Ebijkq(�Sl, δ)2  ≤  CE  \uf8eb  \uf8ed 1  an  �  t∈H1,l  |Ψj( �  Xt)|  sup  |x−xq|<δ  sup  ∥α−αik∥∞,ω<2δ  |ǫ(�St, α + xun) − ǫ(�St, αik + xqun)|  \uf8f6  \uf8f8  2  ≤  CEΨj( �  Xt)2  sup  ∥α−αik∥∞,ω<2δ  sup  |x−xq|<δ  |ǫ(�St, α + xun) − ǫ(�St, αik + xqun)|2 ≤ Cδ2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence {[lijkq, uijkq] : i ≤ N, j ≤ kn, k ≤ R} is a Cδκ bracket of F, whose bracketing number satisfies  N[](Cδκ, F, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥L2(�St)) ≤ N(δ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω)  �  ��  �  N  (C/δ)d  � �� �  R  (n−1/2/δ)  �  ��  �  Mn  kn,  where we used R ≤ (C/δ)d for d = dim(θ0) since θ0 ∈ Θ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then for a generic constant  C > 0,  N[](Gx, F, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥L2(�St)) ≤ CN(x1/κ(G/C)1/κ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω)G−(d+1)/κx−(d+1)/κkn,  ∀x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 4: bound independent blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that U1,f(�Sl) are independent across l and is  mean-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For the envelop G defined in step 2 and some constant ¯  M > 0,  E sup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(�Sl)  ������  ≤ 1  2E sup  g∈F  ������  1  bn  �  l≤bn  g(�Sl)  ������  ≤(i)  b−1/2  n  G  � 1  0  �  1 + log N[](Gx, F, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥L2(�St))dx  ≤  C  √bn  δη  n  � 1  0  �  1 + log N(x1/κ(G/C)1/κ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞) + log  C  n1/2G(d+1)/κx(d+1)/κ + log kndx  ≤(ii)  Cδη  n  √bn  � 1  0  �  1 + p(Hn) log  Cn  x1/κG1/κ + (d + 1) log  C  G1/κx1/κ + log kndx  ≤(iii)  Cδη  n  √bn  � 1  0  �  2 log kn + 2p(Hn) log  Cn  x1/κG1/κ dx  33  ≤(iv)  δη  n  �  Cp(Hn) log n  bn  ,  where (i) follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 of van der Vaart and Wellner (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (ii) follows from As-  sumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (iii) is due to p(Hn) → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We now prove the inequality (iv), which is to show  � 1  0  �  2 log kn + g(x)dx ≤ C  �  p(Hn) log n  where g(x) = 2p(Hn) log  Cn  x1/κG1/κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let A := 2 log kn + 2p(Hn) log  Cn  G1/κ − 2κ−1p(Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We have  log Cn  G → ∞, hence 2κ−1p(Hn) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that log(y) ≤ y − 1 for all y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence  2 log kn + g(x)  =  2 log kn + 2p(Hn) log Cn  G1/κ + 2κ−1p(Hn) log 1  x  ≤  2 log kn + 2p(Hn) log Cn  G1/κ + 2κ−1p(Hn)(1  x − 1)  =  A + 2κ−1p(Hn)x−1 ≤ A + Ax−1 ≤ 2Ax−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The last inequality holds for x < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Thus with n−10 ≤ G, and kn = O(bn),  � 1  0  �  2 log kn + g(x)dx  ≤  √  2A  � 1  0  x−1/2dx ≤ 4  �  2 log kn + 2p(Hn)[log(Cn) + log G−1/κ]  ≤  4  �  2 log kn + 2p(Hn)[log(Cn) + log n10/κ] ≤ C  �  p(Hn) log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Therefore by the Markov inequality, for any ε > 0, with probability at least 1 − ε/4,  sup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(�Sl)  ������  ≤ cn  ε ,  cn = δη  n  �  Cp(Hn) log n  bn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 5: completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1) and step 4,  P  \uf8eb  \uf8edsup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(Sl)  ������  > cn  ε  \uf8f6  \uf8f8 ≤ P  \uf8eb  \uf8edsup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(�Sl)  ������  > cn  ε  \uf8f6  \uf8f8+(bn−1)β(an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We now take an = M log n/2 with M > 0 and bn = [n/(M log n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then (bn − 1)β(an) → 0 for  sufficiently large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also, the requirement in step 4 that p(Hn) = o(bn) holds as long as p(Hn) log n =  o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence with this choice of bn,  sup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(Sl)  ������  = OP  \uf8eb  \uf8edδη  n  �  p(Hn) log2 n  n  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The same rate applies when U1,f is replaced with U2,f following from the same proof of steps 2,3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  34  In addition, |Υ|0 ≤ 2an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence  E sup  f∈E  �����  1  n  �  t∈Υ  f(St) − Ef(St)  �����  ≤  E 1  n  �  t∈Υ  sup  f∈E  |f(St)| ≤ Can  n E max  j≤kn |Ψj(Xt)| sup  α∈Cn  |ǫt(α) − ǫt(α0)|  ≤  Cδη  n log n  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Together, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2) maxj≤kn supα∈Cn | 1  n  �  t Ψj(Xt)(ǫt(St, α) − ǫ(St, α0))| = OP (δη  ndn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  B  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Consistency  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (Consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose kn  n + Q(πnα0) = O(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also suppose Pen(h) is lower semi-  compact on (Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω) and Q(α) is lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then ∥�α − α0∥∞,ω = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The proof of this lemma does not depend on Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' First we show Pen(�h) = OP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let ρn(α), mn(α) be the n × 1 vectors of ρ(Yt+1, α) and m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let �Σ−1  n  be the diagonal matrix  of �Σ(Xt)−1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By steps 1, 3 of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 below,  λPen(�h)  ≤  Qn(πnα0) + λPen(πnh0) + oP (n−1)  ≤  2  n  �  t  [ �m(Xt, πnα0) − �m(Xt, πnα0)]2�Σ(Xt)−1 + CE �m(Xt, πnα0)2 + λPen(πnh0) + oP(n−1)  ≤  2[ρn(πnα0) − mn(πnα0)]′Pn�Σ−1  n Pn[ρn(πnα0) − mn(πnα0)]  +CEm(Xt, πnα0)2 + λPen(πnh0) + oP (n−1)  ≤  OP (kn  n + Q(πnα0) + λ) = OP (λ)  with the condition that kn  n +Q(πnα0) = O(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So let M0 > 0 be a large constant so that Pen(�h) ≤ M0  with probability arbitrarily close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Now take an arbitrary ǫ > 0, let Bǫ = {α = (θ, h) ∈ An : ∥α − α0∥∞,ω ≥ ǫ, Pen(h) ≤ M0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Be-  cause Pen(h) is lower semicompact on (H0, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω) and Q(α) is lower semicontinuous, minα∈Bǫ Q(α)  exists, that is, there is α∗ ∈ Bǫ so that infα∈Bǫ Q(α) = Q(α∗) > c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' If ∥�α − α0∥∞,ω > ǫ, then  Q(�α) ≥ infα∈Bǫ Q(α) > c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Meanwhile, by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1) (to be proved below),  c0 ≤ Q(�α) ≤ Q(πnα0) + λn|Pen(πnh0) − Pen(�h)| + OP (knd2  n + ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  But the right hand side is oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence we must have ∥�α − α0∥∞,ω = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  35  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  The proof depends on some important technical lemmas, one of which is the stochastic equicontinuity  of ǫ(St, α) = ρ(Yt+1, α) − m(Xt, α), given by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We divide the proof in the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let Dn be the sieve space used to estimate  m(X, α), and  �m(X, α) = arg min  �m∈Dn  n  �  t=1  (m(Xt, α) − �m(Xt))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We show the following steps:  step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Show that for c, C > 0, uniformly in α ∈ An,  cE �m(Xt, α)2 ≤ 1  n  n  �  t=1  �m(Xt, α)2 ≤ CE �m(Xt, α)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  To prove it, we shall apply an empirical identifiability result that first proved by Huang (1998)  for the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' case and then extended by Chen and Christensen (2015) to more general setting with  a much simpler proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We note that �m(·, α) ∈ Dn := {g(x) = �kn  j=1 πjΨj(x) : ∥g∥∞,ω < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let  Ψn be the n × kn matrix of the linear sieve bases, and let A := 1  nEΨ′  nΨn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose the linear sieve  satisfies: λmin(A) > c and ∥ 1  nΨ′  nΨn − A∥ = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then ∥A−1/2 1  nΨ′  nΨnA−1/2 − I∥ = oP(1), so the  conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 of Chen and Christensen (2015) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We then apply this lemma  to reach that  sup  α∈An  | 1  n  �  t �m(Xt, α)2 − E �m(Xt, α)2|  E �m(Xt, α)2  ≤ sup  g∈Dn  | 1  n  �  t g(Xt)2 − Eg(Xt)2|  Eg(Xt)2  = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  This then leads to the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Show that  sup  α∈An  1  n  n  �  t=1  [ �m(Xt, α) − �m(Xt, α)]2 = OP (knd2  n),  d2  n := p(Hn) log2 n  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let ǫ(St, α) = ρ(Yt+1, α) − m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also let Pn = Ψn(Ψ′  nΨn)−1Ψ′  n and ¯ǫn(α) be the n × 1  vector of ǫ(St, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We then have  sup  α∈An  1  n  n  �  t=1  [ �m(Xt, α) − �m(Xt, α)]2 = sup  α∈An  1  n¯ǫn(α)′Pn¯ǫn(α) = OP (1) sup  α  ∥ 1  nΨ′  n¯ǫn(α)∥2  ≤  OP (kn) sup  α max  j≤kn | 1  n  n  �  t=1  Ψj(Xt)ǫ(St, α)|2 = OP (knd2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  36  The last bound is given by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Show that supα∈An E[ �m(Xt, α) − m(Xt, α)]2 = O(ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let �mn(α) and mn(α) respectively be the n × 1 vectors of �m(Xt, α) and m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also let  mn(α) = Ψnbα + rα where rα is the sieve approximation error and bα is the sieve coefficient to  approximate mn(X, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then �mn(α) = Pnmn(α) and  sup  α∈An  E[ �m(Xt, α) − m(Xt, α)]2 = 1  n sup  α∈An  Emn(α)′(I − Pn)mn(α) = 1  n sup  α∈An  Er′  α(I − Pn)rα  ≤  1  n sup  α E∥rα∥2 = OP (ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  After achieving the above three steps, then we have (since �Σ(Xt)−1 and Σ(Xt)−1 are bounded  away from zero)  Qn(�α)  ≥  c  n  �  t  �m(Xt, �α)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5c  n  �  t  �m(Xt, �α)2 − c  n  �  t  [ �m(Xt, �α) − �m(Xt, �α)]2  ≥(i)  cE �m(Xt, �α)2 − OP (knd2  n) ≥(ii) cEm(Xt, �α)2 − OP (knd2  n + ϕ2  n) ≥ Q(�α) − OP (knd2  n)  Qn(πnα0)  ≤  C  n  �  t  �m(Xt, πnα0)2 ≤ 2C  n  �  t  �m(Xt, πnα0)2 + 2C  n  �  t  [ �m(Xt, πnα0) − �m(Xt, πnα0)]2  ≤(iii)  CE �m(Xt, πnα0)2 + OP (knd2  n) ≤(iv) CEm(Xt, πnα0)2 + OP (knd2  n + ϕ2  n)  ≤  Q(πnα0) + OP (knd2  n + ϕ2  n)  where (i) (iii) follow from steps 1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (ii) (iv) follow from step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence Qn(�α) + λnPen(�h) ≤ Qn(πnα0) + λnPen(πnh0) + oP (n−1) implies  Q(�α) ≤ Q(πnα0) + λn|Pen(πnh0) − Pen(�h)| + OP (knd2  n + ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  Now by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2,  ∥�α − α0∥2 ≤ C∥πnα0 − α0∥2 + OP (λn + knd2  n + ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence ∥�α − πnα0∥ ≤ ∥�α − α0∥ + ∥πnα0 − α0∥ ≤ C∥πnα0 − α0∥ + OP (√λn + √kndn + ϕn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Thus  ∥�α − α0∥∞,ω  ≤  ∥�α − πnα0∥∞,ω + ∥πnα0 − α0∥∞,ω  ≤  OP (∥πnα0 − α0∥∞,ω + ωn(∥πnα0 − α0∥ +  �  λn +  �  kndn + ϕn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose  (a) E maxj≤kn Ψj(Xt)2 supα∈An ρ(Yt+1, α)2 ≤ C2  37  (b) There are κ > 0 and C > 0 so that EΨj(Xt)2 sup∥α1−α2∥∞,ω<δ |ǫ(St, α1) − ǫ(St, α2)|2 ≤ Cδ2κ  holds for any δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (c) p(Hn) → ∞ and p(Hn) log n = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then supα maxj≤kn | 1  n  �n  t=1 Ψj(Xt)ǫ(St, α)| = OP (  �  p(Hn) log2 n  n  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let E := {ǫ(·, α)Ψj : α ∈ An, j ≤ kn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 1: construct blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This step is the same as that of the proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 2: the envelop function for U1,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that Ef = 0 for f ∈ E and that �St and St are  identically distributed within each block H1,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By Cauchy-Schwarz,  E sup  f∈E  U1,f(�Sl)2  ≤  E sup  f∈E  \uf8eb  \uf8ed 1  an  �  t∈H1,l  f(�St)  \uf8f6  \uf8f8  2  ≤ 1  an  �  t∈H1,l  E sup  f∈E  f(St)2  ≤  2E max  j≤kn Ψj(Xt)2 sup  α∈An  ρ(Yt+1, α)2 ≤ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let F = {U1,f : f ∈ E} and let F := max{n−10, supf∈E |U1,f|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then both supf∈E |U1,f| and F are  envelope functions of F, and  n−10 ≤ G := ∥F∥L2(St) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 3: the bracketing number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We aim to apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 of van der Vaart and Wellner  (1996) to bound  1  bn  �  l≤bn anbnn−1U1,f(�Sl), which requires bounding the bracketing number of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  To do so, suppose h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', hN is a δ-cover of Hn under the norm ∥h∥∞,ω and N := N(δ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', θR is a δ-cover of Θ and R := N(δ, Θ, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥) (the Euclidean norm in Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Here N(δ, A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=') denotes  the covering number for space A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then for any f = ǫ(, α)Ψj ∈ E, there are Ψj and αik = (θk, hi) so that ∥α − αik∥∞,ω ≤  ∥h − hi∥∞,ω + ∥θ − θk∥ ≤ 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let fijk = ǫ(·, αik)Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We have  sup  f=ǫ(·,α)Ψj:∥α−αik∥∞,ω<2δ  |U1,f(�Sl) − U1,fijk(�Sl)| ≤  sup  f=ǫ(·,α)Ψj:∥α−αik∥∞,ω<2δ  | 1  an  �  t∈H1,l  f(�St) − fijk(�St)|  ≤  1  an  �  t∈H1,l  |Ψj( �  Xt)|  sup  ∥α−αik∥∞,ω<2δ  |ǫt(�St, α) − ǫt(�St, αik)| := bijk(�Sl, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then U1,f ∈ [lijk, uijk], where lijk := U1,fijk − bijk(, δ) and uijk = U1,fijk + bijk(, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition,  E[uijk − lijk]2  ≤  4Ebijk(�Sl, δ)2  ≤  CE  \uf8eb  \uf8ed 1  an  �  t∈H1,l  |Ψj( �  Xt)|  sup  ∥α−αik∥∞,ω<2δ  |ǫt(�St, α) − ǫt(�St, αik)|  \uf8f6  \uf8f8  2  38  ≤  CEΨj( �  Xt)2  sup  ∥α−αik∥∞,ω<2δ  |ǫ(�St, α) − ǫ(�St, αik)|2 ≤ Cδ2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence {[lijk, uijk] : i ≤ N, j ≤ kn, k ≤ R} is a Cδκ bracket of F, whose bracketing number satisfies  N[](Cδκ, F, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥L2(�St)) ≤ N(δ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω)  �  ��  �  N  (C/δ)d  � �� �  R  kn,  where we used R ≤ (C/δ)d for d = dim(θ0) since θ0 ∈ Θ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then for a generic constant  C > 0,  N[](Gx, F, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥L2(�St)) ≤ CN(x1/κ(G/C)1/κ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞,ω)G−d/κx−d/κkn,  ∀x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 4: bound independent blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that U1,f(�Sl) are independent across l and is  mean-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For the envelop G defined in step 2 and some constant ¯  M > 0,  E sup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(�Sl)  ������  ≤ 1  2E sup  g∈F  ������  1  bn  �  l≤bn  g(�Sl)  ������  ≤(i)  b−1/2  n  G  � 1  0  �  1 + log N[](Gx, F, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥L2(�St))dx  ≤  C  √bn  � 1  0  �  1 + log N(x1/κ(G/C)1/κ, Hn, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥∞) + log  C  Gd/κxd/κ + log kndx  ≤(ii)  C  √bn  � 1  0  �  1 + p(Hn) log  Cn  x1/κG1/κ + d log  C  G1/κx1/κ + log kndx  ≤(iii)  C  √bn  � 1  0  �  2 log kn + 2p(Hn) log  Cn  x1/κG1/κ dx ≤(iv)  �  Cp(Hn) log n  bn  ,  where (i) follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 of van der Vaart and Wellner (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (ii) follows from As-  sumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (iii) is due to p(Hn) → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (iv) follows from the same proof as that of Proposition  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 5: completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By an inequality similar to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1) and step 4,  P  \uf8eb  \uf8edsup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(Sl)  ������  > cn  ε  \uf8f6  \uf8f8 ≤ P  \uf8eb  \uf8edsup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(�Sl)  ������  > cn  ε  \uf8f6  \uf8f8+(bn−1)β(an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We now take an = M log n/2 with M > 0 and bn = [n/(M log n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then (bn − 1)β(an) → 0 for  sufficiently large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also, the requirement in step 4 that p(Hn) = o(bn) holds as long as p(Hn) log n =  39  o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence with this choice of bn,  sup  f∈E  ������  1  bn  �  l≤bn  anbnn−1U1,f(Sl)  ������  = OP  \uf8eb  \uf8ed  �  p(Hn) log2 n  n  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The same rate applies when U1,f is replaced with U2,f following from the same proof of steps 2,3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition, |Υ|0 ≤ 2an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence  E sup  f∈E  �����  1  n  �  t∈Υ  f(St) − Ef(St)  �����  ≤  2E 1  n  �  t∈Υ  sup  f∈E  |f(St)| ≤ Can  n E max  j≤kn |Ψj(Xt)| sup  α  |ǫ(St, α)| ≤ C log n  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Together, maxj≤kn supα∈An | 1  n  �  t Ψj(Xt)ǫ(St, α)| = OP  ��  p(Hn) log2 n  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  C  Proofs for Section 4  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Local quadratic approximation  Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (LQA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let Cn = {α + xun : |x| < Cn−1/2, α ∈ An, ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤  C¯δ2  n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose for un = v∗  n/∥v∗  n∥, there are C > 0, so that  (a) √n¯δn∥�Σn − Σn∥ = o(1), ϕ2  n¯δ2  n + knd2  nδ2η  n + √kndnδη  n¯δn = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (b)  1  √n∥(I − Pn)Σ−1  n  dmn(α)  dα  [un]∥ +  1  √n∥(I − Pn)dmn(α)  dα  [un]∥ = OP (ϕn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (c) kn supα∈Cn  1  n  �  t[dm(Xt,α)  dα  [un] − dm(Xt,α0)  dα  [un]]2 = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (d) conditions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (e) supτ∈(0,1) supα∈Cn E  �  d2m(Xt,α0+τ(α−α0))  dτ 2  |  �2  = o(n−1) and  (f) E supα∈Cn sup|τ|≤Cn−1/2 1  n  �  t  �  d2  dτ 2 m(Xt, α + τun)|  �2  = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then  sup  α∈Aosn  sup  |x|≤Cn−1/2 |Qn(α + xun) − Qn(α) − An(α(x))| = oP(n−1),  where  (a1) An(α(x)) := 2x[n−1/2Zn + ⟨un, α − α0⟩] + Bnx2  (a2) Bn = 1  n  dmn(α0)  dα  [un]′Σ−1  n  dmn(α0)  dα  [un] →P 1, and  (a3) Zn →d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  40  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let �Qn(α) = 1  n  �  t ℓ(Xt, α)2�Σ(Xt)−1, and  ℓ(x, α) := �m(x, α) + �m(x, α0),  �m(x, α) := Ψ(x)′(Ψ′  nΨn)−1Ψ′  nmn(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 1: expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By assumption, �Qn(α) is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So we shall prove the LQA for  �Qn(α) via the mean value theorem, and show that �Qn(α)−Qn(α) is “small” locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Indeed, Lemma  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 shows that supα∈Cn |Qn(α) − �Qn(α)| = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We write f(s) := �Qn(α+sxun) and by the second order mean value theorem, for some s ∈ (0, 1),  �Qn(α + xun) − �Qn(α) = f ′(0) + 1  2f ′′(s) = 2xG(α) + x2Bx + x2Dx,  G(α)  :=  1  n  �  t  ℓ(Xt, α)�Σ(Xt)−1 d �m(Xt, α)  dα  [un]  Bx  :=  1  n  �  t  �d �m(α + sxun)  dα  [un]  �2  �Σ(Xt)−1  Dx  :=  1  n  �  t  ℓ(α + sxun)�Σ(Xt)−1 d2  dτ 2 �m(α + τxun)|τ=s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 shows that uniformly Dx = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence sup|x|≤Cn−1/2 x2|Dx| = oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 2: convergence of Bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let dmn(α)  dα  [un] and ρn be the n × 1 vectors of dm(Xt,α)  dα  [un] and  ρ(Yt+1, α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also let ∥v∥2  Σ := v′Σ−1  n v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Write Bn := 1  n∥dmn(α0)  dα  [un]∥2  Σ = OP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Uniformly for α(x), s,  Bx − Bn  ≤  1  n∥d �mn(α + sxun)  dα  [un]∥2  �Σn − 1  n∥d �mn(α0)  dα  [un]∥2  �Σn  + 1  n∥d �mn(α0)  dα  [un]∥2  �Σn − 1  n∥dmn(α0)  dα  [un]∥2  �Σn  + 1  n∥dmn(α0)  dα  [un]∥2  �Σn − 1  n∥dmn(α0)  dα  [un]∥2  Σn = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence sup|x|≤Cn−1/2 |Bx − Bn|x2 = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' To show that Bn →P 1, we have  Bn = ⟨un, un⟩ +  � 1  n  dmn(α0)  dα  [un]′Σ−1  n  dmn(α0)  dα  [un] − ⟨un, un⟩  �  = ⟨un, un⟩ + oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt,α0)  dα  [v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then for each v,  ∥v∥2  =  Var( 1  √n  �  t  Zt) = Var(Zt) + 2  n  �  s>t  EZtE(Zs|σs(X)) = Var(Zt) = ⟨v, v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence ⟨un, un⟩ = ⟨v∗  n, v∗  n⟩∥v∗  n∥−2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence Bn = 1 + oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 3: expansion of G(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We have supα∈Cn  1  √n∥mn(α)′Pn∥+supα∈Cn  1  √n∥mn(α)∥ = OP (¯δn)  41  and (√n¯δn + √kn)∥�Σn − Σn∥ = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='G(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nmn(α)′Pn�Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='d �mn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nρ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nPn�Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='d �mn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nmn(α)′PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='d �mn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nρ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nPnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='d �mn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + oP (n−1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nmn(α)′PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nρ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nPnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nmn(α)′PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n (Pn − I)dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nρ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nPnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n (Pn − I)dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + oP (n−1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nmn(α)′PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nρ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nPnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + OP (ϕn¯δn) + oP (n−1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nmn(α)′Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nρ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='nPnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dmn(α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='dα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='[un] + oP (n−1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='=(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='⟨un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α − α0⟩ + 1  nρ′  nPnΣ−1  n  dmn(α)  dα  [un] + oP(n−1/2)  =(b)  ⟨un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α − α0⟩ + 1  nρn(α0)′Σ−1  n  dmn(α0)  dα  [un]  �  ��  �  1  √nZn  +oP (n−1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  where (a) follows from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (b) is due to  �  Eρn(α0)′Pnρn(α0)  �  sup  α∈Cn  1  n  �  t  [dm(Xt, α)  dα  [un] − dm(Xt, α0)  dα  [un]]2  ≤  �  EtrPnΣ(Xt)−1  �  sup  α∈Cn  1  n  �  t  [dm(Xt, α)  dα  [un] − dm(Xt, α0)  dα  [un]]2 = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 4: weak convergence of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' It then remains to show Zn →d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that  Zn =  1  √n  �  t  Zt∥v∗  n∥−1,  Zt = ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt, α0)  dα  [v∗  n],  where un = v∗  n/∥v∗  n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' When s > t, we have Zt ∈ σs(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence E(ZtZs|σs(X)) = ZtE(Zs|σs(X)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Thus  Var( 1  √n  �  t  Zt)  =  Var(Zt) + 2 1  n  �  s>t  EE(ZtZs|σs(X)) = Var(Zt)  =  E Var  �  ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt, α0)  dα  [v∗  n]  ����σt(X)  �  =  EΣ(Xt)−2  �dm(Xt, α0)  dα  [v∗  n]  �2  Var(ρ(Yt+1, α0)|σs(X))  =  ⟨v∗  n, v∗  n⟩ = ∥v∗  n∥2  where we used Var(ρ(Yt+1, α0)|σs(X)) = Σ(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  42  Next, it is assumed that there is some ζ > 0,  E|Zt∥v∗  n∥−1|2+ζ ≤ CE|ρ(Yt+1, α0)|2+ζ  ����  dm(Xt, α0)  dα  [un]  ����  2+ζ  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition, Zt is strictly stationary, satisfying the β-mixing condition (Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let α(n)  denote the α-mixing coefficient (the strong mixing coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We have that, by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3,  α(n) ≤ 1  2β(n) ≤ C exp(−cn) for some c, C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence  ∞  �  n=1  α(n)ζ/(2+ζ) ≤ C  ∞  �  n=1  exp(−cζn/(2 + ζ)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7 of Ibragimov (1962), Zn →d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let �Qn(α) = 1  n  �  t ℓ(Xt, α)2�Σ(Xt)−1, and  ℓ(x, α) := �m(x, α) + �m(x, α0),  �m(x, α) := Ψ(Xt)′(Ψ′  nΨn)−1Ψ′  nmn(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Suppose knd2  nδ2η  n + √kndnδη  n¯δn = o(n−1) and  1  √n∥mn(πnα) − mn(α)∥¯δn ≤ o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then for Cn = {α + xun : α ∈ An : ∥α − α0∥∞,ω ≤ Cδn, Q(α) ≤ C¯δ2  n, |x| ≤ Cn−1/2},  (i)  sup  α∈Cn  |Qn(α) − �Qn(α)| = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii)  sup  α∈Cn  |Qn(α) − Qn(πnα)| = oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) Recall that ǫt(α) = ρ(Yt+1, α) − m(Xt, α) and mn(α), ¯ǫn(α) and ρn(α) are n × 1 vectors  of m(Xt, α), ǫt(α) and ρ(Yt+1, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also write α(x) := α + xun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Qn(α + xun) − �Qn(α + xun) = 1  n  �  t  [ �m(Xt, α(x))2 − ℓ(Xt, α(x))2]�Σ(Xt)−1  =  1  n[¯ǫn(α + xun) − ¯ǫn(α0)]′Pn�Σ−1  n Pn[¯ǫn(α + xun) − ¯ǫn(α0) + 2mn(α + xun) + 2ρn(α0)]  ≤  OP (1) 1  n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥2 + OP (1) 1  n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥∥Pnmn(α + xun)∥  +OP(1) 1  n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥∥Pnρn(α0)∥  ≤  OP (d2  1 + d1 × d2 + d1 × d3)  d1  :=  1  √n∥Pn[¯ǫn(α + xun) − ¯ǫn(α0)∥  d2  :=  1  √n∥Pnmn(α + xun)∥,  d3 :=  1  √n∥Pnρn(α0)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  43  We shall respectively calculate d1 ∼ d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1, d1 = OP (√kndnδη  n) uniformly in α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  As for d2, by steps 1 and 3 in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1, uniformly in α(x),  d2  2 ≤ 1  n  �  t  �m(Xt, α(x))2 ≤ CE �m(Xt, α(x))2 ≤ C(ϕ2  n + Em(Xt, α(x))2) ≤ C¯δ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, d2  3 = OP (kn  n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Together, Qn(α+xun)− �Qn(α+xun) ≤ OP (knd2  nδ2η  n +√kndnδη  n¯δn) = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Let mn(α) and �mn(α) respectively be the n × 1 vectors of m(Xt, α) and �m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' First,  1  √n∥ �mn(πnα) − �mn(α)∥ ≤  1  √n∥mn(πnα) − mn(α)∥ ≤ OP (µn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Second, 1  n∥ �mn(α)∥2 ≤ OP (¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Third,  1  √n∥ �mn(α0)∥ = OP (1)  �  1  nρn(α0)′Pnρn(α0) = OP (  �  kn  n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence for �Qn(α) = 1  n[ �mn(α) + �mn(α0)]′�Σ−1  n [ �mn(α) + �mn(α0)], we have  �Qn(α) − �Qn(πnα)  ≤  OP (1) 1  √n∥ �mn(πnα) − �mn(α)∥  � 1  √n∥ �mn(πnα) − �mn(α)∥ +  1  √n∥ �mn(α)∥ + 1  √n∥ �mn(α0)∥  �  ≤  OP (µn¯δn) = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, by part (i) | �Qn(α) − Qn(α)| = oP(n−1) uniformly in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose supτ∈(0,1) supα∈Cn E  �  d2m(Xt,α0+τ(α−α0))  dτ 2  |  �2  = o(n−1) and  E supα∈Cn sup|τ|≤Cn−1/2 1  n  �  t  �  d2  dτ 2 m(Xt, α + τun)|  �2  = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then uniformly for α ∈ Cn,  (i) supα∈Cn sup|s|≤1,|x|≤Cn−1/2  ��� 1  n  �  t ℓ(Xt, α + sxun)�Σ(Xt)−1 d2  dτ 2 �m(Xt, α + τxun)|τ=s  ��� = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) supα∈Cn  √n| 1  nmn(α)′Σ−1  n  dmn(α0)  dα  [un] − ⟨un, α − α0⟩| = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) We have that  ��� 1  n  �  t ℓ(Xt, α + sxun)�Σ(Xt)−1 d2  dτ 2 �m(Xt, α + τxun)|τ=s  ���  2  ≤ OP (1)AB where  A := 1  n  �  t  ℓ(Xt, α + sxun)2,  B := 1  n  �  t  d2  dτ 2 �m(Xt, α + τxun)|2  τ=s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let mn and ρn denote the n × 1 vectors of m(Xt, ·) and ρ(Yt+1, α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Uniformly for α ∈ Cn,  A ≤ 2  n∥Pnmn(α + sxun)∥2 + 2  n∥Pnρn∥2 = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We have B ≤ OP (1)E supα∈Cn sup|τ|≤Cn−1/2 | d2  dτ 2 m(Xt, α + τun)|2 = OP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  44  (ii) By the second order mean value theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' for some ξ ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  1  nmn(α)′Σ−1  n  dmn(α0)  dα  [un] = 1  n  �  t  [m(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α) − m(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)]Σ(Xt)−1 dm(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)  dα  [un]  =  1  n  �  t  f(Xt) − Ef(Xt) + E[m(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α) − m(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)]Σ(Xt)−1 dm(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)  dα  [un]  =  1  n  �  t  f(Xt) − Ef(Xt) + Edm(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)  dα  [α − α0]Σ(Xt)−1 dm(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)  dα  [un]  +1  2Ed2m(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0 + τ(α − α0))  dτ 2  |τ=ξΣ(Xt)−1 dm(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)  dα  [un]  =  1  n  �  t  f(Xt) − Ef(Xt) + ⟨un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α − α0⟩ + o(n−1/2) = ⟨un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α − α0⟩ + oP (n−1/2)  where f(Xt) = [m(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α) − m(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' α0)]Σ(Xt)−1 dm(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='α0)  dα  [un] and the last equality follows from  sup  f∈En  | 1  √n  �  t  (f(Xt) − Ef(Xt))| = oP (1)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  with En := {m(Xt, α)Σ(Xt)−1 dm(Xt,α0)  dα  [un] : α ∈ Cn} and that m(Xt, α0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By the Riesz representation Theorem, there is v∗  n ∈ cl{An − α0}  dφ(α0)  dα  [�α − α0] = ⟨v∗  n, �α − α0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Next, we show √n⟨un, �α − α0⟩ →d N(0, 1), or more precisely, for Zn →d N(0, 1),  Zn + √n⟨un, �α − α0⟩ = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 implies that for any ǫ > 0, there is C > 0 so that with probability at least  1 − ǫ, �α ∈ Aosn := {α ∈ An : Q(α) ≤ C¯δ2  n, ∥α − α0∥∞,ω ≤ Cδn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We now condition on this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  By Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,  sup  α∈Aosn  sup  |x|≤Cn−1/2 |Qn(α + xun) − Qn(α) − An(α(x))| = oP (n−1),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3)  where An(α(x)) := 2x[n−1/2Zn + ⟨un, α − α0⟩] + Bnx2 with Bn = OP (1), Zn →d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Write  un = (uγ, uh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Now let ∆n be such that sup|x|≤Cn−1/2 |Pen(πn(�h + xuh)) − Pen(�h)| = OP (∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  En := λnPen(πn(�h + xuh)) − λnPen(�h) = OP (λn∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  45  Now by definition, πn(�α + xun) ∈ An, hence  0  ≤  Qn(πn(�α + xun)) − Qn(�α) + En  ≤  Qn(�α + xun) − Qn(�α) + En + |Qn(�α + xun) − Qn(πn(�α + xun))|  ≤  Qn(�α + xun) − Qn(�α) + En + oP (n−1)  ≤  2x[n−1/2Zn + ⟨un, �α − α0⟩] + Bnx2 + En + oP (n−1),  where the third inequality follows from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 and the last inequality follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  By the assumption λn∆n = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence there is ηn = o(n−1), so that  0 ≤ x[n−1/2Zn + ⟨un, �α − α0⟩] + Bnx2 + OP (ηn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  From n1/2ηn = o(n−1/2), we can find ǫn → 0+ so that n1/2ηn ≪ ǫn ≪ n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Set x ∈ {ǫn, −ǫn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Multiply by (2ǫn)−1n1/2 on both sides,  −1  2  √nBnǫn ≤ Zn + √n⟨un, �α − α0⟩ + OP (ηnǫ−1  n n1/2) ≤ 1  2  √nBnǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We have ηnǫ−1  n n1/2 + √nBnǫn = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Therefore we reach Zn + √n⟨un, �α − α0⟩ = oP(1), which  implies √n⟨un, �α − α0⟩ = −Zn + oP (1) →d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, let ζn = ∥v∗  n∥n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Apply Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 with α = �α and un = v∗  n/∥v∗  n∥,  ζ−1  n (φ(�α) − φ(α0))  =  ζ−1  n  dφ(α0)  dα  [�α − α0] + oP (1)  =  ζ−1  n  dφ(α0)  dα  [�α − α0,n] + ζ−1  n  dφ(α0)  dα  [α0,n − α0] + oP (1)  =  √n⟨un, �α − α0,n⟩ + oP (1)  =  √n⟨un, �α − α0⟩ + oP(1) →d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  where in the last equality we used √n⟨un, α0,n − α0⟩ = 0 because α0,n is the projection (under ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='∥)  of α0 onto span{An} and un ∈span{An}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We divide the proof in the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 1: decompose �γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Write σ2 := Var  �  1  √n  �  t Wt  �  + ∥v∗  n∥2, which will be shown to be  the asymptotic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also write bn(α) and ¯bn(α) respectively as the n × 1 vectors of b(St, α) :=  46  l(h(Wt))ρ(Yt+1, α) and ¯b(Xt, α) := E(l(h(Wt))ρ(Yt+1, α)|σt(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  �γ − γ  =  [φn(α0) − φ(α0)] + [φ(�α) − φ(α0)] + 1  n  n  �  t=1  (Γ(Xt) − �Γt)ρ(Yt+1, �α) + a1,  a1  :=  φn(�α) − φ(�α) − [φn(α0) − φ(α0)]  φn(α)  =  1  n  �  t  l(h(Wt)) − Γ(Xt)ρ(Yt+1, α)  φ(α)  =  Eφn(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Bounding a1 is based on the stochastic equicontinuity of φn − φ, established in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3, which  yields a1 = OP (dnδη  n) = oP (σn−1/2) by the assumption that dnδη  n = o(σn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 2: decompose �Γ(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We have Γ(Xt) = ¯b(Xt, α0)Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let �α ∈ Cn denote the  estimated α0 used in defining �Γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  �Γt = Ψ(Xt)′(Ψ′  nΨn)−1Ψ′  nbn(�α)�Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We then achieve the following decomposition:  1  n  �n  t=1(�Γt − Γ(Xt))ρ(Yt+1, �α) = 1  nbn(�α)′Pn�Σ−1  n ρn(�α) − 1  n¯bn(α0)′Σ−1  n ρn(�α) = a2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' + a8 where  a2  :=  1  n[bn(�α) − ¯bn(�α)]′Pn�Σ−1  n ρn(�α)  a3  :=  1  n[¯bn(�α) − ¯bn(α0)]′Pn�Σ−1  n ρn(�α)  a4  :=  1  n  ¯bn(α0)′(Pn − I)(�Σ−1  n − Σ−1  n )ρn(�α)  a5  :=  1  n  ¯bn(α0)′(Pn − I)Σ−1  n (ρn(�α) − mn(�α))  a6  :=  1  n  ¯bn(α0)′(Pn − I)Σ−1  n mn(�α)  a7  :=  1  n  ¯bn(α0)′(�Σ−1  n − Σ−1  n )(ρn(�α) − ρn(α0))  a8  :=  1  n  ¯bn(α0)′(�Σ−1  n − Σ−1  n )ρn(α0)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4)  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3 shows a2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' + a7 = OP (δη  n supx |�Σ(x) − Σ(x)| + √kndnδη  n + ϕ2  n), which is oP(σn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The bound for a8 = oP (σn−1/2) is from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence  1  n  n  �  t=1  (�Γt − Γ(Xt))ρ(Yt+1, �α) = oP (σn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 3: Complete proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By the same proof of that of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,  φ(�α) − φ(α0)  =  ∥v∗  n∥⟨un, �α − α0⟩ + oP(∥v∗  n∥n−1/2)  =  −∥v∗  n∥n−1/2Zn + oP (∥v∗  n∥n−1/2)  47  =  − 1  n  �  t  Zt + oP(∥v∗  n∥n−1/2)  Zt  :=  ρ(Yt+1, α0)Σ(Xt)−1 dm(Xt, α0)  dα  [v∗  n]  φn(α0) − φ(α0)  =  1  n  n  �  t=1  Wt − EWt,  Wt := l(h0(Wt)) − Γ(Xt)ρ(Yt+1, α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Putting together, �γ − γ = [φn(α0) − φ(α0)] + [φ(�α) − φ(α0)] + oP (σn−1/2), and  [φn(α0) − φ(α0)] + [φ(�α) − φ(α0)] = 1  n  n  �  t=1  Wt − EWt − Zt + oP (σn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Next, Wt − EWt − Zt is strictly stationary, satisfying the strong mixing condition (Assumption  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3) with �∞  n=1 α(n)ζ/(2+ζ) ≤ C �∞  n=1 exp(−cζn/(2 + ζ)) < ∞ for any constant ζ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition,  E  ��(Wt − EWt − Zt)σ−1��2+ζ ≤ CE  ��Wtσ−1��2+ζ + CE  ��Zt∥v∗  n∥−1��2+ζ  ≤  CE  ����ρ(Yt+1, α0)dm(Xt, α0)  dα  [un]  ����  2+ζ  + CE |ρ(Yt+1, α0)|2+ζ < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7 of Ibragimov (1962),  √nσ−1 [φn(α0) − φ(α0) + φ(�α) − φ(α0)] → N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5)  This implies the asymptotic normality of �γ − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3 (for Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Recall that bn(α) and ¯bn(α) are the n × 1 vectors of  l(h(Wt))ρ(Yt+1, α) and E(l(h(Wt))ρ(Yt+1, α)|σt(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose  (a) supx |Γ(x)|2 + supw supHn l(h(w))2 < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (b) |l(h1(w)) − l(h2(w))| ≤ C|h1(w) − h2(w)| uniformly for all h1, h2 ∈ Hn,and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (c) E supα∈Cn |l(h(Wt)) − l(h0(Wt))|2 ≤ Cδ2η  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (d) E supα∈Cn(ρ(Yt+1, α) − ρ(Yt+1, α0))2 = Cδ2η  n for some η, C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (e) For some κ, C > 0 , E sup∥α1−α∥∞,ω<δ |ǫt(α1) − ǫt(α)|2 ≤ Cδ2κ for all δ > 0 and α, α1 ∈  cl{a + xb : a, b ∈ An, x ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (f)  1  √n∥¯bn(α0)′(Pn − I)∥ = OP (ϕn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then for ¯ǫn(α) as the n × 1 vector of ρ(Yt+1, α) − m(Xt, α),  48  (i) supα1,α2∈Cn∪{α0} |φn(α1) − φ(α1) − [φn(α2) − φ(α2)]| = OP (dnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) supCn | 1  n¯bn(α0)′(�Σ−1  n − Σ−1  n )[ρn(α) − ρn(α0)]| = OP (δη  n) supx |�Σ(x) − Σ(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) supCn  1  √n∥Pn�Σ−1  n ρn(α)∥ = OP (supx ∥�Σ(x) − Σ(x)∥ + dnδη  n + ¯δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iv) supCn  1  n¯bn(α0)′(Pn − I)Σ−1  n ¯ǫn(α) = OP (√kndnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (v) supCn  1  n[bn(α) − ¯bn(α)]′Pn�Σ−1  n ρn(α) + supCn  1  n[¯bn(α) − ¯bn(α0)]′Pn�Σ−1  n ρn(α)  = OP (δη  n supx ∥�Σ(x) − Σ(x)∥ + dnδ2η  n + √kndnδη  n¯δn +  �  kn/n¯δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (vi) supCn  1  n¯bn(α0)′(Pn − I)(�Σ−1  n − Σ−1  n )ρn(α) = OP (ϕn supx ∥�Σ(x) − Σ(x)∥)  (vii) supCn  1  n¯bn(α0)′(Pn − I)Σ−1  n mn(α) = OP (ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) First recall ǫ(St, α) = ρ(Yt+1, α) − m(Xt, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  a  :=  sup  α1,α2∈Cn∪{α0}  | 1  n  n  �  t=1  Γ(Xt)[ρ(Yt+1, α1) − ρ(Yt+1, α2)] − EΓ(Xt)[ρ(Yt+1, α1) − ρ(Yt+1, α2)]|  =  sup  α1,α2∈Cn∪{α0}  | 1  n  n  �  t=1  Γ(Xt)[ǫ(St, α1) − ǫ(St, α2)]|  ≤  2  sup  α∈Cn∪{α0}  | 1  n  n  �  t=1  Γ(Xt)[ǫ(St, α) − ǫ(St, α0)]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  b  :=  sup  α1,α2∈Cn∪{α0}  | 1  n  n  �  t=1  l(h1(Wt)) − l(h2(Wt)) − E[l(h1(Wt)) − l(h2(Wt))]|  ≤  2  sup  α∈Cn∪{α0}  | 1  n  n  �  t=1  l(h(Wt)) − l(h0(Wt)) − E[l(h(Wt)) − l(h0(Wt))]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Note E supα∈Cn Γ(Xt)2[ǫ(St, α) − ǫ(St, α0)]2 ≤ CE supα∈Cn[ǫ(St, α) − ǫ(St, α0)]2 ≤ Cδ2η  n , η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then the convergence of a and b follow from the same argument of that of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 with  Ψj(Xt) replaced with Γ(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Term b follows from the same proof of this Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We reach  a + b = OP (dnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Therefore supα1,α2∈Cn∪{α0} |φn(α1) − φ(α1) − [φn(α2) − φ(α2)]| ≤ a + b = OP (dnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) First, E supα∈Cn[ρ(Yt+1, α) − ρ(Yt+1, α0)]2 ≤ O(δ2η  n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This implies  1  √n∥ρn(α) − ρn(α0)∥ =  OP (δη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The target of interest is then bounded by  ∥�Σn − Σn∥ 1  √n∥ρn(α) − ρn(α0)∥ = OP (δη  n) sup  x ∥�Σ(x) − Σ(x)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) First, write �mΣ(Xt, α) := Ψ(Xt)′(Ψ′  nΨn)−1Ψ′  nΣ−1  n mn(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then step 1 of the proof of  49  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 carries over, leading to  sup  Cn  1  n∥PnΣ−1  n mn(α)∥2 = sup  Cn  1  n  �  t  �mΣ(Xt, α)2 ≤ C sup  Cn  E �mΣ(Xt, α)2  ≤  C sup  Cn  E[ �mΣ(Xt, α) − m(Xt, α)Σ(Xt)−1]2 + C sup  Cn  Em(Xt, α)2 = OP (¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Also, for ¯ǫn(α) := ρn(α)−mn(α), the first inequality below follows from the same proof of Proposition  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n (¯ǫn(α) − ¯ǫn(α0))∥ ≤ OP (dnδη  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥Pn(�Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n − Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n )(ρn(α) − ρn(α0))∥ ≤ OP (δη  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n) sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='x ∥�Σ(x) − Σ(x)∥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n (ρn(α) − ρn(α0))∥ ≤ sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n (¯ǫn(α) − ¯ǫn(α0))∥ + sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n mn(α)∥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='OP (dnδη  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n + ¯δn)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥Pn�Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n (ρn(α) − ρn(α0))∥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥Pn(�Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n − Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n )(ρn(α) − ρn(α0))∥ + sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='Cn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥PnΣ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n (ρn(α) − ρn(α0))∥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='OP (δη  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n) sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='x ∥�Σ(x) − Σ(x)∥ + OP (dnδη  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n + ¯δn)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='√n∥Pn�Σ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='n ρn(α0)∥ = OP (1) sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='x ∥�Σ(x) − Σ(x)∥ + OP (  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='kn/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Together, supCn  1  √n∥Pn�Σ−1  n ρn(α)∥ ≤ supCn  1  √n∥Pn�Σ−1  n (ρn(α) − ρn(α0))∥ +  1  √n∥Pn�Σ−1  n ρn(α0)∥ whose  final rate is OP (supx ∥�Σ(x) − Σ(x)∥ + dnδη  n + ¯δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iv) First, 1  n¯bn(α0)′(Pn −I)Σ−1  n ¯ǫn(α0) = OP (ϕnn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Next, from the proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,  sup  Cn  1  n  ¯bn(α0)′(Pn − I)Σ−1  n [¯ǫn(α) − ¯ǫn(α0)]  ≤  sup  Cn  1  n∥Ψ′  nΣ−1  n [¯ǫn(α) − ¯ǫn(α0)]∥ + sup  Cn  1  n  ¯bn(α0)′Σ−1  n [¯ǫn(α) − ¯ǫn(α0)] = OP (  �  kndnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  So supCn  1  n¯bn(α0)′(Pn − I)Σ−1  n ¯ǫn(α) = OP (√kndnδη  n + ϕnn−1/2) = OP (√kndnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (v) The same proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 carries over to here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So  sup  α∈Cn  1  n∥Ψ′  n(¯bn(α) − bn(α)) − Ψ′  n(¯bn(α0) − bn(α0))∥ = OP (  �  kndnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition, 1  n∥Ψ′  n(¯bn(α0) − bn(α0))∥ = OP (√knn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This implies  sup  Cn  1  √n[bn(α) − ¯bn(α)]′Pn ≤ OP (1) sup  α∈Cn  1  n∥Ψ′  n(¯bn(α) − bn(α))∥ = OP (dnδη  n + n−1/2)  �  kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  50  Also,  1  n∥¯bn(α) − ¯bn(α0)∥2  ≤  OP (1)E sup  Cn  [E|ρ(Yt+1, α) − ρ(Yt+1, α0)||σt(X)]2 + OP (1) sup  Cn  E|l(h) − l(h0)|2  ≤  OP (δ2η  n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence supCn  1  n[bn(α)−¯bn(α)]′Pn�Σ−1  n ρn(α) = OP (√kndnδη  n+  �  kn/n)(supx ∥�Σ(x)−Σ(x)∥+dnδη  n +¯δn)  and supCn  1  n[¯bn(α) − ¯bn(α0)]′Pn�Σ−1  n ρn(α) = OP (supx ∥�Σ(x) − Σ(x)∥δη  n + dnδ2η  n + ¯δnδη  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So the  final rate of the sum of the two is δη  n supx ∥�Σ(x) − Σ(x)∥ + dnδ2η  n + √kndnδη  n¯δn +  �  kn/n¯δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (vi) (vii) The proof is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  D  Proofs for Section 5  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 shows the following LQA:  sup  α∈Cn  sup  |x|≤Cn−1/2 |Qn(α + xun) − Qn(α) − An(α(x))| = oP (n−1)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  where An(α(x)) := 2x[n−1/2Zn + ⟨un, α − α0⟩] + Bnx2 with Bn = 1 + oP(1), Zn →d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We  respectively provide lower and upper bounds for Qn(�αR) − Qn(�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 1: lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' To apply the LQA, we need to first show that �αR ∈ Cn with a high  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In fact, there is πR  n α0 ∈ AR  n so that  Qn(�αR) + λnPen(�hR) ≤ Qn(πR  n α0) + λnPen(πR  n h0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Given the above inequalities, the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 carries over, establishing that �αR ∈ Cn with  a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We now condition on this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence by (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1), uniformly for all |x| ≤ Cn−1/2,  Qn(�αR + xun) − Qn(�αR)  =  2x[n−1/2Zn + ⟨un, �αR − α0⟩] + Bnx2 + oP(n−1)  =  2xn−1/2Zn + Bnx2 + oP (n−1),  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2)  where the second equality follows from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Next, we note one technical difficulty that the  inequality Qn(�α) + λnPen(�α) ≤ Qn(α) + λnPen(α) may not hold for α = �αR + xun, as An is a  nonlinear space so �αR + xun is not necessarily in An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Nevertheless, we can apply this inequality for  α = πn(�αR + xun), and show that |Qn(πn(�αR + xun)) − Qn(�αR + xun)| is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Specifically, by  51  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3,  Qn(�α) − Qn(�αR + xun)  ≤  λnPen(πn(�αR + xun)) − λnPen(�α) + Qn(πn(�αR + xun)) − Qn(�αR + xun)  ≤  oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3)  Thus (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3) imply Qn(�αR)−Qn(�α) ≥ −2xn−1/2Zn−Bnx2−oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Take x = −ZnB−1  n n−1/2  which maximizes −2xn−1/2Zn − Bnx2, then  Qn(�αR) − Qn(�α) ≥ Z2  nB−1  n n−1 − oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 2: upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Fix x∗ determined as in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1, this lemma shows that x∗ =  n−1/2ZnB−1  n  + oP (n−1/2) and that |Qn(πR  n (�α + x∗un)) − Qn(�α + x∗un)| = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence by (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1)  again,  Qn(�αR) − Qn(�α)  ≤  Qn(πR  n (�α + x∗un)) − Qn(�α) + λn(Pen(πR  n (�α + x∗un)) − Pen(�αR))  =  Qn(�α + x∗un) − Qn(�α) + oP (n−1)  =  2x∗n−1/2[Zn + n1/2⟨un, �α − α0⟩] + Bnx∗2 + oP (n−1)  =  Bnx∗2 + oP (n−1) = Z2  nB−1  n n−1 + oP (n−1),  where the third equality is due to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 that Zn + √n⟨un, �α − α0⟩ = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 3: matching bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Together, we have  Sn(φ0) = n(Qn(�αR) − Qn(�α)) = B−1  n Z2  n + oP (1) →d χ2  1  given that Bn →P 1 proved in Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 (for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose  (a) supα∈Cn  1  n  �n  t=1[m(Xt, πnα) − m(Xt, α)]2 = OP (µ2  n) and  supα∈Cn,φ(α)=φ0  1  n  �n  t=1[m(Xt, πR  n (α + xun)) − m(Xt, α + xun)]2 = OP (µ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (b) µn¯δn = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (c) t → φ(α + tun) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then  (i) ⟨un, �αR − α0⟩ = oP (n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) sup|x|≤Cn−1/2 |Qn(πn(�αR + xun)) − Qn(�αR + xun)| = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  52  (iii) there is x∗ so that φ(�α + x∗un) = φ0 and |Qn(πR  n (�α + x∗un)) − Qn(�α + x∗un)| = oP(n−1)  and x∗ = n−1/2ZnB−1  n  + oP(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) Note that φ(�αR) − φ(α0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,  ���dφ(α0)  dα  [�αR − α0]  ��� = o(∥v∗  n∥n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By  the Riesz representation Theorem,  dφ(α0)  dα  [�αR − α0] = dφ(α0)  dα  [�αR − α0,n] + dφ(α0)  dα  [α0,n − α0] = ∥v∗  n∥⟨un, �αR − α0⟩ + o(∥v∗  n∥n−1/2)  with the definition un = v∗  n/∥v∗  n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) The proof is the same as that of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) First, we prove there is x∗ so that φ(�α+x∗un) = φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Define F(x) := ⟨v∗  n, α−α0⟩+x∥v∗  n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also  define R(x) := φ(α+xun)−φ(α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1, there is a positive sequence bn = o(∥v∗  n∥n−1/2),  uniformly for all α ∈ Cn, for all x ≤ Cn−1/2,  |R(x) − F(x)| ≤ bn  Now fix some r such that |r−⟨v∗  n, α−α0⟩| ≤ C∥v∗  n∥n−1/2 and define x1 = (r−⟨v∗  n, α−α0⟩−2bn)∥v∗  n∥−1  and x2 = (r − ⟨v∗  n, α − α0⟩ + 2bn)∥v∗  n∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This ensures that F(x1) + 2bn = F(x2) − 2bn = r and  |x1| + |x2| ≤ Cn−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Therefore,  R(x1) ≤ F(x1) + bn < r,  R(x2) ≥ F(x2) − bn > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence there is x∗ between x1, x2 so that R(x∗) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In the above proof, suppose r = 0 and  α = �α are admitted, then φ(�α + x∗un) = φ(α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' To show the admissibility, we note (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2) that  n−1/2Zn + ∥v∗  n∥−1⟨v∗  n, �α − α0⟩ = oP (n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence indeed, for any ǫ > 0, there is C > 0,  |⟨v∗  n, α − α0⟩| = ∥v∗  n∥n−1/2|Zn| + oP (∥v∗  n∥n−1/2) ≤ C∥v∗  n∥n−1/2  with probability at least 1 − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Now |x∗ − n−1/2Zn| ≤ |x1 − n−1/2Zn| + |x2 − n−1/2Zn| ≤ 2|bn|  ∥v∗n∥ + oP (n−1/2) = oP (n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, the proof of |Qn(πR  n (�α + x∗un)) − Qn(�α + x∗un)| = oP (n−1) is the same as part (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  53  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' As in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1, we respectively provide lower and upper bounds for 1  n �Sn(φ0) =  Ln(�αR, φ0) − Ln(�α, �γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that Ln(�α, �γ) = Qn(�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let  g1  =  (�φ(�αR) − γ0)2�Σ−1  2  g2  =  φ(�αR) − φ(α0)  g3  =  [φn(α0) − φ(α0)] + [φ(�α) − γ0]  g4  =  [φn(α0) − γ0 + g2]2�Σ−1  2  g5  =  φn(α0) − γ0 − ∥v∗  n∥n−1/2Zn  g6  =  n−1/2Zn + ∥v∗  n∥−1g2  Also note that �αR ∈ Cn with a high probability, by Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We now condition on this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 1: lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Due to λnPen(�αR + xun) − λnPen(�α) = oP (n−1) and �αR ∈ An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' so  uniformly for all |x| ≤ Cn−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Ln(�α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' �γ) − Ln(�αR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' φ0) = Qn(�α) − Qn(�αR) − g1 + oP (n−1)  ≤(a)  Qn(πn(�αR + xun)) − Qn(�αR + xun) + Qn(�αR + xun) − Qn(�αR) − g1 + oP (n−1)  =(b)  Qn(�αR + xun) − Qn(�αR) − g1 + oP (n−1)  =(c)  2x[n−1/2Zn + ⟨un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' �αR − α0⟩] + x2 − g1 + oP (n−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  =(d)  2x[n−1/2Zn + ∥v∗  n∥−1 dφ(α0)  dα  [�αR − α0]] + x2 − g1 + oP (n−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  =(e)  2x[n−1/2Zn + ∥v∗  n∥−1g2] + x2 − g1 + oP (n−1)  =(f)  2x[n−1/2Zn + ∥v∗  n∥−1g2] + x2 − g4  �  ��  �  F (x)  +oP(n−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  where in (a) we used Qn(�α) ≤ Qn(πn(�αR + xun));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (b) follows from |Qn(πn(�αR + xun)) − Qn(�αR +  xun)| ≤ oP (n−1) following the same proof of that of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1(ii);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (c) is from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (d) is from  the Riesz representation: (⟨v∗  n, α0 − α0,n⟩ = 0)  dφ(α0)  dα  [�αR − α0] = dφ(α0)  dα  [�αR − α0,n] + dφ(α0)  dα  [α0,n − α0] = ⟨v∗  n, �αR − α0,n⟩ + oP (n−1/2∥v∗  n∥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (e) is from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (f) is from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We choose x = x∗ to minimize F(x) on the right hand side, leading to the choice x∗ =  −[n−1/2Zn + ∥v∗  n∥−1g2] = −g6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We shall verify that |x∗| = OP (n−1/2) in Step 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose for  now this is true, then we have obtained the lower bound: 1  n �Sn(φ0) ≥ −F(x∗) − oP(n−1), where  −F(x∗) = [n−1/2Zn + ∥v∗  n∥−1g2]2 + g4 = g2  6 + g4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  54  Step 2: upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Uniformly for all |x| ≤ Cn−1/2,  Ln(�αR, φ0) − Ln(�α, �γ)  ≤  Ln(πn(�α + xun), φ0) − Ln(�α, �γ) + λnPen(πn(�α + xun)) − λnPen(�αR) + oP (n−1)  ≤(g)  Ln(�α + xun, φ0) − Ln(�α, �γ) + oP(n−1)  =  Qn(�α + xun) − Qn(�α) + (�φ(�α + xun) − φ0)2�Σ−1  2  + oP(n−1)  =(h)  x2 + 2x[n−1/2Zn + ⟨�α − α0, un⟩] + (�φ(�α + xun) − φ0)2�Σ−1  2  + oP(n−1)  =(i)  x2 + (�φ(�α + xun) − γ0)2�Σ−1  2  + oP(n−1)  =(j)  x2 + [x∥v∗  n∥ + g3]2�Σ−1  2  �  ��  �  G(x)  +oP (n−1)  where (g) follows from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 and that λnPen(πn(�α + xun)) − λnPen(�αR) = oP (n−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (h) is  from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) is from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (j) is from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We choose x = τ ∗ to minimize G(x), leading  to the choice τ ∗ = −g3∥v∗  n∥(∥v∗  n∥2 + �Σ2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' It is easy to see that |τ ∗| = OP (n−1/2), following this  argument: from the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, g3 = OP (σn−1/2), and σ2 = Σ2 + ∥v∗  n∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So provided that  �Σ2 − Σ2 = oP(1)Σ2,  |τ ∗| = OP (n−1/2)  �  Σ2 + ∥v∗n∥2∥v∗  n∥  ∥v∗n∥2 + �Σ2  = OP (n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Thus τ ∗ is admitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then we have obtained the upper bound: 1  n �Sn(φ0) ≤ G(τ ∗) + oP(n−1), where  G(τ ∗) =  g2  3  ∥v∗n∥2 + �Σ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Step 3: matching bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We now show that the lower and upper bounds match, that is, −F(x∗) = G(τ ∗)+oP(n−1), which  requires analyzing g2 = φ(�αR) − φ(α0) and g6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' First, Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3 yields, uniformly in |x| ≤ Cn−1/2,  (�φ(�αR + xun) − γ0)2�Σ−1  2  − (�φ(�αR) − γ0)2�Σ−1  2  = H(x) + oP (n−1)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4)  where H(x) = �Σ−1  2 x2∥v∗  n∥2 + 2x�Σ−1  2 ∥v∗  n∥[�φ(α0) − γ0 + g2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Next, the basic inequality yields  Ln(�αR, φ0) ≤ Ln(πn(�αR + xun), γ0) + oP (n−1) ≤ Ln(�αR + xun, γ0) + oP(n−1)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5)  where the first inequality follows from with the assumption that λnPen(�αR + xun) − λnPen(�αR) =  oP (n−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' the second inequality follows from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Uniformly for |x| ≤ Cn−1/2,  Qn(�αR + xun) − Qn(�αR)  =  oP (n−1) + x2 + 2x[n−1/2Zn + ⟨un, �αR − α0⟩]  =  oP (n−1) + x2 + 2x[n−1/2Zn + g2∥v∗  n∥−1],  55  where ⟨un, �αR − α0⟩ = ∥v∗  n∥−1 dφ(α0)  dα  [�αR − α0] + oP (n−1/2) = ∥v∗  n∥−1g2 + oP(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This along with  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5) give rise to,  0  ≤  x2 + 2x[n−1/2Zn + g2∥v∗  n∥−1] + H(x) + oP(n−1)  =  (1 + ∥v∗  n∥2)x2 + 2x[n−1/2Zn + g2∥v∗  n∥−1 + ∥v∗  n∥(�φ(α0) − γ0 + g2)] + oP (n−1)  =  x2(1 + �Σ−1  2 ∥v∗  n∥2) + 2x[n−1/2Zn + g2∥v∗  n∥−1 + �Σ−1  2 ∥v∗  n∥(�φ(α0) − γ0 + g2)] + oP (n−1)  =  x2(1 + �Σ−1  2 ∥v∗  n∥2) + 2x[g6 + �Σ−1  2 ∥v∗  n∥(φn(α0) − γ0 + g2)] + oP (n−1),  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6)  where in the last equality, �Σ−1  2 ∥v∗  n∥(�φ(α0) − φn(α0)) = oP (n−1/2), from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2:  |�Σ−1  2 ∥v∗  n∥(�φ(α0) − φn(α0))| ≤ |�Σ−1  2 ∥v∗  n∥ 1  n  n  �  t=1  (Γ(Xt) − �Γt)ρ(Yt+1, α0)| = oP (n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6) implies there is some ¯ηn = oP (n−1) so that  x2(1 + �Σ−1  2 ∥v∗  n∥2) + 2x[g6 + �Σ−1  2 ∥v∗  n∥(φn(α0) − γ0 + g2)] + ¯ηn ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7)  We now derive some important intermediate results from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' First, let  Cn := min{∥v∗  n∥Σ−1/2  2  , ∥v∗  n∥2  ∗Σ−1  2 , ∥v∗  n∥σΣ−1  2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  It is known that (Cn+C2  n)/(1+�Σ−1  2 ∥v∗  n∥2) ≤ 2 because ∥v∗  n∥ ≤ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So ¯ηn = oP (n−1)C2  n/(1+�Σ−1  2 ∥v∗  n∥2),  implying ¯ηnn1/2C−1  n  ≪ n−1/2Cn/(1+ �Σ−1  2 ∥v∗  n∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence there is a positive sequence ǫn = OP (n−1/2)  so that ¯ηnn1/2C−1  n  ≪ ǫn ≪ n−1/2Cn/(1+�Σ−1  2 ∥v∗  n∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence ¯ηn/ǫn+ǫn(1+�Σ−1  2 ∥v∗  n∥2) = oP (n−1/2Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Take x = ±ǫn and divide by 2ǫn on (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We reach four intermediate results:  g6 + �Σ−1  2 ∥v∗  n∥(φn(α0) − γ0 + g2) = OP (¯ηn/ǫn + ǫn(1 + �Σ−1  2 ∥v∗  n∥2)) = oP (n−1/2Cn) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8)  (1 + �Σ−1  2 ∥v∗  n∥2)g6 + �Σ−1  2 ∥v∗  n∥g5 = oP (n−1/2Cn)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='9)  g4 = (oP (n−1/2Cn) − g6)2�Σ2∥v∗  n∥−2  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='10)  g6 = OP (n−1/2)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='11)  where (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8) follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='7) with x = ±ǫn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' the left hand sides of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='9) are equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='10) is from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='8) and the definition of g4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='11) is from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='9), g5 = OP (σn−1/2) and that  oP (n−1/2Cn) = σ2Σ−1  2 OP (n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also, the proof of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='11) does not rely on the conclusion of Step  1, so it verifies that |x∗| = OP (n−1/2), a claim used in step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We are now ready to match the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' From (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='10) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='11),  −F(x∗)  =  g2  6 + g4 = g2  6 + (oP (n−1/2Cn) − g6)2�Σ2∥v∗  n∥−2 = g2  6(1 + �Σ2∥v∗  n∥−2) + oP (n−1)  56  =(k)  [oP (n−1/2Cn) − �Σ−1  2 ∥v∗  n∥g5]2  (1 + �Σ−1  2 ∥v∗n∥2)2  (1 + �Σ2∥v∗  n∥−2) + oP (n−1)  =  g2  5  �Σ2 + ∥v∗n∥2 + oP(n−1) =(l)  g2  3  �Σ2 + ∥v∗n∥2 + oP (n−1) = G(τ ∗) + oP (n−1),  where (k) is from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (l) is from the fact that (due to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2))  |g2  3 − g2  5|  ≤  |φ(�α) − φ(α0) + ∥v∗  n∥n−1/2Zn|OP (n−1/2σ)  ≤  ����  dφ(α0)  dα  [�α − α0] + ∥v∗  n∥n−1/2Zn  ���� OP (n−1/2σ) + oP(σ2n−1)  =  ���⟨�α − α0, un⟩ + n−1/2Zn + oP (n−1/2)  ��� OP (∥v∗  n∥n−1/2σ) + oP(σ2n−1) = oP (σ2n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Thus we have proved that the upper and lower bounds match up to oP (n−1), implying  �Sn(φ0) = nG(τ ∗) + oP (1) = ng2  3  �σ2 + oP (1) →d χ2  1  where the convergence in distribution follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose (δη  n supx |�Σ(x)−Σ(x)|+√kndnδη  n+ϕ2  n) = oP (n−1/2 min{1, σ−1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  In addition, suppose (1 + ∥v∗  n∥) supα∈Cn |φ(πnα) − φ(α)| = o(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Write �φ(α) := 1  n  �n  t=1[l(h(Wt)) − �Γtρ(Yt+1, α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  (i) ∥�αR − α∥∞,ω = OP (δn), Q(�αR) ≤ OP (¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) supα1,α2∈Cn[�φ(α1) − �φ(α2)] − [φ(α1) − φ(α2)] = OP (δη  n supx |�Σ(x) − Σ(x)| + √kndnδη  n + ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) |Ln(πn(�α + xun), γ0) − Ln(�α + xun, γ0)| = oP (n−1)  (iv) |Ln(πn(�αR + xun), γ0) − Ln(�αR + xun, γ0)| = oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) The inequality Ln(�αR, φ0) + λnPen(�hR) ≤ Ln(πnα0, γ0) + λnPen(πnh0) implies  Qn(�αR) ≤ Qn(πnα0) + Fn(πnα0) + OP (λ) = O(¯δ2  n) + Fn(πnα0)  where Fn(α) := (�φ(α) − γ0)′�Σ−1  2 (�φ(α) − γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We now bound Fn(πnα0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Note that  �φ(πnα0) − γ0  =  1  n  n  �  t=1  l(πnh0(Wt)) − El(h0(Wt)) − 1  n  n  �  t=1  (�Γt − Γ(Xt))ρ(Yt+1, πnα0)  − 1  n  n  �  t=1  [Γ(Xt)ρ(Yt+1, πnα0) − EΓtρ(Yt+1, α0)] − EΓt[m(Xt, α0) − m(Xt, πnα0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  57  The first term is bounded by OP (n−1/2)+E[l(πnh0(Wt))−l(h0(Wt))];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' the second term is bounded by  OP (¯δn), following from the same argument as those for (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' the third and fourth terms are bounded  by OP (n−1/2)+EΓ(Xt)[m(Xt, πnα0)−m(Xt, α0)] ≤ OP (  �  Q(πnα0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence �φ(πnα0)−γ0 = OP (¯δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  This implies Fn(πnα0) = OP (¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' This yields Qn(�αR) = OP (¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then from the proof of Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1,  Q(�αR) ≤ CE �m(Xt, �αR)2 + OP (¯δ2  n) ≤ C 1  n  �  t  �m(Xt, �αR)2 + OP (¯δ2  n) ≤ Qn(�αR) + OP (¯δ2  n) = OP (¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  It also implies ∥�αR − πnα0∥ ≤ ∥�αR − α0∥ + ∥πnα0 − α0∥ = OP (¯δn), and hence ∥�αR − α0∥∞,ω ≤  ∥πnα0 − α0∥∞,ω + ∥�αR − πnα0∥∞,ω = ∥πnα0 − α0∥∞,ω + OP (ωn(¯δn)) = OP (δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Let a1 = [(φn(α1) − φ(α1)) − (φn(α2) − φ(α2))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We have  [�φ(α1) − �φ(α2)] − [φ(α1) − φ(α2)] = a1 + 1  n  n  �  t=1  (Γ(Xt) − �Γt)ρ(Yt+1, α1) + 1  n  n  �  t=1  (�Γt − Γ(Xt))ρ(Yt+1, α2)  ≤  OP (δη  n sup  x |�Σ(x) − Σ(x)| +  �  kndnδη  n + ϕ2  n),  where the first inequality follows from bounds for (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4) and Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) Let α = �α+xun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By the same proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1(ii), Qn(πnα)−Qn(α) = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Next,  �φ(α) − γ0  =  a1(α) + a3 + a4(α) − 1  n  �  t  (�Γt − Γ(Xt))ρ(Yt+1, α),  a1(α)  :=  φn(α) − φ(α) − [φn(α0) − φ(α0)]  a3  :=  φn(α0) − φ(α0)  a4(α)  :=  φ(α) − φ(α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3 and the same proof for bounding (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4),  a1− 1  n  �  t  (�Γt−Γ(Xt))ρ(Yt+1, α) = OP (δη  n sup  x |�Σ(x)−Σ(x)|+  �  kndnδη  n+ϕ2  n) = oP (n−1/2 min{1, σ−1}),  where the last equality follows from the assumption (δη  n supx |�Σ(x) − Σ(x)| + √kndnδη  n + ϕ2  n) =  oP (n−1/2 min{1, σ−1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' The same bound holds when α is replaced with πnα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Meanwhile, by the  proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, a3 = OP (σn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  To bound a4(α), first note that  ∥πn(�α + xun) − α0∥2 ≤ CEm(Xt, πn(�α + xun))2 ≤ CE[m(Xt, πn(�α + xun)) − m(Xt, �α + xun)]2  +CE[m(Xt, �α + xun) − m(Xt, �α)]2 + CQ(�α) ≤ OP (µ2  n + ¯δ2  n)  ∥(�α + xun) − α0∥2 ≤ OP (¯δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  58  So by Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1, and that ⟨v∗  n, α0 − α0,n⟩ = 0,  a4(�α + xun)  ≤  |φ(�α + xun) − φ(α0)| ≤  ����  dφ(α0)  dα  [�α + xun − α0]  ����  ≤  ����  dφ(α0)  dα  [�α + xun − α0,n]  ���� +  ����  dφ(α0)  dα  [α0,n − α0]  ����  =  oP (∥v∗  n∥)n−1/2 + |⟨�α − α0, v∗  n⟩ + x⟨un, v∗  n⟩| = OP (∥v∗  n∥n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  a4(πn(�α + xun))  ≤  |φ(πn(�α + xun)) − φ(α0)| ≤  ����  dφ(α0)  dα  [πn(�α + xun) − α0]  ����  ≤  ����  dφ(α0)  dα  [πn(�α + xun) − α0,n]  ���� +  ����  dφ(α0)  dα  [α0,n − α0]  ����  =  oP (∥v∗  n∥)n−1/2 + |⟨πn(�α + xun) − α0, v∗  n⟩|  ≤  oP (∥v∗  n∥)n−1/2 + ∥πn(�α + xun) − α0∥∥v∗  n∥  ≤  OP (¯δn)∥v∗  n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Also, by the proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1(ii), Qn(πn(�α + xun)) − Qn(�α + xun) = oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' with  α = �α + xun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' |�φ(α) − γ0| ≤ oP(n−1/2 min{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' σ−1}) + OP (∥v∗  n∥n−1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' and cn := |φ(πnα) − φ(α)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Ln(πn(�α + xun),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' γ0) − Ln(�α + xun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' γ0) = Qn(πnα) − Qn(α)  +(�φ(πnα) − γ0)′�Σ−1  2 (�φ(πnα) − γ0) − (�φ(α) − γ0)′�Σ−1  2 (�φ(α) − γ0)  =  oP (n−1) + (�φ(πnα) − �φ(α))′ �Σ−1  2 (�φ(πnα) − �φ(α)) + 2(�φ(πnα) − �φ(α))′�Σ−1  2 (�φ(α) − γ0)  =  oP (n−1) + OP (1)|φ(πnα) − φ(α)|2 + OP (1)|φ(πnα) − φ(α)||�φ(α) − γ0|  ≤  oP (n−1) + OP (c2  n) + oP (n−1/2 min{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' σ−1})cn + OP (∥v∗  n∥n−1/2cn) = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iv) The proof is the same for part (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Write νn := δη  n supx |�Σ(x) − Σ(x)| + √kndnδη  n + ϕ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose  νn = oP (n−1/2σ−1), (pn + νn)¯δn∥v∗  n∥ = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then Uniformly for |x| ≤ Cn−1/2, for b(x) := �φ(α0) − γ0 + φ(�αR) − φ(α0),  (i) (�φ(�α + xun) − γ0)2�Σ−1  2  = [x∥v∗  n∥ + g3]2�Σ−1  2  + oP (n−1)  (ii) (�φ(�αR + xun) − γ0)2�Σ−1  2  − (�φ(�αR) − γ0)2�Σ−1  2  = x2∥v∗  n∥2�Σ−1  2  + 2x∥v∗  n∥b(x)�Σ−1  2  + oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) (�φ(�αR) − γ0)2�Σ−1  2  = [φn(α0) − γ0 + φ(�αR) − φ(α0)]2�Σ−1  2  + oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (i) Let g3 = [φn(α0) − φ(α0)] + [φ(�α) − φ(α0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  �φ(�α + xun) − γ0  =  φ(�α + xun) − φ(�α) + g3 + b1 + b2  b1  =  [�φ(�α + xun) − �φ(�α)] − [φ(�α + xun) − φ(�α)]  59  b2  =  �γ − γ0 − g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We now work with φ(�α + xun) − φ(�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1 and the Riesz representation,  φ(�α + xun) − φ(�α)  =  φ(�α + xun) − φ(α0) − [φ(�α) − φ(α0)]  =  dφ(α0)  dα  [�α − α0 + xun] − dφ(α0)  dα  [�α − α0]  =  ⟨v∗  n, �α − α0 + xun⟩ − ⟨v∗  n, �α − α0⟩ + oP (∥v∗  n∥)n−1/2  =  ⟨v∗  n, un⟩x = x∥v∗  n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='12)  Hence �φ(�α + xun) − γ0 = x∥v∗  n∥ + g3 + b1 + b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also note that g3 = OP (σn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Together  (�φ(�α + xun) − γ0)2�Σ−1  2  is bounded by  [x∥v∗  n∥ + g3]2�Σ−1  2  + oP(n−1) + OP (b2  1 + b2  2) + OP (b1 + b2)(∥v∗  n∥ + σ)n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  By Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, b1 = OP (νn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' By the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, b2 = OP (νn) = oP (n−1/2σ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So the  above is bounded by (σ ≥ ∥v∗  n∥),  [x∥v∗  n∥ + g3]2�Σ−1  2  + oP (n−1) + OP (νnn−1/2)(∥v∗  n∥ + σ) = [x∥v∗  n∥ + g3]2�Σ−1  2  + oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) (�φ(�αR + xun) − γ0)2�Σ−1  2  − (�φ(�αR) − γ0)2�Σ−1  2  equals  ∆1 := (�φ(�αR + xun) − �φ(�αR))2�Σ−1  2  + 2(�φ(�αR + xun) − �φ(�αR))�Σ−1  2 (�φ(�αR) − γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The same argument as in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='12) yields �φ(�αR + xun) − �φ(�αR) = x∥v∗  n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Meanwhile,  �φ(�αR) − γ0  =  �φ(α0) − γ0 + φ(�αR) − φ(α0) + [�φ(�αR) − �φ(α0)] − [φ(�αR) − φ(α0)]  �  ��  �  =OP (νn) by Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='13)  �φ(�αR + xun) − �φ(�αR)  =  OP (µn) + φ(�αR + xun) − φ(�αR) = ∥v∗  n∥x + OP (νn),  φ(�αR) − φ(α0)  =  dφ(α0)  dα  [�αR − α0] = ⟨�αR − α0, v∗  n⟩  ≤  ∥�αR − α0∥∥v∗  n∥ ≤ C  �  Q(�αR)∥v∗  n∥  ≤  OP (¯δn∥v∗  n∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='14)  Hence with �φ(α0) − γ0 = OP (n−1/2σ), and ̟nσ = O(1)  ∆1  =  x2∥v∗  n∥2�Σ−1  2  + oP (n−1) + 2[x∥v∗  n∥][�φ(α0) − γ0 + φ(�αR) − φ(α0) + OP (µn)]�Σ−1  2  =  x2∥v∗  n∥2�Σ−1  2  + 2x∥v∗  n∥[�φ(α0) − γ0 + φ(�αR) − φ(α0)]�Σ−1  2  + oP (n−1)  +OP (µn)n−1/2∥v∗  n∥  60  =  x2∥v∗  n∥2�Σ−1  2  + 2x∥v∗  n∥[�φ(α0) − γ0 + φ(�αR) − φ(α0)]�Σ−1  2  + oP (n−1),  (iii) Define  z1  :=  (�φ(�αR) − γ0)2�Σ−1  2  z2  :=  [�φ(α0) − γ0 + φ(�αR) − φ(α0)]2�Σ−1  2  z3  :=  [φn(α0) − γ0 + φ(�αR) − φ(α0)]2�Σ−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  First the proof for bounding (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='4) can be simplified to yield  |�φ(α0) − φn(α0)| ≤ 1  n  n  �  t=1  (�Γt − Γ(Xt))ρ(Yt+1, α0)  =  1  n  ¯bn(α0)′(�Σ−1  n − Σ−1  n )ρn(α0) + OP (δη  n)[sup  x ∥�Σ(x) − Σ(x)∥ +  �  kn  n ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  √z2 + √z3 = OP (n−1/2σ + ¯δn∥v∗  n∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  By (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='13), and with the assumption (νn + pn)¯δn∥v∗  n∥ = oP (n−1),  |z1 − z2|  ≤  OP (ν2  n) + OP (µn)|�φ(α0) − γ0 + φ(�αR) − φ(α0)|  =  oP(n−1) + OP (νn)(n−1/2σ + ¯δn∥v∗  n∥) = oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  |z2 − z3|  ≤  OP (1)|�φ(α0) − φn(α0)|(√z2 + √z3) = oP (n−1) + OP (pn¯δn∥v∗  n∥) = oP(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence z1 − z3 = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  E  Verifying conditions for RL, NPIV and NPQIV in Section 6  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Reinforcement learning model: proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Recall that Qπ denotes the true Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let Xt = (St, At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We have  m(Xt, h) = E(Rt|Xt) − h(St, At) + γE  ��  x∈A  π(x|St+1)h(St+1, x)  ����St, At  �  dx  In addition, for dm  dh [v] defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2), ∥v∥2 := E  � dm  dh [v]  �2 Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  The Bellman equation implies m(Xt, Qπ) = 0 so for all h ∈ Hn,  m(Xt, h) = m(Xt, h) − m(Xt, Qπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence m(Xt, h) = dm  dh [h − Qπ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  ∥h − Qπ∥2 = E  �dm  dh [h − Qπ]  �2  Σ(Xt)−1 = Em(Xt, h)2Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  61  This shows condition (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For condition (ii), it is also easy to see:  Em(Xt, πnα0)2Σ(Xt)−1 = E  �dm  dh [πnQπ − Qπ]  �2  Σ(Xt)−1 = ∥πnQπ − Qπ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For condition (i), let Tt = Ψj(Xt)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also,  ρ(Yt+1, h) = Rt − h(St, At) + γK(h),  K(h) =  �  x∈A  π(x|St+1)h(St+1, x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  and ǫ(St, h1) − ǫ(St, h2) = γK(h1) − γK(h2) − γE[K(h1) − K(h2)|St, At].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Now  |K(h1) − K(h2)| ≤ ∥h1 − h2∥∞,ωM(St+1),  M(St+1) :=  �  π(x|St+1)(1 + x2 + S2  t+1)ω/2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Uniform in j, with E maxj≤kn Ψj(Xt)4 < ∞, and EM(St+1)4 < ∞,  ETt  sup  ∥h1−h∥∞,ω<δ  |ǫ(St, h1) − ǫ(St, h)|2 ≤ 4γ2  sup  ∥h1−h∥∞,ω<δ  ∥h1 − h∥2  ∞,ωC ≤ Cδ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For (ii), Let T1 := maxj≤kn Ψj(Xt)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also ER4  t < C, ET1R2  t < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Since suph∈Hn ∥h∥2  ∞,ω < C,  ET1 suph∈Hn h(St, At)2 ≤ ET1(1+|St|2+|At|2)ω∥h∥2  ∞,ω < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also ET1 suph K(h)2 ≤ EM(St+1)2 suph ∥h∥2  ∞,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Hence ET1 suph∈Hn ρ(Yt+1, h)2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For (iii), the pathwise derivative of m is given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2), for C :=  �  E(1 + |St|2 + |At|2)ω + EM(St+1)2�  ,  ∥h − Qπ∥2 ≤ CE[h − Qπ]2 + CE|E(K(h) − K(Qπ)|St, At)|2 ≤ C∥h − Qπ∥∞,ω  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We note that for any h ∈ Hn ∪ {Qπ},  dm(Xt, h)  dh  [un] = γ  �  x∈A  E [π(x|St+1)un(St+1, x)|St, At] dx − un(St, At),  which does not depend on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also, for any h, τ, v, because of the linearity,  d2  dτ 2 m(Xt, h + τv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For condition (i), let r := 2 + ζ, a := |ρ(Yt+1, Qπ)|2+ζ, b :=  ���dm(Xt,Qπ)  dh  [un]  ��� then we have  b ≤ |γEtEπun(St+1, A)|+|un(St, At)| where Et = E(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='|St, At) and Eπ is with respect to the distribution  π(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='|St+1) for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let dπ := EEπ|un(St+1, A)|2r and d := E|un(St, At)|2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  Ea2 ≤ C + E|Rt|4+2ζ < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Also, d + dπ ≤ C because EEπ|un(St+1, A)|2r + E|v∗  n(St, At)|2r ≤ ∥v∗  n∥2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence  E|ρ(Yt+1, Qπ)|2+ζ  ����  dm(Xt, Qπ)  dh  [un]  ����  2+ζ  + E|ρ(Yt+1, Qπ)|2+ζ ≤ C(Ea2)1/2 �  d1/2 + d1/2  π  + 1  �  < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  62  Conditions (ii)(iii)(iv) are trivially satisfied because of the linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For condition (v), let T2 = maxj≤kn Ψj(Xt)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Recall that for h ∈ Cn, h = h1 + xun where  ∥h1 − Qπ∥∞,ω < Cδn and |x| ≤ Cn−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Hence  ET2 sup  h∈Cn  (ρ(Yt+1, h) − ρ(Yt+1, Qπ))2 ≤ CET2 sup  h∈Cn  |h(Xt) − Qπ(Xt)|2 + ET2 sup  h∈Cn  γ2[K(h) − K(Qπ)]2  ≤  �  CET2(1 + ∥Xt∥2)ω + γ2ET2M(St+1)2�  sup  h∈Cn  ∥h − Qπ∥2  ∞,ω  ≤  O(δ2  n + n−1) ≤ Cδ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  NPIV model: proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  In this case m(Xt, h) = E(h0(Wt) − h(Wt)|σt(X)) and ǫ(St, α) = Ut + h0(Wt) − h(Wt) − E(h0(Wt) −  h(Wt)|σt(X)), dm(Xt,α)  dh  [v] = E(v(Wt)|σt(X)), and  d2  dτ 2 m(Xt, h + τv) = 0 because of the linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We sequentially verify conditions in Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 This assumption follows immediately from  ∥α1 − α2∥2 = E (E(h1(Wt) − h2(Wt)|σt(X)))2 Σ(Xt)−1 = E[m(Xt, h1) − m(Xt, h2)]2Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 (i) Uniformly in j ≤ kn, for Mt := (1 + |Wt|2)ω,  E[Ψj(Xt)2 + 1]  sup  ∥α−α1∥∞,ω<δ  |ǫ(St, α1) − ǫ(St, α)|2  ≤  E[Ψj(Xt)2 + 1]  sup  ∥h−h1∥∞,ω<δ  [h1(Wt) − h(Wt) − E(h1(Wt) − h(Wt)|σt(X))]2  ≤  4E[Ψj(Xt)2 + 1] [Mt + E(Mt|σt(X))]  sup  ∥h−h1∥∞,ω<δ  ∥h1 − h∥2  ∞,ω ≤ Cδ2  given that E[Ψj(Xt)2 + 1]Mt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) Suppose E maxj≤kn Ψj(Xt)2[(1 + |Wt|2)ω + U 2  t ] < ∞,  E max  j≤kn Ψj(Xt)2 sup  α∈An  ρ(Yt+1, α)2  ≤  2E max  j≤kn Ψj(Xt)2 sup  α∈An  [h0(Wt) − h(Wt)]2 + 2E max  j≤kn Ψj(Xt)2U 2  t ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) We have ∥α1 − α2∥2 ≤ C∥α1 − α2∥2  ∞,ωE(1 + |Wt|2)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2  63  For (i) we have  E|ρ(Yt+1, α0)|2+ζ  ����  dm(Xt, α0)  dα  [un]  ����  2+ζ  = E|Ut|2+ζ|E(v∗  n(Wt)|Xt)|2+ζ∥v∗  n∥−(2+ζ) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For (ii)(iii), we have  d2  dτ 2 m(Xt, h + τv) = 0 for any h and v inside H0 ∪ Hn because of the  linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For (iv), we have supα∈Cn  1  n  �  t[dm(Xt,α)  dα  [un] − dm(Xt,α0)  dα  [un]]2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  For (v), let A := maxj≤kn Ψj(Xt)2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For h ∈ Cn, we know there is hn ∈ Hn and |x| ≤ Cn−1/2  so that  h = hn + xun,  ∥hn − h0∥∞,ω ≤ Cδn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Because EAun(W)2 < C, hence  EA sup  α∈Cn  (ρ(Yt+1, h) − ρ(Yt+1, α0))2 = EA sup  α∈Cn  (h(Wt) − h0(Wt))2  ≤  2EA sup  Cn  |hn(Wt) − h0(Wt)|2 + Cn−1EAun(W)2  ≤  Cδ2  n + Cn−1 ≤ Cδ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' For notational simplicity, write Γt = Γ(Xt), Σt = Σ(Xt), �Σt = �Σ(Xt)  and ρt = ρ(Yt+1, α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Using �Σ−1  t  − Σ−1  t  = �Σ−1  t (Σt − �Σt)Σ−1  t , the triangular inequality yields  1  n  �  t  ΓtΣt(�Σ−1  t  − Σ−1  t )ρt ≤ | 1  n  �  t  ΓtΣt(�Σ−1  t  − Σ−1  t )(�Σt − Σt)Σ−1  t ρt|  +| 1  n  �  t  Γt(�Σt − Σt)Σ−1  t ρt|  ≤  | 1  n  �  t  Γt(�Σt − Σt)Σ−1  t ρt| + OP (1) 1  n  �  t  |(�Σt − Σt)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Note �Σt = �A′  nΨn(Ψ′  nΨn)−1Ψ(Xt) where �An is a n × 1 vector of �ρ2  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also let (An, E(An|X), Gn, Un)  respectively be n × 1 vectors of (ρ2  t , Σt, gt, ut) where gt = ΓtΣ−1  t ρt and ut = ρ2  t − E(ρ2  t|σt(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let Jt  be the t th element of (I − Pn)E(An|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  We have ( 1  √n∥ �An − An∥)2 ≤ C 1  n  �  t(�ρt − ρt)2ρ2  t + C 1  n  �  t(�ρt − ρt)4 = OP (δ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' In addition, let  D be the diagonal matrix of ΓtΣ−1  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  1  √n∥PnGn∥ = OP ( 1  √n  �  ρ′nDPnDρn) = OP (  �  kn/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So  we have the following decomposition  1  n  �  t  Γt(�Σt − Σt)Σ−1  t ρt = 1  n[ �A′  nPn − E(An|X)]Gn = a1 + a2 + a3,  1  n  �  t  |(�Σt − Σt)|2 = 1  n∥Pn �An − E(An|X)∥2 ≤ C(a4 + a5 + a6)  a1  =  1  nE(An|X)′(Pn − I)Gn = 1  n  �  t  JtΓtΣ−1  t ρt = OP ( 1  √n)  �  EJ2  t Γ2  t Σ−1  t  64  =  OP (n−1/2  �  EJ2  t ) = OP (  �  ϕ2n/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  a2  =  1  nU ′  nPnGn ≤ OP (1)∥ 1  nU ′  nΨn∥∥ 1  n  �  t  Ψ(Xt)ΓtΣ−1  t ρt∥ = OP (kn  n )  a3  =  1  n[ �An − An]′PnGn ≤  1  √n∥ �An − An∥ 1  √n∥PnGn∥ ≤ OP (δ2  n + kn  n ) = OP (δ2  n)  a4  =  1  n∥(I − Pn)E(An|X)∥2 = OP (ϕ2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  a5  =  1  n∥PnUn∥2 ≤ OP (1)∥ 1  nU ′  nΨn∥2 = OP (kn  n )  a6  =  1  n∥Pn( �An − An)∥2 = OP (δ2  n)  Putting together, 1  n  �  t ΓtΣt(�Σ−1  t  − Σ−1  t )ρt = OP (pn) where pn = ϕ2  n + kn  n + δ2  n ≤ Cδ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3  NPQIV model: proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='3  In this model m(Xt, α) = P(Ut < h−h0|σt(X))−̟ where Ut = Yt−h0(Wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Suppose the conditional  distribution of Ut given (Xt, Wt) is absolutely continuous with density function fUt|σt(X),Wt(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Then  the derivative is defined as  dm(Xt, α)  dh  [v] = E(fUt|σt(X),Wt(h(Wt) − h0(Wt))v(Wt)|σt(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Verifying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Let  At(h)  :=  � 1  0  fUt|σt(X),Wt (x(h(Wt) − h0(Wt))) dx  Bt(v, h)  :=  E {At(v)[h(Wt) − h0(Wt)]|σt(X)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then m(Xt, h) = Bt(h, h), Em(Xt, h)2Σ(Xt)−1 = EBt(h, h)2Σ(Xt)−1 and ∥α−α0∥2 = EBt(h0, h)2Σ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  This assumption then follows from the condition that c2EBt(h, h)2Σ(Xt)−1 ≤ EBt(h0, h)2Σ(Xt)−1 ≤  c1EBt(h, h)2Σ(Xt)−1 for all ∥h − h0∥ < ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 (i) Let Aj := [Ψj(Xt)2 + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Fix any α = h ∈ An,  EAj  sup  ∥α−α1∥∞,ω<δ  |ǫ(St, α1) − ǫ(St, α)|2  ≤  2EAj  sup  ∥α−α1∥∞,ω<δ  |ρ(Yt+1, α1) − ρ(Yt+1, α)|2 + 2EAj  sup  ∥α−α1∥∞,ω<δ  |m(Xt, α1) − m(Xt, α)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  On one hand,  EAj  sup  ∥α−α1∥∞,ω<δ  |m(Xt, α1) − m(t, α)|2  65  ≤  2EAj  sup  ∥h−h1∥∞,ω<δ  P(h(Wt) − h0(Wt) ≤ Ut ≤ h1(Wt) − h0(Wt)|X)21{h1(Wt) > h(Wt)}  +2EAj  sup  ∥h−h1∥∞,ω<δ  P(h1(Wt) − h0(Wt) ≤ Ut ≤ h(Wt) − h0(Wt)|X)21{h(Wt) > h1(Wt)}  ≤  2EAj sup  u  fUt|σt(X),Wt(u)2(1 + |Wt|2)ω  sup  ∥h−h1∥∞,ω<δ  ∥h1 − h2∥2  ∞,ω  ≤  2EAj sup  u fUt|σt(X),Wt(u)2(1 + |Wt|2)ωδ2 ≤ Cδ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  On the other hand, for notational simplicity, write a = h(Wt) − h0(Wt), and a1 = h1(Wt) − h0(Wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Then ∥h − h1∥∞,ω < δ implies |a − a1| ≤ δ(1 + |Wt|2)ω/2 := gt(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' So  EAj  sup  ∥α−α1∥∞,ω<δ  |ρ(Yt+1, α1) − ρ(Yt+1, α)|2  ≤  EAj  sup  ∥h−h1∥∞,ω<δ  1{a ≤ Ut ≤ a1}1{a1 > a} + EAj  sup  ∥h−h1∥∞,ω<δ  1{a1 ≤ Ut ≤ a}1{a > a1}  ≤  EAj  �  sup  h1:∥h−h1∥∞,ω<δ  1{a ≤ Ut ≤ a1}fUt|σt(X),Wt(u)du1{a1 > a}  +EAj  �  sup  h1:∥h−h1∥∞,ω<δ  1{a1 ≤ Ut ≤ a}fUt|σt(X),Wt(u)du1{a > a1}  ≤  EAj  � a+gt(δ)  a  fUt|σt(X),Wt(u)du1{a1 > a} + EAj  � a  a−gt(δ)  fUt|σt(X),Wt(u)du1{a > a1}  ≤  2 sup  u fUt|σt(X),Wt(u)δEAj(1 + |Wt|2)ω/2 ≤ Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (ii) We have  E max  j≤kn Ψj(Xt)2 sup  α∈An  ρ(Yt+1, α)2 ≤ CE max  j≤kn Ψj(Xt)2 < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  (iii) Because Bt(h0, h)2 ≤ E  �  At(h0)2(1 + W 2  t )ω|σt(X)  �  ∥h − h0∥2  ∞,ω, we have  ∥h − h0∥2 ≤ EBt(h0, h)2Σ(Xt)−1 ≤ ∥h − h0∥2  ∞,ωEAt(h0)2(1 + W 2  t )ωΣ(Xt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Trivially |ρ(y, h)| + |m(x, h)| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Also  dm(Xt, α)  dh  [un] = E(fUt|σt(X),Wt(0)un(Wt)|σt(X)) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  So E|ρ(Yt+1, α0)|2+ζ ���dm(Xt,α0)  dα  [un]  ���  2+ζ  + E|ρ(Yt+1, α0)|2+ζ < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let f ′  Ut|σt(X),Wt denote the first derivative of fUt|σt(X),Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' We  have  d2  dτ 2 m(Xt, h + τv) = E[f ′  Ut|σt(X),Wt(h(Wt) − h0(Wt) + τv(Wt))v(Wt)2|σt(X)]  66  Hence  E sup  α∈Cn  sup  |τ|≤Cn−1/2  1  n  �  t  � d2  dτ 2 m(Xt, α + τun)|  �2  ≤  E sup  α∈Cn  sup  x  sup  |τ|≤Cn−1/2 E  �  f  ′2  Ut|σt(X),Wt(h(Wt) − h0(Wt) + τv(Wt))un(Wt)4|σt(X) = x  �  ≤  sup  u,x,w f  ′2  Ut,x,w(u)E[un(Wt)4|σt(X)] < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  sup  τ∈(0,1)  sup  α∈Cn  E  � d2  dτ 2 m(Xt, α0 + τ(α − α0))  �2  ≤  sup  τ∈(0,1)  sup  α∈Cn  E  �  E[f ′  Ut|σt(X),Wt(τ(h − h0))(h − h0)2|σt(X)]  �2  ≤ sup  α∈Cn  E  �  E(h − h0)2|σt(X)  �2  ≤  sup  h∈Cn  sup  w |h(w) − h(w)|4 ≤ O(δ4  n) = o(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let g1 := h(Wt) − h0(Wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  kn sup  α∈Cn  1  n  �  t  [dm(Xt, α)  dα  [un] − dm(Xt, α0)  dα  [un]]2  ≤  kn sup  α∈Cn  1  n  �  t  [E(fUt|σt(X),Wt(g1) − fUt|σt(X),Wt(0))un(Wt)|σt(X)]2  ≤  knL sup  α∈Cn  1  n  �  t  E(g2  1|σt(X))E(un(Wt)2|σt(X))  ≤  Ckn sup  α∈Cn  1  n  �  t  (E(h(Wt) − h0(Wt))2|σt(X)) = O(knδ2  n) = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Verifying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='2 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Let A = maxj≤kn Ψj(Xt)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  EA sup  h∈Cn  (ρ(Yt+1, h) − ρ(Yt+1, h0))2  ≤  EA sup  h∈Cn  1{−|h − h0| < Ut < |h − h0|} ≤ EA1{− sup  h∈Cn  |h − h0| < Ut < sup  h∈Cn  |h − h0|}  =  EA  � suph∈Cn |h−h0|  − suph∈Cn |h−h0|  fUt|σt(X),Wt(u)du  ≤  2EA sup  u fu|σt(X),Wt(u) sup  Cn  |h(Wt) − h0(Wt)| ≤ O(δn)EA(1 + Wt)ω/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Finally, Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='6 is naturally satisfied in the NPQIV model where �Σ(Xt) = Σ(Xt) =  ̟(1 − ̟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  67  References  Ai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
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+page_content=' Swiss  Finance Institute, Research Paper Series 20–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
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+page_content=' Journal  of Econometrics 188 447–465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
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+page_content=' and Ludvigson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
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+page_content=' Land of addicts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' an empirical investigation of  habit-based asset pricing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Journal of Applied Econometrics 24 1057–1093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
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+page_content=', Babuschkin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Chung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Mathieu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Jaderberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Czarnecki,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Dudzik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Huang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Georgiev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=', Powell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Alphastar:  Mastering the real-time strategy game starcraft ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' DeepMind blog 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' and Barron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Information-theoretic determination of minimax rates  of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content=' Annals of Statistics 1564–1599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
+page_content='  71' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AyT4oBgHgl3EQfSveK/content/2301.00092v1.pdf'}
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+Computer Anxiety: Supporting the Transition
+from Desktop to Mobile
+Thiago Donizetti dos Santos
+Federal University of ABC (UFABC)
+Santo André, SP, Brazil
+thiagods05@gmail.com
+Vagner Figueredo de Santana
+IBM Research
+São Paulo, SP, Brazil
+vagsant@br.ibm.com
+ABSTRACT
+Computer Anxiety is a phenomenon studied in multiple contexts
+and, in the actual COVID-19 scenario, it is gaining more and more
+importance as it impacts technology adoption and autonomy. Peo-
+ple with Computer Anxiety (PwCA) might feel intimidated, afraid
+of feeling embarrassed or scared of damaging computers, even
+before the actual interaction. Thus, supporting the detection of
+Computer Anxiety at scale has the potential to support the tech-
+nology industry to cope with this challenge. This position paper
+presents a user study involving 39 elderly participants in an inves-
+tigation on the feasibility of using interaction events common to
+desktop and smartphones to predict different levels of Computer
+Anxiety. Moreover, it also proposes research directions about the
+role of smartphones in the context of Computer Anxiety for elderly
+people as a mean of supporting good first user experiences with
+technology, meaningful daily use, privacy, and feeling safe even
+when doing mistakes. We expect this position paper motivates prac-
+titioners, designers, and developers to consider Computer Anxiety
+as one of the existing barriers when creating mobile applications
+for elderly people.
+CCS CONCEPTS
+• Human-centered computing → Field studies.
+KEYWORDS
+Computer Anxiety; Aging; Older Adults; Accessibility; Usability;
+User Experience; smartphones; mobile
+ACM Reference Format:
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana. 2021. Com-
+puter Anxiety: Supporting the Transition from Desktop to Mobile. , 7 pages.
+1
+INTRODUCTION
+The use of smartphones in daily activities is increasing in a fast pace.
+The ownership of smartphones by adults, in US, grew from 35%
+to 81% in the period between 2011 and 2019 [6]. Smartphones are
+generally used to make and receive calls, to access the internet, to
+text, to access social media and services (e.g., such as food delivery,
+transport, and mobility). Bearing in mind the ownership of devices,
+the rate of people owning smartphones drops from 96% (for the
+Permission to make digital or hard copies of part or all of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for third-party components of this work must be honored.
+For all other uses, contact the owner/author(s).
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13,
+2021, Yokohama, Japan
+© 2021 Copyright held by the owner/author(s).
+group aging between 18 and 29 years) to 53% (for the group aging
+65+ years) [6]. This difference shows that elderly people are increas-
+ingly using smartphones, but have not yet adopted this technology
+in the same way as young adults, which may indicate the existence
+of aspects impacting the adoption of smartphones by elderly users.
+Since a variety of services and content are currently available
+online, the difficulties faced by elderly people while using smart-
+phones may impact their quality of life. Through the use of mobile
+applications, one could order food, get a driver using a transport
+app, travel alone using maps, communicate with family members
+using instant messaging, learn new things on e-learning platforms
+or just browse photos and other contents on the social media appli-
+cations. Such services foment autonomy and help to avoid common
+stereotypes of dependency and limitations.
+One phenomenon that can help in the understanding of issues
+faced by the elderly people when using new technologies is Com-
+puter Anxiety (CA). CA can be defined in terms of affective factors
+such as intimidation, fear, apprehension, hostility, and worries that
+one will be embarrassed, will look stupid, or thinks she/he could
+damage the computer [15]. Although generally related to the use of
+computers, CA can also impact the use of other electronic devices
+and previously was also called “Technophobia” [3]. CA is related
+to technology acceptance [36] and it is generally related to biologi-
+cal changes such as blood pressure, heart rate and electrodynamic
+responses that occur while a person is using a device [30]. CA
+symptoms can occur during the interaction with the system and
+even before it, affecting the perceived ease of use and acting as a
+barrier, impacting the system accessibility as well [35]. Although
+CA can affect people of all ages, the literature shows that CA is
+more present in older groups [7, 12, 35]. Adding this to the rela-
+tionship between CA and technology acceptance, elderly people
+may face problems to use mobile devices and, when they use it, CA
+could make the system difficult to use, make them perform poorly
+on tasks or fail to achieve their goals while using the device.
+In this context, this position paper presents results from a user
+study involving 39 elderly participants aiming at exploring the
+feasibility of predicting CA from interaction events common to
+desktop and smartphones, mapped here as a regression problem.
+In addition, it also discusses the role of smartphones in supporting
+people with CA (PwCA) in the process of learning how to use
+technology, first experience, daily use, and autonomy. Thus, the
+following research questions were defined to guide the study: (1)
+Is it possible to predict different levels of CA using interaction events
+common to desktop and smartphones? and (2) How can smartphones
+be used to support the adoption of new technologies by PwCA? The
+work is organized as follows: the section 2 presents the related
+work, the section 3 details the user study, the section 4 shows the
+arXiv:2301.02201v1  [cs.HC]  5 Jan 2023
+
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana
+results, section 5 discusses the role of smartphone for PwCA and
+section 6 concludes.
+2
+RELATED WORK
+CA is also called Computerphobia, Computer Apprehension, and
+Technophobia in the literature [3]. Rosen et al. [32] pointed out the
+following methods and questionnaires to measure CA:
+• Computer Anxiety Index (CAIN) examines avoidance of,
+caution with, negative attitudes toward, and disinterest in
+computers [23].
+• Computer Attitude Scale (CAS) assesses computer liking,
+confidence, and anxiety through a Likert attitude-measurement
+format [21].
+• Attitudes Toward Computers Questionnaire (ATCQ) as-
+sesses attitudes towards computer appreciation, usage, and
+societal impact [31].
+• Computer Anxiety Rating Scale (CARS) assesses behav-
+ioral, cognitive, and affective components related to technol-
+ogy use [15, 32].
+• Mobile Computer Anxiety Scale (MCAS) assesses anxiety
+regarding mobile computer using a 38-item Likert scale [39].
+The literature presents multiple factors associated with CA. In
+sum, PwCA usually have less experience in using computers, have
+low Computer Self-efficacy (CSE)1, take too long to accomplish
+tasks, perform worse when compared to other users, have negative
+beliefs about computer/skills, or negative bodily sensations previ-
+ous/during the interaction with a computer [35]. Earlier studies find
+a strong relationship between age and CA levels, showing evidence
+that CA is more present in groups with older people and that they
+have more CA than younger ones [7, 12, 27]. This can be related
+to the pace in which technology advances and to the fact that 48%
+of older adults report that they usually need someone else to set
+up a new electronic device or show them how to use it [5]. In this
+scenario, mobile accessibility has potential to support PwCA in
+increasing CSE, reducing negative beliefs and worries of using or
+damaging the device in front of other people.
+CA is also present in a few acceptance models. The Technology
+Acceptance Model (TAM) is one example. It uses the CA as a compo-
+nent that changes the perceived ease of use2. So, when considering
+the adoption of new technology by elderly people, CA should be
+taken into account, from design to personalization features.
+The influence of CA was investigated in contexts such as accep-
+tance of e-learning tools, e-gov, new technologies and health-care
+systems [8, 14, 17, 18, 22, 29, 36]. These studies stated that:
+• PwCA tend to prefer traditional classes instead of e-learning
+systems and computer-based tests;
+• PwCA usually perform worse on virtual classes when com-
+pared to people without CA;
+• PwCA have more difficulties in accepting new technologies;
+• Older PwCA face difficulties using home telehealth services
+and to learn how to use smartphones.
+1Computer Self-efficacy (CSE) is the belief one has in his/her own abilities to perform
+a task in the computer [9]
+2Perceived ease of use is defined as the degree to which a person believes that using a
+particular system would be free of effort [1]
+Considering support offered to PwCA, the literature presents
+that instructional/technical support reduce CA in the context of
+e-learning systems [13, 24, 26]. Finally, smartphones have potential
+to provide instructional support in a privacy respecting way, sup-
+porting the user-technology dialog, reducing worriers associated
+to trial and error inherent to learning.
+3
+METHOD
+This section details how the user study was run, including its materi-
+als, procedure, setup, experiment design, and data analysis planned.
+The goal of the study was to collect data related to questionnaires
+to identify different CA levels and collect detailed interaction data
+while participants performed tasks on a website in order to detect
+CA. Moreover, this study also aimed at exploring interaction data
+common to desktop and smartphone and at understanding the role
+of smartphone usage for PwCA. The types of data captured will be
+detailed in the Data Analysis section. Before the main experiment,
+a pilot was performed in order to assess the user study plan as
+whole. The pilot included 4 elderly participants, recruited the same
+way the participants of the main experiment (detailed in the next
+section). The pilot achieved its goals in assessing protocol adopted,
+duration, and data capture procedure. Next, we detail the method
+followed in the main experiment.
+3.1
+Participants
+Elderly people may face difficulties in staying up-to-date with tech-
+nology and, since they have not used computers since childhood,
+many of them face CA even when performing a simple task to
+others age groups [35]. Hence, the target-audience considered in
+this work is elderly people.
+The participants of the experiment were recruited from a list
+of registered people at the elderly center of the city of São Paulo,
+Brazil, called Reference Centre for Citizenship of Elderly (CRECI@).
+São Paulo is the biggest city in Latin America, with a population
+of approximately 11.2 million people in the last census (2010) and
+the current estimate is of 12.2 million people3; the population of
+elderly people is approximately 11.9%4. Before recruiting partici-
+pants, a partnership was signed and the proper process for ethics
+committee was followed at the Federal University of ABC (process #
+2.808.392 and CAEE: 94704418.8.0000.5594), detailing the materials,
+procedure, questionnaires, data to be collected, and analysis to be
+performed.
+Moreover, only those who had never taken computer classes
+offered by CRECI@ were invited as potential participants, since
+results from the literature point that computer classes might reduce
+CA [35], which could result in a bias.
+3.2
+Materials
+Questionnaires about CA, use of smartphones and computers skills
+were applied (Appendix A). In order to isolate CA from other co-
+morbidities, questionnaires to assess cognitive abilities and levels
+of depression were also applied. Only participants with low levels
+of depression and those who do not present signals of dementia or
+3https://cidades.ibge.gov.br/brasil/sp/sao-paulo/panorama
+4http://produtos.seade.gov.br/produtos/retratosdesp/view/index.php?
+temaId=1&indId=4&locId=3550308
+
+Computer Anxiety: Supporting the Transition
+from Desktop to Mobile
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+cognitive deficits had their data considered in the analyses. These
+two metrics were considered in the exclusion criteria, detailed in
+the procedure. The questionnaires applied are listed below:
+• Technology use and profile: Has questions about the par-
+ticipant’s age, educational level, and frequency of use of
+computers and smartphones. This questionnaire was applied
+to give an overview of how participants use technology on
+a daily basis (Appendix A).
+• Mini Mental: A cognitive screening test used for adults and
+the elderly to evaluate orientation, memory and attention,
+naming ability, obedience to verbal and writing commands,
+free writing of a sentence, and copying a complex drawing
+(two intersecting polygons). It is currently the most used test
+for this type of assessment in the world [25]. The rationale
+for using Mini Mental was to identify comorbidity to CA.
+• Geriatric Depression Scale (GDS): GDS is a scale with 30
+yes/no questions used for screening depression in elderly
+people [2, 40]. The rationale for using GDS was also to iden-
+tify comorbidity to CA.
+• Computer Anxiety Rating Scale (CARS): CARS has nine-
+teen questions in a five points Likert scale ranging from
+strongly disagree to strongly agree. It assesses the behavioral,
+cognitive and affective components related to technology
+use [15, 32]. The rationale for using CARS is that it is the
+most referenced questionnaire for screening CA [35].
+• Computer Self-Efficacy (CSE): CSE is a ten items scale used
+to assess Computer Self-efficacy [9, 38]. The rationale for
+using CSE was to cross check the results from this study
+with results from the literature that show that CSE has a
+strong but inverse relationship with CA.
+• System Usability Scale (SUS): A five points Likert scale ques-
+tionnaire with ten items. It is often used to assess the per-
+ceived usability [4]. The rationale for using it was to compare
+perceived usability of the website and the CARS values.
+In order to capture the interaction data, the participants used a
+desktop computer including a interaction logger and internet access.
+The interaction logger used was the open source logger called User
+Test Logger5. The User Test Logger captures all JavaScript events
+such as mouse movements, clicks, keys pressed, etc., and generates
+a raw log file where each line represents an event and information
+about when, where, and what is related to the event triggered [34].
+3.3
+Procedure
+The experiment was structured into three steps: pre-test, test, and
+post-test. The steps are detailed next.
+3.3.1
+Pre-test. First, screening tests for cognitive deficit, depres-
+sion, and literacy were applied to identify participants whose scores
+fall outside the inclusion criteria. The Mini Mental presents a score
+indicating good cognitive capacity relating the answer points ob-
+tained and years of education of the participant as follows:
+• No formal education: ≤ 21 points;
+• 1 to 5 years of formal education: ≤ 24 points;
+• 6 to 11 years of formal education: ≤ 26 points;
+• 12+ years of formal education: ≤ 27 points
+5https://github.com/IBM/user-test-logger
+For GDS screening test, a score of four points or less on the scale
+indicates low levels of depression. Hence, the exclusion criterion
+was: 𝐺𝐷𝑆 ≥ 5 points. After the tests considered in the exclusion
+criteria, CARS and CSE were applied.
+3.3.2
+Test. The tasks were performed individually on a computer
+with the User Test Logger installed. First, each participant was asked
+to access SESC homepage (Figure 1). The Social Service of Com-
+merce (SESC) is a private entity maintained by the entrepreneurs of
+the trade in goods, tourism, and services. SESC aims to provide the
+welfare and quality of life to workers in this sector and their fami-
+lies 6. SESC offers many activities for elderly people such as courses,
+sports, art exhibition and culture-related events in multiple units in
+the metropolitan area of São Paulo. The tasks were defined aiming
+to encourage participants to search online for activities offered by
+SESC and others services in the city, hoping to help them to see the
+internet as a tool which they can use as a means of improving their
+quality of life. The same way they do at CRECI@. In addition, the
+tasks were structured to be as familiar and as close to real tasks as
+possible. The tasks read out loud to participants were the following:
+(1) Search for an event, class, or activity he/she might be inter-
+ested;
+(2) Find the address of the unit where the chosen activity/event
+is offered;
+(3) Find the route to the unit.
+Figure 1: Homepage of the SESC’s website.
+There was no maximum time limit for the tasks. Thus, the task
+duration depended on participants saying whether they finished or
+gave up on the task. Finally, Thinking-Aloud Protocol [20] was used
+to understand the rationale of users while performing the tasks.
+3.3.3
+Post-test. In order to evaluate the perceived usability and
+any relationship with CA levels, they were asked to answer the
+SUS questionnaire.
+3.4
+Data analysis
+According to the exclusion criteria defined, the data from partic-
+ipants who did not score the points required by Mini Mental or
+scored five or more points on GDS were removed from the data set
+to be analyzed. The resulting data set combined data from the inter-
+action logger, questionnaires, and thinking-aloud protocol. Since
+the logged data were in raw format, all data captured were pro-
+cessed in order to extract usage metrics. The following metrics were
+considered having in mind they could also be applied in a mobile
+interaction setting: time to perform the task, number of clicks and
+double-clicks, click duration, typing velocity, and total time typing.
+6https://www.sescsp.org.br/
+
+3 Pagina Inicial - Sesc SP - Mozilla Firefox
+Debugging with Firefox Developer T X
+S Pagina Inicial - Sesc SP
++
+@ https://www.sescsp.org.br
+Q Pesquisar
+国三不
+Um Portal para cada um!
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+OPORTUNIDADES
++ CREDENCIAL PLENA
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+indios pela defesa de seus territorios e os desafios de
+construir o futuro mediante contextos impostos pela
+sociedade nao indigenaCHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana
+Finally, all metrics and the questionnaires scores were combined
+in a single comma-separated-values (CSV) file used to perform
+the data analysis. This CSV file is the main data source for the
+regression analysis performed to predict CARS values based only
+on interaction data, which could allow its use at scale.
+4
+RESULTS
+The experiment included 39 participants, but data from 8 partici-
+pants were not considered in the analysis due to exclusion crite-
+ria. Thus, considering the data from the remaining 31 participants
+(51.61% of males, 48.39% of females). The participants’ ages ranged
+between 62 and 87 years (𝑥 = 72.84). Regarding the computer usage,
+25.81% of the participants reported that they do not own a computer
+and do not have frequent access to computers, while 12.90% do not
+own a computer, but use it at lanhouses or those available in public
+places; 61.29% reported owning computers. Regarding frequency
+of use, 61.29% reported that rarely (less than once a month) use a
+computer, 16.13% reported that use it sometimes (more than once
+a month), 6.45% reported that usually (more than once a week)
+use it, and 16.13% reported that always (everyday) use computers.
+When considering ownership and use of smartphones, 87.10% of
+the participants reported that they own smartphones and 48.39%
+reported that they always use it. The education level of the par-
+ticipants varies from 0 to 15 years of formal education (𝑥 = 10.42)
+(Figure 2).
+Figure 2: Distribution of years of formal education.
+The obtained scores for CARS ranged from 20 to 59 (𝑥 = 42.19).
+Based on [10], the maximum and minimum scores were used to
+divide the data into 3 groups as follows: 𝐶𝐴𝑅𝑆_𝑟𝑎𝑛𝑔𝑒 : 59 − 20 = 39
+and 𝐺𝑟𝑜𝑢𝑝_𝑟𝑎𝑛𝑔𝑒 : 𝐶𝐴𝑅𝑆_𝑟𝑎𝑛𝑔𝑒/3 = 13.
+• No CA: CARS < 33 (6 participants);
+• Moderate CA: 33 ≤ CARS < 46 (14 participants);
+• High CA: CARS ≥ 46 (11 participants).
+Considering the three CA groups, Figure 3 shows the presence of
+CA considering the participants’ age. It can be seen that participants
+in the no CA group are among the youngest. The age of these
+participants ranged from 63 to 78 years old (𝑥 = 68.43, 𝜎 = 5.68).
+The age of the participants in the moderate CA group ranged from
+62 to 83 years old (𝑥 = 73.92, 𝜎 = 5.60). And, for the high CA group,
+the age of the participants ranged from 63 to 87 (𝑥 = 74.36, 𝜎 = 6.77),
+showing that high levels of CA are present over almost the entire
+age range covered in the study. Mann-Whitney non-parametric test
+shows that age was different between no CA and moderate CA
+groups (p-value = 0.03); no significant difference was found in other
+pairwise group comparisons.
+Figure 3: Age distributions for different CA groups.
+Bearing in mind the use of smartphones, Figure 4 shows smart-
+phone ownership and different uses by different CA groups. It can
+be seen that high CA group uses smartphones more for calls and less
+for leisure and other communication activities (e.g., games, music,
+video, instant messages, and social networks). On the other hand,
+people in the no CA group use smartphone heavily to access social
+network, instant messages, and internet. Analyzing the ownership
+rate by CA groups, it can be seen that 73% of the participants with
+high CA own smartphones, while the ownership is greater in the
+moderate CA group (92%) and reaches 100% for the no CA group.
+Similarly, considering the frequency of use of computers, people
+who rarely use computers are the ones with greater CA levels. 82%
+of the participants with high CA use it rarely, while it is 61.5% of
+the participants of the moderate CA and 28% of the no CA group.
+Figure 4: Smartphone ownership and use by CA groups.
+Bearing in mind task completion, 23 (74.19%) participants found
+an activity and completed the first task. Four participants found an
+activity that was not of interest to them or was offered by a unit
+far from their home, but they did not find another one after that.
+The remaining four gave up without finding an activity. For the
+Task 2, nine out of 23 (39.13%) participants found the address of
+the unit where the selected activity is offered and 14 participants
+found only the name of the unit. Regarding the Task 3, six out of
+nine (66.67%) participants found the map available on the site, but
+all of them failed to find the route to the unit. Only two out of
+nine (22.22%) participants figured out how to put the starting point
+address on the map-based UI. In sum, task completion dropped from
+74.19% (task 1), to 39.13% (task 2), and to 0% (task 3); 22.22% (6.45%,
+considering 31 participants) partially completed the last task. This
+can be related to task difficulty, fatigue effects, and task dependence.
+
+Education levels (in years)
+12
+ participants
+10
+Number of 
+2
+0
+4
+6
+8
+10
+11
+13
+14
+15
+Education yearsAge vs. CARS groups
+85
+80 
+e
+75
+70 
+65
+。
+No CARS
+Moderate CARS
+High CARSSmartphone use x Computer Anxiety Leve
+Music/Video
+Message
+Games
+Call
+Instant Message
+Social networks
+Internet
+Smartphone ownership
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+%06
+100%
+ No CA
+ Moderate CA
+High CAComputer Anxiety: Supporting the Transition
+from Desktop to Mobile
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+However, after triangulating these results with thinking-aloud data,
+it was possible to identify that participants faced difficulties with
+the map-based UI. For instance, participant 11 said: “It doesn’t say
+where it is. I didn’t like SESC.”, participant 35 said: “Why don’t you
+have the address on the about the unit page?” and participant 42
+said: “It will take me a long time to find it (address)”. While using
+the map, for instance, participant 22 said: “What should I do here?
+I have never used it (map) before”; participants are numbered from
+1-4 for the pilot and 5-43 for the experiment.
+Table 1 summarizes the time taken by each group to complete
+the tasks, showing the mean time and standard deviation by CA
+group. It can be seen that the group of participants with high CA
+had a mean shorter task time than the other groups in some tasks.
+This might be related to the fact that PwCA usually gave up more
+because they feel frustrated, lost, or think that they would not be
+able to finish the task. This also shows the relationship between
+high CA and low CSE, as identified in previous studies [35] and
+[11]. This can be exemplified by the participants quotes as: “I think
+I will have difficulty in this task”, “This is difficult”, “I’m lost, I don’t
+know what to do”, and “I don’t know how to find it”.
+Task 1 in sec.
+Task 2 in sec.
+Task 3 in sec.
+Group
+𝑥 (𝜎)
+𝑥 (𝜎)
+𝑥 (𝜎)
+High CA
+411.18 (230.72)
+599.67 (220.30)
+398.00 (262.19)
+Mod. CA
+647.90 (666.46)
+450.13 (336.52)
+560.50 (30.50)
+No CA
+525.14 (331.39)
+587.00 (390.73)
+399.67 (375.05)
+Table 1: Average task time by group and standard deviations.
+Although the interaction data was collected during the use of a
+desktop computer, in this study we explore metrics which can be
+captured in a mobile setting as well, namely: task time, numbers
+of clicks and double clicks, mean click duration (interval between
+pressing and releasing), typing velocity and total time typing. All
+metrics were normalized for the regression analysis. Prior to fit-
+ting the regression model, a random oversampling algorithm was
+applied7 addressing the minority values and a train-test split of
+80% / 20% was applied. Figure 5 shows Random Forest regression
+predictions for CA values (y-axis) vs. CARS values in the test set
+(x-axis). The obtained regressor has a mean squared error (MSE) of
+22.21 and 𝑅2 = 0.84. The high MSE value is due to errors related to
+predictions for lower CA scores. This pessimist prediction would
+show that users need more support than they would actually need,
+so such regressor might be useful for indicating when support for
+PwCA could be applied.
+5
+DISCUSSION
+Although there are few studies reporting no significant relationship
+between age and CA [16, 19, 28], there are also evidences that older
+people manifest more CA than younger ones [7, 12, 27, 33, 37].
+Results suggest that younger participants were in the no CA group,
+while the older were in the moderate CA group. For high CA group,
+results suggest that there were participants with high CA almost
+in the whole age range considered, but it still shows a greater
+concentration among the older ones (median = 74 years, 𝑥 = 74.2,
+7https://imbalanced-learn.org/stable/over_sampling.html
+Figure 5: Regression test results of CARS scores prediction.
+𝑠𝑖𝑔𝑚𝑎 = 7.12). These results are similar to other findings in the
+literature, indicating that the CA is more present in the older groups.
+Moreover, unlike [30], the results suggest that previous experience
+may impact CA levels, since people who rarely use computers and
+smartphones were the ones with greater CA levels.
+The difficulty in adopting new technologies is suggested by re-
+sults in Figure 4. Although there is a high rate of PwCA owning
+smartphones, the most frequent reported use is making calls. Thus,
+the participants use the smartphone, but they use the same way
+they used to do with the old phones: making calls. Moreover, the no
+CA group presented the same rate (57%) when using smartphones
+to make calls, using instant message and social networks apps. In
+contrast, moderate and high CA groups use more to make calls
+than to any of the other functions analyzed. Also they use more
+to make calls then the no CA group and presented a lower rate of
+ownership of smartphones than the no CA group. The increasing
+rate of smartphone ownership and the decreasing rate of use of
+different smartphone functions show possible impacts of CA on
+the behavior of the elderly regarding technology. The second most
+frequent use is instant message apps, even though there is also a
+difference between groups for this use. The use of this application
+could be related to the presence of functions such as the ability to
+make calls using the app or using voice messages. They reported to
+be used to make calls and that the use of voice messages is easier for
+them, since they generally have difficulties using the keyboard of
+the smartphones to write and may have difficulty in reading due to
+the size of the screen and letters. These findings suggest that high
+levels of CA may affect people of all ages, but it is more present
+among older ones due to factors such as lack of practice or lack of
+knowledge about recent features that could improve UX.
+The results regarding task completion and time taken to complete
+tasks show how CA levels might affect task performance. Besides
+the low task completion rate for the high CA group, this group
+took less time trying to complete the Task 1, showing that PwCA
+usually gave up more. The implications for HCI researchers in this
+aspect are related to the design of shorter and simpler tasks and to
+improve user experience, accessibility and usability, since high CA
+levels impact negatively the perceived easy of use and CSE, as they
+may feel frustrated or lost when facing problems to achieve their
+goals during the interaction.
+
+Random Forest Regression
+50 
+30 
+区
+21
+31
+50
+CARSCHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana
+CA is also related to specific situations since it tends to arise or
+be stronger during the first use of a device [17]. Thus, in addition
+to promoting the contact of the elderly with new technologies, it is
+important to promote a good first experience. It means an experi-
+ence free of effort, that helps the user to feel safe and unafraid of
+making mistakes. A bad experience, during which the user feels lost
+or makes mistakes, can reinforce the fears of PwCA. Consequently,
+this can increase their CA levels, making them believe they are
+unable to use it and prevent them from trying again.
+The results of the Random Forest regression show that CA in-
+fluences on how users interact with the system. Although the data
+belongs to desktop computer interaction domain, the result sug-
+gests that such approach should be explored in the mobile settings
+as well, given that the interaction events selected are common to
+desktop and smartphones. The prediction of higher CARS values
+could trigger personalization features and additional support, for
+instance. Smartphones have a myriad of sensors that could be used
+for such personalization features and here we advocate the use of
+the following metrics as a starting point: task time, numbers of
+clicks and double clicks, mean click duration, typing velocity and
+total time typing.
+This paper defends the idea that the use of smartphones by the
+elderly can bring benefits such as autonomy, access to content and
+services and communication. However, CA may create barriers
+which prevent elderly from enjoying these benefits. Therefore, we
+argue that further studies are needed regarding the influences of
+levels of CA in acceptance of smartphones and apps by the elderly
+people. On his work about the development of a mobile computer
+anxiety scale (MCAS), [39] argues that MCAS is related to three
+distinct components: (1) traditional CA construct; (2) Internet anxi-
+ety construct and (3) special factors making up the mobile anxiety
+construct (e.g., equipment limitation). The limitations of mobile
+equipment listed by [39] and that elderly people report as being
+problematic for them are: small screens and small multi-function
+key pads; lower display resolution; unfriendly user-interfaces; and
+graphical limitations. Thus, these limitations should be considered
+when developing new technologies for elderly people as well. Fur-
+thermore, the importance of the first experience is found in the
+literature and reported in the interviews conducted in this study.
+They reported having purchased or been presented with a new
+smartphone and feeling lost or afraid to use it. So, it is important
+that the device has a simple, accessible and usable interface. Another
+common factor reported, is the fear of making mistakes, looking
+stupid or breaking the device. In this sense, it is important that the
+system provides a safe environment, which asks for confirmation
+for important (dangerous) actions. And, in case the user makes a
+mistake, the system must provide ways to recover from the error,
+returning to the previous state without difficulty.
+In Brazil, families often have a computer to be shared by family
+members. It can make elderly people afraid of breaking what be-
+longs to the family or afraid of losing some important data if they
+do something wrong. The smartphone, on the other hand, is seen
+as an personal device. According to the participants’ reports, this
+brings greater freedom to learn how to use through trial and error.
+Besides that, as it is mobile, it has the advantage that elderly people
+can avoid to use it in front of other people. This can help dealing
+with the fear of not knowing how to use it or making mistakes in
+front of younger people.
+Finally, we believe a research agenda about the role of smart-
+phones for elderly people in the context of CA should address the
+following research questions: (1) How to detect CA during the use
+of mobile phones at scale? (2) How to provide (first) good user
+experiences for this population? (3) How to create a secure envi-
+ronment for PwCA to recover from mistakes? (4) How to combine
+technology use with learning in order to increase CSE? (5) How
+CA on the desktop relates to CA on smartphones?
+6
+CONCLUSION
+This position paper discussed how elderly people use smartphones
+in a specific region of São Paulo, Brazil, and shows how CA is
+present in this sample of the population. Our findings suggest
+that higher CA levels are prevalent on higher age and CA impacts
+how users interact with technologies. In addition, results indicate
+that the behavior of elderly users when performing tasks can be
+negatively impacted not only because of age-related factors, but
+also by the CA levels. The results indicating the preference of some
+applications over others by elderly people indicate the need for
+further studies on why some technologies still present barriers
+for PwCA. Moreover, the shorter task time obtained by the high
+CA group and the fact that they usually gave up when feel lost
+shows the importance of shorter and simpler tasks. The differences
+between groups regarding ownership of smartphones show that
+CA may impact on the technology adoption. In addition, the results
+showing the preference for known functions can be important to
+designers and developers to consider when developing new systems,
+since the inclusion of functions considered easy to use may increase
+the system adoption and improve user experience by reducing
+frustration.
+Finally, tackling technology adoption by elderly people may im-
+prove their quality of life, since the use of smartphones and the
+wide variety of applications and services may help they achieve
+independence, make their lives more comfortable, promoting their
+better participation in the community and allowing access to the
+most diverse online content. The presented user test shows that
+metrics such as task time, number of clicks, click duration, typing
+speed and total time typing can support the prediction of different
+CARS scores (𝑅2=0.84). In the current context of the COVID-19 pan-
+demics, promoting autonomy, communication and leisure activities
+became core goals for any technology and here we emphasize that
+by understanding CA and considering it in design and development
+phases mobile apps have the potential to change the live of PwCA.
+ACKNOWLEDGMENTS
+We thank the CRECI@ for all the support. This study was financed
+in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
+Superior - Brasil (CAPES) - Finance Code 001.
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+A
+TECHNOLOGY USE AND PROFILE
+(1) Do you have a computer available at home? If not, do you use
+a computer somewhere else (e.g., at work or in the lanhouse)?
+(2) How often do you use a computer?
+A: [ ] Rarely
+[ ] Sometimes
+[ ] Usually
+[ ] Always
+(3) What do you usually do on computer?
+(4) Do you own a smartphone?
+(5) How often do you use smartphones?
+A: [ ] Rarely
+[ ] Sometimes
+[ ] Usually
+[ ] Always
+(6) What do you usually do on smartphone?
+Internet: [ ] Yes
+[ ] No
+Social networks: [ ] Yes
+[ ] No
+Instant Message: [ ] Yes
+[ ] No
+Call: [ ] Yes
+[ ] No
+Games: [ ] Yes
+[ ] No
+Message: [ ] Yes
+[ ] No
+Music / Video: [ ] Yes
+[ ] No
+
diff --git a/FtE0T4oBgHgl3EQfRAB3/content/tmp_files/load_file.txt b/FtE0T4oBgHgl3EQfRAB3/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf,len=635
+page_content='Computer Anxiety: Supporting the Transition  from Desktop to Mobile  Thiago Donizetti dos Santos  Federal University of ABC (UFABC)  Santo André, SP, Brazil  thiagods05@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='com  Vagner Figueredo de Santana  IBM Research  São Paulo, SP, Brazil  vagsant@br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='com  ABSTRACT  Computer Anxiety is a phenomenon studied in multiple contexts  and, in the actual COVID-19 scenario, it is gaining more and more  importance as it impacts technology adoption and autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Peo-  ple with Computer Anxiety (PwCA) might feel intimidated, afraid  of feeling embarrassed or scared of damaging computers, even  before the actual interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Thus, supporting the detection of  Computer Anxiety at scale has the potential to support the tech-  nology industry to cope with this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This position paper  presents a user study involving 39 elderly participants in an inves-  tigation on the feasibility of using interaction events common to  desktop and smartphones to predict different levels of Computer  Anxiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Moreover, it also proposes research directions about the  role of smartphones in the context of Computer Anxiety for elderly  people as a mean of supporting good first user experiences with  technology, meaningful daily use, privacy, and feeling safe even  when doing mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' We expect this position paper motivates prac-  titioners, designers, and developers to consider Computer Anxiety  as one of the existing barriers when creating mobile applications  for elderly people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  CCS CONCEPTS  Human-centered computing → Field studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  KEYWORDS  Computer Anxiety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Aging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Older Adults;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Accessibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Usability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  User Experience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' smartphones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' mobile  ACM Reference Format:  Thiago Donizetti dos Santos and Vagner Figueredo de Santana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Com-  puter Anxiety: Supporting the Transition from Desktop to Mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' , 7 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  1  INTRODUCTION  The use of smartphones in daily activities is increasing in a fast pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The ownership of smartphones by adults, in US, grew from 35%  to 81% in the period between 2011 and 2019 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Smartphones are  generally used to make and receive calls, to access the internet, to  text, to access social media and services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=', such as food delivery,  transport, and mobility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Bearing in mind the ownership of devices,  the rate of people owning smartphones drops from 96% (for the  Permission to make digital or hard copies of part or all of this work for personal or  classroom use is granted without fee provided that copies are not made or distributed  for profit or commercial advantage and that copies bear this notice and the full citation  on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13,  2021, Yokohama, Japan  © 2021 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  group aging between 18 and 29 years) to 53% (for the group aging  65+ years) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This difference shows that elderly people are increas-  ingly using smartphones, but have not yet adopted this technology  in the same way as young adults, which may indicate the existence  of aspects impacting the adoption of smartphones by elderly users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Since a variety of services and content are currently available  online, the difficulties faced by elderly people while using smart-  phones may impact their quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Through the use of mobile  applications, one could order food, get a driver using a transport  app, travel alone using maps, communicate with family members  using instant messaging, learn new things on e-learning platforms  or just browse photos and other contents on the social media appli-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Such services foment autonomy and help to avoid common  stereotypes of dependency and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  One phenomenon that can help in the understanding of issues  faced by the elderly people when using new technologies is Com-  puter Anxiety (CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' CA can be defined in terms of affective factors  such as intimidation, fear, apprehension, hostility, and worries that  one will be embarrassed, will look stupid, or thinks she/he could  damage the computer [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Although generally related to the use of  computers, CA can also impact the use of other electronic devices  and previously was also called “Technophobia” [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' CA is related  to technology acceptance [36] and it is generally related to biologi-  cal changes such as blood pressure, heart rate and electrodynamic  responses that occur while a person is using a device [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' CA  symptoms can occur during the interaction with the system and  even before it, affecting the perceived ease of use and acting as a  barrier, impacting the system accessibility as well [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Although  CA can affect people of all ages, the literature shows that CA is  more present in older groups [7, 12, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Adding this to the rela-  tionship between CA and technology acceptance, elderly people  may face problems to use mobile devices and, when they use it, CA  could make the system difficult to use, make them perform poorly  on tasks or fail to achieve their goals while using the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  In this context, this position paper presents results from a user  study involving 39 elderly participants aiming at exploring the  feasibility of predicting CA from interaction events common to  desktop and smartphones, mapped here as a regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  In addition, it also discusses the role of smartphones in supporting  people with CA (PwCA) in the process of learning how to use  technology, first experience, daily use, and autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Thus, the  following research questions were defined to guide the study: (1)  Is it possible to predict different levels of CA using interaction events  common to desktop and smartphones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' and (2) How can smartphones  be used to support the adoption of new technologies by PwCA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The  work is organized as follows: the section 2 presents the related  work, the section 3 details the user study, the section 4 shows the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='02201v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='HC]  5 Jan 2023  CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan  Thiago Donizetti dos Santos and Vagner Figueredo de Santana  results, section 5 discusses the role of smartphone for PwCA and  section 6 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  2  RELATED WORK  CA is also called Computerphobia, Computer Apprehension, and  Technophobia in the literature [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Rosen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' [32] pointed out the  following methods and questionnaires to measure CA:  Computer Anxiety Index (CAIN) examines avoidance of,  caution with, negative attitudes toward, and disinterest in  computers [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Computer Attitude Scale (CAS) assesses computer liking,  confidence, and anxiety through a Likert attitude-measurement  format [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Attitudes Toward Computers Questionnaire (ATCQ) as-  sesses attitudes towards computer appreciation, usage, and  societal impact [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Computer Anxiety Rating Scale (CARS) assesses behav-  ioral, cognitive, and affective components related to technol-  ogy use [15, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Mobile Computer Anxiety Scale (MCAS) assesses anxiety  regarding mobile computer using a 38-item Likert scale [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The literature presents multiple factors associated with CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In  sum, PwCA usually have less experience in using computers, have  low Computer Self-efficacy (CSE)1, take too long to accomplish  tasks, perform worse when compared to other users, have negative  beliefs about computer/skills, or negative bodily sensations previ-  ous/during the interaction with a computer [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Earlier studies find  a strong relationship between age and CA levels, showing evidence  that CA is more present in groups with older people and that they  have more CA than younger ones [7, 12, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This can be related  to the pace in which technology advances and to the fact that 48%  of older adults report that they usually need someone else to set  up a new electronic device or show them how to use it [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In this  scenario, mobile accessibility has potential to support PwCA in  increasing CSE, reducing negative beliefs and worries of using or  damaging the device in front of other people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  CA is also present in a few acceptance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The Technology  Acceptance Model (TAM) is one example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It uses the CA as a compo-  nent that changes the perceived ease of use2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' So, when considering  the adoption of new technology by elderly people, CA should be  taken into account, from design to personalization features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The influence of CA was investigated in contexts such as accep-  tance of e-learning tools, e-gov, new technologies and health-care  systems [8, 14, 17, 18, 22, 29, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' These studies stated that:  PwCA tend to prefer traditional classes instead of e-learning  systems and computer-based tests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  PwCA usually perform worse on virtual classes when com-  pared to people without CA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  PwCA have more difficulties in accepting new technologies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Older PwCA face difficulties using home telehealth services  and to learn how to use smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  1Computer Self-efficacy (CSE) is the belief one has in his/her own abilities to perform  a task in the computer [9]  2Perceived ease of use is defined as the degree to which a person believes that using a  particular system would be free of effort [1]  Considering support offered to PwCA, the literature presents  that instructional/technical support reduce CA in the context of  e-learning systems [13, 24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Finally, smartphones have potential  to provide instructional support in a privacy respecting way, sup-  porting the user-technology dialog, reducing worriers associated  to trial and error inherent to learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3  METHOD  This section details how the user study was run, including its materi-  als, procedure, setup, experiment design, and data analysis planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The goal of the study was to collect data related to questionnaires  to identify different CA levels and collect detailed interaction data  while participants performed tasks on a website in order to detect  CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Moreover, this study also aimed at exploring interaction data  common to desktop and smartphone and at understanding the role  of smartphone usage for PwCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The types of data captured will be  detailed in the Data Analysis section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Before the main experiment,  a pilot was performed in order to assess the user study plan as  whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The pilot included 4 elderly participants, recruited the same  way the participants of the main experiment (detailed in the next  section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The pilot achieved its goals in assessing protocol adopted,  duration, and data capture procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Next, we detail the method  followed in the main experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='1  Participants  Elderly people may face difficulties in staying up-to-date with tech-  nology and, since they have not used computers since childhood,  many of them face CA even when performing a simple task to  others age groups [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Hence, the target-audience considered in  this work is elderly people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The participants of the experiment were recruited from a list  of registered people at the elderly center of the city of São Paulo,  Brazil, called Reference Centre for Citizenship of Elderly (CRECI@).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  São Paulo is the biggest city in Latin America, with a population  of approximately 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='2 million people in the last census (2010) and  the current estimate is of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='2 million people3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' the population of  elderly people is approximately 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='9%4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Before recruiting partici-  pants, a partnership was signed and the proper process for ethics  committee was followed at the Federal University of ABC (process #  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='392 and CAEE: 94704418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='5594), detailing the materials,  procedure, questionnaires, data to be collected, and analysis to be  performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Moreover, only those who had never taken computer classes  offered by CRECI@ were invited as potential participants, since  results from the literature point that computer classes might reduce  CA [35], which could result in a bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='2  Materials  Questionnaires about CA, use of smartphones and computers skills  were applied (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In order to isolate CA from other co-  morbidities, questionnaires to assess cognitive abilities and levels  of depression were also applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Only participants with low levels  of depression and those who do not present signals of dementia or  3https://cidades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='ibge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='br/brasil/sp/sao-paulo/panorama  4http://produtos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='seade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='br/produtos/retratosdesp/view/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  temaId=1&indId=4&locId=3550308  Computer Anxiety: Supporting the Transition  from Desktop to Mobile  CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan  cognitive deficits had their data considered in the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' These  two metrics were considered in the exclusion criteria, detailed in  the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The questionnaires applied are listed below:  Technology use and profile: Has questions about the par-  ticipant’s age, educational level, and frequency of use of  computers and smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This questionnaire was applied  to give an overview of how participants use technology on  a daily basis (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Mini Mental: A cognitive screening test used for adults and  the elderly to evaluate orientation, memory and attention,  naming ability, obedience to verbal and writing commands,  free writing of a sentence, and copying a complex drawing  (two intersecting polygons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It is currently the most used test  for this type of assessment in the world [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The rationale  for using Mini Mental was to identify comorbidity to CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Geriatric Depression Scale (GDS): GDS is a scale with 30  yes/no questions used for screening depression in elderly  people [2, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The rationale for using GDS was also to iden-  tify comorbidity to CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Computer Anxiety Rating Scale (CARS): CARS has nine-  teen questions in a five points Likert scale ranging from  strongly disagree to strongly agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It assesses the behavioral,  cognitive and affective components related to technology  use [15, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The rationale for using CARS is that it is the  most referenced questionnaire for screening CA [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Computer Self-Efficacy (CSE): CSE is a ten items scale used  to assess Computer Self-efficacy [9, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The rationale for  using CSE was to cross check the results from this study  with results from the literature that show that CSE has a  strong but inverse relationship with CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  System Usability Scale (SUS): A five points Likert scale ques-  tionnaire with ten items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It is often used to assess the per-  ceived usability [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The rationale for using it was to compare  perceived usability of the website and the CARS values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  In order to capture the interaction data, the participants used a  desktop computer including a interaction logger and internet access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The interaction logger used was the open source logger called User  Test Logger5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The User Test Logger captures all JavaScript events  such as mouse movements, clicks, keys pressed, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=', and generates  a raw log file where each line represents an event and information  about when, where, and what is related to the event triggered [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='3  Procedure  The experiment was structured into three steps: pre-test, test, and  post-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The steps are detailed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='1  Pre-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' First, screening tests for cognitive deficit, depres-  sion, and literacy were applied to identify participants whose scores  fall outside the inclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The Mini Mental presents a score  indicating good cognitive capacity relating the answer points ob-  tained and years of education of the participant as follows:  No formal education: ≤ 21 points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  1 to 5 years of formal education: ≤ 24 points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  6 to 11 years of formal education: ≤ 26 points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  12+ years of formal education: ≤ 27 points  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='com/IBM/user-test-logger  For GDS screening test, a score of four points or less on the scale  indicates low levels of depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Hence, the exclusion criterion  was: 𝐺𝐷𝑆 ≥ 5 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' After the tests considered in the exclusion  criteria, CARS and CSE were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='2  Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The tasks were performed individually on a computer  with the User Test Logger installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' First, each participant was asked  to access SESC homepage (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The Social Service of Com-  merce (SESC) is a private entity maintained by the entrepreneurs of  the trade in goods, tourism, and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' SESC aims to provide the  welfare and quality of life to workers in this sector and their fami-  lies 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' SESC offers many activities for elderly people such as courses,  sports, art exhibition and culture-related events in multiple units in  the metropolitan area of São Paulo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The tasks were defined aiming  to encourage participants to search online for activities offered by  SESC and others services in the city, hoping to help them to see the  internet as a tool which they can use as a means of improving their  quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The same way they do at CRECI@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In addition, the  tasks were structured to be as familiar and as close to real tasks as  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The tasks read out loud to participants were the following:  (1) Search for an event, class, or activity he/she might be inter-  ested;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  (2) Find the address of the unit where the chosen activity/event  is offered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  (3) Find the route to the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Figure 1: Homepage of the SESC’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  There was no maximum time limit for the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Thus, the task  duration depended on participants saying whether they finished or  gave up on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Finally, Thinking-Aloud Protocol [20] was used  to understand the rationale of users while performing the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='3  Post-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In order to evaluate the perceived usability and  any relationship with CA levels, they were asked to answer the  SUS questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='4  Data analysis  According to the exclusion criteria defined, the data from partic-  ipants who did not score the points required by Mini Mental or  scored five or more points on GDS were removed from the data set  to be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The resulting data set combined data from the inter-  action logger, questionnaires, and thinking-aloud protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Since  the logged data were in raw format, all data captured were pro-  cessed in order to extract usage metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The following metrics were  considered having in mind they could also be applied in a mobile  interaction setting: time to perform the task, number of clicks and  double-clicks, click duration, typing velocity, and total time typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  6https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='sescsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='br/  3 Pagina Inicial - Sesc SP - Mozilla Firefox  Debugging with Firefox Developer T X  S Pagina Inicial - Sesc SP  +  @ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='sescsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='br  Q Pesquisar  国三不  Um Portal para cada um!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Login  Esqueci a senha I Cadastre-se  Para ver os destaques da programagao de acordo com seus interesses  fEntrar com Facebook  ou  email  senha  ok  online, cadastre-se aqui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  OPORTUNIDADES  + CREDENCIAL PLENA  Meu perfil  Sesc  SAO PAULO  O que voce procura?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Projeto Sawe  Encontros no Sesc Ipiranga debatem a luta dos  indios pela defesa de seus territorios e os desafios de  construir o futuro mediante contextos impostos pela  sociedade nao indigenaCHI ’21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Workshop on Designing Interactions for the Ageing Populations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' May 08–13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 2021,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Yokohama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Japan  Thiago Donizetti dos Santos and Vagner Figueredo de Santana  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' all metrics and the questionnaires scores were combined  in a single comma-separated-values (CSV) file used to perform  the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This CSV file is the main data source for the  regression analysis performed to predict CARS values based only  on interaction data, which could allow its use at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  4  RESULTS  The experiment included 39 participants, but data from 8 partici-  pants were not considered in the analysis due to exclusion crite-  ria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Thus, considering the data from the remaining 31 participants  (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='61% of males, 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='39% of females).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The participants’ ages ranged  between 62 and 87 years (𝑥 = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Regarding the computer usage,  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='81% of the participants reported that they do not own a computer  and do not have frequent access to computers, while 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='90% do not  own a computer, but use it at lanhouses or those available in public  places;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='29% reported owning computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Regarding frequency  of use, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='29% reported that rarely (less than once a month) use a  computer, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='13% reported that use it sometimes (more than once  a month), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='45% reported that usually (more than once a week)  use it, and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='13% reported that always (everyday) use computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  When considering ownership and use of smartphones, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='10% of  the participants reported that they own smartphones and 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='39%  reported that they always use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The education level of the par-  ticipants varies from 0 to 15 years of formal education (𝑥 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='42)  (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Figure 2: Distribution of years of formal education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The obtained scores for CARS ranged from 20 to 59 (𝑥 = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Based on [10], the maximum and minimum scores were used to  divide the data into 3 groups as follows: 𝐶𝐴𝑅𝑆_𝑟𝑎𝑛𝑔𝑒 : 59 − 20 = 39  and 𝐺𝑟𝑜𝑢𝑝_𝑟𝑎𝑛𝑔𝑒 : 𝐶𝐴𝑅𝑆_𝑟𝑎𝑛𝑔𝑒/3 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  No CA: CARS < 33 (6 participants);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Moderate CA: 33 ≤ CARS < 46 (14 participants);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  High CA: CARS ≥ 46 (11 participants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Considering the three CA groups, Figure 3 shows the presence of  CA considering the participants’ age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It can be seen that participants  in the no CA group are among the youngest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The age of these  participants ranged from 63 to 78 years old (𝑥 = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='43, 𝜎 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The age of the participants in the moderate CA group ranged from  62 to 83 years old (𝑥 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='92, 𝜎 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' And, for the high CA group,  the age of the participants ranged from 63 to 87 (𝑥 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='36, 𝜎 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='77),  showing that high levels of CA are present over almost the entire  age range covered in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Mann-Whitney non-parametric test  shows that age was different between no CA and moderate CA  groups (p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='03);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' no significant difference was found in other  pairwise group comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Figure 3: Age distributions for different CA groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Bearing in mind the use of smartphones, Figure 4 shows smart-  phone ownership and different uses by different CA groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It can  be seen that high CA group uses smartphones more for calls and less  for leisure and other communication activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=', games, music,  video, instant messages, and social networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' On the other hand,  people in the no CA group use smartphone heavily to access social  network, instant messages, and internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Analyzing the ownership  rate by CA groups, it can be seen that 73% of the participants with  high CA own smartphones, while the ownership is greater in the  moderate CA group (92%) and reaches 100% for the no CA group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Similarly, considering the frequency of use of computers, people  who rarely use computers are the ones with greater CA levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 82%  of the participants with high CA use it rarely, while it is 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='5% of  the participants of the moderate CA and 28% of the no CA group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Figure 4: Smartphone ownership and use by CA groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Bearing in mind task completion, 23 (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='19%) participants found  an activity and completed the first task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Four participants found an  activity that was not of interest to them or was offered by a unit  far from their home, but they did not find another one after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The remaining four gave up without finding an activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' For the  Task 2, nine out of 23 (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='13%) participants found the address of  the unit where the selected activity is offered and 14 participants  found only the name of the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Regarding the Task 3, six out of  nine (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='67%) participants found the map available on the site, but  all of them failed to find the route to the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Only two out of  nine (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='22%) participants figured out how to put the starting point  address on the map-based UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In sum, task completion dropped from  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='19% (task 1), to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='13% (task 2), and to 0% (task 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='22% (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='45%,  considering 31 participants) partially completed the last task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This  can be related to task difficulty, fatigue effects, and task dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Education levels (in years)  12  participants  10  Number of  2  0  4  6  8  10  11  13  14  15  Education yearsAge vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' CARS groups  85  80  e  75  70  65  。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  No CARS  Moderate CARS  High CARSSmartphone use x Computer Anxiety Leve  Music/Video  Message  Games  Call  Instant Message  Social networks  Internet  Smartphone ownership  0%  10%  20%  30%  40%  50%  60%  70%  80%  %06  100%  No CA  Moderate CA  High CAComputer Anxiety: Supporting the Transition  from Desktop to Mobile  CHI ’21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Workshop on Designing Interactions for the Ageing Populations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' May 08–13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 2021,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Yokohama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Japan  However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' after triangulating these results with thinking-aloud data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  it was possible to identify that participants faced difficulties with  the map-based UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' For instance, participant 11 said: “It doesn’t say  where it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' I didn’t like SESC.”, participant 35 said: “Why don’t you  have the address on the about the unit page?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' and participant 42  said: “It will take me a long time to find it (address)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' While using  the map, for instance, participant 22 said: “What should I do here?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  I have never used it (map) before”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' participants are numbered from  1-4 for the pilot and 5-43 for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Table 1 summarizes the time taken by each group to complete  the tasks, showing the mean time and standard deviation by CA  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It can be seen that the group of participants with high CA  had a mean shorter task time than the other groups in some tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  This might be related to the fact that PwCA usually gave up more  because they feel frustrated, lost, or think that they would not be  able to finish the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This also shows the relationship between  high CA and low CSE, as identified in previous studies [35] and  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This can be exemplified by the participants quotes as: “I think  I will have difficulty in this task”, “This is difficult”, “I’m lost, I don’t  know what to do”, and “I don’t know how to find it”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Task 1 in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Task 2 in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Task 3 in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Group  𝑥 (𝜎)  𝑥 (𝜎)  𝑥 (𝜎)  High CA  411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='18 (230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='72)  599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='67 (220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='30)  398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='00 (262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='19)  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' CA  647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='90 (666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='46)  450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='13 (336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='52)  560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='50 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='50)  No CA  525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='14 (331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='39)  587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='00 (390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='73)  399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='67 (375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='05)  Table 1: Average task time by group and standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Although the interaction data was collected during the use of a  desktop computer, in this study we explore metrics which can be  captured in a mobile setting as well, namely: task time, numbers  of clicks and double clicks, mean click duration (interval between  pressing and releasing), typing velocity and total time typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' All  metrics were normalized for the regression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Prior to fit-  ting the regression model, a random oversampling algorithm was  applied7 addressing the minority values and a train-test split of  80% / 20% was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Figure 5 shows Random Forest regression  predictions for CA values (y-axis) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' CARS values in the test set  (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The obtained regressor has a mean squared error (MSE) of  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='21 and 𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The high MSE value is due to errors related to  predictions for lower CA scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This pessimist prediction would  show that users need more support than they would actually need,  so such regressor might be useful for indicating when support for  PwCA could be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  5  DISCUSSION  Although there are few studies reporting no significant relationship  between age and CA [16, 19, 28], there are also evidences that older  people manifest more CA than younger ones [7, 12, 27, 33, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Results suggest that younger participants were in the no CA group,  while the older were in the moderate CA group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' For high CA group,  results suggest that there were participants with high CA almost  in the whole age range considered, but it still shows a greater  concentration among the older ones (median = 74 years, 𝑥 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='2,  7https://imbalanced-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='org/stable/over_sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='html  Figure 5: Regression test results of CARS scores prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  𝑠𝑖𝑔𝑚𝑎 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' These results are similar to other findings in the  literature, indicating that the CA is more present in the older groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Moreover, unlike [30], the results suggest that previous experience  may impact CA levels, since people who rarely use computers and  smartphones were the ones with greater CA levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The difficulty in adopting new technologies is suggested by re-  sults in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Although there is a high rate of PwCA owning  smartphones, the most frequent reported use is making calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Thus,  the participants use the smartphone, but they use the same way  they used to do with the old phones: making calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Moreover, the no  CA group presented the same rate (57%) when using smartphones  to make calls, using instant message and social networks apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In  contrast, moderate and high CA groups use more to make calls  than to any of the other functions analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Also they use more  to make calls then the no CA group and presented a lower rate of  ownership of smartphones than the no CA group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The increasing  rate of smartphone ownership and the decreasing rate of use of  different smartphone functions show possible impacts of CA on  the behavior of the elderly regarding technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The second most  frequent use is instant message apps, even though there is also a  difference between groups for this use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The use of this application  could be related to the presence of functions such as the ability to  make calls using the app or using voice messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' They reported to  be used to make calls and that the use of voice messages is easier for  them, since they generally have difficulties using the keyboard of  the smartphones to write and may have difficulty in reading due to  the size of the screen and letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' These findings suggest that high  levels of CA may affect people of all ages, but it is more present  among older ones due to factors such as lack of practice or lack of  knowledge about recent features that could improve UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The results regarding task completion and time taken to complete  tasks show how CA levels might affect task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Besides  the low task completion rate for the high CA group, this group  took less time trying to complete the Task 1, showing that PwCA  usually gave up more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The implications for HCI researchers in this  aspect are related to the design of shorter and simpler tasks and to  improve user experience, accessibility and usability, since high CA  levels impact negatively the perceived easy of use and CSE, as they  may feel frustrated or lost when facing problems to achieve their  goals during the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Random Forest Regression  50  30  区  21  31  50  CARSCHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan  Thiago Donizetti dos Santos and Vagner Figueredo de Santana  CA is also related to specific situations since it tends to arise or  be stronger during the first use of a device [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Thus, in addition  to promoting the contact of the elderly with new technologies, it is  important to promote a good first experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It means an experi-  ence free of effort, that helps the user to feel safe and unafraid of  making mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' A bad experience, during which the user feels lost  or makes mistakes, can reinforce the fears of PwCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Consequently,  this can increase their CA levels, making them believe they are  unable to use it and prevent them from trying again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  The results of the Random Forest regression show that CA in-  fluences on how users interact with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Although the data  belongs to desktop computer interaction domain, the result sug-  gests that such approach should be explored in the mobile settings  as well, given that the interaction events selected are common to  desktop and smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The prediction of higher CARS values  could trigger personalization features and additional support, for  instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Smartphones have a myriad of sensors that could be used  for such personalization features and here we advocate the use of  the following metrics as a starting point: task time, numbers of  clicks and double clicks, mean click duration, typing velocity and  total time typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  This paper defends the idea that the use of smartphones by the  elderly can bring benefits such as autonomy, access to content and  services and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' However, CA may create barriers  which prevent elderly from enjoying these benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Therefore, we  argue that further studies are needed regarding the influences of  levels of CA in acceptance of smartphones and apps by the elderly  people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' On his work about the development of a mobile computer  anxiety scale (MCAS), [39] argues that MCAS is related to three  distinct components: (1) traditional CA construct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' (2) Internet anxi-  ety construct and (3) special factors making up the mobile anxiety  construct (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=', equipment limitation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The limitations of mobile  equipment listed by [39] and that elderly people report as being  problematic for them are: small screens and small multi-function  key pads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' lower display resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' unfriendly user-interfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' and  graphical limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Thus, these limitations should be considered  when developing new technologies for elderly people as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Fur-  thermore, the importance of the first experience is found in the  literature and reported in the interviews conducted in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  They reported having purchased or been presented with a new  smartphone and feeling lost or afraid to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' So, it is important  that the device has a simple, accessible and usable interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Another  common factor reported, is the fear of making mistakes, looking  stupid or breaking the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In this sense, it is important that the  system provides a safe environment, which asks for confirmation  for important (dangerous) actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' And, in case the user makes a  mistake, the system must provide ways to recover from the error,  returning to the previous state without difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  In Brazil, families often have a computer to be shared by family  members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' It can make elderly people afraid of breaking what be-  longs to the family or afraid of losing some important data if they  do something wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The smartphone, on the other hand, is seen  as an personal device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' According to the participants’ reports, this  brings greater freedom to learn how to use through trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Besides that, as it is mobile, it has the advantage that elderly people  can avoid to use it in front of other people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This can help dealing  with the fear of not knowing how to use it or making mistakes in  front of younger people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Finally, we believe a research agenda about the role of smart-  phones for elderly people in the context of CA should address the  following research questions: (1) How to detect CA during the use  of mobile phones at scale?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' (2) How to provide (first) good user  experiences for this population?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' (3) How to create a secure envi-  ronment for PwCA to recover from mistakes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' (4) How to combine  technology use with learning in order to increase CSE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' (5) How  CA on the desktop relates to CA on smartphones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  6  CONCLUSION  This position paper discussed how elderly people use smartphones  in a specific region of São Paulo, Brazil, and shows how CA is  present in this sample of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Our findings suggest  that higher CA levels are prevalent on higher age and CA impacts  how users interact with technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In addition, results indicate  that the behavior of elderly users when performing tasks can be  negatively impacted not only because of age-related factors, but  also by the CA levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The results indicating the preference of some  applications over others by elderly people indicate the need for  further studies on why some technologies still present barriers  for PwCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Moreover, the shorter task time obtained by the high  CA group and the fact that they usually gave up when feel lost  shows the importance of shorter and simpler tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The differences  between groups regarding ownership of smartphones show that  CA may impact on the technology adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In addition, the results  showing the preference for known functions can be important to  designers and developers to consider when developing new systems,  since the inclusion of functions considered easy to use may increase  the system adoption and improve user experience by reducing  frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Finally, tackling technology adoption by elderly people may im-  prove their quality of life, since the use of smartphones and the  wide variety of applications and services may help they achieve  independence, make their lives more comfortable, promoting their  better participation in the community and allowing access to the  most diverse online content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' The presented user test shows that  metrics such as task time, number of clicks, click duration, typing  speed and total time typing can support the prediction of different  CARS scores (𝑅2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In the current context of the COVID-19 pan-  demics, promoting autonomy, communication and leisure activities  became core goals for any technology and here we emphasize that  by understanding CA and considering it in design and development  phases mobile apps have the potential to change the live of PwCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank the CRECI@ for all the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' This study was financed  in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível  Superior - Brasil (CAPES) - Finance Code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  REFERENCES  [1] Fazil Abdullah, Rupert Ward, and Ejaz Ahmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content='  [2] Osvaldo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Almeida and Shirley A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Almeida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Confiabilidade da versão  brasileira da Escala de Depressão em Geriatria (GDS) versão reduzida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Arquivos  Computer Anxiety: Supporting the Transition  from Desktop to Mobile  CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan  de Neuro-Psiquiatria 57 (06 1999), 421 – 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='scielo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='br/scielo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  script=sci_arttext&pid=S0004-282X1999000300013&nrm=iso  [3] Mustafa Baloğlu and Vildan Çevik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Multivariate effects of gender, ownership,  and the frequency of use on computer anxiety among high school students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Computers in Human Behavior 24, 6 (2008), 2639–2648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content=' SUS-A quick and dirty usability scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Tech Adoption Climbs Among Older Adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' http://  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='pewinternet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='org/2017/05/17/tech-adoption-climbs-among-older-adults/  [6] Pew Research Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Mobile Fact Sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='pewresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content=' Information Processing & Management 45, 2 (2009), 200–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content='(Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content=' Interaction design  and usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Berlin: Springer (Lecture notes in computer science, 4550) (2007),  812–817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content=' Experiencing Computer Anxiety Later  in Life: The Role of Stereotype Threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' In International Conference on Human  Aspects of IT for the Aged Population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content=' Evaluation of a Web-based introductory psychology course: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' Learning  and satisfaction in on-line versus lecture courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+page_content='  A  TECHNOLOGY USE AND PROFILE  (1) Do you have a computer available at home?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=' If not, do you use  a computer somewhere else (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content=', at work or in the lanhouse)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  (2) How often do you use a computer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  A: [ ] Rarely  [ ] Sometimes  [ ] Usually  [ ] Always  (3) What do you usually do on computer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  (4) Do you own a smartphone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  (5) How often do you use smartphones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  A: [ ] Rarely  [ ] Sometimes  [ ] Usually  [ ] Always  (6) What do you usually do on smartphone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
+page_content='  Internet: [ ] Yes  [ ] No  Social networks: [ ] Yes  [ ] No  Instant Message: [ ] Yes  [ ] No  Call: [ ] Yes  [ ] No  Games: [ ] Yes  [ ] No  Message: [ ] Yes  [ ] No  Music / Video: [ ] Yes  [ ] No' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfRAB3/content/2301.02201v1.pdf'}
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+arXiv:2301.03804v1  [quant-ph]  10 Jan 2023
+Quantum mechanics and quantum field theory.
+Algebraic and geometric approaches.
+Квантовая механика и квантовая теория поля.
+Алгебраический и геометрический подходы.
+A. Schwarz
+Department of Mathematics
+University of California
+Davis, CA 95616, USA,
+schwarz @math.ucdavis.edu
+I. Frolov
+Department of Mathematics,
+National Research Nuclear University MEPhI
+(Moscow Engineering Physics Institute)
+115409, Kashirskoe shosse 31, Moscow, Russia
+frolovi55@mail.ru
+Аннотация
+This is a non-standard exposition of main notions of quantum mechanics and quantum
+field theory that also includes some recent results. It is based on algebraic approach where
+the starting point is an associative algebra with involution and states are defined as positive
+linear functionals on this algebra and on geometric approach where the starting point is a
+set of states considered as a convex subset of linear space. The exposition does not depend
+on textbooks in quantum mechanics.
+Standard formulas for quantum probabilities are derived from decoherence. This derivation
+allows us to go beyond quantum theory in geometric approach. Particles are defined as
+elementary excitations of ground state (and quasiparticles as elementary excitations of
+any translation invariant state). It follows from this definition that the notion of identical
+particles is very natural. The scattering of particles is analyzed in the framework of
+generalization of Haag-Ruelle theory. The conventional scattering matrix does not work for
+quasiparticles (and even for particles if the theory does not have particle interpretation).
+The analysis of scattering in these cases is based on the notion of inclusive scattering
+matrix, closely related to inclusive cross-sections. It is proven that the conventional scattering
+matrix can be expressed in terms of Green functions (LSZ formula) anf inclusive scattering
+matrix can be expressed in terms of generalized Green functions that appear in the
+Keldysh formalism of non-equilibrium statistical physics. It is shown that generalized
+Green functions and inclusive scattering matrices appear also in the formalism of L-
+functionals that can be identified with positive functionals on Weyl or Clifford algebras.
+The derivation of the expression of the evolution operator and other physical quantities
+in terms of functional integrals is based on the notion of symbol of operator; these arguments
+1
+
+can be applied also in geometric approach. This result can be used, in particular, to give
+a simple derivation of diagram technique for generalized Green functions.
+The notion of inclusive scattering matrix makes sense in geometric approach (but it
+seems that one cannot give a definition of conventional scattering matrix in this situation).
+The geometric approach is used to show that quantum mechanics and its generalizations
+can be considered as classical theories where our devices are able to measure only a part
+of observables.
+This text is based on first ten lectures of the course taught by A. Schwarz in the Spring
+of 2022; see www.mathnet.ru for lectures (in Russian) and slides (in English).
+Keywords Inclusive scattering matrix; generalized Green function, geometric approach
+2
+
+Содержание
+1
+Лекция 1
+4
+1.1
+Введение . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.1.1
+Выпуклые множества . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.1.2
+Квантовая теория. Геометрический подход . . . . . . . . . . . . .
+5
+1.1.3
+Алгебраический подход к квантовой теории . . . . . . . . . . . . .
+5
+2
+Лекция 2
+12
+2.1
+Квантовая механика как деформация классической механики. Алгебра
+Вейля . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+2.2
+Квадратичные гамильтонианы . . . . . . . . . . . . . . . . . . . . . . . .
+16
+2.3
+Стационарные состояния
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+2.4
+Пространство Фока . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+2.5
+Гамильтонианы сохраняющие число частиц . . . . . . . . . . . . . . . . .
+21
+2.6
+Представления алгебры Вейля . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+3
+Лекция 3
+25
+3.1
+Алгебра Клиффорда и алгебра Грассманна . . . . . . . . . . . . . . . . .
+25
+3.2
+Представления алгебры Клиффорда . . . . . . . . . . . . . . . . . . . . .
+28
+3.3
+Статистическая физика . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+32
+4
+Лекция 4
+36
+4.1
+Адиабатическое приближение. Декогерентность . . . . . . . . . . . . . .
+36
+4.2
+L-функционалы
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+43
+5
+Лекция 5
+45
+5.1
+Функциональные интегралы
+. . . . . . . . . . . . . . . . . . . . . . . . .
+45
+5.2
+L-функционалы и функциональные интегралы . . . . . . . . . . . . . . .
+50
+6
+Лекция 6
+55
+6.1
+Солитоны как аналоги частиц
+. . . . . . . . . . . . . . . . . . . . . . . .
+55
+6.2
+Частицы и квазичастицы . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+57
+7
+Лекция 7
+64
+7.1
+Одночастичные и многочастичные состояния . . . . . . . . . . . . . . . .
+64
+7.2
+Рассеяние; in- и out-состояния . . . . . . . . . . . . . . . . . . . . . . . . .
+66
+8
+Лекция 8
+74
+8.1
+Связь с локальной квантовой теорией поля . . . . . . . . . . . . . . . . .
+74
+8.2
+Функции Грина. Связь с матрицей рассеяния.
+. . . . . . . . . . . . . . .
+77
+9
+Лекция 9
+85
+9.1
+Обобщенные функции Грина . . . . . . . . . . . . . . . . . . . . . . . . .
+85
+9.2
+Матрица Мёллера
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+85
+9.3
+Матрица рассеяния. Формула LSZ
+. . . . . . . . . . . . . . . . . . . . . .
+90
+9.4
+Инклюзивная матрица рассеяния . . . . . . . . . . . . . . . . . . . . . . .
+92
+10 Лекция 10
+94
+10.1 Удаление лишних состояний . . . . . . . . . . . . . . . . . . . . . . . . . .
+94
+10.2 Квантовая механика из классической механики . . . . . . . . . . . . . . .
+94
+3
+
+1
+Лекция 1
+1.1
+Введение
+В обычном изложении квантовой механики мы живем в гильбертовом пространстве и
+рассматриваем операторы в этом пространстве. Самосопряженные операторы отвеча-
+ют наблюдаемым. Это подход, которым пользуются физики почти всегда, но он имеет
+свои недостатки. Я буду говорить про другие подходы. Это прежде всего, алгебраиче-
+ский подход, где исходным пунктом является алгебра наблюдаемых — ассоциативная
+алгебра с инволюцией, в которой самосопряженные элементы является наблюдаемы-
+ми. Этот подход, по-моему, почти столь же стар, как и сама квантовая механика.
+Кроме того, я буду говорить про геометрический подход, в котором исходная точка
+— это множество состояний [1-7]. Этот подход я предложил пару лет тому назад, и
+он много более общий чем алгебраический подход.
+Основная вещь, которая, по-моему, недостаточно подчеркивается в обычном изло-
+жении квантовой механики (чуть больше говорится в квантовой теории поля) это то,
+что понятие частицы не является основным в квантовой механике. Это понятие произ-
+водное. Частицы — это элементарные возбуждения основного состояния. Квазичасти-
+цы (тоже важное понятие) — это элементарные возбуждения любого трансляционно
+инвариантного состояния. Основное понятие, которое есть в физике элементарных
+частиц — это понятие матрицы рассеяния и о нем я буду больше всего говорить. Это
+понятие тесно связано с понятием сечения рассеяния.
+Кроме того, я буду говорить о понятии инклюзивной матрицы рассеяния, тесно
+связанном с понятием инклюзивного сечения рассеяния. Матрицы рассеяния выража-
+ются через функции Грина известной формулой, принадлежащей Леману, Симанчику
+и Циммерману, а инклюзивные матрицы рассеяния выражаются через обобщённые
+функции Грина, которые появились впервые в неравновесной статистической физике
+в формализме Келдыша.
+1.1.1
+Выпуклые множества
+Перед тем как переходить к физике, я хочу сказать несколько слов про выпуклые
+множества, которые у меня будут появляться многократно.
+Выпуклое множество C — это подмножество векторного пространства, которое
+вместе с каждыми двумя точками содержит отрезок, соединяющий эти точки. Важ-
+но то, что в выпуклом множестве можно рассматривать смесь точек этого множества.
+Если взять несколько точек множества и каждой точке приписать неотрицательное
+число так, чтобы сумма чисел равнялась единице, то тогда сумма точек ei ∈ C с коэф-
+фициентами pi ≥ 0, � pi = 1 также будет принадлежать выпуклому множеству. Эта
+сумма � piei ∈ C называется смесью точек множества с вероятностями pi. Можно
+рассматривать числа pi как веса, и тогда эта сумма будет представлять центр тя-
+жести. Другое важное понятие — это понятие крайней точки выпуклого множества.
+Крайняя точка — это такая точка, которая не лежит внутри никакого отрезка с кон-
+цами, принадлежащими множеству. У многогранника крайние точки — это вершины.
+У шара крайние точки — это сфера, ограничивающая этот шар.
+Я всегда буду считать, что в том векторном пространстве, которое я рассматри-
+ваю, есть какая-то топология, есть понятие непрерывности и, тем самым, понятие
+замкнутого множества. Я буду предполагать, что все выпуклые множества, которые
+я рассматриваю, замкнуты, и тогда можно рассматривать смесь не только конечно-
+го числа точек, но и смесь счетного числа точек. В последнем случае сумму нужно
+считать бесконечной.
+4
+
+Также можно рассматривать смесь точек любого подмножества выпуклого мно-
+жества, если на подмножестве задано распределение вероятностей. Если, скажем, на
+сфере, ограничивающей шар, задано распределение вероятности с плотностью ве-
+роятности ρ(λ), то можно рассматривать смесь исходя из этого распределения ве-
+роятности: просто вместо суммы нужно брать интеграл. Если λ — это параметры,
+описывающие точки на сфере, ω(λ) — это точки сферы, тогда можно рассматривать
+такую смесь, беря вместо суммы интеграл
+�
+ω(λ)ρ(λ)dλ. Если выпуклое множество
+компактно, то каждая точка является смесью его крайних точек.
+1.1.2
+Квантовая теория. Геометрический подход
+В двух словах я повторю то, что мне нужно из квантовой механики. Начну с того, что
+для меня понятие состояния в квантовой механике — это понятие матрицы плотно-
+сти. Матрица плотности — это самосопряженный оператор K, который положительно
+определен и у которого след равен единице: trK = 1. Множество матриц плотности
+выпукло, а его крайние точки называются чистыми состояниями. Эти чистые состо-
+яния отвечают векторам в гильбертовом пространстве. Каждому нормированному
+вектору Ψ соответствует матрица плотности, которая определяется как ортогональ-
+ная проекция на этот вектор:
+KΨ(x) = ⟨x, Ψ⟩Ψ.
+Нужно заметить что если два вектора пропорциональны (Ψ′ = λΨ), то соответ-
+ствующие матрицы плотности совпадают: KΨ′ = KΨ.
+У матрицы плотности K существует базис из собственных векторов ei с неотри-
+цательными собственными значениями pi сумма которых равна 1. Можно считать,
+что каждому из этих векторов соответствует чистое состояние, а матрица плотности
+— это смесь этих чистых состояний. В том представлении, в котором матрица диаго-
+нальна, ее диагональные элементы обозначим как pi. Поскольку след равен единице,
+их сумма равна единице: � pi = 1. Поскольку матрица положительно определена,
+эти диагональные элементы неотрицательны: pi ≥ 0. Следовательно, можно сказать,
+что матрица плотности — это смесь чистых состояний с вероятностями pi.
+В обычных курсах все это рассказывается в обратном порядке начиная с чистых
+состояний. Матрицы плотности определяются как смешанные состояния.
+Я обсудил один способ представления матрицы плотности как смеси чистых состо-
+яний. На самом деле, это можно делать бесконечным множеством разных способов.
+В геометрическом подходе исходной точкой является множество состояний. Пред-
+полагается, что это ограниченное замкнутое выпуклое подмножество топологического
+линейного пространства. Оказывается, что этих предположений достаточно для того,
+чтобы построить содержательную теорию.
+1.1.3
+Алгебраический подход к квантовой теории
+В то время как в геометрическом подходе основное — это пространство состояний,
+в алгебраическом подходе основное — это алгебра наблюдаемых A. Напомню, что
+алгебра — это векторное пространство, в котором можно умножать элементы с дис-
+трибутивным законом, но еще я потребую, чтобы в этой алгебре была инволюция.
+Типичный пример алгебры (для меня основной) — это алгебра ограниченных опера-
+торов в гильбертовом пространстве. В этой алгебре есть инволюция A → A∗, которая
+отвечает переходу к сопряженному оператору. Она обладает тем свойством, что, пе-
+рейдя дважды к сопряженному оператору, мы возвращаемся к исходному: A∗∗ = A.
+Если берать сопряженный к произведению оператор, это будет опять же произведе-
+ние, но в обратном порядке: (AB)∗ = B∗A∗. Кроме того, инволюция антилинейна. В
+5
+
+алгебре операторов это простые свойства, а для произвольных ассоциативных алгебр
+с инволюцией (∗-алгебр) это — аксиомы.
+Я всегда предполагаю, что в алгебре есть какая-то топология, в которой все опе-
+рации непрерывны. Я обычно про эти топологии говорить не буду, во-первых, по-
+тому, что это требует времени, а, во-вторых, потому, что разные топологии могут
+быть одинаково разумны. В некоторых ситуациях более удобны одни, в других си-
+туациях — другие. Того, что есть топология обычно бывает мало. Иногда требуется,
+чтобы была норма, в которой алгебра является банаховым пространством, тогда тре-
+буется, чтобы норма произведения была меньше или равна чем произведение норм
+(||AB|| ≤ ||A|| · ||B||) (это определение банаховой алгебры). Иногда требуется нечто
+большее, например, чтобы алгебра была C∗ -алгеброй.(Это значит, что норма произ-
+ведения A∗A равна квадрату нормы оператора A, то есть ||A∗A|| = ||A||2). Я обычно
+не буду уточнять, какая именно топология выбрана. Я хочу подчеркнуть, что при
+рассмотрении гомоморфизма или автоморфизма алгебры я буду всегда считать, что
+он согласован с инволюцией.
+Если есть алгебра с инволюцией, то самосопряженные элементы (A = A∗) отве-
+чают физическим величинам. Сами по себе самосопряженные элементы алгебры не
+образуют. Произведение самосопряженных элементов — это не обязательно самосо-
+пряженный элемент. Этот недостаток алгебраического подхода было замечен еще в
+30-е годы, что привело к понятию йордановой алгебры. Йордан заметил, что хотя
+произведение самосопряженных элементов не является самосопряженным , но если
+взять антикоммутатор A ◦ B = AB + BA, где A и B — самосопряженные, этот ан-
+тикоммутатор снова будет самосопряжённым. Он аксиоматизировал эту операцию.
+Теория йордановых алгебр была построена в 30-е годы, в основном, в знаменитой
+работе Йордана, Вигнера и фон Нойманна. Но хотя йорданова алгебра — очень кра-
+сивый и действительно полезный во многих частях математики объект, в физике до
+сих пор он не получил большого применения. Сейчас он естественно появился в гео-
+метрическом подходе и, может быть, снова вернется в физику.
+Если начинать со ∗-алгебры (с алгебры с инволюцией), можно определить понятие
+состояния: состояние — это линейный функционал ω на алгебре A, удовлетворяющий
+условию неотрицательности на элементах вида A∗A:
+ω(A∗A) ≥ 0.
+Состояния, отличающиеся численным множителем, отождествляются. Часто удобно
+рассматривать только состояния, удовлетворяющие условию ω(1) = 1 (нормирован-
+ные состояния).
+Теперь можно определить понятие математического ожидания в данном состоя-
+нии. Для функции f(A) от A ∈ A математическое ожидание задается формулой:
+⟨f(A)⟩ω = ω(f(A)).
+Для C∗-алгебры можно определить f(A) для любой непрерывной функции от само-
+сопряженного элемента A; тогда, зная математическое ожидание для всех непрерыв-
+ных функций, можно определить понятие распределения вероятностей физической
+наблюдаемой в нормированном состоянии (состоянии, для которого ω(1) = 1).
+Одного понятия состояния мало: нужно еще понятие эволюции состояния, потому
+что целью физики, как и любой науки — делать предсказания. Физик прежде всего
+рассматривает задачу: если есть какое-то начальное состояние, нужно предсказать,
+что будет потом.
+В алгебраическом подходе для того, чтобы определить эволюцию, нужно прежде
+всего рассмотреть группу Aut автоморфизмов алгебры. Напомним, что автоморфиз-
+мы A всегда должны коммутировать с инволюцией, и тогда эта группа автоморфиз-
+6
+
+мов, естественно действующая на линейных функционалах, переводит положитель-
+ные функционалы в положительные (состояния в состояния). В любом подходе к
+квантовой теории состояния должны зависеть от времени. Должен быть оператор
+эволюции U(t), который переводит состояние в начальный момент в состояние ω(t) в
+какой-то другой момент времени t. Тогда ω(t) = U(t)ω(0).
+В алгебраическом подходе можно считать, что операторы U(t) являются автомор-
+физмами алгебры; они порождают обозначаемые точно так же операторы, действую-
+щие на состояния. Как и в обычной квантовой механике, есть картина Шредингера,
+где эволюционирует состояние, и есть картина Гейзенберга, где эволюционирует опе-
+ратор. Эти две картины эквивалентны:
+ω(t)(A) = ω(A(t)).
+Наблюдать динамику состояния по времени при неподвижном элементе алгебры, это
+то же самое, что наблюдать динамику элемента алгебры A при неизменном состоянии.
+В физике оператор эволюции обычно вычисляется исходя из уравнения движе-
+ния, описывающих тот же оператор эволюции, но за бесконечно малое время. Если
+есть инвариантность относительно сдвига времени, то можно утверждать, что опе-
+ратор, описывающий изменение за бесконечно малый промежуток времени, сам от
+времени не зависит. Уже было сказано, что изменение за конечное время должно
+быть автоморфизмом алгебры, значит, изменения за бесконечно малые промежутки
+времени — это инфинитезимальные автоморфизмы. Зная инфинитезимальный авто-
+морфизм A, будем решать уравнение движения dV/dt = AV. Решение выражается
+через экспоненту от инфинитезимального автоморфизма eAt ∈ A. В результате по-
+лучаем однопараметрическую группу автоморфизмов, состоящую из преобразований
+вида eAt (операторов эволюции). (Физики в курсах квантовой механики пишут обыч-
+но мнимую единицу в экспоненте — я ее не пишу, но, конечно, никакой разницы в
+этом нет). Решение уравнения движения имеет вид V (t) = eAtV (0).
+Я не дал формального определения инфинитезимального автоморфизма. Одно
+из возможных формальных определений: инфинитезимальный автоморфизм — это
+касательный вектор к кривой в группе автоморфизмов в единичном элементе этой
+группы. Можно потребовать несколько больше: чтобы эта кривая была однопарамет-
+рической подгруппой.
+Важно отметить, что инфинитезимальный автоморфизм является дифференци-
+рованием алгебры. Это означает, что он должен удовлетворять правилу Лейбница:
+применяя его к произведению xy нужно сначала применить его к первому сомно-
+жителю, оставляя второй неизменным, потом — ко второму сомножителю, первый
+оставив неизменным A(xy) = (Ax)y + x(Ay). Это следует мгновенно из самого опре-
+деления автоморфизма и самого определения инфинитезимального автоморфизма.
+Если есть инфинитезимальный автоморфизм, то 1 + tA при малом t — это уже авто-
+морфизм. (Точнее, 1+tA плюс нечто высшего порядка по t — это есть автоморфизм).
+Если применить определение автоморфизма, как раз получится правило Лейбница.
+Обратно, если есть дифференцирование, то есть если выполнено правило Лейб-
+ница, и еще к тому же оно согласованно с инволюцией, то есть, выполняется условие
+(Ax)∗ = A(x∗), то можно надеяться, что это инфинитезимальный автоморфизм. Для
+того чтобы проверить, что это так, нужно написать уравнение движения
+dU/dt = AU,
+где U(t) — это элемент алгебры A. Если это уравнение имеет решение с начальным
+условием U(0) = 1, то A является инфинитезимальным автоморфизмом — анало-
+гом гамильтониана. Я буду говорить, что A это “гамильтониан”. Если алгебра конеч-
+номерна, то можно применить теорему существования решений дифференциальных
+7
+
+уравнений. В этом случае дифференцирование и инфинитезимальный автоморфизм
+— это одно и то же. Поскольку алгебра в физике бесконечномерна, то это в общем
+случае не так. Не всякое дифференцирование определяет инфинитезимальный ав-
+томорфизм. Нужно, чтобы уравнение движения имело решение, то есть,чтобы от-
+вечающая этому уравнению эволюция существовала. Поэтому есть различие между
+дифференцированием и инфинитезимальным автоморфизмом. Легко проверить, что
+дифференцирования образуют алгебру Ли. Это же верно для дифференцирований,
+которые согласованы с инволюцией. Можно считать, что дифференцирования согла-
+сованные с инволюцией образуют алгебру Ли группы автоморфизмов Aut(A). Для
+случая бесконечномерных групп понятие алгебры Ли не очень хорошо определено, но,
+тем не менее, это важное понятие, которое очень хорошо работает во многих случаях.
+Я рассматривал случай, когда уравнение движения не зависит от времени, но,
+вообще говоря, это не обязательно. “Гамильтониан” может зависеть от времени, и
+тогда уравнение движения для операторов эволюции имеет вид:
+dU
+dt = A(t)U(t).
+Если оператор A(t) не зависит от t, то операторы эволюции образуют однопарамет-
+рическую группу:
+U(t + τ) = U(t)U(τ).
+Матрице плотности K отвечает линейный функционал ω(A) = trKA на алгеб-
+ре ограниченных операторов; этот функционал удовлетворяет условию ω(A∗A) ≥ 0.
+(Легко понять, что это условие вытекает из положительной определенности оператора
+K). Эволюция матрицы плотности, как это рассказывается в курсе квантовой механи-
+ки, описывается уравнением, в котором в правой части стоит коммутатор с самосопря-
+женным оператором. Это уравнение имеет вид dK/dt = H(K), где H(K) = [ ˆH, K]/iℏ,
+ˆH — это обычный гамильтониан, а H — это “гамильтониан”.
+Здесь введены следующие обозначения: операторы в гильбертовом пространстве
+— операторы с крышечкой, а операторы на матрицах — без крышечки. По теореме
+Стоуна самосопряженные операторы в гильбертовом пространстве (не обязательно
+ограниченные) отвечают однопараметрическим подгруппам группы унитарных опе-
+раторов. Унитарные операторы можно считать действующими на матрицах плот-
+ности. В теореме Стоуна требуется, чтобы подгруппы были непрерывны в сильном
+смысле. (Я не буду пояснять, что это такое — мне это не понадобится). Если са-
+мосопряженный оператор ограничен, то соответствующая однопараметрическая под-
+группа дифференцируема в смысле сходимости по норме. В дальнейшем я не буду
+обращать внимания на эти тонкости.
+Я хочу еще пару слов сказать про связь алгебраического подхода со стандартным
+подходом, основанным на гильбертовых пространствах, и объяснить, почему алгебра-
+ический подход лучше. Пусть у нас имеется сохраняющее инволюцию представление
+алгебры A операторами в гильбертовом пространстве H. Иными словами, рассмотрим
+сохраняющий инволюцию гомоморфизм алгебры A в алгебру операторов. Обозначим
+оператор, отвечающий элементу A алгебры A через ˆA, тогда каждому нормирован-
+ному вектору Φ ∈ H отвечает нормированное же состояние ω алгебры A по формуле
+ω(A) = ⟨ ˆAΦ, Φ⟩.
+Более того, каждой матрице плотности K отвечает состояние по формуле ω(A) =
+tr(K ˆA). Иными словами, можно получать состояния алгебры из векторов в гиль-
+бертовом пространстве. Возникает естественный вопрос: все ли состояния алгебры
+можно так получить? Ответ на этот вопрос положительный. Всякое состояние может
+8
+
+быть представлено вектором в гильбертовом пространстве, и это причина того, что
+физики в состоянии работать все время в гильбертовом пространстве.
+Почему это во многих случаях неудобно? Дело в том, что исходя из алгебры наблю-
+даемых невозможно ограничиться одним гильбертовым пространством — в разных
+ситуациях придется рассматривать разные гильбертовы пространства. Например, в
+статистической физике рассматриваются равновесные состояния. Каждое равновес-
+ное состояние лежит в своем гильбертовом пространстве. Это не всегда удобно.
+Одним гильбертовым пространством, как правило, можно обойтись в квантовой
+теории поля, потому что там обычно рассматривается гильбертово пространство, в
+котором лежит основное состояние. Его элементы отвечают возбуждениям основного
+состояния, которые только и нужны в квантовой теории поля. По этой причине в
+квантовой теории поля обычно можно обойтись одним гильбертовым пространством,
+но это не всегда — в квантовой электродинамике этого сделать нельзя: там электрон
+одевается облаком мягких фотонов, которые выводят из изначального гильбертова
+пространства.
+Сейчас я собираюсь доказать, что каждое состояние алгебры с инволюцией пред-
+ставляется вектором из гильбертова пространства, которое я собираюсь построить.
+Построение такого гильбертова пространства осуществляется конструкцией Гельфанда
+– Наймарка – Сигала (GNS-конструкция), которую я сейчас объясню. Для каждой
+алгебры A я построю предгильбертово пространство E. ( Здесь мне будет удобно
+работать с предгильбертовыми пространствами.) Я построю представление алгебры
+операторами в этом предгильбертовом пространстве таким образом, что некоторому
+циклическому вектору, который я обозначаю буквой θ ∈ E, будет отвечать состояние
+ω(A) = ⟨ ˆAθ, θ⟩. То, что вектор циклический означает, что всякий другой вектор мож-
+но из него получить с помощью операторов из алгебры (все векторы имеют вид ˆAθ,
+где A ∈ A).
+Конструкция, которую я буду объяснять, однозначна с точностью до эквивалент-
+ности, как будет видно из построения. Для ее построения я предположу, что такое
+представление у меня уже есть, и определю в алгебре скалярное произведение по
+формуле
+⟨A, B⟩ = ω(B∗A).
+Зная это скалярное произведение в алгебре, я могу вычислить произведение векторов
+ˆAθ и ˆBθ просто перекидывая ˆB на первый сомножитель, а дальше пользуюсь тем,
+что гомоморфизм, сохраняет инволюцию, я могу с этой пары операторов перекинуть
+крышечку на B∗A. Это как раз и есть ω(B∗A):
+⟨ ˆAθ, ˆBθ⟩ = ⟨ ˆB∗ ˆAθ, θ⟩ = ⟨ �
+(B∗A)θ, θ⟩ = ω(B∗A).
+В результате получилось так, что если я знаю ω(B∗A), то могу вычислить скаляр-
+ное произведение ˆAθ и ˆBθ. Так как вектор θ циклический, каждый вектор простран-
+ства имеет вид ˆAθ. Это условие цикличности. Теперь я могу сказать, что имеется
+отображение ν : A → E, которое переводит A в ˆAθ, и это отображение сюрьективное
+(то есть, на). Отсюда следует, что E получается из A факторизацией. Нужно профак-
+торизовать по всем векторам, которые дают 0 в скалярном произведении с любыми
+другими векторами (нулевым векторам).
+Чтобы дать теперь окончательный ответ на вопрос: как построить E исходя из
+алгебры A и ω, нужно просто взять алгебру A, ввести в ней скалярное произведе-
+ние по формуле ω(B∗A) и профакторизовать по нулевым векторам. Получится пред-
+гильбертово пространство. (Факторизуя по нулевым векторам, можно снова опре-
+делить скалярное произведение в фактор-пространстве). В результате получилась
+эта конструкция, которую я, по существу, вывел, а не придумал. Я увидел, что она
+9
+
+обязательно должна быть такой, потому что исходя из того, что вектор θ цикли-
+ческий, я сделал две вещи: во-первых, построил предгильбертово пространство, а,
+во-вторых, доказал, что моя конструкция по существу единственна, ничего другого
+сделать нельзя. Я ее вывел из цикличности. Это рассуждение (конструкция Гель-
+фанда–Наймарка–Сигала) — самый важный элемент из того, что я в этой лекции
+собираюсь рассказывать. Этой конструкцией я буду пользоваться много раз. Вместо
+предгильбертова пространства я могу рассматривать его пополнение ¯E — гильберто-
+во пространство (тогда вектор θ будет циклическим в более слабом смысле: векторы
+вида Aθ будут плотны в ¯E).
+Для иллюстрации возьмем какое-то стационарное состояние (состояние, которое
+не меняется при эволюции) и применим к нему GNS- конструкцию. Тогда получится
+некоторое гильбертово пространство и циклический вектор в нем, который тоже будет
+стационарным, не будет зависеть от времени.
+Утверждение: если исходить из стационарного состояния, то в новом гильбертовом
+пространстве, полученном с помощью данной конструкции будет действовать группа
+унитарных операторов, отвечающих операторам эволюции U(t).
+Это очень просто понять. Дело в том, что GNS-конструкция использовала в каче-
+стве скалярного произведения ω(B∗A). Но это скалярное произведение инвариантно
+относительно операторов U(t) потому что ω инвариантно (то есть, не изменяется
+оператором эволюции). Раз скалярное произведение инвариантно, то операторы U(t)
+спускаются в унитарные операторы ˆU(t). Операторы ˆU(t) образуют однопараметри-
+ческую группу. У нее есть генератор (инфинитезимальный автоморфизм) ˆH, и это
+то, что в физике называется гамильтонианом. (На самом деле это не совсем так, пото-
+му что в физике гамильтониан считается самосопряженным оператором, и для этого
+нужно поставить мнимую единицу в определении: ˆU(t) = e−i ˆ
+Ht).
+Я говорю, что состояние ω — это основное состояние, если спектр оператора ˆH
+неотрицателен. Обращаю внимание, что основное состояние будет иметь нулевую
+энергию при таком определении. Это оказывается согласованным со стандартным
+определением. Если применить GNS-конструкцию к алгебре ограниченных опера-
+торов и состоянию, отвечающему собственному вектору гамильтониана с собствен-
+ным значением E, то тогда генератором группы ˆU(t) построенной с помощью GNS-
+конструкции будет ˆH − E. В квантовой теории поля всегда говорят, что мы будем
+отсчитывать энергию от основного состояния, и не будем обращать внимание на тот
+бесконечный вклад, который присутствует в буквальном подходе. Алгебраическая
+GNS-конструкция делает это сама по себе.
+Я все время говорил, что рассматриваю алгебраический подход в квантовой ме-
+ханике. Этот подход прекрасно работает и в классической механике. Чтобы пока-
+зать это, будем работать в гамильтоновом формализме и тогда можно повторить
+те же самые рассуждения. Чистое состояние описывается обобщенными импульсами
+p = (p1, ..., pn) и обобщенными координатами q = (q1, ..., qn), представляющими точки
+в 2n-мерном пространстве, которое называется фазовым пространством. Это — чи-
+стое состояние, но, точно так же как в квантовой механике, можно рассматривать и
+смешанные состояния.
+Смешанное состояние — это распределение вероятностей на фазовом простран-
+стве или положительная мера на фазовом пространстве (считается, что мера все-
+го пространства равна единице). Все эти распределения вероятностей образуют вы-
+пуклое множество D. Чистые состояния — это крайние точки этого множества. Чи-
+стое состояние — это распределение вероятностей, которое сосредоточено ровно в
+одной точке. Ему отвечает плотность распределения вероятностей, которая является
+дельта-функцией. Любая функция является суперпозицией дельта-функций (иными
+словами, любую функцию можно представить как интеграл от дельта-функций). Это
+10
+
+означает, что всякому распределению вероятностей на фазовом пространстве отве-
+чает распределение вероятностей на чистых состояниях, а чистые состояния можно
+отождествить с крайними точками пространства всех состояний.
+В рассматриваемом случае, соответствующем в классической механике, всякое со-
+стояние может быть представлено единственным образом как смесь чистых состо-
+яний. Это отличает классическую механику от квантовой механики, где состояние
+может быть представлено как смесь чистых состояний множеством разных спосо-
+бов. Позже я расскажу, как квантовую механику можно получить из классической
+механики, если дозволить себе рассматривать не все наблюдаемые. Естественно с фи-
+зической точки зрения предположить, что наши приборы не могут измерить всё, что
+есть на свете. Мое утверждение: если есть такая ситуация, то классическая механика
+может превратиться в квантовую механику.
+Дальше я напомню хорошо известное уравнение движения Гамильтона
+dq
+dt = ∂H
+∂p , dp
+dt = −∂H
+∂q
+и уравнение Лиувилля для плотности распределения вероятностей, которое записы-
+вается в терминах скобок Пуассона как:
+d
+dtρ(p, q, t) = {H, ρ(p, q, t)}.
+Скобки Пуассона при этом определяются формулой
+{f, H} = −∂f
+∂p
+∂H
+∂q + ∂f
+∂q
+∂H
+∂p .
+Для того чтобы определить эволюцию U(t) нужно будет решить уравнение d
+dtU(t) =
+LU(t), где Lρ = {H, ρ}. Это уравнение эквивалентно уравнению Лиувилля. Для того,
+чтобы в этом убедиться, нужно просто проверить, что на чистых состояниях оно сво-
+дится к гамильтоновым уравнениям. В классической механике, точно так же как в
+квантовой, можно следить как развиваются наблюдаемые (а не состояния). Наблюда-
+емые - это действительные функции f(p, q) на фазовом пространстве. Легко можно
+вывести из уравнений Гамильтона, что это развитие происходит в соответствии с
+уравнением
+d
+dtf(p(t), q(t)) = −∂f
+∂p
+∂H
+∂q + ∂f
+∂q
+∂H
+∂p
+или
+d
+dtf(p(t), q(t)) = {f, H},
+где в правой части стоит скобка Пуассона. Теперь мы видим, что находимся в той же
+ситуации как квантовая механика или, наоборот, квантовая механика находится в той
+же ситуации, как и классическая механика. У нас есть наблюдаемые. Они образуют
+алгебру A. Эти наблюдаемые вы можете перемножать. Есть понятие инволюции.
+(Инволюция — это просто комплексное сопряжение). Каждому состоянию ω отвечает
+линейный функционал на наблюдаемых A: нужно взять интеграл от функции по
+распределению вероятностей
+ω(f) =
+�
+fω.
+(Распределение вероятностей — это мера, по которой можно проинтегрировать). Этот
+функционал удовлетворяет условию положительности : ω(f) ≥ 0 при f ≥ 0 потому
+что для каждой положительной функции будет неотрицательный ответ. Функции ти-
+па A∗A, несомненно, положительны (просто квадрат модуля), так что классическая
+механика входит как маленький кусочек в алгебраический подход к квантовой меха-
+нике, но есть разница: в классической механике алгебра наблюдаемых коммутативна.
+11
+
+2
+Лекция 2
+2.1
+Квантовая механика как деформация классической
+механики. Алгебра Вейля
+Теперь я попытаюсь объяснить как математик мог бы вывести квантовую механику из
+классической. Он знает, что квантовая механика в пределе малой постоянной Планка
+сводится к классической. Это означает, что квантовая механика получается дефор-
+мацией — маленьким изменением классической механики. Должно быть семейство
+алгебр Aℏ, которое зависит от постоянной Планка ℏ. Зависимость эта непрерывная и
+дифференцируемая. При равной нулю постоянной Планка ℏ = 0 должна получаться
+классическая механика, то есть, должна получаться коммутативная алгебра с произ-
+ведением A · B. Будем считать (это не обязательно, как мы увидим вскоре), что все
+эти алгебры определены в том же самом векторном пространстве, то есть, сложение
+и умножение на число независимы от постоянной Планка, а умножение элементов
+алгебры зависит от постоянной Планка A·ℏ B. Теперь рассмотрим коммутатор в этой
+алгебре, зависящей от постоянной Планка:
+[A, B]ℏ = A ·ℏ B − B ·ℏ A
+Наше основное требование — чтобы этот коммутатор обращался в ноль при обраще-
+нии в ноль постоянной Планка : ℏ = 0. Будем считать, что зависимость от постоянной
+Планка является гладкой. Это означает, что коммутатор представляется как линей-
+ная по ℏ часть и нечто более высокого порядка:
+[A, B]ℏ = i{A, B}ℏ + O(ℏ2).
+То, что линейно по ℏ обозначено как фигурная скобка, умноженная на мнимую еди-
+ницу; я докажу, что фигурная скобка — это и есть скобка Пуассона. Точнее, она имеет
+те же свойства, которые есть у скобки Пуассона. Это означает, что новая операция
+есть дифференцирование (удовлетворяет правилу Лейбница):
+{A · B, C} = {A, C} · B + A · {B, C}
+и, кроме того, удовлетворяет аксиомам алгебры Ли. Нетрудно доказать, что это так.
+Cвойства обычного коммутатора в ассоциативной алгебре:
+[A, B] = −[B, A],
+[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0,
+[AB, C] = [A, C]B + A[B, C]
+должны удовлетворяться при каждой постоянной Планка ℏ. Разложим все эти ра-
+венства по постоянной Планка. Во втором равенстве нужно разлагать до второго
+порядка, а в остальных — до первого достаточно. Если приравнять члены первого
+порядка по ℏ, то как раз получаются нужные свойства. То есть, мы убедились, что
+в пределе из квантовой механики получается классическая механика. Коммутатор
+переходит в скобку Пуассона.
+Давайте посмотрим: можно ли пойти наоборот. Мы видели как из квантовой меха-
+ники получается классическая. А теперь мы интересуемся: можно ли из классической
+механики получить квантовую? Для этого я сначала опишу все возможные скобки
+Пуассона в том случае, когда алгебра A является алгеброй полиномиальных функций
+на каком-то векторном пространстве с координатами (u1, ..., un). Если она является
+12
+
+алгеброй полиномов, то для того, чтобы вычислить, как работает скобка Пуассона,
+нужно знать только какова скобка Пуассона у этих координат. Это обусловлено тем,
+что любая другая функция — это полином, а у меня есть свойство, которое позволяет
+вычислять скобку Пуассона от произведения. Полином — это линейная комбинация
+произведений координат, и поэтому скобку двух полиномов можно вычислить. Ре-
+зультат вычисления таков:
+{A, B} = 1
+2σkl(u) ∂A
+∂uk
+∂B
+∂ul,
+(1)
+где σkl(u) обозначает скобку Пуассона координат uk, ul.
+Наоборот, можно проверить, что если σ является антисимметрической и незави-
+сящей от u, то это выражение будет удовлетворять условиям, наложенным на скобку
+Пуассона. Это как раз ситуация, которая возникает, когда мы имеем дело со стан-
+дартной скобкой Пуассона в фазовом пространстве.
+Теперь я могу задать вопрос: как продеформировать скобку Пуассона, чтобы по-
+лучилось то, что мне нужно в квантовой механике. Эта задача непростая. Она была
+решена сравнительно недавно Концевичем. Но в том случае, когда матрицы σ не
+зависят от u, то есть, когда скобка Пуассона двух координат не зависит от u, то сде-
+лать это очень легко. Именно, я определю алгебру Aℏ как ассоциативную алгебру с
+генераторами ˆuk, связанными соотношением:
+ˆukˆul − ˆulˆuk = iℏσkl.
+(2)
+Я просто переписал то условие, которое у меня было наложено на скобку Пуас-
+сона координат. Это, конечно, можно было сделать и в том случае, когда σ зависит
+от u, но тогда неизвестно: получилась бы ассоциативная алгебра или нет. Если же σ
+не зависит от u, то получается ассоциативная алгебра, которая называется алгеброй
+Вейля. Если мы исходим из полиномов, то это — единственный (с точностью до вы-
+бора генераторов) способ продеформировать скобку Пуассона. Я ввожу инволюцию
+в алгебре Вейля считая генераторы ˆuk самосопряженными.
+Мы получили коммутационные соотношения, которые несколько в другом виде
+хорошо известны из квантовой механики. Чтобы это показать, я потребую, чтобы
+матрица σ была обратима. Тогда эту матрицу можно упростить. Это антисимметрич-
+ная матрица и, в буквальном смысле, ее диагонализовать нельзя, однако ее можно
+записать в подходящем базисе как блочно-диагональную матрицу, состоящую из дву-
+мерных блоков. Если этим воспользоваться, то заменив систему генераторов, можно
+свести коммутационные соотношения в алгебре Вейля к хорошо известным в кванто-
+вой механике коммутационным соотношениям:
+ˆpkˆpl = ˆplˆpk,
+ˆqkˆql = ˆqlˆqk,
+ˆpkˆql − ˆqlˆpk = ℏ
+i δl
+k.
+Это — коммутационные соотношения для координат и импульсов в стандартной кван-
+товой механике. Они называются каноническими коммутационными соотношениями.
+Вместо самосопряженных генераторов можно взять другие генераторы, которые
+не самосопряжены, но сопряжены друг с другом и удовлетворяют соотношениям
+ˆakˆal = ˆalˆak,
+ˆa∗
+kˆa∗
+l = ˆa∗
+l ˆa∗
+k,
+ˆakˆa∗
+l − ˆa∗
+l ˆak = ℏδkl.
+Здесь ˆak =
+1
+√
+2(ˆqk + iˆpk), ˆa∗
+k =
+1
+√
+2(ˆqk − iˆpk).
+Так обозначаются операторы рождения и уничтожения, но, пока что, это у меня
+— формальный математический объект. Я их ввел формально и написал для них
+коммутационные соотношения. Эти коммутационные соотношения тоже называются
+каноническими коммутационными соотношениями.
+13
+
+Могу ли я теперь сказать, что полученная алгебра Aℏ — это деформация ком-
+мутативной алгебры? Формально — не могу, потому что, когда я определял понятие
+деформации, то потребовал, чтобы все эти алгебры были определены на одном и том
+же пространстве, иначе было бы трудно рассматривать все одновременно. Моя ком-
+мутативная алгебра состояла из полиномов, а эта алгебра состоит неизвестно из чего.
+Но ее тоже можно сделать состоящей из полиномов. Это делается очень просто.
+Построенная алгебра порождена элементами ˆqk и ˆpk, то есть, ее элементы явля-
+ются суммами мономов, составленных из генераторов ˆqk и ˆpk. Благодаря наличию
+коммутационного соотношения я могу сдвинуть все генераторы ˆqk налево а ˆpk на-
+право (можно и наоборот) и потом снять крышечки с них. Тогда получится обычный
+полином и я могу сказать, что элемент из моей алгебры представлен полиномом, кото-
+рый называется q-p-символом. Тогда алгебра определена на пространстве полиномов.
+Это не очень хорошее представление, потому что оно не согласовано с инволюцией.
+Тем не менее оно тоже очень полезно во многих случаях.
+Если исходить из операторов ˆak, ˆa∗
+k, то можно использовать ту же самую идею:
+сдвинуть ˆa∗
+k налево, ˆak — направо и получится то, что называется нормальной фор-
+мой элемента вейлевской алгебры. Теперь опять можно снять крышечки и получить
+полином, который называется символом Вика. Физики термин “символ Вика” не ис-
+пользуют, но слово “нормальная форма” они используют все время. Как легко видеть,
+переход к нормальной форме согласован с инволюцией.
+Мы можем рассматривать алгебру Вейля с бесконечным числом образующих. До
+сих пор параметр k у нас считался дискретным (хотя количество uk или ak с ин-
+дексом k могло быть бесконечным), но можно считать этот параметр непрерывным.
+Например, рассматривать алгебру с генераторами ˆa(k), ˆa∗(l) и соотношениями
+ˆa(k)ˆa(l) = ˆa(l)ˆa(k),
+ˆa∗(k)ˆa∗(l) = ˆa∗(l)ˆa∗(k),
+ˆa(k)ˆa∗(l) − ˆa∗(l)ˆa(k) = ℏδ(k, l).
+В этом случае вместо символа Кронекера возникает непрерывный его аналог: δ−функция.
+Так как функция δ(k, l) — это обобщенная функция, генераторы алгебры ˆa(k), ˆa∗(l)
+нужно также рассматривать как обобщенные функции. Обобщенная функция — это
+такая функция, которая имеет смысл только под знаком интеграла. Имеют смысл
+только элементы ˆa(f) =
+�
+f(k)ˆa(k)dk и ˆa∗(g) =
+�
+g(l)ˆa∗(l)dl, представляющие собой
+формальные интегралы. Эти элементы линейно зависят соответственно от f и g и
+удовлетворяют коммутационным соотношениям:
+ˆa(f)ˆa(g) = ˆa(g)ˆa(f),
+ˆa∗(f)ˆa∗(g) = ˆa∗(g)ˆa∗(f),
+ˆa(f)ˆa∗(g) − ˆa∗(g)ˆa(f) = ℏ⟨f, ¯g⟩.
+Для того, чтобы эти соотношения имели смысл, нужно чтобы было определено ска-
+лярное произведение ⟨f, ¯g⟩. Поскольку скалярное произведение зависит от g анти-
+линейно, то в нем стоит комплексное сопряжение от g. Я обычно буду считать, что
+индекс k дискретен, но следует понимать, что все можно перенести на случай непре-
+рывного индекса.
+Я уже говорил, что элементы алгебры Вейля можно представлять полиномами,
+например, с помощью понятия виковского символа, который определяется с помощью
+сдвига всех генераторов ˆa∗
+k налево и снятием крышечек.
+Виковские символы согласованы с определением инволюции. (Инволюции в ал-
+гебре отвечает комплексное сопряжение полиномов.)
+Переход к символам — это операция, которая тесно связана с операцией квантова-
+ния. Что такое квантование? Пусть имеется классический гамильтониан и мы хотим
+14
+
+получить его квантовый аналог. Если гамильтониан зависит от uk, то при простой за-
+мене u на ˆu возникает проблема, в каком порядке ставить эти генераторы? В классике
+не важен порядок: u1u2 и u2u1 — это одно и то же, но если переходить к квантовой
+механике, к алгебре Вейля, то все зависит от порядка. Это – то, что по-английски
+называется «ordering ambiguity» и означает, что нет однозначной процедуры кванто-
+вания. Ее можно сделать однозначной выбрав понятие символа, но символов разных
+много.
+Есть такие случаи, когда квантовый гамильтониан имеет естественное определе-
+ние. Это, например, стандартная в классической механике ситуация, когда гамиль-
+тониан делится на кинетическую и потенциальную энергии. Кинетическая энергия
+зависит только от импульсов, а потенциальная энергия — только от координат, и
+тогда можно ставить шапочки и все будет абсолютно однозначно потому что кван-
+товые импульсы между собой коммутируют и квантовые координаты между собой
+коммутируют.
+Для того, чтобы написать уравнение движения есть стандартный способ: в класси-
+ческих уравнениях движения на месте скобок Пуассона нужно писать коммутаторы:
+∂ˆuk
+dt = i[ ˆH, ˆuk].
+В этой формуле я считаю, что ℏ = 1. В дальнейшем постоянная Планка ℏ всегда
+будет приниматься равной единице, если не сказано обратное.
+Уравнение движения осмыслено, если гамильтониан ˆH является элементом ал-
+гебры Вейля. Я уже объяснял, что в уравнении движения должна стоять операция,
+которая удовлетворяет правилу Лейбница. Такая операция называется дифференци-
+рованием. Коммутатор вида Dh(a) = [h, a] как раз удовлетворяет правилу Лейбница
+[h, ab] = [h, a]b + a[h, b] для любой алгебры и поэтому пока ˆH является элементом
+алгебры Вейля — все замечательно. Единственное, что нехорошо – это то, что он
+очень часто не является элементом алгебры Вейля в случае бесконечного количества
+степеней свободы. Типичный пример — гамильтониан вида
+ˆH =
+�
+ǫkˆa∗
+kˆak.
+Когда число индексов бесконечно, это бесконечная сумма. В алгебре Вейля ниче-
+го похожего нет. Этот гамильтониан не принадлежит алгебре Вейля. Тем не менее,
+можно формально взять коммутатор гамильтониана ˆH и ˆak. Получается уравнение
+движения, которое еще встретится неоднократно:
+dˆak
+dt = −iǫkˆak,
+dˆa∗
+k
+dt = iǫkˆa∗
+k.
+Таким образом, в случае бесконечного числа степеней свободы гамильтониан — это
+не элемент алгебры Вейля, а просто формальное выражение вида
+ˆH =
+�
+Γm,n(k1, ..., km, l1, ..., ln)ˆa∗
+k1...ˆa∗
+kmˆal1...ˆaln,
+которое само по себе смысла оператора и смысла элемента алгебры Вейля не имеет,
+но, тем не менее, имеет смысл под знаком коммутатора в уравнениях движения. Это
+происходит не всегда, но есть очень простые условия, когда оно точно имеет смысл.
+Когда берется коммутатор операторов ˆak или ˆa∗
+k с произведением, следует прокомму-
+тировать с каждым элементом этого произведения. При коммутировании возникнет
+символ Кронекера, а в коммутатор войдут только те коэффициенты, у которых ин-
+декс совпадает с индексом у операторов, с которыми происходит коммутирование.
+Если у всякого индекса в гамильтониане есть только конечное число ненулевых ко-
+эффициентов содержащих этот индекс, то уравнение движения имеет смысл.
+15
+
+2.2
+Квадратичные гамильтонианы
+Перейдем к рассмотрению квадратичных гамильтонианов вида
+H(u) = 1
+2Hklukul.
+Здесь упорядочение не играет никакой роли, потому что при смене порядка появит-
+ся константа. Эта константа несущественна, поскольку гамильтониан используется
+только под знаком коммутатора, где константы исчезают. Классические уравнения
+движения и квантовые уравнения движения совершенно одинаковы. Более того, ес-
+ли мы умеем решать классические уравнения движения, то сразу же умеем решать
+и квантовые уравнения движения, потому что все уравнения движения линейны, а
+разница между классической и квантовой механикой возникает только тогда, когда
+перемножаются операторы. Тут же перемножать ничего не нужно, так что все очень
+просто.
+Эту же задачу можно сделать еще проще, а именно, можно гамильтониан упро-
+стить. Если предположить, что гамильтониан положительно определен и матрица
+Hkl невырождена, то можно представить гамильтониан как сумму квадратов.
+Остается еще проблема с матрицей σ. Ее тоже можно сделать проще, сохраняя при
+этом представление гамильтониана как суммы квадратов. Для того чтобы сохранить
+это свойство, следует брать только ортогональные преобразования. Заметим, что σ
+— антисмметричная матрица (так как она появилась из алгебры Вейля). Если анти-
+симметричную матрицу умножить на мнимую единицу получится матрица, которая
+отвечает самосопряженному оператору и ее можно диагонализовать. Эта диагонали-
+зация происходит в комплексной области, но можно сказать, что вместе с каждым
+собственным вектором будет и комплексно сопряженный собственный вектор. Мож-
+но рассмотреть картинку, которая содержит два комплексно сопряженных вектора,
+взять в ней действительную и мнимую части и получить представление матрицы σkl
+в блочно-диагональном виде с двумерными блоками. Эти двумерные блоки будут яв-
+ляться антисимметричными матрицами, поэтому гамильтониан примет форму суммы
+гамильтонианов вида
+ˆH = 1
+2(ˆp2 + ǫ2ˆq2),
+где [ˆp, ˆq] = 1
+i . Это чрезвычайно важное упрощение, которое всегда можно сделать
+для положительного квадратичного гамильтониана. Я это рассказывал для случая
+конечного числа степеней свободы. Важно заметить, что для случая бесконечного
+числа степеней свободы это все тоже правильно. Единственная проблема заключает-
+ся в том, что при диагонализации самосопряженного оператора кроме дискретного
+спектра может появиться непрерывный спектр. Я к этому еще вернусь. Теперь рас-
+смотрим гамильтониан
+ˆH =
+� 1
+2(ˆp2
+k + ǫ2
+k(ˆqk)2)
+и будем решать соответствующие ему уравнения движения
+dˆpk
+dt = −ǫ2
+kˆqk,
+dˆqk
+dt = ˆpk.
+Это стандартные уравнения движения осциллятора точно такие же, как в классике.
+Их можно решить десятками способов, но самый простой способ — это ввести новые
+переменные
+ˆak =
+1
+√
+2(√ǫkˆqk + iˆpk
+√ǫk
+),
+ˆa∗
+k =
+1
+√
+2(√ǫkˆqk − iˆpk
+√ǫk
+).
+16
+
+В таком случае уравнения движения
+dˆak
+dt = −iǫkˆak,
+dˆa∗
+k
+dt = iǫkˆa∗
+k,
+которые отвечают гамильтониану
+ˆH =
+�
+ǫkˆa∗
+kˆak,
+(3)
+имеют очень простое решение:
+ˆak(t) = e−itǫkˆak(0),
+ˆa∗
+k(t) = eitǫkˆa∗
+k(0).
+В случае бесконечного числа степеней свободы в силу спектральной теоремы тоже
+можно считать, что все диагонально, но вместо суммы возникнет интеграл и тогда
+гамильтониан будет иметь вид:
+ˆH =
+�
+dλǫ(λ)ˆa∗(λ)ˆa(λ).
+В физике λ — это обычно совокупность непрерывных и дискретных индексов, а ин-
+теграл включает как обычное интегрирование, так и суммирование по дискретному
+индексу. В частном случае когда теория трансляционно инвариантна, мы считаем, что
+операторы ˆa∗(x) и ˆa(x) зависят от координат x, которые можно сдвигать не меняя
+гамильтониана. Это значит, что гамильтониан имеет вид
+ˆH =
+�
+dxdyǫ(x − y)ˆa∗(x)ˆa(y),
+где коэффициент зависит только от разности x − y. (В рассматриваемом выражении
+могут быть также дискретные индексы — по ним нужно проводить суммирование.)
+Можно перейти к импульсному представлению (взять преобразование Фурье). То-
+гда гамильтониан примет вид:
+ˆH =
+�
+dkǫ(k)ˆa∗(k)ˆa(k).
+Мы будем все время рассматривать трансляционно инвариантные гамильтонианы, и
+эта формула будет существенно использоваться.
+2.3
+Стационарные состояния
+Теперь кратко обсудим стационарные (не зависящие от времени) состояния. Если
+операторы эволюции обозначить U(t), то для стационарного состояния выполняется
+условие U(t)ω = ω.
+Если работать в формализме матриц плотности K, то, учитывая что уравнения
+движения для матрицы плотности записывается как коммутатор с гамильтонианом
+ˆH, можно сделать вывод что матрица плотности представляет стационарное состоя-
+ние, если она коммутирует с гамильтонианом. В частности, если матрица плотности
+является функцией оператора ˆH, то состояние стационарно.
+Если говорить не о матрице плотности, а о векторе состояния, то стационарное со-
+стояние удовлетворяет условию ˆHΨ = EΨ, то есть стационарное состояние является
+собственной функцией гамильтониана. Гамильтониан имеет смысл оператора энер-
+гии, а E — это уровень энергии. Вектор Ψ с течением времени просто умножается на
+множитель:
+Ψ(t) = ˆU(t)Ψ = e−itEΨ ∼ Ψ.
+17
+
+Вектор меняется — состояние не меняется.
+В алгебраическом подходе гамильтониан может быть формальным выражением,
+но можно применить GNS-конструкцию к какому-то стационарному состоянию ω и
+получить гильбертово пространство, в котором есть унитарные операторы, описыва-
+ющие эволюцию (сдвиг по времени) и есть генератор группы временных трансляций,
+который имеет смысл гамильтониана — оператора энергии. Его собственные значе-
+ния имеют смысл уровней энергии возбуждений состояния ω. Точнее говоря, такая
+интерпретация будет совершенно правильной в том случае, когда само ω стационар-
+но и трансляционно инвариантно (инвариантно как относительно пространственных,
+так и временных трансляций).
+Проиллюстрировать это можно следующим образом: трансляционно инвариант-
+ное состояние можно представить в виде горизонтальной прямой. У него обычно бес-
+конечная энергия, но все отсчитывается от этой бесконечной энергии. (В алгебраи-
+ческой квантовой механике никакой бесконечной энергии нет — просто надо отсчи-
+тывать энергию от энергии этого состояния, считая ее равной нулю). Возбуждение
+нужно воспринимать как сосредоточенный в конечной области горбик на этой гори-
+зонтальной прямой, и тогда имеет смысл понятие энергии. Эта энергия отсчитывается
+уже от энергии трансляционно инвариантного состояния.
+Важное и простое замечание, которое в курсе квантовой механики объясняется
+в несколько менее общей ситуации, заключается в следующем. Рассмотрим класси-
+ческий гамильтониан H(u), у которого есть минимум в невырожденной критической
+точке. Это означает, что квадратичная часть в разложении Тейлора положительно
+определена; тогда нет нулевых мод, квантовый гамильтониан в первом приближении
+квадратичен; в соответствующих координатах он будет иметь вид:
+ˆH =
+�
+dλǫ(λ)ˆa∗(λ)ˆa(λ) + ...,
+где обозначенные как ... члены начинаются с кубических по ˆa∗, ˆa (линейных членов
+быть не может потому что мы находимся в критической точке). Члены высшего по-
+рядка по ˆa∗, ˆa также являются членами высшего порядка по постоянной Планка. По
+крайней мере, в квазиклассике можно этими членами пренебречь. Если же мы работа-
+ем не в квазиклассическом приближении, здесь можно поставить какой-то множитель
+и рассматривать теорию возмущений по этому множителю.
+2.4
+Пространство Фока
+Перейдем теперь к рассмотрению представлений алгебры Вейля (или, что то же —
+представлений канонических коммутационных соотношений). Среди этих представ-
+лений есть одно замечательное, самое простое, которое называется представлением
+Фока, а пространство, в котором оно живет, называется пространством Фока. Опре-
+деляется фоковское представление просто: в нем существует циклический вектор |0⟩,
+который уничтожается всеми операторами ˆak:
+ˆak|0⟩ = 0.
+Это условие с точностью до эквивалентности однозначно определяет представление.
+Попробуем сейчас выписать в конкретной форме то как оно устроено.
+Вектор |0⟩ — циклический. Определение цикличности зависит от того, живем ли
+мы в предгильбертовом или в гильбертовом пространстве. Если мы живем в пред-
+гильбертовом пространстве, то, применяя элементы алгебры к циклическому вектору,
+можно получить все векторы. Если мы живем в гильбертовом пространстве это не со-
+всем так: нужно разрешить взятие предела. То есть, взять еще замыкание множества
+18
+
+тех векторов, которые получаются с помощью алгебраических операций. Давайте по-
+смотрим, что будет здесь? Оператор ˆak уничтожает |0⟩, поэтому действовать им на
+|0⟩ не надо — ничего нового не получится. Нужно действовать только операторами
+рождения ˆa∗
+k:
+(ˆa∗
+k1)n1...(ˆa∗
+ks)ns|0⟩.
+(4)
+Здесь применены операторы рождения много раз. (Конечное число раз, так как в
+алгебре Вейля существуют только конечные комбинации этих операторов.)
+Теперь спрашивается: можем ли мы получить что-нибудь новое, если применим к
+этому выражению оператор ˆak? Ответ: нет, не получим, потому что можно пользуясь
+коммутационными соотношениями переставить оператор ˆak так, чтобы он действо-
+вал на вакуум. Тогда оператор ˆak исчезнет в силу наложенного условия. Поэтому
+только выражения вида (4) и их линейные комбинации принадлежат пространству
+Фока, если пользоваться буквальным определением цикличности без взятия предела.
+Дальше следует заметить, что все выписанные состояния (4) являются собственными
+векторами любого гамильтониана формы ˆH = � ǫkˆa∗
+kˆak с собственными значениями,
+равными � nkǫk.
+Как это проверить? Легко посчитать коммутатор гамильтониана (3), с которым
+мы сейчас работаем, с оператором ˆa∗
+l :
+ˆHˆa∗
+l = ˆa∗
+l ˆH + ǫlˆa∗
+l .
+Чтобы вычислить действие гамильтониана ˆH на (4) достаточно, пользуясь этим
+соотношением, передвинуть гамильтониан на |0⟩. Несколько более простое формаль-
+ное рассуждение таково: известно, как операторы ˆa∗
+k изменяются с течением времени
+— просто умножаются на численную экспоненту. По этой причине динамика вектора
+состояния сводится к умножению на экспоненту с некоторым показателем. Динамика
+здесь ровно такая, какая требуется для стационарного состояния.
+Мы получили ортогональный (но не ортонормированный) базис фоковского про-
+странства, который состоит из собственных векторов (4).
+Все полученные формулы можно применить для случая многомерного гармони-
+ческого осциллятора. Там операторы ˆak, ˆa∗
+k называются операторами рождения и
+уничтожения квантов. В кристалле атомы как-то между собой взаимодействуют, но
+кристалл находится в стационарном состоянии близком к основному состоянию. По
+крайней мере, в первом приближении кристалл описывается квадратичным гамильто-
+нианом. Для квантов в этой ситуации есть другое название: фононы — кванты звука.
+В общем случае мы имеем дело с системой невзаимодействующих бозонов. Опера-
+торы ˆak, ˆa∗
+k называются операторами рождения и уничтожения частиц, а числа nk в
+формуле для уровней энергии � nkǫk называются числами заполнения.
+Если мы хотим, чтобы фоковское пространство было гильбертовым простран-
+ством, нужно взять пополнение.
+Есть свои преимущества в том и в другом подходе. В предгильбертовом простран-
+стве рассматриваемые операторы определены везде. В гильбертовом пространстве —
+это неограниченные операторы, которые не всюду определены, но есть возможность
+рассматривать пределы.
+Предгильбертово фоковское пространство может быть представлено как простран-
+ство полиномов. В самом деле, есть следующая формула для базиса:
+(ˆa∗
+k1)n1...(ˆa∗
+ks)ns|0⟩.
+Чтобы получить полином, я предлагаю вычеркнуть |0⟩ и снять крышечку. Получится
+некоторый моном.
+Линейная комбинация таких мономов — это полином. То есть, каждому элементу
+фоковского пространства (которое до пополнения мы считаем предгильбертовым)
+19
+
+можно сопоставить полином. Легко посчитать, что скалярное произведение в таком
+представлении в виде полиномов будет даваться формулой:
+⟨F, G⟩ =
+�
+da∗daF(a∗)G(a∗)∗e−a∗a,
+(5)
+где F(a∗) и G(a∗) — это полиномы, отвечающие разным векторам. Обращаю внима-
+ние, что G(a∗) стоит со звездочкой. Звездочка, примененная к a дважды — это опять
+a, и поэтому здесь уже будет полином от a.
+Проверим, что скалярное произведение записывается в виде (5). Это можно посчи-
+тать, но этого даже не нужно делать. Дело в том, что фоковское пространство одно-
+значно определяется наличием циклического вектора, который уничтожается всеми
+операторами уничтожения при том, что операторы рождения и уничтожения удовле-
+творяют нужным коммутационным соотношениям. В пространстве полиномов от a∗
+k
+я определяю оператор ˆa∗
+k просто как умножение на a∗
+k, а оператор ˆak — как диф-
+ференцирование по a∗
+k. Легко понять, что умножение и дифференцирование как раз
+удовлетворяют необходимым коммутационным соотношениям. Если умножить на a∗
+k,
+а потом продифференцировать и применить правило Лейбница, то как раз получится
+то что нужно.
+Таким образом, мы имеем коммутационные соотношения, есть и циклический век-
+тор, который просто равен единице. Полином равный единице обнуляется введенными
+операторами уничтожения и является правильным кандидатом на |0⟩, цикличность
+тоже имеет место. Осталось проверить, что ˆak и ˆa∗
+k сопряжены друг к другу. (Это
+нужно, чтобы правильно работала инволюция.) Действительно, в формуле (5) можно
+применить интегрирование по частям и убедиться, что действительно для этого ска-
+лярного произведения умножения и дифференцирования сопряжены друг другу. В
+итоге все свойства фоковского представления выполнены, и поэтому не нужно срав-
+нивать исходное скалярное произведение с новым: оно обязательно будет таким же
+(с точностью до численного множителя).
+До сих пор мы имели дело с полиномами. Но я хочу все-таки иметь возможность
+работать в гильбертовом пространстве. Для этого нужно пополнить пространство
+полиномов, и тогда кроме полиномов появятся и другие функции. Они будут голо-
+морфными функциями от a∗
+k, но это будут не все голоморфные функции, а только
+те, что имеют конечную норму в нашем скалярном произведении.
+Важно отметить, что все эти рассуждения применимы и к бесконечному числу
+степеней свободы. Хотя там интеграл бесконечномерен, но, все равно, он оказывается
+хорошо определенным.
+Есть еще один способ описания фоковского пространства. Дело в том, что по-
+линомы связаны с симметрическими функциями от дискретного аргумента. Хорошо
+известно, что если имеется квадратичная форма, то всегда можно считать, что коэф-
+фициенты этой квадратичной формы симметричны по индексу. Если есть кубичный
+полином, можно считать, что коэффициент зависит от трех индексов, но, опять же,
+можно наложить условие симметрии. Его нужно наложить, если вы хотите иметь
+однозначное представление. Для полиномов высших степеней ситуация аналогична.
+Поэтому можно считать что есть единственное представление для всякого элемента
+предгильбертова фоковского пространства в форме:
+�
+n
+�
+k1,...,kn
+fn(k1, ..., kn)ˆa∗
+k1...ˆa∗
+kn|0⟩,
+где n = 0, 1, 2, ..., а коэффициенты симметричны по индексам. Это значит, что фоков-
+ское пространство представимо как последовательность симметрических функций от
+20
+
+растущего числа переменных:
+f0, f1(k), f2(k1, k2), f3(k1, k2, k3), ... .
+Пока мы работаем с полиномами, должно быть только конечное количество сим-
+метричных функций от k1, k2, k3, .... В этом пространстве можно взять пополнение,
+и тогда придется рассматривать последовательность функций f0, f1(k), ..., которые
+должны удовлетворять единственному условию: норма этой бесконечной последова-
+тельности конечна. Обычно эта последовательность записывается как столбик (фо-
+ковский столбик). Эта запись имеет смысл и в том случае, когда k — непрерывный
+параметр. В лекциях по квантовой механике фоковское пространство обычно опреде-
+ляется именно как пространство состоящее из столбиков симметрических функций.
+2.5
+Гамильтонианы сохраняющие число частиц
+В квантовой теории важную роль играет оператор
+ˆN =
+�
+ˆa∗
+kˆak =
+�
+dλˆa∗(λ)ˆa(λ),
+который представляет собой сумму по всем k (или интеграл, если есть непрерывный
+индекс). Этот оператор имеет физический смысл числа частиц или числа квантов,
+когда имеем дело с осцилляторами. Название не важно, а важно, что если есть соб-
+ственный вектор X оператора числа частиц с каким-то собственным значением N,
+то операторы a∗
+k, действуя на X, увеличивают число частиц на единицу, а ak, на-
+оборот, уменьшают на единицу (рождают частицу или уничтожают частицу). Это
+значит, что оператор, сохраняющий число частиц, должен содержать одинаковое ко-
+личество операторов рождения и уничтожения. Будем рассматривать квадратичные
+гамильтонианы обладающие этим свойством.
+Квадратичные гамильтонианы сохраняющие число частиц очень важны потому,
+что вблизи состояния с минимальной энергией (основного состояния) они играют
+основную роль. Возьмем квадратичный гамильтониан, который сохраняет число ча-
+стиц:
+ˆH =
+�
+dxdyA(x, y)ˆa∗(x)ˆa(y) = ⟨a∗, Aa⟩.
+В нем есть произведения типа a∗a, а произведений типаa∗a∗, aa, — нет. Если у опе-
+ратора A есть только дискретный спектр, то можно записать гамильтониан в виде:
+ˆH =
+�
+ǫkˆa∗
+kˆak,
+где ˆak =
+�
+dxφk(x)ˆa(x) отвечает собственной функции φk(x) оператора A.
+Давайте теперь возьмем в качестве x и y векторы в евклидовом пространстве и
+будем считать что A записан в виде: A = − 1
+2m∆ + ˆU, который хорошо известен из
+обычной квантовой механики. Это тот оператор, который получается из квантования
+классического гамильтониана вида p2
+2m +U(x). В таком случае гамильтониан ˆH описы-
+вает систему невзаимодействующих нерелятивистских тождественных бозонов. Если
+добавить сохраняющие число частиц неквадратичные члены, получится гамильтони-
+ан системы взаимодействующих нерелятивистских тождественных бозонов.
+Хочу обратить внимание что содержание этой лекции — это, в каком-то смыс-
+ле, объяснение того, как математик мог бы угадать квантовую механику, но не уга-
+дал. Угадали, конечно, физики. Смотрите, какова была логика. Математик знает,
+что наблюдаемые в классической теории — это просто алгебра функций на фазовом
+21
+
+пространстве. Он знает, что квантовая механика — это несколько продеформирован-
+ная классическая механика: все подчиняется классическим законам, но в некоторых
+ситуациях должны быть поправки. Исходя из этого он говорит: давайте я эту ком-
+мутативную алгебру продеформирую: ассоциативность оставлю, но введу некомму-
+тативность. Простейшая деформация, которая есть — это алгебра Вейля. Математик
+угадал, что нужно жить с алгеброй Вейля, а после этого он понимает, что в алгебре
+Вейля нужно рассматривать простейший гамильтониан, который описывает жизнь
+около основного состояния. Это — квадратичный гамильтониан. Уровни энергии это-
+го гамильтониана даются формулой:
+�
+ǫkˆnk,
+где ˆnk = 0, 1, 2, ... — числа заполнения. Эта формула описывает уровни энергии си-
+стемы невзаимодействующих тождественных частиц. То есть, в появлении тожде-
+ственных частиц никакой тайны нет. С точки зрения математика, именно они долж-
+ны возникнуть. Наоборот, нетождественных частиц может не быть, а тождественные
+всегда есть, потому что простейший квадратичный гамильтониан уже описывает тож-
+дественные частицы.
+В нерелятивистском случае одночастичные уровни энергии ǫk должны получаться
+при квантовании классического одночастичного гамильтониана
+p2
+2m + U(x). Добавив
+неквадратичные члены, мы приходим к гамильтониану системы нерелятивиских тож-
+дественных бозонов.
+2.6
+Представления алгебры Вейля
+Теперь перейдем к вопросу, как на самом деле устроены другие представления алгеб-
+ры Вейля и есть ли они вообще? Прежде всего отметим, что другие представления,
+несомненно, есть: можно взять прямую сумму двух представлений, и это будет новое
+представление. Так как фоковское представление, как нетрудно видеть, неприводимо,
+(внутри него нет никакого другого представления), то основной вопрос заключает-
+ся в том, какие есть неприводимые представления? Ответ на этот вопрос такой: в
+случае когда рассматриваются представления алгебры Вейля с конечным числом ге-
+нераторов (случай коечного числа степеней свободы), то единственное (с точностью
+до эквивалентности) неприводимое представление — это фоковское. Его можно запи-
+сывать в разных видах. Есть, например, представление, в котором оператор импульса
+реализуется как дифференцирование, а оператор координаты — как умножение, но
+оно эквивалентно фоковскому. Там тоже можно ввести операторы ˆa∗, ˆa.
+В общем случае ответ на этот вопрос трудно дать, потому что вопрос плохо сфор-
+мулирован. Дело в том, что в гильбертовом пространстве операторы ˆa∗
+k, ˆak имеют
+какую-то область определения. Она — не все гильбертово пространство. Можно об-
+ласть определения изменить немножко, а оператор оставить тем же самым и спросить
+себя: это то же самое представление или другое? С формально математической точки
+зрения — другое. На самом деле, конечно, то же самое. При работе с неограниченны-
+ми операторами эта проблема всегда возникает. Ее можно решить, но лучше иметь
+дело с ограниченными операторами. Это можно сделать.
+Можно рассмотреть экспоненты:
+Vα = eiαk ˆuk,
+в показателе которых стоит линейная комбинация самосопряженных операторов ˆuk,
+удовлетворяющих коммутационным соотношениям:
+ˆukˆul − ˆulˆuk = iσk,l.
+22
+
+Коэффициенты в экспоненте Vα выбраны тдействительными, тогда в показателе сто-
+ит оператор вида мнимая единица, умноженная на самосопряженный оператор. Тогда
+то, что получилось — это оператор унитарный, а унитарный оператор ограничен. С
+такими операторами удобно жить. Имеет место соотношение:
+VαVβ = e−i 1
+2 ασβVα+β.
+(6)
+В этом легко убедиться, если использовать формулу:
+eXeY = eX+Y e
+1
+2C,
+которая верна, если коммутатор операторов X и Y представляет собой число, которое
+здесь обозначено буквой C:
+[X, Y ] = C.
+В рассматриваемом нами случае ровно такая ситуация, и поэтому математики,
+если они не хотят иметь дело с неограниченными операторами, работают с этими
+унитарными операторами и с экспоненциальной формой коммутационных соотноше-
+ний (6). Это — экспоненциальная форма алгебры Вейля. С точки зрения физика это
+— та же самая алгебра, с которой мы имели дело, с точки зрения математика, это не
+совсем так.
+Существует единственное неприводимое представление алгебры Вейля в случае
+конечного числа степеней свободы. Я дам доказательство этого факта, не пользуясь
+экспоненциальной формой алгебры Вейля. Оно не вполне строгое, но, с точки зрения
+физика, оно вполне приемлемо.
+Рассуждение такое: рассмотрим уже упомянутый оператор числа частиц ˆN =
+� ˆa∗
+kˆak. В случае конечного числа генераторов это хороший оператор. Возьмем соб-
+ственный вектор этого оператора и начнем применять к нему операторы уничтоже-
+ния. (Математик спросит: откуда вы знаете, что есть такой собственный вектор, но
+физик, наверное, это делать не будет). Если все операторы уничтожения дают ноль,
+я скажу, что это как раз тот самый фоковский вакуум, который нам нужен. Если
+есть такой оператор, который не дает ноль, тогда я применю оператор уничтожения
+еще раз и еще раз и еще раз и буду делать это до тех пор, пока у меня не получится
+такой вектор |0⟩, который обнуляется всеми операторами уничтожения: ak|0⟩ = 0.
+Такой вектор обязательно есть, потому что оператор числа частиц положительно
+определен, а операторы уничтожения уменьшают число частиц. После этого я возьму
+то подпредставление, в котором он лежит. Я применю к вектору |0⟩ все операторы
+ˆa∗
+k много раз и, когда я возьму линейные комбинации получившихся выражений, у
+меня получится пространство, которое инвариантно относительно всех операторов
+рождения и уничтожения. Это — подпредставление моего представления и оно будет,
+несомненно, фоковским представлением потому, что там есть вектор, который уни-
+чтожается всеми ˆak и этот вектор циклический (я все могу получить из него). Отсюда
+все получается.
+В случае бесконечного числа генераторов эти рассуждения неприменимы. Я сей-
+час объясню, как построить пример представления, которое не является фоковским,
+но, тем не менее является хорошим представлением. Я возьму фоковское простран-
+ство и возьму новые операторы, которые я буду обозначать буквой ˆAk. Это старые
+операторы минус число:
+ˆAk = ˆak − fk,
+ˆA∗
+k = ˆa∗
+k − ¯fk
+(для каждого ˆak я вычитаю свое число). Рассматривая коммутационные соотношения,
+которые мне нужны для алгебры Вейля, я сразу увижу, что все они удовлетворяют-
+ся. Таким образом, я снова получил представление канонических коммутационных
+соотношений, представление алгебры Вейля.
+23
+
+Теперь я попытаюсь решить уравнение ˆAkΘ = 0. Его решения — это собственные
+функции операторов ˆak. Мы с этими векторами встретимся много раз. Они называ-
+ются иногда пуассоновскими векторами. Легко понять, что ответ будет представлен
+экспонентой
+Θ = efˆa∗|0⟩
+где fˆa∗ = � fkˆa∗
+k. Если рассматривать представление векторов из фоковского про-
+странства с помощью функций, легко увидеть, что это действительно собственный
+вектор всех операторов ˆak.
+Легко вычислить норму пуассоновского вектора и скалярное произведение двух
+пуассоновских векторов.
+Скалярное произведение задается интегралом; этот интеграл гауссов и легко бе-
+рется. Норма Θ будет конечной, если сумма � |fk|2 конечна. Если норма конечна, то-
+гда вектор Θ принадлежит фоковскому пространству и представление эквивалентно
+фоковскому представлению, но если норма бесконечна, то это не так и в представле-
+нии нет вектора, который уничтожается операторами ˆak. Оно не является фоковским
+раз нет такого вектора и следовательно это представление — пример того, что есть
+неприводимые представления, которые неэквивалентны фоковскому представлению.
+В таком представлении нет даже подпредставления эквивалентного фоковскому. В
+случае конечного числа степеней свободы � |fk|2 всегда конечна, и там эти рассуж-
+дения ничего не дают.
+Теперь я хочу обобщить эту конструкцию. Я рассмотрю оператор ˆAk, определен-
+ный несколько по-другому. Раньше я просто добавлял fk, а тут я еще возьму линейные
+преобразования:
+ˆAk = Φl
+kˆal + Ψl
+kˆa∗
+l + fk
+ˆA∗
+k = ¯Ψl
+kˆal + ¯Φl
+kˆa∗
+l + ¯fk
+и потребую, чтобы новые операторы ˆAk, ˆA∗
+k тоже удовлетворяли каноническим ком-
+мутационным соотношением. Это накладывает некоторые условия на коэффициенты.
+Такое преобразование называется линейным каноническим преобразованием. Опера-
+торы ˆAk, ˆA∗
+k определяют новое представление алгебры и, опять же, если вектор Θ,
+являющийся решением уравнения ˆAkΘ = 0, принадлежит фоковскому пространству,
+то тогда линейное каноническое преобразование эквивалентно фоковскому представ-
+лению. Такое каноническое преобразование называется собственным. Если же эти
+условия не выполнены, то это преобразование не эквивалентно фоковскому представ-
+лению. Линейные канонические преобразования часто бывают полезны. Их аналоги
+в фермионном случае называются боголюбовскими преобразованиями. Они важны в
+теории сверхпроводимости.
+24
+
+3
+Лекция 3
+3.1
+Алгебра Клиффорда и алгебра Грассманна
+У вейлевской алгебры есть близкие родственники, которые определяются теми же
+самыми формулами, как и вейлевская алгебра, только вместо коммутаторов стоят
+антикоммутаторы:
+ukul + uluk = 2hkl,
+где hkl — обратимая матрица. Эта формула описывает алгебру Клиффорда.
+Есть еще другое понятие — алгебра Грассмана, которая получится, если hkl счи-
+тать равным нулю:
+ukul + uluk = 0.
+В клиффордовской алгебре требуется, чтобы справа была обратимая матрица. Грас-
+сманова алгебра не является частным случаем клиффордовской — здесь в правой
+части стоит ноль. Элементы грассмановой алгебры — это тоже полиномы, но только,
+как говорят, от антикоммутирующих переменных. Для клиффордовой алгебры верно
+все то же, что было сказано для алгебры Вейля почти буквально. Просто всюду нуж-
+но заменять коммутаторы на антикоммутаторы. Единственное отличие состоит в том,
+что в то время как все сказанное в прошлой лекции о невзаимодействующих бозонах,
+в алгебре Клиффорда будет относиться к невзаимодействующим фермионам.
+Начнем с грассмановой алгебры. Это ассоциативная алгебра с единицей и с анти-
+коммутирующими генераторами ǫ1, ..., ǫn:
+ǫiǫj = −ǫjǫi.
+(7)
+Она обозначается Λn.
+Каждый элемент грассмановой алгебры можно однозначным образом записать
+как сумму мономов по ǫi так, чтобы в каждом мономе индексы возрастали: ω = α +
+�
+i αiǫi+�
+i<j αijǫiǫj+...+�
+i1<...<ik αi1,...,ikǫi1...ǫik+...+α1,...,nǫ1...ǫn. Это очевидно: мы
+можем воспользоваться отношениями антикоммутации, чтобы переставить меньшие
+индексы налево, а двух совпадающих индексов не может быть. (Если положить i = j,
+получится, что квадрат равен нулю. Я напоминаю, что все алгебры рассматриваются
+над комплексными числами и на 2 можно делить.)
+Грассманова алгебра, так же, как и обычная алгебра полиномов, Z-градуированная.
+Это значит, что есть понятие степени: степень каждого монома — это число генерато-
+ров в этом мономе; ясно, что при умножении степени складываются, так что в этом
+смысле все так же как у обычных полиномов. Следующее замечание: можно получить
+другое однозначное представление элементов грассманновой алгебры не упорядочи-
+вая индексы, а наложив требование антисимметричности коэффициентов. Это точно
+так же как в обычных полиномах — генераторы можно переставлять и поэтому мож-
+но считать, что коэффициенты симметричны. Здесь же можно переставлять, но со
+знаком минус и поэтому можно считать, что коэффициенты антисимметричны.
+Я уже говорил, что есть понятие степени (Z-градуировки). Важнее всего то, что из
+этой Z -градуировки можно получить Z2-градуировку сказав, что существуют четная
+и нечетная части, четные и нечетные элементы. Есть то, что натянуто на четные
+мономы Λeven
+n
+= �
+k≥0 Λ2k
+n и то, что натянуто на нечетные мономы Λodd
+n
+= �
+k≥0 Λ2k+1
+n
+.
+Это разбиение на четную и нечетную части здесь более важно, чем степень, потому
+что именно оно заведует умножением.
+Четный элемент коммутирует с чем угодно, и это понятно, потому что при пе-
+рестановке двух генераторов возникает знак минус, но если это проделать дважды,
+25
+
+то минус на минус даст плюс и ничто не изменится. Два нечетных элемента между
+собой антикоммутируют.
+Можно сказать, что элемент грассмановой алгебры — это полином от антиком-
+мутирующих элементов ǫ1, ..., ǫn или, иными словами, это функция антикоммутиру-
+ющих переменных, но она автоматически полиномиальна, потому что есть только
+конечное число мономов (в случае, когда есть конечное число антикоммутирующих
+переменных). Аналогия с полиномами — это важная идея, потому что она подсказы-
+вает, что должен быть анализ в грассмановой алгебре, и, действительно, такой анализ
+есть. Можно определить дифференцирование по переменной ∂i = ∂/∂ǫi. Просто надо
+вычеркнуть эту переменную. Если переменной ǫi в мономе нет, то эта производная
+равна нулю. Вычеркивая переменную ǫi нужно учесть, что переменная может сто-
+ять где угодно. Для случая антикоммутирующих переменных есть понятия левой
+производной и правой производной. Будем для определенности рассматривать левую
+производную; это означает, что перед тем как вычеркивать следует соответствующий
+элемент переместить налево — поставить на первое место:
+∂i(ǫiǫi1...ǫin) = ǫi1...ǫin
+(8)
+∂i(ǫi1...ǫin) = 0.
+(9)
+Понятие производной связано с правилом Лейбница. Здесь тоже есть правило
+Лейбница, но только с той разницей, что появляется знак, который связан с переста-
+новками, о которых уже было сказано:
+∂i(ωρ) = (∂iω) · ρ + (−1)¯ω · ω · ∂iρ,
+(10)
+где ω, ρ ∈ Λn, а ω обладает определенной четностью ¯ω, то есть ω является либо
+четной (¯ω = 0) либо нечетной (¯ω = 1). Это называется градуированным правилом
+Лейбница. Если выполнено это правило, говорят про нечетное дифференцирование,
+если выполнено обычное правило Лейбница, говорят про четное дифференцирование.
+Есть и понятие интегрирования
+�
+: Λn → C. Интеграл любого монома не макси-
+мальной степени дает ноль, а интеграл монома максимальной степени — плюс или
+минус единицу:
+�
+ǫi1...ǫikdnǫ = 0
+if k < n,
+�
+ǫ1...ǫndnǫ = 1.
+Я выбрал обозначение, при котором плюс единица получается тогда, когда эти гене-
+раторы упорядочены в возрастающем порядке. Легко понять, что интеграл от произ-
+водной всегда равен нулю:
+�
+(∂iω)dnǫ = 0.
+(11)
+Это происходит потому, что в производной не может быть члена максимальной сте-
+пени — при дифференцировании уменьшается количество генераторов. Из этого и из
+правила Лейбница можно вывести правило интегрирования по частям:
+�
+(∂iω) · ρdnǫ = −(−1)¯ω
+�
+ω · ∂iρdnǫ.
+Хочу заметить, что, рассматривая грассманову алгебру Λn, мы фиксировали си-
+стему образующих ǫ1, ..., ǫn. Естественным образом, можно перейти к другим образу-
+ющим (это — аналог замены переменных) и тогда изменится все — изменится понятие
+дифференцирования, изменится понятие интегрирования. Для таких преобразований
+26
+
+существует аналог теоремы о дифференцировании сложной функции, аналог якоби-
+ана, о которых я много говорить не буду.
+Я хочу рассмотреть только частный случай, когда замена переменных линейна:
+˜ǫi =
+�
+Ai
+jǫj.
+(12)
+Легко понять, что в интеграле появится детерминант при замене переменных точно
+так же, как при обычной замене переменных с той только разницей, что там, где
+стоял детерминант, здесь стоит детерминант в минус первой степени.
+Следующее замечание состоит в том, что в гладкую функцию f, которая зависит
+от действительной переменной x ∈ R, можно подставить четный элемент грассма-
+новой алгебры ω, и выражению f(ω) можно придать смысл. В грассмановой алгебре
+всякий элемент может быть представлен в виде суммы ω = a+ν, где a — число, а ν —
+нильпотентный элемент (т.е. νk = 0 для некоторого k). Возьмем разложение функции
+f(a + ν) в ряд Тейлора по нильпотентной части:
+f(ω) = f(a) + f ′(a)
+1!
+ν + ... + f l(a)
+l!
+νl + ...
+(13)
+и заметим, что из-за нильпотентности этот ряд Тейлора имеет конечное число членов
+— остается только полином, и это имеет четкий смысл.
+В частности, можно рассмотреть экспоненту. Нужно заметить, что для экспоненты
+имеются обычные правила. (Четные элементы между собой коммутируют, и поэтому
+все формальные рассуждения можно проводить для экспоненты четного элемента
+грассмановой алгебры так же, как и для числа.) В качестве примера рассмотрим
+экспоненту квадратичного выражения:
+eλ1ǫ1ǫ2+λ2ǫ3ǫ4+...+λkǫ2k−1ǫ2k = eλ1ǫ1ǫ2...eλkǫ2k−1ǫ2k =
+(1 + λ1ǫ1ǫ2)...(1 + λkǫ2k−1ǫ2k),
+отсюда следует, что
+�
+eλ1ǫ1ǫ2+...+λkǫ2k−1ǫ2kd2kǫ = λ1...λk.
+(14)
+В более общем случае, когда ω = 1
+2
+� aijǫiǫj, можно привести антисимметричную
+невырожденную матрицу a к блочно-диагональному виду путем замены переменных.
+Используя правило замены переменных, можно вычислить гауссов интеграл:
+�
+eωdnǫ = (det a)1/2.
+(15)
+Ответ такой же, как и в обычном случае, но только там, где был детерминант
+в степени −1/2 здесь будет детерминант в степени +1/2. Точнее говоря, совпаде-
+ние имеет место с точностью до множителя, который может быть убран с помощью
+соответствующей нормировки меры.
+Следующее замечание заключается в том, что можно рассматривать функции од-
+новременно от коммутирующих и от антикоммутирующих переменных. Это просто
+означает, что в разложениях по антикоммутирующим переменным коэффициенты
+можно считать функциями от коммутирующих переменных. Можно рассматривать
+либо полиномиальные функции, либо гладкие функции. Мы полугаем функции на
+суперпространстве. Это просто слова — вы говорите, что суперпространство это та-
+кое пространство, на которой заданы функции (эти функции я определил), а точек
+у суперпространства в таком формальном определении нет. На самом деле, мож-
+но определить понятие точки суперпространства — это важно и удобно. Об этом я
+несколько слов скажу, но не буду здесь давать никаких точных определений.
+27
+
+Есть понятие супермногообразия. Мы можем рассматривать полиномиальные или
+гладкие функции коммутирующих переменных x1, ..., xm и антикоммутирующих пе-
+ременных ǫ1, .., ǫn — элементов алгебры C[x1, ..., xm] ⊗ Λn или алгебры C∞(Rm) ⊗ Λn.
+Кроме того, могут быть некоторые уравнения: на коммутирующие и антикоммутиру-
+ющие переменные можно наложить условия. Важно чтобы в условии все слагаемые
+были одинаковой четности. Например, может быть такое условие: x1 + ǫ1ǫ2 = 0. По-
+лучаем алгебраические супермногообразия. Обычные алгебраические многообразия
+задаются полиномиальными уравнениями в обычном линейном пространстве. Здесь
+есть коммутирующие переменные и полиномиальные уравнения.
+Представим себе, что коэффициенты этих уравнений являются целыми числами.
+Если есть такие уравнения, то можно считать переменные xi какими угодно: действи-
+тельными, комплексными, элементами какого-то кольца, элементами поля вычетов по
+модулю n. Это все имеет смысл.
+Можно рассматривать алгебраическое многообразие над разными полями или над
+разными кольцами, то есть, можно говорить, что есть точки алгебраического много-
+образия над каким-то полем, каким-то кольцом. Например, если рассмотреть, ска-
+жем, окружность мнимого радиуса x2 + y2 + 1 = 0, то действительных точек у такого
+объекта нет, а над комплексными числами точки есть. Когда рассматриваются су-
+пермногообразия, возникает такая же ситуация. Можно рассматривать переменные
+в уравнении принадлежащими некоторой грассмановой алгебре (любой). И тогда по-
+явится понятие Λ-точки супермногообразия, где Λ — любая грассманнова алгебра.
+Нужно только позаботиться о том, чтобы вместо четной переменной подставлялся
+четный элемент грассманновой алгебры, а вместо нечетной переменной нечетный.
+Иными словами, нужно чтобы четность сохранялась при введении Λ-точек.
+Понятие Λ-точки очень удобно. Оно, например, позволяет дать очень простое
+определение понятия супералгебры Ли. Супералгебра Ли — это Z2-градуированная
+алгебра с некоторым обобщением правила Якоби. Запоминать это обобщение не на-
+до. Просто нужно сказать, что это обобщение должно быть таким, чтобы Λ-точки
+супералгебры Ли образовывали обыкновенные алгебры Ли. Подробно я об этом го-
+ворить не буду. То, что я рассказывал — это маленький кусочек того, что называется
+суперматематикой.
+3.2
+Представления алгебры Клиффорда
+Следующие рассуждения, как я уже обещал, будут в точности повторять рассуж-
+дения предыдущей лекции, только вместо “Вейль” я буду говорить “Клиффорд” и
+вместо слова “коммутатор” я буду говорить “антикоммутатор”.
+Согласно определению алгебры Клиффорда, антикоммутатор генераторов явля-
+ется симметричной невырожденной матрицей hkl = hlk:
+ˆukˆul + ˆulˆuk = 2hkl.
+Эти же соотношения встречаются при рассмотрении уравнений Дирака: гамма-матрицы
+являются генераторами алгебры Клиффорда.
+Будем рассматривать алгебру Клиффорда, в которой есть инволюция, предпола-
+гая при этом, что генераторы ˆak, ˆa∗
+k получаются друг из друга с помощью инволюции
+и удовлетворяют равенствам:
+ˆakˆal + ˆalˆak = 0, ˆa∗
+kˆa∗
+l + ˆa∗
+l ˆa∗
+k = 0, ˆakˆa∗
+l + ˆa∗
+l ˆak = δkl,
+которые называются каноническими антикоммутационными соотношениями. Они по-
+лучаются из канонических коммутационных соотношений заменой коммутаторов на
+28
+
+антикоммутаторы. Отмечу, что, представленные здесь антикоммутационные соотно-
+шения — это в точности антикоммутационные соотношения, которые получаются,
+если в алгебре Грассмана взять дифференцирования и умножения на генераторы.
+Можно рассматривать алгебры Клиффорда с бесконечным числом генераторов
+или генераторы ˆa(k), ˆa(k)∗, зависящие от непрерывного параметра и удовлетворяю-
+щие соотношениям:
+ˆa(k)ˆa(l) = −ˆa(l)ˆa(k),
+ˆa∗(k)ˆa∗(l) = −ˆa∗(l)ˆa∗(k),
+ˆa(k)ˆa∗(l) + ˆa∗(l)ˆa(k) = δ(k, l).
+— все то же самое, что и для алгебры Вейля, не стоит повторяться.
+В алгебре Клиффорда, определяемой с помощью генераторов ˆak, ˆa∗
+k, есть понятие
+нормальной формы. Точно так же как и в случае алгебры Вейля генераторы ˆa∗
+k уво-
+дятся налево, ˆak уводятся направо, но есть небольшая особенность. Раньше, получая
+нормальную форму, я мог определить виковский символ, просто снимая крышечки
+и рассматривая ak, a∗
+k как комплексные переменные. Здесь этого нельзя сделать —
+получится ноль, но я могу сделать то же самое, говоря при этом, что, снимая кры-
+шечки, я получаю антикоммутирующие переменные. Иными словами, из элементов
+алгебры Клиффорда я могу получить полином от антикоммутирующих переменных,
+произнося все те же слова, только вставляя при этом приставку “анти”.
+Рассмотрим формальный гамильтониан
+ˆH =
+�
+Γm,n(k1, ..., km, l1, ..., ln)ˆa∗
+k1...ˆa∗
+kmˆal1...ˆaln.
+Гамильтониан всегда нужно считать четным элементом. Когда количество перемен-
+ных бесконечно, гамильтониан обычно не является элементом алгебры Клиффорда,
+но коммутаторы, тем не менее имеют смысл, если наложить те же условия, что и для
+алгебры Вейля.
+Определение фоковского представления алгебры Клиффорда в точности повто-
+ряет данное для алгебры Вейля: требуется, чтобы существовал циклический вектор
+|0⟩, для которого выполнено условие ˆak|0⟩ = 0. В силу условия цикличности можно
+получить все что угодно, применяя к вектору |0⟩ операторы ˆa∗
+k. Все состояния будут
+линейными комбинациями мономов вида Π(ˆa∗
+k)nk|0⟩.
+Единственная разница заключается в том, что показатели nk в этих мономах могут
+быть только ноль или единица (nk = 0, 1) потому что ˆa∗2
+k = 0. Это — то, что назы-
+вается принципом Паули. Опять же, элементы этого базиса являются собственными
+векторами любого гамильтониана вида � ǫkˆa∗
+kˆak и собственные значения даются точ-
+но той же формулой, как и в бозонном случае: � nkǫk.
+Показатели nk называются числами заполнения; операторы ˆa∗
+k является опера-
+торами рождения, а ˆak — операторами уничтожения по тем же причинам, что и в
+бозонном случае. Можно сказать, что гамильтониан � ǫkˆa∗
+kˆak описывает невзаимо-
+действующие фермионы.
+Теперь вспомним, что в бозонном случае для получения представления элемен-
+тов фоковского пространства полиномами я в мономах снимал крышечки и получал
+полином от комплексных переменных. Сейчас я хочу сделать то же самое: снять кры-
+шечки и получить полином от антикоммутирующих переменных. В этом случае все
+будет абсолютно то же. Форма скалярного произведения абсолютно такая же, только
+интегрирование будет вестись по антикоммутирующим переменным. Как и в случае
+алгебры Вейля, легко проверить, что в этом представлении оператор ˆa∗
+k действует
+как умножение, ˆak действует как дифференцирование, и это дает правильные анти-
+коммутационные соотношения.
+29
+
+Единственное, что осталось проверить, это то, что в скалярном произведении
+⟨F, G⟩ =
+�
+da∗daF(a∗)G(a∗)∗e−a∗a
+умножения и дифференцирования являются сопряженными операторами. Это можно
+сделать, применяя интегрирование по частям, с той лишь разницей, что при вычис-
+лениях следует применять градуированное правило Лейбница.
+Если рассматривать одни полиномы мы получим предгильбертово пространство,
+но можно взять пополнение. В случае конечного числа степеней свободы брать по-
+полнение не надо — там только полиномы и есть, а в случае бесконечного числа
+степеней свободы, если мы хотим работать в гильбертовом пространстве, фоковское
+пространство нужно пополнить.
+В фоковском пространстве векторы можно представить в виде суммы мономов с
+антисимметрическими коэффициентами:
+�
+n
+�
+k1,...,kn
+fn(k1, ..., kn)ˆa∗
+k1...ˆa∗
+kn|0⟩
+в то время как для алгебры Вейля коэффициенты симметрические. В пополненном
+фоковском пространстве эти суммы бесконечны. Другими словами, точка фоковско-
+го пространства представляется последовательностью антисимметрических функций
+тогда как в алгебре Вейля она была последовательностью симметрических функций.
+Это стандартное представление из учебников квантовой механики. Такое представ-
+ление работает также в случае, когда k является непрерывным параметром.
+Можно, опять же, рассмотреть оператор числа частиц:
+ˆN =
+�
+ˆa∗
+kˆak =
+�
+dλˆa∗(λ)ˆa(λ)
+и снова ˆa∗
+k увеличивает число частиц на единицу, ˆak — уменьшает.
+Перейдем теперь к рассмотрению операторов, которые сохраняют число частиц,
+как и в нерелятивистской квантовой механике, где гамильтониан сохраняет число
+частиц. Самый простой гамильтониан — квадратичный гамильтониан.
+ˆH =
+�
+dxdyA(x, y)ˆa∗(x)ˆa(y) = ⟨a∗, Aa⟩.
+Если оператор A имеет дискретный спектр, то, диагонализуя его, мы получим опе-
+ратор
+ˆH =
+�
+ǫkˆa∗
+kˆak,
+где ǫk, φk(x) — собственные значения и собственные функции оператора A:
+ˆak =
+�
+dxφk(x)ˆa(x).
+В прошлой лекции было сказано, что, взяв A = − 1
+2m∆ + ˆU, мы получаем си-
+стему невзаимодействующих нерелятивистских бозонов. Имея тот же оператор, но
+используя канонические антикоммутационные соотношения, мы получаем систему
+невзаимодействующих нерелятивистских фермионов. Единственная разница заклю-
+чается в том, что нужно считать, что этот оператор действует на многокомпонентные
+функции от x. Он имеет точно такой же вид, но есть еще и дискретный индекс.
+Какие канонические соотношения нужно брать — зависит от того, как действу-
+ет на волновые функции ортогональная группа. Действие ортогональной группы на
+дискретные индексы определяет спин частицы. В случае полуцелого спина нужно
+30
+
+квантовать с помощью алгебры Клиффорда, а в случае целого спина — с помощью
+алгебры Вейля. По-другому можно сказать, что если представление неприводимое, то
+все определяется числом индексов. Если четное число, то мы имеем дело с алгеброй
+Вейля, если нечетное — с алгеброй Клиффорда.
+Для того чтобы описывать взаимодействующие частицы в нерелятивистской кван-
+товой механике, нужно добавить члены более высокого порядка с равным количе-
+ством операторов рождения и уничтожения; тогда эти члены сохраняют число ча-
+стиц.
+В случае конечного числа степеней свободы существует единственное неприводи-
+мое представление алгебры Клиффорда, и оно изоморфно фоковскому представле-
+нию. Доказательство неприводимости фоковского представления и его единственно-
+сти такое же, как и в случае алгебры Вейля с той лишь разницей, что представленное
+доказательство будет строгим. В предыдущем случае оно было нестрогое, потому что
+операторы ˆak и ˆa∗
+k в случае алгебры Вейля являются неограниченными операторами,
+а здесь все эти операторы ограничены. Это легко понять, потому что выполняется
+соотношение ˆakˆa∗
+k + ˆa∗
+kˆak = 1, в котором оба слагаемые положительно определены, их
+сумма равна единице и, следовательно, каждый из операторов ограничен.
+Если мы имеем бесконечное число степеней свободы, то число операторов ˆak и
+ˆa∗
+k бесконечно. Можно рассмотреть каноническое преобразование, введя новые гене-
+раторы, которые по-прежнему удовлетворяют антикоммутационным соотношениям.
+Здесь очень легко построить пример канонического преобразования. Дело в том, что
+условия антикоммутативности симметричны относительно ˆak и ˆa∗
+k. Если поменять
+местами операторы рождения и уничтожения, то в коммутаторе изменится знак, а
+в антикоммутаторе ничего не изменится и, поэтому, если просто поменять местами
+операторы рождения и уничтожения, появится снова представление алгебры Клиф-
+форда, и нужно задать вопрос: будет ли это представление эквивалентно исходному?
+Ответ: не всегда.
+Дальнейшие рассуждения точно такие же, как и в бозонном случае. Возьмем в
+пространстве Фока операторы, определенные формулой ˆAk = ˆak для k ∈ I, ˆAk = ˆa∗
+k
+для k /∈ I. Эти операторы подчиняются каноническим антикоммутационным соот-
+ношениям, следовательно, они задают представление алгебры Клиффорда. Для того
+чтобы это представление было фоковским, у него должен быть циклический вектор
+Θ который обнуляется всеми новыми операторами уничтожения: ˆAkΘ = 0.
+Очень легко найти решение этого уравнения. Нужно подействовать на |0⟩ теми
+операторами ˆa∗
+k, которые изменилиcь (k /∈ I). Напомню, изменившиеся операторы
+ˆa∗
+k стали операторами уничтожения. Теперь если взять моном с операторами ˆa∗
+k, то
+действуя на этот моном любым оператором ˆa∗
+k из этого множества, получим ноль
+потому что операторы ˆa∗
+k будут появляться дважды с одним и тем же индексом. Тех
+же операторов ˆak, которые не менялись в этом произведении нет — они тоже дают
+ноль, потому что их можно перенести на |0⟩. Таким образом, у нас получился моном
+Θ, который удовлетворяет соотношению ˆAkΘ = 0.
+Дальше все зависит от того, сколько операторов было изменено. Если их чис-
+ло конечно — все в порядке, будет получен конечный моном. Если же изменилось
+бесконечное число операторов, будет получено что-то, что фоковскому пространству
+совершенно не принадлежит — моном бесконечной степени. Новое представление не
+будет эквивалентно фоковскому, так как у него нет циклического вектора, который
+нам нужен. Таким образом, мы получили пример неэквивалентных представлений.
+Это рассуждение восходит к Дираку. Дираковское море подразумевает заполнение
+бесконечного количества состояний, в результате чего получается неэквивалентное
+представление антикоммутационных соотношений.
+31
+
+Перейдем к рассмотрению линейных канонических преобразований вида:
+ˆAk = Φl
+kˆal + Ψl
+kˆa∗
+l ,
+ˆA∗
+k = ¯Ψl
+kˆal + ¯Φl
+kˆa∗
+l .
+В отличие от вейлевского случая, здесь нельзя добавлять численные слагаемые, но
+все остальное то же самое. Если потребовать, чтобы новые операторы удовлетворяли
+каноническим антикоммутационным соотношениям, мы получим новое представле-
+ние алгебры Клиффорда. Это — то что называется боголюбовским преобразованием.
+Вот, более или менее все, что я хотел рассказать про алгебру Клиффорда. В заклю-
+чение я еще хотел добавить, что алгебра Клиффорда тесно связана с представлениями
+ортогональной группы. Из нее мгновенно получается то, что называется спинорным
+представлением ортогональной группы.
+3.3
+Статистическая физика
+Переходя к новой теме, напомню некоторые положения из статистической физики.
+Как в классической, так и в квантовой механике есть понятие равновесного состо-
+яния и в обоих случаях оно определяется как состояние с максимальной энтропией
+при заданных условиях. Выражение “заданные условия” может иметь разные смыслы
+в разных ситуациях, но условие максимума энтропиии при заданных условиях долж-
+но быть выполнено. Если состояние представлено матрицей плотности, то энтропия
+состояния записывается формулой
+S = −TrK log K.
+Если матрица плотности диагональна, то диагональные элементы матрицы pi ин-
+терпретируются как вероятности, и эта формула дает обычное классическое выраже-
+ние для энтропии:
+S = −
+�
+pi log pi.
+Если задан гамильтониан ˆH, то мы можем зафиксировать среднюю энергию. Мож-
+но зафиксировать допустимый интервал энергии — тогда это будет микроканониче-
+ское распределение, но давайте зафиксируем среднюю энергию E = Tr ˆHK (тогда
+мы получим каноническое распределение). Максимизируя энтропию при заданной
+средней энергии, увидим, что матрица плотности, на которой достигается максимум
+энтропии, дается формулой:
+e−β ˆ
+H
+Z
+.
+(16)
+Константу Z мы можем вычислить, потому что матрица плотности по определению
+должна иметь след равный единице и поэтому знаменатель должен быть равен следу
+оператора e−β ˆ
+H:
+Z = Tre−β ˆ
+H.
+Это выражение носит название статистической суммы.
+Если спектр ˆH дискретен с уровнями Ei, то
+Z =
+�
+e−βEi.
+Физический смысл β — это обратная температура: β =
+1
+T . Из выражения для ста-
+тистической суммы видно, что при β → ∞ или, что то же самое, T → 0, вклад в
+это выражение вносит только член с минимальной энергией, то есть, член отвечаю-
+щий основному состоянию, которое мы считаем невырожденным. Таким образом, в
+32
+
+случае нулевой температуры получается чистое состояние — единственное основное
+состояние.
+Вывод выписанной формулы (16) для состояния с максимальной энтропией ис-
+пользует метод Лагранжа для случая когда зафиксирована средняя энергия Tr ˆHK =
+E, а кроме того, как мы знаем, что след матрицы плотности равен единице TrK = 1.
+Все это нужно включить в число условий. Можно стандартным способом посчитать
+вариацию выражения
+L = −TrK log K − βTr( ˆHK − E) − ζ(TrK − 1)
+и найти стационарные точки:
+− log K − 1 − β ˆH − ζ = 0.
+Единственная тонкость заключается в том, что нужно использовать формулу:
+δTrφ(K) = Trφ′(K)δK.
+(17)
+Достаточно проверить эту формулу для случая φ(K) = Kn. Когда рассматривается
+вариация функции Kn, то в силу некоммутативности будет n членов, но когда вы
+берете след, то все эти члены становятся одинаковыми и получается ответ, соответ-
+ствующий (17).
+Следующее замечание заключаются в том, что хотя статистическая сумма Z сама
+по себе не имеет физического смысла, но через нее более или менее все выражается.
+Тесно связаны с логарифмом статсуммы выражения для средней энергии
+E = ¯H = − 1
+β
+∂ log Z
+∂β
+и энтропии
+S = βE + log Z.
+Есть еще важная характеристика, которая называется свободной энергией. Она
+определяется формулой F = E − TS и выражается через статсумму так:
+F = −T log Z.
+Свободная энергия удобна тем, что вместо нахождения максимума энтропии можно
+искать минимум свободной энергии. Это сразу следует из метода множителей Лагран-
+жа.
+Как правило физические величины можно получать так: берем какой-то гамиль-
+тониан, к нему что-то прибавляем и смотрим что получится в пределе, когда то, что
+было прибавлено, стремится к нулю. При этом следует заметить, что когда гамиль-
+тониан ˆH чуточку меняется:
+ˆH(λ) = ˆH + λA + . . . ,
+то изменение статистической суммы
+Z(λ) = Z + (−β)λTrAe−β ˆ
+H + . . . .
+управляется средним значением этой добавки.
+Оператор A — это тоже физическая величина, и она заведует изменением стат-
+суммы. Изменение статсуммы выражается в терминах среднего значения (математи-
+ческого ожидания) этой физической величины:
+¯A = TrAe−β ˆH
+Z
+.
+33
+
+(Для этого выражения я буду использовать еще альтернативное обозначение ⟨A⟩).
+Перейдем теперь к основной нашей цели – к тому, что называется корреляцион-
+ной функцией. Корреляционная функция — это просто среднее значение произведе-
+ния некоторых физических величин ⟨A1 · · · An⟩. Можно считать также, что эти фи-
+зические величины зависят от времени — являются гейзенберговскими операторами
+и удовлетворяют гейзенберговским уравнениям. Тогда выражение ⟨A1(t1) · · · An(tn)⟩
+тоже называется корреляционной функцией. Можно рассматривать корреляционные
+функции по любому состоянию — не обязательно состоянию равновесия, но если речь
+идет о состоянии равновесия, я пишу в качестве индекса значение обратной темпера-
+туры: ⟨A1(t1) . . . An(tn)⟩β.
+Здесь я хочу заметить, что все сказанное про статсумму и связанные с ней вещи
+применимо очень часто в случае конечного числа степеней свободы, а в случае бес-
+конечного числа степеней свободы применимо достаточно редко. В нерелятивистской
+квантовой механике применимо, когда мы находимся в конечном объеме. В беско-
+нечном объеме это не работает даже в нерелятивистской квантовой механике. Очень
+важно понимать, что на самом деле даже в нерелятивистской квантовой механике, но
+в бесконечном объеме следует рассматривать статсумму сначала в конечном объеме, а
+после этого — перейти к переделу бесконечного объема в корреляционных функциях.
+Нельзя работать сразу в бесконечном объеме.
+Еще я должен заметить, что при переходе к бесконечному объему обычно берется
+предел, при котором сохраняется не число частиц, а плотность числа частиц. Иными
+словами, при переходе к пределу число частиц меняется пропорционально объему, и
+тогда плотность числа частиц остается постоянной.
+Предположим теперь, что набор корреляционных функций в бесконечном объеме
+получен — что с ним делать? Нет никакого гильбертова пространства, в котором опре-
+делены эти операторы A в бесконечном объеме, но есть корреляционные функции.
+Обычно в таком случае по этим корреляционным функциям строится гильбертово
+пространство, применяя некий аналог конструкции GNS. Физики это обычно делают
+неявно. Они просто говорят “теперь у нас есть равновесное состояние — его можно
+представить вектором в гильбертовом пространстве или матрицей плотности в гиль-
+бертовом пространстве и там действуют эти операторы A”. На самом деле, нужно
+применять конструкцию, которая в аксиоматической квантовой теории поля называ-
+ется “reconstruction theorem”. Там роль корреляционных функций играют функции
+Вайтмана.
+Перейду теперь к вопросу: как можно поступать в алгебраическом подходе с рав-
+новесными состояниями в ситуации, когда невозможно использовать принцип макси-
+мума энтропии? Здесь можно применить то, что называется условием Кубо — Мар-
+тина — Швингера (KMS):
+⟨A(t)B⟩β = ⟨BA(t + iβ)⟩β.
+(18)
+Это условие на корреляционные функции наблюдаемых A и B в равновесном состо-
+янии. Его легко вывести в случае конечномерного гильбертова пространства, потому
+что в этом случае все хорошо определено. В таком случае есть оператор Гейзенберга, в
+определении которого участвует eiHt, есть матрица плотности, в которой фигурирует
+e−βH, но время можно считать комплексным и тогда, когда комплексное число явля-
+ется действительным, получается операторы эволюции, а в случае когда оно является
+чисто мнимым и Im(t) > 0, то тогда будет получена матрица плотности равновесного
+состояния (с точностью до множителя). Это важное замечание, которое заключается
+в том, что можно пытаться жить в комплексном времени, заменяя время веществен-
+ное временем комплексным или чисто мнимым. Это – то, что называется словом
+“виков поворот”. Не любое комплексное время годится, потому что если в показателе
+34
+
+степени будет стоять положительное выражение, то это будет вещь в конечномерном
+случае вполне хорошая, а в бесконечномерном случае — плохая. Однако корреляци-
+онную функцию ⟨BA(t)⟩β можно продолжить аналитически в полосу 0 ≤ Im(t) ≤ β
+и за счет этого условие KMS будет иметь смысл.
+Условие KMS не использует понятие энтропии — нужно знать только корреляци-
+онные функции. Оно работает также в бесконечном объеме. Можно рассматривать
+условие KMS как определение равновесного состояния. Равновесное состояние, как я
+его определял раньше, практически почти всегда является единственным, а здесь по-
+является простор для того, чтобы равновесное состояние было неединственным. Эта
+неединственность равновесного состояния связана с наличием фазовых переходов.
+Условие KMS является заменой условия максимума энтропии в случае алгебраиче-
+ской квантовой теории.
+Разберем теперь примеры.
+Самый простой пример — квадратичный гамильтониан.
+Положительно определенный квадратичный гамильтониан можно привести к ви-
+ду:
+ˆH =
+�
+ǫiˆa∗
+i ˆai,
+описывающему невзаимодействующие бозоны. Невзаимодействующие бозоны с точки
+зрения формальной — это то же самое, что многомерный гармонический осциллятор,
+для которого легко можно посчитать матрицу плотности равновесного состояния, ста-
+тистическую сумму, среднее значение энергии и т.д.. Статсумма просто распадается
+на произведение статсумм для разных значений k. Для каждого k можно просумми-
+ровать геометрическую прогрессию, а потом взять произведение
+Z = Π
+1
+1 − eβǫi .
+Среднее значение энергии вычисляется по формуле:
+E = ¯H =
+�
+ǫi¯ni,
+где ¯ni = (eβǫi − 1)−1 — средние числа заполнения.
+Для случая фермионов существенной разницы нет:
+Z = Π(1 + eβǫi),
+¯H =
+�
+ǫi¯ni,
+где ¯ni = (eβǫi +1)−1. Основная разница в том, что в выражении для чисел заполнения
+стоит не минус, а плюс и поэтому средние числа заполнения всегда не превышают
+единицы.
+В заключение добавлю, что, как я говорил, все корреляционные функции можно
+получать с помощью дифференцирования статистической суммы, но удобнее диф-
+ференцировать ее логарифм, который совпадает с точностью до множителя со сво-
+бодной энергией. При этом дифференцировании получаются не сами корреляционные
+функции, а то, что называется усеченными корреляционными функциями. Что это та-
+кое? Если рассматривать, например, корреляционные функции двух операторов, это
+среднее значение ⟨AB⟩ , а если дифференцировать свободную энергию, то получится
+⟨AB⟩ − ⟨A⟩⟨B⟩. Грубо говоря, вычитается часть, отвечающая меньшему количеству
+операторов. Такие функции будут мне важны позже.
+35
+
+4
+Лекция 4
+4.1
+Адиабатическое приближение. Декогерентность
+Эту лекцию я начну с объяснения того, что происходит, когда гамильтониан ˆH(t)
+зависит от времени, но меняется медленно (адиабатически). Я предполагаю, что все
+уровни энергии En(t) различны и зависят от t непрерывно и даже дифференцируе-
+мо. Соответствующие собственные векторы я обозначаю φn(t). Я предполагаю, что
+зависящий от времени вектор φn(t) меняется медленно и его производной по t мож-
+но пренебречь. Мы покажем, что если начать с собственного вектора гамильтониана
+ˆH(t = 0), то во время эволюции, управляемой медленно меняющимся гамильтони-
+аном ˆH(t) он остается собственным, но это будет собственный вектор переменного
+гамильтониана. При эволюции будет возникать фазовый множитель αn(t):
+ˆU(t)φn(0) = e−iαn(t)φn(t),
+(19)
+который определяется уравнением:
+dαn(t)
+dt
+= En(t).
+(20)
+Это легко получить — нужно просто продифференцировать предыдущее равенство с
+учетом правила Лейбница, пренебрегая при этом производной от φn(t). Это рассуж-
+дение чуточку неаккуратно, потому что я предположил, что φn(t) с течением времени
+меняется медленно, что не очевидно, потому что собственный вектор определен неод-
+нозначно (только с точностью до множителя). Можно ли подобрать множитель так,
+чтобы это условие выполнялось — это не так ясно.
+Проведем более аккуратное рассмотрение. Будем считать, что гамильтониан ˆH(g)
+зависят от какого-то параметра или многих параметров, которые я обозначил буквой
+g. Предположим, что собственные векторы φn(g) и собственные значения En(g) глад-
+ко зависят от g. Будем считать, что параметр g зависит от времени таким образом,
+что производной от g по t можно пренебречь. Стандартный выбор таков: фиксиру-
+ется g(t) и строится семейство ga(t) = g(at), где a → 0. Очевидным образом, на
+таких функциях условие малости производной по t выполняется. В таком случае все
+приведенные выше рассуждения становится вполне аккуратным.
+Для того чтобы обобщить эти рассуждения на матрицы плотности, следует просто
+заметить, что зависимость матрицы плотности K(t) от времени определяется урав-
+нением
+dK
+dt = H(t)K(t) = 1
+iℏ[ ˆH(t), K(t)],
+где стоит коммутатор с гамильтонианом ˆH(t). Этот коммутатор и есть оператор H(t)
+(без крышечки), действующий на K.
+Теперь нужно рассмотреть собственные векторы ψmn(t) оператора H(t). Их лег-
+ко найти. Это будут просто проекции на собственные векторы гамильтониана ˆH(t),
+которые я обозначил как φn(t) :
+ψmn(t)x = ⟨x, φn(t)⟩φm(t).
+В представлении где оператор ˆH(t) диагонален это будут матрицы, у которых стоит
+единица в позиции (m, n), а остальные элементы нулевые. В этом случае в точности
+такое же рассуждение, которое было приведено выше, определяет эволюцию U(t)
+этих собственных векторов:
+U(t)ψmn(0) = e−iβmn(t)ψmn(t),
+dβmn(t)
+dt
+= Em(t) − En(t).
+(21)
+36
+
+Это легко проверяется. Нужно продифференцировать по t и пренебречь производной
+вектора ψmn(t). Очень важное замечание состоит в том, что когда m = n можно
+считать, что фаза обращается в ноль: βmm = 0.
+Представим то же самое в несколько других обозначения, а именно, возьмем мат-
+рицу плотности и запишем ее как сумму по собственным векторам с какими-то коэф-
+фициентами kmn:
+K =
+�
+kmnψmn.
+Вместо того, чтобы рассматривать эволюцию собственных векторов ψmn рассмотрим
+эволюцию коэффициентов kmn(t). Формулы те же самые — коэффициенты kmn(t)
+получают фазовые множители, а в показателе стоит βmn(t):
+kmn(t) = e−iβmn(t)kmn,
+βmn(t) =
+� t
+0
+(Em(τ) − En(τ))dτ.
+Если адиабатический гамильтониан ˆH(t) таков, что в момент T он возвращается к
+тому, что был в нулевой момент времени: ˆH(T) = ˆH(0), то диагональные элементы
+матрицы K не меняются, но меняются недиагональные элементы — они умножаются
+на фазовый множитель.
+Фиксируем теперь гамильтониан ˆH. Можно себе представить молекулу или атом
+или что-нибудь побольше — любую квантовую систему и эта система живет в каком-то
+внешнем мире. Будем считать, что внешнее окружение просто меняет гамильтониан,
+который управляет молекулой; новый гамильтониан ˆH(t) может зависеть от време-
+ни, но мы считаем, что он меняется медленно. Можно себе представить, что около
+молекулы довольно далеко от нее пролетает какая-то космическая частица. В таком
+случае есть электрическое поле, которое возникает от этой частицы. Это означает,
+что гамильтониан из-за космической частицы изменился. Если эта частица пролетает
+достаточно далеко, то можно считать, что изменение адиабатично.
+Мой любимый пример: вы проводите эксперимент, а в соседней комнате кто-то
+решил подогреть свой завтрак и включил микроволновку. Тогда у вас тоже возникает
+какое-то электрическое поле, которое, несомненно, можно считать адиабатическим.
+Другой пример: мы знаем, что живем в мире, где есть микроволновое космиче-
+ское излучение, которое тоже порождает некоторое электромагнитное поле. Очень
+маленькое, но тем не менее оно есть. Этих адиабатических возмущений мы не знаем,
+но мы знаем, что на диагональные элементы матрицы плотности адиабатическое воз-
+мущение не оказывает влияния, а у недиагональных элементов матрицы плотности
+появляются фазовые множители, которые мы, естественно, не знаем, потому что не
+знаем возмущения.
+То, что я сказал, можно интерпретировать другим способом. Можно рассмотреть
+линейные комбинации вида α0φ0 + α1φ1 двух или большего количества собственных
+векторов гамильтониана ˆH = ˆH(0). При рассмотрении эволюции этого состояния бу-
+дут появляться фазовые множители: αk(t) = e−iEktαk. Эти фазовые множители по-
+являются всегда, они предсказуемы, но если наложено адиабатическое возмущение,
+то тогда фазовые множители становятся непредсказуемыми, а абсолютные значения
+коэффициентов αk(t) остаются постоянными. Раньше два собственных вектора коге-
+рентно менялись с течением времени (пока не было адиабатического возмущения), а
+сейчас эта когерентность исчезла. Это — то, что называется декогерентностью.
+Теперь я хочу объяснить как из этих очень простых соображений появляется
+стандартный рецепт теории измерений. Будем считать, что та же самая молекула
+взаимодействует с окружающей средой и есть случайное адиабатическое возмуще-
+ние ˆH(t) гамильтониана рассматриваемой системы ˆH. Это означает что есть гамиль-
+тониан, который зависит от каких-то параметров λ ∈ Λ, по которым есть некото-
+рое распределение вероятности. Будем считать, что это адиабатическое возмущение
+37
+
+действует в период времени от 0 до T и тогда, как я уже говорил, коэффициенты
+kmn матрицы плотности Kλ(T) в представлении ˆH получают фазовые множители
+Kλ(T) = � Cmn(λ, T)kmnψmn. Фазовые множители Cmn(λ, T) равны единице для
+диагональных элементов и не равны единице для остальных элементов. Поскольку
+гамильтониан случайный, матрицу плотности следует усреднять по этому возмуще-
+нию, то есть, у недиагональных элементов нужно усреднить этот фазовый множи-
+тель. Совершенно ясно, что при усреднении этих фазовых множителей получается
+нечто по модулю меньшее единицы. Наложив некоторые условия, легко проверить,
+что среднее от недиагональных матричных элементов будет равно нулю.
+Формальное доказательство заключается в следующем. Я уже говорил, что мы
+можем включить наш гамильтониан в некоторое семейство гамильтонианов ˆH(g), где
+g принадлежит некоторому множеству параметров обозначенному через Λ (g ∈ Λ).
+Будем считать, что все эти возмущения таковы, что g(0) = 0, g(1) = 0 а зависимость
+гамильтониана от времени определяется формулой ˆH(g(t)). Я определяю адиабати-
+ческий гамильтониан следующим образом:
+ˆHα(t) = ˆH(g(αt)),
+где α → 0. При этом растянется время. Если раньше оно изменялось от нуля до
+единицы, то сейчас — от нуля до T = α−1. Если обозначить через En(g) собственные
+значения гамильтониана ˆH(g), то значения фазовых множителей вида e−iβmn(t) при
+t = T будут определяться следующей формулой:
+βmn =
+� T
+0
+dτ(Em(g(ατ)) − En(g(ατ))).
+Проведя замену переменных ατ = τ ′, можно 1/α вынести за знак интеграла:
+βmn = 1
+α
+� 1
+0
+dτ ′(Em(g(τ ′)) − En(g(τ ′))).
+Воспользуемся теперь леммой Римана – Лебега, которая утверждает, что если
+ищется среднее значение экспоненты eikx с большим, стремящемся к бесконечности
+показателем и среднее берется с какой-то сколько-нибудь приличной функцией (точ-
+ное высказывание — она должна быть абсолютно интегрируемой), то тогда искомое
+среднее в пределе будет стремиться к нулю:
+�
+eikxρ(x)dx → 0
+при k → ∞.
+Если распределение вероятностей по λ не слишком плохое (с приличной плотно-
+стью распределения вероятности), то коэффициенты kmn матрицы плотности исчез-
+нут, если m ̸= n. В результате матрица плотности K становится диагональной после
+введения случайных адиабатических возмущений.
+Обозначим теперь диагональные матричные элементы буквами pn. В обычном
+подходе pn — это вероятности различных состояний. У нас же так и получилось:
+усредненная матрица плотности — это смесь чистых состояний с вероятностями pn.
+Таким образом, получилась обычная формула для вероятностей разных чистых со-
+стояний в данном смешанном состоянии. Из этих соображений получаются обычные
+формулы теории измерений.
+Отмечу, что обычно говорится, что если имеет место взаимодействие с классиче-
+ской системой, то матрица плотности становится диагональной. В терминах волновых
+функций это называется коллапсом. В приведенных рассуждениях нет классической
+системы.
+38
+
+Таким образом коллапс волновой функции можно вывести из случайного адиаба-
+тического взаимодействия. Здесь нет постоянной Планка — есть только адиабатика.
+До сих пор мы рассматривали обычную квантовую механику, а теперь рассмот-
+рим геометрический подход к квантовой теории. Я уже говорил, что есть алгебраиче-
+ский подход, где исходная точка это алгебра наблюдаемых — ассоциативная алгебра
+с инволюцией. Собственно наблюдаемыми являются самосопряженные элементы ал-
+гебры. Состояния определяются как положительные линейные функционалы на этой
+алгебре, то есть, понятие состояния вторично. Давайте будем считать, что понятие
+состояния первично, что есть некоторое множество состояний и подумаем, что от него
+нужно потребовать.
+Первым делом я хочу потребовать, чтобы у меня было понятие смеси состояний
+— не смешанного состояния, а смеси состояний, чтобы любые состояния можно бы-
+ло смешивать с некоторыми вероятностями, с некоторыми весами. Это совершенно
+необходимое требование и, в общем-то, оно не имеет отношения к физике. Можно, на-
+пример, в экономике или в теории игр рассматривать смешанные стратегии. Для того
+чтобы можно было смешивать, множество должно быть выпуклым. Будем считать,
+что выпуклое множество — это подмножество некоторого векторного пространства
+и тогда понятие смешивания будет определено очевидным образом: если xi — точки
+выпуклого множества, pi — неотрицательные числа, сумма которых равна единице,
+то смесь этих точек дается формулой ¯x = � pixi (числаpi рассматриваются как веса
+или вероятности).
+Что нужно еще потребовать? Если мы смешиваем конечное количество состояний,
+то ничего больше не надо. Но если мы хотим смешивать бесконечное количество со-
+стояний, то необходимо понятие предела, а для этого нужны две вещи. Во-первых,
+нужна топология в векторном пространстве L, в котором лежит выпуклое множе-
+ство C0. Оно должно быть топологическим векторным пространством. Для простоты
+предположу, что это банахово пространство (нормированное полное пространство).
+Во-вторых нужно потребовать, чтобы пределы принадлежали тому же самому мно-
+жеству — чтобы множество было замкнутым. При выполнении этих условий можно
+смешивать любое бесконечное множество состояний и поэтому мы требуем, чтобы
+множество состояний было замкнутым выпуклым множеством.
+Еще потребуем, чтобы множество было ограниченным. Это, по существу, един-
+ственное требование, которое нужно для того чтобы развивать теорию.
+Также мне нужно иметь понятие оператора эволюции σ(t), который переводит
+состояние в момент времени ноль в состоянии какой-то момент времени t. (Ведь что
+самое главное не просто в физике — в науке? Нужно уметь предсказывать будущее.)
+Что мне нужно потребовать от этого оператора эволюции? Для каждого состояния я
+должен получить другое состояние, то есть, оператор эволюции должен отображать
+множество состояний в себя. Кроме того, я считаю, что t может быть отрицательным;
+тогда оператор эволюции будет обратимым пробразованием множества состояний.
+Будем считать, что оператор эволюции — это линейное преобразование. Точнее,
+мы считаем, что он может быть продолжен как линейный оператор на векторное
+пространство L, включающее множество C0. Введем понятие группы автоморфизмов
+множества состояний C0. Это группа линейных операторов в объемлющем простран-
+стве L, которые обратимы и отображают множество C0 на себя. Естественно считать,
+что оператор эволюции будет автоморфизмом. Иногда удобно наложить на оператор
+эволюции дополнительные условия, например, потребовать, чтобы операторы эволю-
+ции принадлежали некоторой подгруппе V группы автоморфизмов. Это необходимо,
+например, в классической механике.
+Дальше нужно написать уравнение движения. Обычно мы считаем, что знаем
+изменение системы за бесконечно малое время. Это называется уравнением движения.
+39
+
+В самом общем виде уравнение движения можно записать так:
+dσ
+dt = H(t)σ(t),
+(22)
+где H(t) это линейный оператор.
+Формально можно сказать, что H(t) определяется из этого уравнения, но в фи-
+зике мы обычно считаем, что оператор H(t) известен и нужно найти оператор эво-
+люции. Назовем оператор H(t) “гамильтонианом” (в кавычках). Если “гамильтониан”
+не зависит от времени, то оператор эволюции — это экспонента от “гамильтониана” :
+σ(t) = exp(Ht). Всё как в обычной квантовой механике, я лишь не поставил мнимую
+единицу. Можно считать это просто определением экспоненты: для независящего от
+времени оператора H в банаховом пространстве экспоненту можно определить как
+решение уравнения (22).
+Из уравнения (22) следует, что “гамильтониан” принадлежит алгебре Ли группы V
+(он является касательным вектором группы V в единичном элементе группы). Группа
+V, вообще говоря, бесконечномерна, поэтому понятие касательного вектора зависит от
+выбора топологии в группе, но я не буду обращать внимание на эти тонкости. Кроме
+того, разумно потребовать, чтобы для не зависящего от времени “гамильтониана”
+уравнение (22) имело решение.
+Теперь у меня есть понятие состояния, у меня есть уравнения движения, но еще
+мне нужно ввести понятие наблюдаемой. Раньше я шел от наблюдаемых к состоя-
+ниям, а сейчас хочу перейти от состояний к наблюдаемым. Что такое наблюдаемая?
+Прежде всего, это должен быть оператор A на который мы наложим те же условия,
+как и на “гамильтониан”. Иными словами, мы требуем, чтобы была определена экс-
+понента σA(t) = exp(At), которую можно рассматривать как однопараметрическую
+подгруппу в V.
+Зафиксируем еще функционал a, который инвариантен относительно операторов
+σA(t) = exp(At). (Это условие эквивалентно условию a(Ax) = 0.) Функционал a опре-
+деляет значения наблюдаемой.
+В частности, в случае обычной квантовой механики V — это группа унитарных
+операторов. Она действует на матрицы плотности по формуле U(K) = ˆUK ˆU −1, где ˆU
+— унитарный оператор. Если ˆA — самосопряженный оператор (не обязательно огра-
+ниченный), то экспоненту ei ˆ
+At можно рассматривать как однопараметрическую под-
+группу группы V. Так получаются все однопараметрические подгруппы непрерывные
+в сильной топологии (теорема Стоуна). Это позволяет отождествить самосопряжен-
+ные операторы с элементами алгебры Ли группы унитарных операторов. Здесь есть
+маленькие трудности, которые заключаются в том, что коммутатор неограниченных
+самосопряженных операторов не всегда определен и поэтому, строго говоря, струк-
+тура алгебры Ли нарушается. Эти трудности возникают всегда и к рассматриваему
+подходу не имеют никакого отношения.
+“Гамильтониан” выражается через самосопряженный оператор ˆH по формуле
+H(K) = 1
+i [ ˆH, K],
+где K — матрица плотности. (Мы считаем, что ℏ = 1).
+Аналогично, любому самосопряженному оператору ˆA мы сопоставляем оператор
+на матрицах плотности по формуле:
+A(K) = 1
+i [ ˆA, K].
+(23)
+Наблюдаемая задается парой ( ˆA, a), где ˆA — самосопряженный оператор, действую-
+щий на матрицах плотности по формуле (23) и функционал на матрицах плотности
+a задается формулой a(K) = tr( ˆAK).
+40
+
+В алгебраическом подходе группа V состоит из автоморфизмов ассоциативной
+алгебры с инволюцией A (∗-алгебры); в частности, самосопряженный элемент A ал-
+гебры A определяет инфинитезимальный автоморфизм (элемент алгебры Ли группы
+V) по формуле αA(X) = i[A, X]. Линейный функционал ω на алгебре называется по-
+ложительным, если ω(X∗X) ≥ 0 для всех X ∈ A; множество всех положительных
+функционалов я обозначаю буквой C.
+Множество состояний C0 состоит из нормированных положительных функциона-
+лов (функционалов, удовлетворяющих условию ω(1) = 1). Это — ограниченное вы-
+пуклое множество; на нем естественным образом действует группа V. Наблюдаемая
+— это пара (A, a), где A — это самосопряженный элемент алгебры A, рассматрива-
+емый как инфинитезимальный автоморфизм, а a-это линейный функционал на C0
+сопоставляющий состоянию ω число a(ω) = ω(A).
+В геометрическом подходе точно так же есть декогерентность. Доказательство
+этого, практически такое же, как и раньше. Я начну с независящего от времени “га-
+мильтониана”, обозначаемого буквой H (без крышечки). Тогда оператор эволюции —
+это просто экспонента σ(t) = etH. Я предположу, что оператор H диагонализуем, то
+есть, существует базис (ψj), состоящий из собственных векторов оператора H:
+Hψj = ǫjψj.
+Я предположил, что оператор эволюции отображает множество состояний в себя и что
+множество состояний является ограниченным. При этих условиях следует ожидать,
+что этот оператор диагонализуем.
+Это легко доказать в конечномерном случае. Если σ(t) = etH — это семейство
+ограниченных операторов в конечномерном пространстве и эти операторы не просто
+ограничены — они ограничены вместе, то есть, норма всех этих операторов ограни-
+чена одним и тем же не зависящим от t числом, то из этого сразу следует две вещи.
+Во-первых, собственные значения этих операторов чисто мнимые, потому что функ-
+ция вида eǫt ограничена только в том случае, если ǫ чисто мнимое. (Асимптотика
+определяется вещественной частью, а мнимая часть дает фазовый множитель; если ǫ
+— действительное число, ограниченность будет только в том случае, когда ǫ = 0.) Во-
+вторых, в случае наличия жордановой клетки размера больше единицы, экспонента
+σ(t) = etH тоже не будет ограниченной. Таким образом, в конечномерном случае во-
+первых все собственные значения чисто мнимые, а во-вторых все жордановы клетки
+одномерны, то есть H диагонализуем.
+В бесконечномерном случае этого недостаточно для диагонализуемости, но тем не
+менее жордановых клеток быть не может и в этом случае. Поэтому нужно ожидать,
+что оператор H диагонализуем. Я это и буду предполагать.
+Теперь я повторяю свои рассуждения. Я включаю мой “гамильтониан” в семейство
+H(g) “гамильтонианов”, с собственными значениями (ǫj(g)) и собственными функци-
+ями (ψj(g)):
+H(g)ψj(g) = ǫj(g)ψj(g),
+удовлетворяющими прежним условиям — базис гладко зависит от g и для значения
+g = 0 получается исходный “гамильтониан”: H(0) = H.
+Дальше я говорю, что ψj — это робастная нулевая мода, если ǫj(g) ≡ 0, то есть,
+собственное значение равно нулю при любом g. Можно было бы сказать, что это —
+устойчивая нулевая мода (она была у H и остается нулевой модой после введения
+H(g)), но слово “устойчивость” занято для других целей.
+Теперь я хочу сделать все то же, что я делал раньше — буду считать, что взаимо-
+действие со средой определяется случайным “гамильтонианом” H(g(t)), где g зависит
+от t, и буду считать, что эти случайные “гамильтонианы” адиабатичны. Они медлен-
+но меняются так, что производной от g по t можно пренебречь. Можно сказать, что
+41
+
+рассматривается адиабатическая эволюция. Собственный вектор при этом остается
+собственным вектором меняющегося “гамильтониана”, но в нем появляется фазовый
+множитель:
+σ(t)ψj = eρj(t)ψj(g(t)),
+где dρj
+dt = ǫj(g(t)). Тут нет мнимой единицы, но ǫj(g) само чисто мнимое. Для дока-
+зательства просто нужно продифференцировать правую часть этого выражения по t
+применяя уравнения движения и пренебрегая производной ˙g(t).
+Фазовый множитель не появляется для робастных нулевых мод, а для неробаст-
+ных появляется. Если же адиабатическое возмущение действовало некоторое конеч-
+ное время, то в конце для робастных нулевых мод просто ничего не изменится. В
+середине эти моды менялись, они становились нулевыми модами другого гамильто-
+ниана, а в конце они приходят к тому, что было: σ(T)ψj = ψj, если g(T) = g(0). У
+всех остальных мод есть фазовые множители.
+Дальше все так же, как в обычной квантовой механике. Для неробастных нулевых
+мод все случайные фазовые факторы после усреднения по случайному возмущению
+обнуляются, а для робастных нулевых мод ничего не меняется.
+В обычной квантовой механике робастные нулевые моды — это диагональные эле-
+менты матрицы плотности в ˆH — представлении. Они, как мы знаем, являются ну-
+левыми модами, потому что все диагональные матрицы коммутируют между собой,
+поэтому здесь применимы предыдущие высказывания.
+Интерпретировать это нужно так: произвольное состояние x ∈ C0 можно разло-
+жить по собственным функциям оператора H, и при этом взаимодействие с окру-
+жающей средой убьет все моды, кроме робастных нулевых мод. Я обозначу через P ′
+тот оператор, который убивает все, кроме робастных нулевых мод, и теперь можно
+сказать, что вся наблюдаемая физика лежит в проекции на робастные нулевые моды
+P ′x ∈ C0. Далее нужно разложить состояния P ′x ∈ C0 по чистым робастным нулевым
+модам P ′x = � piui. Коэффициенты pi в этом разложении должны быть интерпрети-
+рованы как вероятности. В обычной квантовой механике таким образом получаются
+обычные вероятности. В общем случае коэффициент pi — это вероятность чистой
+робастной нулевой моды ui в состоянии x.
+Гамильтониану H отвечает физическая величина (H, h), где функционал h нужно
+отождествить с энергией. Коэффициент pi следует интерпретировать как вероятность
+найти энергию h(ui) в состоянии x. (Предполагается, что все числа h(ui) различны.
+Если некоторые из них совпадают, то чтобы получить вероятность того, что энергия
+равна h, нужно просуммировать все коэффициенты pi, для которых h(ui) = h.)
+Если все нулевые моды робастные, то можно написать простую формулу для опе-
+ратора: P ′ = P, где P — оператор, убивающий все ненулевые моды:
+P = lim
+T→∞
+1
+T
+� T
+0
+σ(t)dt.
+Если в этом интеграле разлагать σ(t) по собственным векторам, то для ненулевых мод
+появляются ненулевые фразовые множители; усредняя их по времени, в результате
+получаем ноль.
+Сейчас я объяснил то, как появляются вероятности для “гамильтонианов”. Они
+интерпретируются как вероятности различных уровней энергии. Повторить те же
+самые рассуждения можно рассматривая и другие наблюдаемые, представляемые в
+виде (A, a), где A ∈ V, a — функционал, для которого a(Ax) = 0. Можно определить
+понятие робастной нулевой моды для любой наблюдаемой.
+Я дам несколько другое, более общее, определение робастной нулевой моды. Когда
+говорится, что x — это робастная нулевая мода, прежде всего, нужно потребовать,
+42
+
+чтобы это была нулевая мода: Ax = 0. Если A слегка меняется (заменяется на близ-
+кий элемент A′ группы V) , то нужно потребовать, чтобы у A′ была нулевая мода
+x′, близкая к исходной нулевой моде x. Дальше все то же самое, что было сказано
+для энергии, можно повторить для произвольной наблюдаемой. Нужно рассмотреть
+проекцию PA на пространство нулевых мод: CA = (KerA) ∩ C = Im(PA), где
+PA = lim
+T→∞
+1
+T
+� T
+0
+dteAt.
+(Я предположил, что все нулевые моды робастные). После этого нужно разложить
+проекцию на нулевые моды по крайним точкам:
+PA(x) =
+�
+piui.
+Коэффициенты pi будут интерпретироваться как вероятности значений a(ui) в состо-
+янии x.
+4.2
+L-функционалы
+Я сейчас объяснил, что есть формализм, в котором начальный объект — это мно-
+жество состояний. Возникает вопрос, который должен задать физик: удобен ли этот
+формализм, удобно ли в этом формализме считать? На этот вопрос я собираюсь сей-
+час ответить, вернувшись уже к обычному формализму квантовой механики. Я буду
+работать в рамках алгебраического подхода, в котором алгеброй является алгебра
+Вейля с генераторами ˆui и соотношениями
+ˆukˆul − ˆulˆuk = iℏσk,l.
+Я хочу ввести понятие L-функционала, определяемого для всякой матрицы плот-
+ности K в любом представлении алгебры Вейля по формуле:
+LK(α) = treiαk ˆukK = trVαK,
+где Vα = eiαk ˆuk = eiαˆu — операторы, которые мы уже рассматривали, αk — это набор
+чисел, отвечающих генераторам алгебры Вейля. Напомню, что алгебру Вейля можно
+задавать генераторами Vα. Если uk — самосопряженные, а αk -действительные, то
+эти генераторы унитарны и удовлетворяют соотношениям
+VαVβ = e−i ℏ
+2ασβVα+β,
+где ασβ = αkσk,lβl.
+Это так называемая экспоненциальная форма алгебры Вейля.
+Важное свойство L-функционалов заключается в том, что они объединяют вме-
+сте все представления алгебры Вейля. Проблема неэквивалентности представлений
+алгебры Вейля в формализме L-функционалов полностью исчезает. В формуле для
+L-функционала унитарный оператор умножается на оператор из класса операторов
+имеющих след, и поэтому след хорошо определен.
+Я могу ввести пространство L всех линейных функционалов на алгебре Вейля.
+Можно считать, что L-функционал определяет линейный функционал на алгебре
+Вейля: LK ∈ L. ( Это вытекает из замечания, что всякий элемент, выраженный через
+генераторы Vα с помощью конечного числа операций сложения и умножения явля-
+ется линейной комбинацией генераторов Vα.) Как легко проверить, функционал LK
+положителен и, кроме того, нормирован (на единичном элементе алгебры он равен
+единице). В дальнейшем я буду отождествлять L-функционалы с положительными
+функционалами на алгебре Вейля.
+43
+
+Определим операции в рассматриваемом пространстве L. Эти операции определе-
+ны для любой алгебры с инволюцией.
+Если имеется алгебра с инволюцией, то на линейных функционалах есть две опе-
+рации, отвечающие элементу A этой алгебры. Можно определить операцию на функ-
+ционале ω(x), где x — элемент алгебры, умножая x справа на A. Другая операция
+получается если умножать слева на A∗. Первую из них я обозначил той же самой
+буквой A, а другую — ˜A :
+(Aω)(x) = ω(xA), ( ˜Aω)(x) = ω(A∗x).
+Зададимся вопросом: остается ли функционал положительным при действии эти-
+ми операторами? Ответ: нет, но если применить комбинацию ˜AA, тогда положитель-
+ные функционалы будут переходить в положительные функционалы (которые не обя-
+зательно нормированы). Напомню, что положительные функционалы должны быть
+неотрицательны на элементах вида x = B∗B. Легко проверить, что при действии опе-
+рации ˜AA происходит преобразование x → A∗xA = (BA)∗xBA. Отсюда видно, что
+положительность сохраняется.
+Это важное высказывание. Мы обозначили через C пространство всех положитель-
+ных функционалов (ненормированных состояний). В нем действует оператор ˜AA.
+Следующее замечание состоит в том, что операторы вида ˜A всегда коммутируют
+с операторами вида A. Это происходит потому, что один умножает слева, а другой —
+справа.
+Далее, если оператор имеет вид A = eitH, тогда ˜A = e−it ˜
+H. Отсюда легко вывести,
+что если уравнения движения записать как
+idσ/dt = ( ˜H − H)σ,
+(24)
+то оператор эволюции будет иметь вид e−it( ˜
+H−H). Согласно предыдущему замеча-
+нию, это выражение может быть представлено в виде ˜AA. То есть если уравнения
+движения записать в виде (24), то σ не выводит за пределы конуса положительных
+функционалов (конуса ненормированных состояний). Это я буду использовать.
+Перейдем теперь к вопросу о виде уравнений движения в случае алгебры Вейля.
+Легко вычислить операторы Vβ и ˜Vβ :
+(VβL)(α) = e+i ℏ
+2 ασβL(α + β), ( ˜VβL)(α) = e+i ℏ
+2ασβL(α − β),
+Запишем ˆH как интеграл
+ˆH =
+�
+dβh(β)Vβ.
+Этот оператор будет самосопряженным если h(−β) = h(β)∗. Введем постоянную
+Планка и сделаем замену 1
+ℏ ˆH → ˆH. Уравнение движения, в котором ˆH играет роль
+гамильтониана примет вид:
+iℏdσ
+dt = ( ˜ˆH − ˆH)σ.
+Это уравнение в L-функционалах можно записать в виде:
+iℏdL
+dt =
+�
+dβh(−β)e+i ℏ
+2 ασβL(α − β) −
+�
+dβh(β)e+i ℏ
+2ασβL(α + β) =
+= −
+�
+dβh(β)(e+i ℏ
+2ασβ − e−i ℏ
+2ασβ)L(α + β).
+В результате приходим к формуле:
+dL
+dt = −
+�
+dβh(β)2 sin(ℏ
+2ασβ)
+ℏ
+L(α + β).
+Из этой формулы ясно, что уравнение движения для L-функционалов имеет пре-
+дел когда ℏ → 0.
+44
+
+5
+Лекция 5
+5.1
+Функциональные интегралы
+Настоящая лекция посвящена, прежде всего, широко применяемым в квантовой ме-
+ханике функциональным интегралам. Я буду рассказывать об основанном на идее
+Березина подходе, который позволяет применять их и в обычном подходе к кванто-
+вой механике, и в геометрическом подходе, и в формализме L-функционалов.
+В квантовой механике физическая величина представляется в виде функциональ-
+ного интеграла — бесконечномерного интеграла, где подынтегральное выражение
+включает экспоненту от действия, умноженную на что-то. Действие — это функ-
+ционал от кривой (точнее, от функции q(τ), график которой мы рассматриваем как
+кривую):
+S[q(τ)] =
+� t
+0
+dτL(q(τ), ˙q(τ)).
+В данном выражении q(τ) стоит в квадратных скобках, чтобы подчеркнуть, что S
+— это не функция, а функционал, который зависит не от точки кривой, а от кривой.
+Матричный элемент ⟨q2| ˆU(t)|q1⟩ оператора эволюции в координатном представле-
+нии: ˆU(t) = e− it ˆ
+H
+ℏ , выражается через функциональный интеграл с подынтегралным
+выражением:
+e
+i
+ℏ S[q(τ)],
+для которого область интегрирования — это кривые q(τ) с заданными начальными и
+конечными значениями q(0) = q1, q(t) = q2.
+Что же означает слово “функциональный интеграл”? Интеграл описанного мной
+типа можно аппроксимировать конечномерными интегралами: под знаком функци-
+онального интеграла заменить обычный интеграл, скажем, на интегральные суммы
+и взять предел. Проблема в том, что в отличие от теории из курса обычного инте-
+грального исчисления, когда способ приближения несущественен, для функциональ-
+ных интегралов это вовсе не так. Кроме того, предел, как правило, не существует: он
+обычно или бесконечный, или нулевой, и нужно предпринимать какие-то действия
+для того, чтобы извлечь из всего этого конечный ответ.
+То, что я собираюсь обсуждать, может быть сделано строгим до некоторого момен-
+та. Строгие вещи можно сделать для гауссовых интегралов, представляющих собой
+интегралы от экспонент квадратичных выражений, возможно, умноженных на поли-
+ном. Для вычисления таких интегралов в конечномерном случае можно воспользо-
+ваться формулой:
+�
+exp( i
+2⟨Ax, x⟩ + i⟨b, x⟩)dx = (det A)− 1
+2 exp(− i
+2⟨A−1b, b⟩).
+(В этой формуле должен стоять постоянный множитель, который мы не пишем, вклю-
+чив его в определение интеграла. Оператор A может быть комплексным, но нужно
+наложить условия, гарантирующие, что интеграл имеет смысл.) Дифференцируя это
+выражение по b, мы придем к выводу, что интеграл вида
+�
+P(x) exp( i
+2⟨Ax, x⟩)dx,
+где P(x) — некоторый полином может быть выражен через детерминанты, а, по край-
+ней мере, для случая эллиптических операторов существует вполне хорошая теория
+45
+
+вычисления таких детерминантов. Опять же, такой детерминант нужно чем-то при-
+ближать и после этого выбрасывать бесконечные члены разложения для логарифма
+детерминанта.
+Тем не менее если имеется подынтегральное выражение вида W(x) = Q(x)+gV (x),
+где Q — квадратичное выражение по x, а константу g, мы считаем малой, то тогда
+нормированный функциональный интеграл вида:
+�
+eW (x)dx
+�
+eQ(x)dx
+можно представить, в виде суммы фейнмановских диаграмм. В более общем случае,
+когда W(x) представляется как квадратичное по x − x0 выражение плюс кусочек,
+который содержит только мономы большей степени по x − x0, теория возмущений
+тоже хорошо определена.
+После этих предварительных слов перейдем к вопросу, как можно построить эти
+функциональные интегралы? Их можно строить для более или менее любой физи-
+ческой величины. Я буду строить функциональный интеграл для экспоненты опера-
+тора, действующего в банаховом (не обязательно в гильбертовом) пространстве или,
+что почти то же, я буду решать уравнение движения:
+dσ
+dt = H(t)σ(t)
+(25)
+и искать оператор эволюции.
+Оператор H(t) в уравнении движения может быть функцией от t, но, для того
+чтобы упростить формулы, я буду считать, что H не зависит от t; это совершенно
+несущественно.
+Давайте теперь то пространство, в котором действует оператор H, обозначим бук-
+вой L и определим понятие символа оператора. Мы встречались с этим понятием. Я
+хочу дать очень общее определение. Символ оператора — это функция, определенная
+на каком-то пространстве. Я буду считать, что на пространстве с мерой, потому что
+хочу интегрировать, но мне ничего не нужно из теории пространств с мерой. Доста-
+точно понимать, что символ оператора — это функция, определенная где-то, где есть
+понятие интегрирования.
+Символы будем выделять подчеркиванием снизу. Основные свойства символов
+следующие. Символ равен тождественно единице, если оператор равен единице. Сим-
+вол A должен зависеть линейно от оператора A. Произведению операторов C = AB
+должна отвечать некоторая операция на символах, которую я обозначу символом ∗,
+то есть C = A ∗ B.
+Теперь я проведу совершенно тривиальные рассуждения, которые сразу приво-
+дят к функциональному интегралу. Я буду использовать стандартную формулу для
+экспоненты:
+σ(t) = etH = lim
+N→∞(1 + tH
+N )N.
+Если я буду рассматривать экспоненту от символа exp t
+N H, то при больших N, в
+первом приближении будет справедлива формула:
+1 + tH
+N = e
+t
+N H + O(N −2).
+Ошибка будет иметь порядок
+1
+N2 , но когда N → ∞, то этой поправкой можно прене-
+бречь, и в результате получается выражение:
+σ(t) = lim
+N→∞ IN(t),
+46
+
+IN(t) = e
+t
+N H ∗ ... ∗ e
+t
+N H
+(N сомножителей).
+Мое утверждение заключается в том, что я вывел представление оператора эволю-
+ции в качестве функционального интеграла. Нужно только дать примеры символов
+и расшифровать, что такое “звездочка”.
+Я хочу подчеркнуть одну вещь, на мой взгляд, важную и физиками недооценен-
+ную. Дело в том, что до сих пор я приводил тривиальные рассуждения, которые могут
+быть сделаны строгими. На самом деле, не нужно говорить про функциональные ин-
+тегралы — просто нужно исследовать функции IN(t). Можно, например, применять
+метод стационарной фазы и получать более или менее те же результаты, что и на
+языке функциональных интегралов, проводя вполне строгие рассуждения.
+Позже я введу большой класс операторов, для которых операция ∗ записывается
+в простом виде. Этот класс операторов включает в себя q-p-символы и виковские
+символы, про которые я уже рассказывал. Сейчас я буду обсуждать их в несколько
+ином виде. Я считаю, что символ — это функция от двух переменных A(α, β). Для
+этого большого класса символов выражение для символа произведения выглядит так:
+C(α, β) =
+�
+dγdγ′A(α, γ)B(γ′, β)ec(α,γ)+c(γ′,β)−c(α,β)−r(γ′,γ),
+(26)
+где c(α, β) и r(α, β) — некоторые функции. Для q-p-символов это просто скалярные
+произведения с точностью до знака:
+c(q, p) = r(q, p) = −ipq.
+(27)
+Позже я вернусь к вопросу как получаются такого типа формулы, а пока я считаю,
+что есть такая формула для символа произведения. Зная формулу для произведения
+двух операторов, можно записать формулу для произведения n операторов A1, ..., AN:
+C(α, β) =
+�
+dγ1dγ′
+1...dγN−1dγ′
+N−1A1(α, γ1)A2(γ′
+1, γ2)...An(γ′
+N−1, β)eρN ,
+где
+ρN = c(α, γ1) + c(γ′
+1, γ2) + ... + c(γ′
+N−1, β) − c(α, β)) − r(γ′
+1, γ1) − ... − r(γ′
+N−1, γN−1).
+В результате получаем произведение символов, умноженное на экспоненту от неко-
+торого выражения, которое обозначено символом ρN. Возвращаясь к выражению
+IN(t), приближающему оператор эволюции, мы можем представить его в виде:
+IN(t) =
+�
+dγ1dγ′
+1...dγN−1dγ′
+N−1eρN exp( t
+N (H(α, γ1) + H(γ′
+1, γ2) + ... + H(γ′
+N−1, β)).
+То, что я рассказал — это очень широкая схема. Есть один совершенно конкретный
+пример — это q-p-символ. Я хочу заметить, что если рассматривать ядро единичного
+оператора в смысле математики (или матрицу единичного оператора, как говорят
+физики), то это — δ-функция: в координатном представлении ⟨q2|1|q1⟩ = δ(q1 − q2),
+а в импульсном представлении ⟨p2|1|p1⟩ = δ(p1 − p2). Мы же хотим, чтобы символ
+единичного оператора был равен единице, а не δ-функции. Это очень просто сде-
+лать. Дело в том, что преобразование Фурье от δ-функции — это константа, поэтому
+для того, чтобы получить символ, дающий для единичного оператора единицу, мы
+должны просто взять преобразование Фурье и помножить на постоянный множитель.
+Раз матрица — это δ-функция от аргумента q1 − q2, значит, возьмем преобразование
+Фурье матричного элемента ⟨q2|A|q1⟩ по этому аргументу:
+Aq−p(q, p) =
+�
+dy⟨y|A|q⟩eip(q−y).
+47
+
+Здесь написано, что q-p-символ (я так назвал этот символ) — это просто преобра-
+зование Фурье матричного элемента по разности между аргументами. Это действи-
+тельно символ. Для единицы это единица. Очевидно, можно взять обратное преоб-
+разование Фурье и выразить матричные элементы ⟨p2|A|p1⟩ через q-p-символ. По-
+скольку мы знаем, как выразить матричный элемент произведения через матричные
+элементы сомножителей, мы можем вычислить, чему равен q-p-символ от произведе-
+ния двух операторов. Ответ дается формулой (26), где функции c(α, β) и r(α, β) —
+просто скалярные произведения.
+Определение q-p-символа, которое было только что дано, отличается от опреде-
+ления, данного раньше. Это определение применимо к любому оператору, лишь бы
+интеграл сходился. Если же A — это дифференциальный оператор, то легко понять,
+что определение q-p-символа, которое я сейчас дал с помощью интеграла, соответ-
+ствует предыдущему: если имеется дифференциальный оператор с полиномиальными
+коэффициентами, то его можно записать как полином от операторов координат qj и
+операторов импульсов ˆpj = 1
+i
+∂
+∂qj . После этого операторы координат нужно сдвинуть
+налево, а операторы импульсов — направо, после чего нужно снять шляпочки с опе-
+раторов. Получится из операторного выражения полиномиальная функция, которую
+я называл q-p-символом.
+Отмечу, что я считал, что постоянная Планка ℏ = 1, но иногда удобно ее оставить
+в формулах.
+Перейдя к обычной квантовой механике, мы обнаружим, что формула для IN
+принимает вид:
+IN(q, p, t) =
+�
+N−1
+�
+1
+dqαdpα exp(i
+N
+�
+1
+pα(qα − qα−1) − it
+N
+N
+�
+1
+H(pα, qα−1)),
+(28)
+где pN = p, q0 = qN = q.
+Таким образом, мы получили оператор эволюции как предел конечнократных ин-
+тегралов.
+На этом вполне разумно было бы остановиться и просто изучать это представле-
+ние, но также можно сказать слова, которые содержат выражение “функциональный
+интеграл”. Для этого следует обратить внимание на то, что выражение, которое стоит
+в экспоненте — это в точности интегральная сумма для некоторого интеграла. Этот
+интеграл хорошо известен физикам — это функционал действия
+S[p(τ), q(τ)] =
+� t
+0
+(p(τ) d
+dtq(τ) − H(p(τ), q(τ))dτ.
+Все уже очень близко к тому, что я хочу. Давайте подойдем еще ближе. То, что
+было написано – это q-p-символ для оператора эволюции. Мы уже знаем, что q-p-
+символ — это просто Фурье преобразование от матричного элемента. В результате
+мы можем сказать, что q-p-символ — это функциональный интеграл от экспоненты
+eiS[p(τ),q(τ)],
+где стоит функционал S от пары функций p(τ), q(τ), удовлетворяющих граничным
+условиям
+p(0) = p(t) = p, q(0) = q(t) = q,
+когда τ изменяется от 0 до t. Далее можно перейти к матричным элементам, сделав
+Фурье — преобразование. В результате получится такой же интеграл, но с другими
+граничными условиями: q(0) = q1, q(t) = q2.
+Чтобы прийти к той формуле, про которую я уже говорил, рассмотрим частный
+случай, когда символ H(p, q) — это сумма квадратичной функции от p (кинетиче-
+ской энергии) и некоторой функции V (q) (потенциальной энергии). Тогда ничего не
+48
+
+стоит проинтегрировать по p — это гауссов интеграл. В результате матричный эле-
+мент оператора эволюции может быть представлен как функциональный интеграл с
+подынтегральным выражением вида экспоненты от функционала действия
+eiS[q(τ)] = ei(
+� t
+0 dτ(T( ˙q(τ))−V (q(τ))).
+Функционал действия при этом представляет собой интеграл от разности кинети-
+ческой и потенциальной энергий. По p(τ) интегрирование уже проведено, и остаётся
+только интеграл по функциям q(τ), удовлетворяющим условиям q(0) = q1, q(t) = q2.
+Таким образом, я вывел функциональный интеграл, с которого я начал и даже
+в более общем виде. Когда кинетическая энергия квадратична по импульсам, это
+вычисление всегда работает.
+В приведенных рассуждениях допущена определенная недосказанность. На самом
+деле, в выражении (28) для IN должен стоять постоянный множитель, стремящийся к
+нулю при N → ∞ (этот множитель происходит от константы, которую мы выбросили,
+когда писали выражение для гауссова интеграла). То есть, здесь выброшен нулевой
+множитель. Это все время делается в функциональных интервалах, потому что на
+самом деле хороший объект — это частное от деления функциональных интегралов.
+Я хочу построить большое количество примеров. Эти примеры обобщают то, что
+Березин называл ковариантными символами.
+Я возьму два банаховых пространства L и L′. Можно взять два сопряженных
+банаховых пространства, но это не обязательно — важно только чтобы между ними
+было невырожденное скалярное произведение (спаривание). Поскольку я хочу, чтобы
+частным случаем было гильбертово пространство, где скалярное произведение ⟨l, l′⟩
+линейно по одному аргументу l ∈ L и антилинейно по другому аргументу l′ ∈ L′, я и
+здесь такое потребую.
+Итак, у меня есть два банаховых пространства, которые почти двойственны друг
+другу в том смысле, что между ними есть невырожденное скалярное произведение.
+После этого я возьму две системы векторов eα ∈ L, где α ∈ M, и e′
+β ∈ L′, где β ∈ M′, в
+этих пространствах. Они не должны быть линейно независимы — это не базисы. Это,
+как говорят, переполненные системы векторов. Мне важно только то, чтобы любой
+вектор можно было выразить через эти векторы как предел линейных комбинаций.
+(Примером такой переполненной системы являются пуассоновы векторы.)
+Теперь я потребую, чтобы, как говорят физики, можно было вставить единицу.
+Это означает, что если я хочу рассмотреть скалярное произведение ⟨l, l′⟩, то нужно
+взять скалярные произведения ⟨l, e′
+µ⟩ и ⟨eλ, l′⟩ и из этих скалярных произведений
+с помощью некоторого интегрирования создать общее скалярное произведение. Это
+всегда можно сделать и даже разными способами. Я буду считать, что такой способ
+зафиксирован:
+⟨l, l′⟩ =
+�
+⟨l, e′
+µ⟩⟨eλ, l′⟩e−r(λ,µ)dλdµ,
+(29)
+где r(λ, µ) — функция на пространстве с мерой M × M′.
+После этого я хочу определить ковариантный символ A(α, β) оператора A, дей-
+ствующего в L (можно рассмотреть L′ с тем же успехом). Этот символ определяется
+простой формулой:
+A(α, β) =
+⟨Aeα, e′
+β⟩
+⟨eα, e′
+β⟩ .
+Мое основное условие выполнено — символ единичного оператора равен едини-
+це. Легко посчитать символ C произведения операторов C = AB, воспользовавшись
+49
+
+соотношением: ⟨ABeα, e′
+β⟩ = ⟨Beα, A∗e′
+β⟩, и формулой (29) для ⟨l, l′⟩, где l = Beα,
+l′ = A∗e′
+β. Введя обозначения ⟨eα, e′
+β⟩ = ec(α,β), получаем следующее выражение:
+C(α, β) =
+�
+dλdµB(α, µ)A(λ, β) exp(−r(λ, µ) − c(α, β) + c(α, µ) + c(λ, β));
+оно совпадает с формулой (26) для символа произведения, которую я постулировал
+ранее.
+Обращаю внимание на то, что моя конструкция невероятно общая. Векторы eα
+и e′
+β могли быть выбраны практически произвольным образом, лишь бы их было
+достаточно много для того, чтобы это была переполненная система.
+Таким образом можно построить громадное количество конкретных примеров.
+Важно лишь, чтобы были достаточно простые выражения для скалярного произве-
+дения ⟨eα, e′
+β⟩ и для функции r(λ, µ).
+Если L = L′ — гильбертово пространство фоковского представления алгебры Вей-
+ля, то можно взять в качестве eα пуассоновы векторы eα = eαˆa∗|0⟩, которые мы уже
+рассматривали. В этом случае c(α, β) = r(α, β) = ⟨α, β⟩. Это просто вычислить, по-
+тому что все интегралы гауссовы.
+Замечу, что помимо гильбертова пространства, соответствующего случаю обыч-
+ной квантовой механики, можно рассматривать и банаховы пространства. Например,
+взять в качестве L и L′ двойственные друг к другу пространства Lp и Lq = (Lp)∗, где
+1
+p + 1
+q = 1.
+5.2
+L-функционалы и функциональные интегралы
+Перейдем теперь к L-функционалам.
+Для L-функционала можно дать два определения, потому что для алгебры Вей-
+ля было дано два определения. Одно из определений алгебры Вейля состояло в том,
+что есть операторы ak и a+
+k , которые подчиняются каноническим коммутационным
+соотношениям, и это была алгебра полиномов этих операторов. В другом определе-
+нии я рассматривал в качестве основных элементов унитарные операторы, которые
+выражены как экспоненты линейных выражений.
+В этом определении красивые формулы будут получаться, если взять:
+в качестве L′ — алгебру Вейля с операторами e′
+α = Vα, представляющими собой
+экспоненты линейных выражений;
+в качестве L — линейные функционалы на алгебре Вейля, а eβ — экспоненты
+линейных выражений.
+Другой вариант определения L-функционала:
+LK(α∗, α) = Tre−αa+eα∗aK,
+(30)
+где αa+ = � αka+
+k и α∗a = � α∗
+kak.
+В первом определении был след от экспоненты линейного выражения, а здесь —
+след от произведения экспонент. Эта разница невелика: они отличаются численным
+множителем, но при таком определении понятие L-функционала я могу сказать, что
+L-функционал — это просто производящий функционал для корреляционных функ-
+ций. Можно разложить по α и α∗, после чего в пределе α = α∗ = 0 будет получен
+след от произведения некоторого полинома по a и a+ на матрицу плотности K. Это
+как раз и есть корреляционная функция (по определению).
+Что можно еще сделать? Можно считать индекс k не дискретным, а непрерывным
+параметром, когда канонические коммутационные соотношения принимают вид:
+[a(k), a+(k′)] = ℏδ(k, k′),
+50
+
+[a(k), a(k′)] = [a+(k), a+(k′)] = 0,
+где k, k′ ∈ M. Для того чтобы объединить дискретный случай с непрерывным в од-
+ном выражении, будем считать, что M — пространство с мерой. В дискретном случае
+здесь стоят сумма и δ-символ Кронекера, а в случае, когда индексы k, k′ непрерыв-
+ны — интеграл и δ-функция. В случае когда α — это квадратично интегрируемая
+функция, выражение для LK хорошо определено. Если бы была экспонента от ли-
+нейного выражения (не произведение экспонент, а экспонента от суммы), то тогда все
+будет конечно, потому что оператор унитарен. Если стоит произведение экспонент, но
+функция α интегрируема с квадратом, то будет тоже все хорошо, потому что просто
+появится дополнительный конечный множитель.
+Перейдем к вопросу о действии алгебры Вейля A на пространстве L-функционалов
+L. Зададимся вопросом, что такое L-функционал? Это, по существу, положительный
+линейный функционал на алгебре Вейля. Как я уже говорил, если рассматривать
+просто произвольный линейный функционал на любой алгебре, то всякий элемент
+алгебры порождает два оператора на пространстве функционалов. Один из них по-
+лучается, если аргумент умножить слева на что-то, а другой получится если умно-
+жить справа на что-то. Когда множитель ставится слева, берется еще и операция
+сопряжения. Таким образом, из каждого элемента алгебры получается два операто-
+ра на функционалах: первый из них был обозначен тем же символом, как и элемент
+алгебры, а другой — этим же символом, но с тильдой.
+Действие алгебры Вейля A на пространстве L-функционалов L реализуется опера-
+торами b и b+, действие которых на функционалы LK сводится к умножению матрицы
+плотности на операторы a+ и a справа:
+b(k)LK = LKa+(k),
+b+(k)LK = LKa(k).
+Нетрудно проверить, что эти операторы удовлетворяют каноническим коммутацион-
+ным соотношениям и могут быть представлены в следующем виде
+b+(k) = −ℏc+
+2 (k) + c1(k),
+b(k) = −c2(k),
+где c+
+i (k) — операторы умножения на α∗
+k для i = 1 и на αk для i = 2, а ci(k) —
+производные, взятые, соответственно, по α∗
+k и αk.
+Альтернативное действие A на L реализуют операторы с тильдой, действие кото-
+рых на функционалы LK сводится к умножению матрицы плотности на операторы a
+и a+ слева:
+˜b(k)LK = La(k)K,
+˜b+(k)LK = La+(k)K.
+Эти операторы также удовлетворяют каноническим коммутационным соотношениям.
+Они могут быть представлены в виде:
+˜b+(k) = ℏc+
+1 (k) − c2(k),
+˜b(k) = c1(k).
+Таким образом, на L-функционалах действует два экземпляра алгебры Вейля.
+Происходит то, что физики называют удвоением полей.
+Рассмотрим теперь формальный гамильтониан ˆH
+ˆH =
+�
+m,n
+�
+ki,lj
+Hm,n(k1, ...km|l1, ..., ln)a+
+k1...a+
+kmal1...aln,
+(31)
+выраженный через операторы рождения и уничтожения и приведенный к нормаль-
+ной форме (то есть, все операторы рождения перемещены налево). Напомню, что в
+теории, которую мы рассматриваем в алгебраическом подходе, формальные гамиль-
+тонианы могут не иметь смысла как операторы, но соответствующие уравнения дви-
+жения могут иметь смысл. Гамильтониану ˆH отвечают два формальных оператора,
+51
+
+действующего на L-функционалах (точнее говоря, на всех линейных функционалах
+на алгебре Вейля):
+ˆH =
+�
+m,n
+�
+ki,lj
+Hm,n(k1, ...km|l1, ..., ln)b+
+k1...b+
+kmbl1...bln,
+(32)
+˜H =
+�
+m,n
+�
+ki,lj
+Hm,n(k1, ...km|l1, ..., ln)˜b+
+k1...˜b+
+km˜bl1...˜bln.
+(33)
+Один из них обозначен тем же символом, другой — символом с тильдой. И теперь
+можно записать уравнение движения для L-функционала: L(α∗, α):
+iℏdL
+dt = HL = ˜HL − ˆHL,
+(34)
+где введено обозначение H = ˜H − ˆH.
+Еще следует заметить, что если рассматривать трансляционно инвариантный га-
+мильтониан, то в импульсном представлении коэффициенты Hm,n будут содержать
+δ-функции δ(k1 + ... + km − l1 − ... − ln), задающие закон сохранения импульса.
+Уравнения для L-функционала имеют то преимущество, что они имеют смысл да-
+же в том случае, когда уравнение движения в фоковском пространстве плохо опреде-
+лено. Дело в том, что в фоковском пространстве обычно все хорошо, когда мы живем
+в конечном объеме, а когда мы живем в бесконечном объеме, то все плохо. Тот эф-
+фект неэквивалентности различных представлений канонических коммутационных
+соотношений, который обсуждался, всегда проявляется, а для L-функционалов, если
+нет ультрафиолетовых расходимостей, то все в порядке.
+Возвращаясь к выписанным ранее выражениям для ˆH и ˜H, можно сказать, что
+мы оказываемся в абсолютно привычной обстановке. Действительно, имеется пред-
+ставление двух алгебр Вейля (можно считать, что это одна алгебра Вейля, только
+большая). В курсе квантовой теории поля обычно все делается по теории возмуще-
+ний. Берется оператор эволюции в представлении взаимодействия и выписывается
+T-экспонента в качестве оператора эволюции в представлении взаимодействия и из
+этого выводится диаграммная техника. В случае L-функционалов ничего принципи-
+ально не изменилось. Существуют те же самые коммутационные соотношения. Можно
+применять ровно ту же технику.
+Единственное, что изменилось — удвоилось количество полей и не стало гиль-
+бертова пространства, но то, что мы жили в гильбертовом пространстве, нигде не
+использовалось и поэтому в формализме L-функционалов вся стандартная техника
+из курса квантовой теории поля работает. Та техника функциональных интегралов,
+о которой я рассказал, тоже работает. Таким образом, формализм L-функционалов
+с точки зрения вычисления ничем не хуже, чем обычный. На самом деле, он лучше.
+Как я объяснял, он уничтожает проблемы с тривиальными объемными расходимостя-
+ми. Также он существенно лучше, если рассматривать адиабатическое приближение.
+Сейчас мы в этом убедимся.
+Давайте рассмотрим семейство “гамильтонианов”. То, что я говорю, относится не
+только к L-функционалам, а вообще к любой ситуации в геометрическом подходе к
+квантовой теории. Например, рассмотрим семейство, отвечающее теории возмущений
+H(g) = H0+gV . Я утверждаю, что если рассматривать выражение ω(g(t)), являюще-
+еся стационарным состоянием для “гамильтониана” H(g(t)), то тогда ω(g(t)) является
+решением уравнения движения для нестационарного “гамильтониана”.
+Действительно, при рассмотрении адиабатического приближения в случае обыч-
+ной квантовой механики стационарное состояние оставалось стационарным с тече-
+нием времени, но поскольку мы жили в гильбертовом пространстве, состояние было
+52
+
+определено только до численного множителя, и на собственном векторе нарастал
+численный (фазовый) множитель. Здесь же этого нет. Этому есть совершенно триви-
+альная причина — просто можно пренебречь производной по времени ˙g(t). Это дает
+возможность написать следующую формулу:
+ω(g) = lim
+α→0 σα(0, −∞)ω(0).
+(35)
+Чтобы ее получить я рассматриваю гамильтониан H0 + ge−α|t|V , в котором введен
+адиабатический множитель, исчезающий на бесконечности. Тогда я могу сказать, что
+если у меня есть какой-то начальный “гамильтониан” H0 (обычно берется свободный,
+но это не обязательно) и имеется стационарное состояние для этого “гамильтониана”,
+то формула (35) описывает стационарное состояние “гамильтониана” с константой g.
+Такая же формула есть в обычной квантовой механике, только там стоит фазовый
+множитель. В обычной квантовой механике данная формула обычно применяется ко-
+гда ω(0) является основным состоянием и тогда ω(g) будет основным состоянием для
+константы связи g. В нашем подходе можно применять эту формулу в несравненно
+более общей ситуации. Например, если гамильтониан трансляционно инвариантен,
+можно применять эту формулу к любому трансляционно инвариантному стационар-
+ному состоянию свободного гамильтониана H0, для которого все такие состояния лег-
+ко вычислить.
+В результате мы получаем стационарное трансляционно инвариантное состояние
+в случае, когда константа связи равна g. Такой подход очень естественен как в равно-
+весной, так и в неравновесной статистической физике. Можно взять в качестве ω(g),
+скажем, равновесное состояние для какой-то температуры и, применив этот процесс,
+получить равновесное состояние, правда, при другой температуре, но с той же эн-
+тропией, потому что адиабатический процесс не меняет энтропии. В неравновесной
+статистической физике есть формализм Келдыша, тесно связанный с формализмом
+L-функционалов. В нем, по существу, применяется эта же формула, но, может быть,
+с несколько меньшим основанием.
+Наряду с оператором эволюции σα(0, −∞) можно при заданном α рассматривать
+оператор эволюции σα(+∞, −∞), представляющий собой адиабатическую S-матрицу.
+Если адиабатический параметр α стремиться к нулю, в формализме L-функционалов
+адиабатическая S-матрица, умноженная на некоторые множители (которые я не хочу
+здесь описывать, но это множители, которые связаны с уровнями энергии) стремится
+к инклюзивной матрице рассеяния. Инклюзивная матрица рассеяния — это основа
+для расчета инклюзивного сечения. Этот вопрос мы обсудим позже. Я только хо-
+чу сказать, что обычная матрица рассеяния в квантовой механике тоже может быть
+получена из адиабатического оператора эволюции. Голые частицы при этом превра-
+щается в одетые, происходит перенормировка волновой функции, но не происходит
+перенормировки константы связи.
+Сейчас я хочу объяснить некие вещи, к которым я потом вернусь с другой точки
+зрения.
+Хорошо известно (я буду доказывать это через некоторое время), что матрица
+рассеяния выражается через функции Грина. Это то, что называется формулой Ле-
+мана — Симанчика — Циммермана. Но если я хочу рассмотреть инклюзивную матри-
+цу рассеяния, я должен рассмотреть то, что можно называть обобщенной функцией
+Грина.
+Обычная функция Грина определяется как хронологическое произведение
+M = T(B∗
+1(x1, t1) . . . B∗
+n(xn, tn)) = T(B∗).
+усредненное по какому-то состоянию. В хронологическом произведении все времена
+53
+
+идут в порядке убывания слева направо. Можно точно так же рассмотреть и анти-
+хронологическое произведение (времена возрастают)
+N = T opp(B1(x′
+1, t′
+1) . . . Bn(x′
+n, t′
+n)) = T opp(B).
+В обобщенных функциях Грина берется хронологическое произведение операто-
+ров, умножается на антихронологическое произведение операторов и затем берется
+среднее по какому-то состоянию:
+Это обобщенная функция Грина в данном состоянии. Именно эти функции Грина
+появляются в формализме Келдыша, и, как сейчас будет объяснено, они же появля-
+ются в формализме L-функционалов.
+Напомню, что в L-функционалах есть операторы без тильды и операторы с тиль-
+дой. Справедлива формула
+(T( ˜BB)ω)(x) = ω(T(B∗)xT opp(B)) = ω(MxN).
+(36)
+В левой части здесь стоит обычное хронологическое произведение, однако операторы
+без тильды действуют, умножая справа. При этом меняется порядок и хронологиче-
+ское произведение превращается в антихронологическое.
+Заметим, что формула (36) справедлива не только для L-функционалов (линей-
+ных функционалов на алгебре Вейля), но и для линейных функционалов на любой
+ассоциативной алгебре с инволюцией.
+Если в выписанном выражении положить x = 1, то в точности получится ω(MN).
+Здесь M определено как хронологическое произведение, а N — как антихронологи-
+ческое произведение, так что T-произведение на L-функционалах переходит в хро-
+нологическое и антихронологическое произведения. Рассматривая эту картинку в
+L-функционалах, я могу применять для вычисления обобщенных функций Грина
+технику вычисления обычных функций Грина. Это дает диаграммную технику для
+обобщенных функций Грина.
+54
+
+6
+Лекция 6
+6.1
+Солитоны как аналоги частиц
+Я собираюсь дать определение квантовых частиц и квазичастиц. Основное утвержде-
+ние состоит в том, что понятие частицы вторично. Я определяю частицу как элемен-
+тарное возбуждение основного состояния. Можно также рассматривать элементарное
+возбуждение любого стационарного трансляционно инвариантного состояния, тогда
+элементарное возбуждение — это квазичастица.
+Прежде всего, я хочу рассказать о классическом аналоге всего этого. Это понятия
+солитона и обобщенного солитона. Рассмотрим трансляционно инвариантный гамиль-
+тониан в бесконечномерном фазовом пространстве, которое состоит из векторнознач-
+ных функций f(x), где x ∈ Rd — это пространственные координаты. Я считаю, что
+пространственные трансляции действуют как сдвиги этих координат, а временные
+определяются гамильтонианом, который коммутирует с пространственными транс-
+ляциями. Предположим, что соответствующее уравнение движения имеет вид
+∂f
+∂t = Af + B(f),
+где A — линейный оператор и B — нелинейная часть. Считаем, что нелинейная часть
+как минимум квадратична; тогда для малых f доминирует линейная часть. В част-
+ности, мы можем сказать, что f ≡ 0 — это решение, и в его окрестности можно
+пренебречь нелинейной частью.
+Теперь мы определяем солитон как решение, которое имеет вид s(x − vt). Соли-
+тон — это такой горбик. Мы можем изображать решение f ≡ 0 как горизонтальную
+прямую, тогда солитон (уединенная волна) — это решение, где этот горбик двигает-
+ся равномерно без изменения формы. Есть еще понятие обобщенного солитона. Это
+горбик, который двигается, с поостоянной средней скоростью но одновременно может
+пульсировать, менять свою форму. Я не буду говорить об этом понятии подробно.
+Я требую, чтобы солитон имел конечную энергию. Для трансляционно инвариант-
+ного решения понятие энергии бессмысленно — для него можно говорить о плотности
+энергии, но я буду считать его энергию равной нулю и отсчитывать энергию солитона
+от энергии трансляционно инвариантного состояния. То, что энергия конечна, озна-
+чает, грубо говоря, что солитон более или менее сосредоточен в какой-то конечной
+области.
+Есть гипотеза, которая была высказана в моей работе с Фатеевым и Тюпкиным
+лет сорок пять тому назад. Она состоит в том, что для очень многих систем и по-
+чти для всех начальных условий с конечной энергией при временах стремящихся к
+плюс или минус бесконечности мы получаем несколько солитонов и еще нечто, что
+удовлетворяет линейному уравнению, по крайней мере, приблизительно. Это хорошо
+известно для интегрируемых систем в случае d = 1; мы предположили, что это вер-
+но без предположения интегрируемости в любой размерности. Не думаю, что нашу
+работу прочел кто-нибудь из математиков, но эта гипотеза также была высказана в
+других работах. В работе Соффера это называется “grand conjecture”, в работе Тао –
+“soliton resolution conjecture”. Тем не менее гипотеза так и остается гипотезой. Пока
+что это остается вне пределов существующей математики за исключением случая,
+когда солитоны отсутствуют изначально. В этом случае можно предполагать, что в
+конце мы асимптотически получим решение линейного уравнения.
+Это утверждение можно обосновать следующим образом. Пусть имеются какие-
+то отличные от нуля начальные условия. В этом случае происходит обычно то, что в
+старых учебниках квантовой механики называлось расплыванием волнового пакета.
+55
+
+То есть если начальные данные были где-то сосредоточены, то дальше они расплы-
+ваются. При этом энергия все-таки сохраняется, поэтому это расплывание приводит
+к тому, что амплитуда волны уменьшается. Если же она действительно все время
+уменьшается, то, как я говорил, в случае, малых амплитуд нелинейной частью можно
+пренебречь и решение нелинейного уравнения приближается к решению линейного.
+Конечно, это не происходит, если в теории есть солитон. Амплитуда солитона по
+определению не меняется. Какой высоты был горбик, таким он и остается, но можно
+думать, что в конце концов останутся какие-то солитоны или обобщенные солито-
+ны, а также хвостик, который приблизительно удовлетворяет линейному уравнению.
+Конечно, никаким доказательством эти рассуждения не являются, но это очень ве-
+роятная гипотеза. Нет сомнения, что это верно не всегда, но естественно думать, что
+это верно в очень многих случаях.
+Раз есть такая картинка, которую я описал, нужно думать, что есть понятие рассе-
+яния солитонов. Есть на свете точно решаемые модели размерности 1+1 (одномерное
+пространство и время); для таких моделей мое утверждение — это теорема. Что про-
+исходит? Сталкиваются два таких горбика, образуется некая каша, а потом снова воз-
+никают те же самые солитоны: это специфика интегрируемых случаев. Стандартная
+ситуация в неинтегрируемых случаях чуточку другая: после столкновения возника-
+ют, возможно, другие солитоны и еще, может быть, то, что я назвал словом хвостик.
+Хвостик — это образование, которое асимптотически стремится к решению линейного
+уравнения.
+Теперь я попытаюсь облечь эти общие рассуждения в некую форму. Обозначим
+пространство возможных начальных данных буквой R, и тогда наложенное условие
+означает для общего случая, что, задав какие-то начальные условия, в конце мы полу-
+чим солитоны и хвостик. Солитоны характеризуются какими-то данными, а хвостик
+— решением линейного уравнения. Формально на математическом языке сформули-
+рованная гипотеза означает, что на плотном множестве начальных данных можно
+определить отображение D+(t) : R → Ras начальных данных в асимптотические
+данные на плюс бесконечности (t → +∞). Я мог бы взять и минус бесконечность:
+D−(t) : R → Ras. Единственное отличие в том, что уже нельзя будет говорить слово
+“начальные данные”.
+Теперь я предполагаю, что есть и обратное отображение, то есть исходя из асимп-
+тотики можно найти начальные условия или, по крайней мере, доказать, что заданная
+асимптотика получается из каких-то начальных условий. То есть, я хочу рассмотреть
+обратные операторы S(t, +∞) = (D+(t))−1 и S(t, −∞) = (D−(t))−1.
+Здесь возникает задача : определить из асимптотических данных решения уравне-
+ния и тем самым начальные данные. Это, по-видимому, нетрудно, но никто этого не
+сделал. Я спросил у Тао, известно ли это? Он сказал, что да, наверное, это нетрудно
+сделать, но на тему о том, что кто-нибудь это сделал, он мне ничего не написал. Я
+думаю, что это интересная и не очень трудная задача: по асимптотическим данным
+построить решение. В квантовом случае решение этой задачи хорошо известно – это
+то, что называется теорией рассеяния Хаага — Рюэля; я буду рассказывать обобщение
+этой теории.
+Теперь я могу определить то, что следует назвать нелинейной матрицей рассеяния:
+S = S(0, +∞)−1S(0, −∞) : Ras → Ras.
+Грубо говоря, мы задаем начальные условия на минус бесконечности и смотрим
+асимптотику на плюс бесконечности. Можно надеяться, что есть предельный переход
+от квантовой матрицы рассеяния к этой нелинейной. Отмечу еще раз, что приведен-
+ные рассуждения — это вещи гипотетические.
+Еще одно маленькое замечание заключается в том, что если теория лоренц-инвариантна,
+то можно применить преобразование Лоренца к солитону и снова получится солитон.
+56
+
+Солитоны ходят семействами — солитоны с разными скоростями. То же верно и для
+преобразований Галилея. Завершая обсуждение, хочу сказать, что классический со-
+литон нужно рассматривать как модель квантовой частицы. В квантовой механике
+понятие частицы — это асимптотическое понятие: если две частицы сталкиваются,
+образуется некая каша, которая потом распадается на частицы.
+Следующие рассуждения еще сильнее подчеркивают аналогию солитонов с кван-
+товыми частицами в том определении, которое я буду давать. Я рассматривал про-
+странство всех начальных данных R. Это симплектическое многообразие. Рассмот-
+рим ситуацию, когда имеются фазовое пространство и гамильтониан, то есть, име-
+ется симплектическое многообразие и оператор эволюции. Также предположим, что
+есть пространственные трансляции, а временные трансляции описываются трансля-
+ционно инвариантным гамильтонианом. Формально математически это означает, что
+на этом симплектическом многообразии M действует коммутативная группа T про-
+странственных и временных трансляций. Теперь возьмем стационарную трансляцион-
+но инвариантную точку m ∈ M этого симплектического многообразия. В предыдущей
+картинке такой точкой было решение f = 0. Это решение трансляционно инвариантно
+и стационарно.
+Определим возбуждение трансляционно инвариантного стационарного состояния
+как состояние с конечной энергией (напомню, что мы считаем энергию трансляционно
+инвариантного состояния равной нулю).
+Теперь я определю элементарное симплектическое многообразие как такое мно-
+гообразие, в котором в координатах Дарбу p, x пространственные сдвиги действует
+просто как сдвиги x → x + a, при этом p остается неизменным. Мы считаем, что
+гамильтониан инвариантен относительно этих трансляций. Это означает, что он за-
+висит только от p. Обозначим его как ǫ(p). Тогда временные сдвиги будут просто
+сдвигами с определенной скоростью x → x+v(p)t, где скорость v рассчитывается по
+формуле v(p) = ∇ǫ(p).
+Предположим теперь, что M представляет собой пространство вектор-функций
+f(x), где x ∈ Rd, а пространственные трансляции действуют как сдвиги x → x+a. Ес-
+ли взять симплектическое вложение элементарного симплектического пространства
+во множество возбуждений в M и потребовать, чтобы вложение коммутировало с
+пространственно-временными трансляциями, то возникнет семейство солитонов. Это
+очень просто объяснить. При вложении точка (p, 0) переходит в некоторую функцию
+sp(x), зависящую от p, а поскольку вложение E → M коммутирует с трансляциями,
+то точка (p, a) будет переходить в сдвинутую функцию sp(x + a). Условие того, что
+отображение E → M коммутирует с временными сдвигами, как раз и будет означать,
+что функция sp(x − v(p)t) удовлетворяет уравнению движения.
+6.2
+Частицы и квазичастицы
+Теперь введем понятие частицы и более общее понятие квазичастицы. Отличие только
+в том, что частица — это возбуждение основного состояния, а квазичастица — это
+возбуждение любого трансляционно инвариантного стационарного состояния. Для
+того, чтобы определить понятие частицы, мне нужны понятия пространственных и
+временных сдвигов.
+В обычной квантовой механике если мы рассматриваем эволюцию, нужно иметь
+понятие временного сдвига Tτ. Я хочу, чтобы были еще пространственные сдвиги
+Ta, которые действовали бы на состояния и коммутировали с временными сдвигами.
+В геометрическом подходе пространство состояний это основная вещь, но здесь мне
+удобно рассматривать ненормированные состояния. Напомню, что в алгебраическом
+подходе состояниями были положительные функционалы, нормированные условием,
+57
+
+что функционал от единицы равен единице. Это условие нормировки я отброшу, и
+в таком случае получается конус, которые я обозначаю буквой C. Состояние теперь
+определено только с точностью до численного множителя. Я все время буду говорить
+про этот конус ненормированных состояний. Пространственно-временные трансляции
+должны действовать на этом конусе.
+Обозначим теперь группу пространственно-временных трансляций как T . В алгеб-
+раическом подходе эта группа должна действовать на самой алгебре A. Симметрия в
+алгебраическом подходе — это автоморфизм алгебры, то есть имеется гомоморфизм
+группы T в группу автоморфизмов алгебры A. Группа автоморфизмов алгебры (и,
+следовательно, группа T ) действует на C (напомню, что мы всегда считаем, что ав-
+томорфизмы согласованы с инволюцией).
+Я ввожу стандартное обозначение, A(τ, x) = TτTxA для элемента A ∈ A сдвинуто-
+го по времени и пространству. Трансляционно-инвариантное стационарное состояние
+ω в алгебраическом подходе удовлетворяет условию ω(A(τ, x)) = ω(A).
+В частности, давайте рассмотрим алгебру Вейля A в координатном представ-
+лении, когда заданы генераторы ˆa∗(x), ˆa(x). Будем считать, что пространственные
+сдвиги просто сдвигают аргумент, а временные сдвиги определяются формальным
+гамильтонианом, который выражен через ˆa∗(x), ˆa(x) с какими-то коэффициентными
+функциями, зависящими только от разностей xi−xj. Тем самым обеспечена трансля-
+ционная инвариантность. Еще я потребую, чтобы коэффициентные функции быстро
+убывали. Тогда уравнение движения будет иметь смысл.
+Я могу сделать преобразование Фурье и перейти к импульсному представлению.
+Тогда аргументом будет k и пространственные сдвиги сведутся к умножению на экс-
+поненты вида exp(±ika), а временные сдвиги будут опять определяться гамильто-
+нианом. Условие, что функции в координатном представлении зависят от разности,
+приводит к появлению δ-функций, отвечающих сохранению импульса, а требование,
+чтобы коэффициентные функции быстро убывали, означает, что функции в импульс-
+ном представлении будут гладкими (после выделения δ-функции).
+В общем случае, когда задан конус состояний, я могу определить понятие трансля-
+ционно инвариантного стационарного состояния как состояния, которое не меняется
+ни при пространственных, ни при временных сдвигах. Это будет для меня основной
+объект. Стандартные примеры такого состояния — основное состояние и равновесное
+состояние.
+Теперь я хочу определить понятие возбуждения трансляционно инвариантного
+стационарного состояния как аналог введенного ранее понятия состояния с конечной
+энергией. Когда солитон уходит на бесконечность, мы его перестаем видеть. Говоря
+формально, если имеется возбуждение σ трансляционно инвариантного состояния
+ω ∈ C и к нему применяется пространственный сдвиг, то в пределе a → ∞ остается
+одно ω ∈ C:
+(Taσ)(A) → const · ω(A).
+Тут еще написана константа, потому, что состояние в конусе определено только с точ-
+ностью до численного множителя — там ненормированные состояния. Такое понятие
+возбуждения — это общее понятие, которое может быть применено как в геометри-
+ческом, так и в алгебраическом подходах. В случае солитонов аналогом этого было
+то, что солитон имеет конечную энергию и тем самым более или менее сосредоточен
+в конечной области.
+В алгебраическом подходе по трансляционно инвариантному стационарному со-
+стоянию можно построить с помощью конструкции GNS предгильбертово простран-
+ство H (я хочу жить в предгильбертовом пространстве). В этом пространстве есть
+циклический вектор θ, соответствующий состоянию ω. Напомню, это означает, что ω
+58
+
+— это положительный функционал
+ω(A) = ⟨ ˆAθ, θ⟩,
+где ˆA — оператор, который отвечает A в H.
+Пространственные и временные трансляции Ta и Tτ спускаются на предгильберто-
+во пространство H как унитарные операторы. Трансляции действуют в самой алгебре
+как автоморфизмы алгебры, а мы строили предгильбертово пространство факторизуя
+алгебру неким образом. Это позволяет спустить эти операторы на H как унитарные
+операторы. Дальше мы определяем операторы энергии и импульса как операторы
+инфинитезимальных трансляций по времени и по пространству:
+Ta = ei ˆPa,
+Tτ = e−i ˆ
+Hτ.
+В алгебраическом подходе элементы предгильбертова пространства H можно отож-
+дествить с возбуждениями состояния ω. Физический смысл GNS – конструкции состо-
+ит в том, что она по некоторому трансляционно инвариантному состоянию позволяет
+построить пространство H, в котором живут возбуждения. Это, собственно говоря,
+объяснение, почему эта конструкция так важна в физике.
+То, что я утверждал, не всегда верно. Мне нужно потребовать, чтобы было то,
+что по-английски называется cluster property, а по-русски — распадение корреляций
+или кластеризация.
+Представим себе ферромагнетик. Если спин имеет какое-то направление в начале
+координат, то такое же направление спина будет везде, по крайней мере, статистиче-
+ски. Это тот случай, когда нет распадения корреляций. В более стандартной ситуации
+на больших расстояниях спин уже не помнит, каков был спин в начале координат. Вот,
+это и есть то, что называется кластеризацией.
+Формально математически это можно сформулировать следующим образом. Если
+взять ω(A(τ, x)B), где A, B — два элемента алгебры, и один из этих элементов сдви-
+гать по пространству, оставляя время постоянным, то тогда произойдет распадение
+этого среднего: limx→∞ ω(A(τ, x)B) = ω(A)ω(B)
+Это простейшая форма распадения корреляций. Позже я сформулирую это более
+формально и в более общем виде. Для меня важно в данный момент еще то, что когда
+есть три оператора B′, A и B и один из этих операторов сдвигается на бесконечность,
+то произойдет распадение:
+lim
+x→∞ ω(B′A(τ, x)B) = ω(A)ω(B′B)
+Эти равенства можно продифференцировать:
+lim
+x→∞
+d
+dτ ω(A(τ, x)B) = d
+dτ (ω(A))ω(B).
+Любой вектор из H может быть представлен в виде Bθ ∈ H и тогда для соответ-
+ствующего этому вектору состояния σ(A) справедлива формула:
+σ(A) = ⟨ ˆA ˆBθ, ˆBθ⟩ = ω(B∗AB).
+(37)
+Теперь я буду сдвигать аргумент у A. Когда A будет уходить на бесконечность,
+произойдет кластеризация, и в пределе x → ∞ получится следующее выражение:
+(Txσ)(A) = σ(A(0, x) = ω(B∗A(0, x)B) → ω(A)ω(B∗B),
+При этом ω(B∗B) можно считать константой, то есть, σ(A) является возбуждени-
+ем в том смысле, в котором я его определил, и поэтому в алгебраическом подходе
+59
+
+все элементы предгильбертова пространства H дают возбуждения. В алгебраическом
+подходе я только такие возбуждения и буду рассматривать. На самом деле, я мог
+бы начать с этого момента — определить понятие возбуждения таким образом: взять
+ω, применить конструкцию GNS и взять элементы предгильбертова пространства H.
+Это будут возбуждения в алгебраическом подходе.
+Я сейчас определю понятие элементарного возбуждения трансляционно инвари-
+антного состояния. Элементарное возбуждение — это то, что называется частицей
+в квантовой теории поля, потому что там рассматривается возбуждение основного
+состояния, а квазичастицы — это элементарные возбуждения любого трансляцион-
+но инвариантного состояния. Поскольку я буду рассматривать оба случая, то я буду
+говорить про элементарные возбуждение, но я могу также употреблять термины “ча-
+стица” или “квазичастица”.
+Как понять, что такое элементарное возбуждение в алгебраическом подходе? В
+алгебраическом подходе мы живем в гильбертовом пространстве, которое получается
+с помощью GNS-конструкции. Я хочу понять, что такое частица в этой ситуации.
+У частицы есть импульс. Бывают и другие квантовые числа — они просто будут
+появляться как дискретные индексы, которые мне никак не мешают. Я скажу, что у
+частицы есть состояние с определенным импульсом и обозначу его как Φ(p). Импульс
+этого состояния равняется p:
+ˆPΦ(p) = pΦ(p)
+(38)
+а энергия этого состояния – некоторая функция ǫ(p), которая называется законом
+дисперсии
+ˆHΦ(p) = ǫ(p)Φ(p).
+(39)
+Заметим, что (38), (39) можно переписать в виде
+TaΦ(p) = eipaΦ(p),
+(40)
+TτΦ(p) = e−iǫ(p)τΦ(p)
+(41)
+Важно отметить, что Φ(p) — это не вектор, а обобщенная векторная функция. При
+заданном значении импульса состояние является ненормированным, поэтому нужно
+рассматривать интеграл этой функции с какой-то пробной функцией φ(p)
+Φ(φ) =
+�
+dpφ(p)Φ(p),
+(42)
+Это будет хорошо определенный вектор. Кроме того, обычно накладывается усло-
+вие нормировки на δ-функцию — я его тоже наложу:
+⟨Φ(p), Φ(p′)⟩ = δ(p − p′)
+(43)
+Если рассматривать настоящие векторы, где аргументом является пробная функция,
+то нормировка на δ-функцию будет означать, что скалярное произведение векторов
+равняется скалярному произведению соответствующих пробных функций
+⟨Φ(φ), Φ(φ′)⟩ = ⟨φ, φ′⟩.
+(44)
+Теперь я хочу определить понятие элементарного пространства h как простран-
+ства пробных функций φa(x), принимающих значения в пространстве Cr. Действие
+пространственных трансляций на пробные функции в x-представлении сводится к
+простому сдвигу аргумента, а в импульсном представлении — к умножению на экс-
+поненту eika. Вывести то, как будут вести себя временные сдвиги, можно исходя из
+60
+
+требования, чтобы временной сдвиг коммутировал с пространственным сдвигом. Это
+будет просто умножение на экспоненту e−iE(k)τ от некоторой эрмитовой матрицы
+E(k) размерности r × r. Я могу диагонализовать эту матрицу, тогда будет просто
+умножение на скалярные фазовые множители. Это значит, что всегда можно ограни-
+читься случаем r = 1,
+Элементарное возбуждение трансляционно инвариантного стационарного состо-
+яния ω можно определить как коммутирующее с пространственными и временны-
+ми трансляциями изометричное отображение σ элементарного пространства h в про-
+странство возбуждений.
+Важно отметить, что для скалярного случая r = 1 не сказано ничего нового.
+Действительно, элементарные возбуждения были определены как функции Φ(p), яв-
+ляющиеся собственными для импульса и энергии (38, 39). Есть еще условия норма-
+лизации (43). Я также говорил, что нужно рассматривать пробные функции φ(p),
+тогда Φ(φ) — это отображение пространства пробных функций в пространство воз-
+буждений. Изометричность этого отображения вытекает из условия нормализации. А
+формулы (38, 39) гарантируют, что трансляции вектора Φ(p) отвечают трансляциям
+функции φ(p) и по пространству, и по времени. Таким образом, в алгебраическом
+подходе при r = 1 можно просто положить σ(φ) = Φ(φ).
+В геометрическом подходе я должен рассматривать в качестве пространств состо-
+яний конусы — там никакой надежды на линейность быть не может. Если мы исходим
+из теории в алгебраическом подходе, то функции φ из элементарного пространства
+можно сопоставить состояние
+(σ′(φ))(A) = ⟨AΦ(φ), Φ(φ)⟩,
+и тогда оказывается, что σ′ — это квадратичное (точнее, эрмитово) отображение эле-
+ментарного пространства в конус, которое коммутирует со всеми трансляциями.
+Это замечание подсказывает, что в геометрическом подходе нужно определять
+элементарное возбуждение как коммутирующее с трансляциями отображение элемен-
+тарного пространства в конус (условие эрмитовости необязательно, но часто удобно).
+Если Φ(φ) в алгебраическом подходе принадлежало пространству H, а θ — цик-
+лический вектор в этом пространстве, то Φ(φ) получается применением какого-то
+элемента из алгебры к циклическому вектору: Φ(φ) = B(φ)θ. Тогда можно легко
+убедиться в справедливости формулы
+σ′(φ) = L(φ)ω,
+(45)
+где L(φ) = ˜B(φ)B(φ).
+Напомню, что в алгебраическом подходе на пространстве функционалов были
+определены два вида операторов: один отвечает умножению аргумента слева, дру-
+гой отвечает умножению аргумента справа. На один из них я поставил волну.
+Существование оператора L(φ), удовлетворяющего соотношению (45), удобно вклю-
+чить в определение элементарного возбуждения в геометрическом подходе.
+Я уже говорил, что при построении теории рассеяния важна только трансляцион-
+ная инвариантность, но если, скажем, мы имеем дело с лоренц-инвариантной теорией,
+естественно думать, что, по крайней мере, вакуумный вектор лоренц-инвариантен.
+Тогда в пространстве возбуждений действует вся группа Пуанкаре, и элементарное
+пространство будет являться представлением этой группы. Отмечу, что в локальной
+квантовой теории поля лоренц-инвариантные частицы определяются как неприводи-
+мые представления группы Пуанкаре. Здесь тоже нужно требовать, чтобы представ-
+ление было неприводимо или, по крайней мере, было суммой неприводимых.
+Сделаем теперь следующее замечание. Пусть в нерелятивистской квантовой ме-
+ханике задан трансляционно инвариантный гамильтониан. В этом случае нет по-
+61
+
+тенциального поля, гамильтониан инвариантен относительно преобразований Гали-
+лея, а энергия элементарного возбуждения задается привычной формулой: ǫ(p) =
+p2/2m + const.
+Если же взять оператор ˆa∗(p) и применить к трансляционно инвариантному фо-
+ковскому вакууму |0⟩, то получится элементарное возбуждение фоковского вакуума
+Φ(p) = ˆa∗(p)|0⟩.
+Это частица, но кроме такой частицы есть еще другие частицы, которые тоже удо-
+влетворяют наложенным условиям, например, связанные состояния.
+Что такое связанное состояние? Гамильтониан действует на состояния с любым
+числом частиц. Возьмем n частиц и отделим движение центра инерции. Если рас-
+смотреть спектр гамильтониана в этом пространстве, то там могут оказаться норми-
+рованные состояния. Они называются связанными состояниями.
+Говоря формально, если не отделять центр инерции и суммарный импульс равен
+нулю, в выражении для элементарного возбуждения будет присутствовать δ-функция
+от суммы импульсов:
+�
+dp1...dpnΨ(p1, ...pn)δ(p1 + ... + pn)ˆa∗(p1)...ˆa∗(pn)|0⟩.
+Пусть теперь это состояние может двигаться. У него может быть импульс; легко
+понять, что такое движущееся связанное состояние будет описываться обобщенной
+функцией
+Φ(p) =
+�
+dp1...dpnΨ(p1, ...pn)δ(p − p1 − ... − pn)ˆa∗(p1)...ˆa∗(pn)|0⟩,
+которая удовлетворяет всем выписанным условиям. То есть, это элементарное воз-
+мущение в моем смысле, и для меня такие связанные состояния ничем не хуже, чем
+элементарные возбуждения с n = 1. Кстати говоря, в свое время была такая про-
+блема: как рассматривать рассеяние составных частиц? В общую теорию, которую я
+буду излагать, это все вполне укладывается.
+Обращаю внимание, что в определении элементарного возбуждения — одноча-
+стичного состояния Φ(φ) = B(φ)θ я использовал только понятие пространственных и
+временных трансляций. Если же я хочу определить понятие рассеяния, то мне нужно
+определить двухчастичные состояния, а также многочастичные состояния.
+Для того чтобы понять, как разумно определить понятие двухчастичного состо-
+яния, я хочу рассмотреть то, как изменяется одночастичное состояние со временем.
+Я рассматриваю одночастичное состояние и смотрю за динамикой пробной функ-
+ции из h. Напоминаю, что у меня действие оператора эволюции на пробные функции
+соответствуют действию оператора эволюции на векторы:
+TτΦ(φ) = Φ(Tτφ).
+Таким образом при эволюции одночастичного состояния происходит умножение проб-
+ной функции из h на матричную экспоненту
+(Tτφ)(k) = e−iE(k)τφ(k).
+Если я считаю для простоты, что пробные функции принимают значения в комплекс-
+ных числах, то пробная функция просто умножается на фазовый множитель:
+(Tτφ)(k) = e−iǫ(k)τφ(k).
+62
+
+(В более общем случае я просто могу разложить на одномерные собственные про-
+странства — все будет то же самое, так что это — не ограничение.) Переходя от
+импульсного пространства к координатному, я должен сделать преобразование Фу-
+рье. В результате получается, что сдвиг по времени в координатном пространстве
+дается интегралом
+(Tτφ)(x) =
+�
+dkeixk−iǫ(k)τφ(k).
+При больших |τ| фазовый множитель велик. Я могу применить метод стационарной
+фазы, и тогда у меня возникнут уравнения:
+x
+τ = ∇ǫ(k)
+(46)
+Ясно, что нужно рассматривать только ситуацию, когда φ(k) ̸= 0.
+Теперь я определю множество U как окрестность множества точек x, где выпол-
+нено условие (46) с τ = 1. После этого я замечу, что вне множества τU уравнение (46)
+не имеет решения, поэтому функция (Tτφ)(x очень мала при x /∈ τU.Я говорю, что
+τU — это существенный носитель волновой функции рассматриваемого состояния в
+координатном представлении для больших |τ|. Более аккуратно я расскажу об этом
+в следующей лекции.
+63
+
+7
+Лекция 7
+7.1
+Одночастичные и многочастичные состояния
+В прошлой лекции мы определили элементарное пространство h как пространство
+пробных функций φa(x), где пространственные перемещения действуют, сдвигая ар-
+гумент. Пробные функции принимают значения в Cr. Для определенности мы пред-
+полагаем, что пробные функции принадлежат пространству S гладких быстро убы-
+вающих функций.
+В импульсном представлении пространственные сдвиги действуют как умножение
+на eika, а временные сдвиги как умножение на e−iE(k)τ. (Это следует из предположе-
+ния, что временные сдвиги являются унитарными операторами, коммутирующими с
+пространственными перемещениями.) Здесь E(k) обозначает эрмитову r ×r матрицу.
+Диагонализуя матрицу E(k), мы можем свести общий случай к случаю r = 1.
+Напомню теперь, что такое элементарное возбуждение. Элементарное возбужде-
+ние трансляционно инвариантного стационарного состояния ω (квазичастица) зада-
+ется отображением σ из h в пространство возбуждений. Это отображение должно
+коммутировать с трансляциями (и пространственными, и временными).
+Нужно еще сказать, что такое пространство возбуждений. В алгебраическом под-
+ходе — это предгильбертово пространство H, которое получается с помощью кон-
+струкции Гельфанда, Наймарка и Сигала (GNS), где мы исходим из некоторого со-
+стояния, являющегося трансляционно инвариантным и по пространственным, и по
+временным переменным (то есть, из стационарного трансляционно инвариантного
+состояния). Оно представляется циклическим вектором, обозначенным буквой θ.
+Обсуждаемое отображение σ дает некоторый вектор σ(φ),который в прошлой лек-
+ции был обозначен Φ(φ). Оно является отображением в пространство H, и это значит,
+что существует элемент B(φ) из алгебры, применив который к циклическому вектору,
+получаем наш вектор
+σ(φ) = B(φ)θ.
+(47)
+Я предполагаю еще, что отображение σ изометрично.
+Хочу подчеркнуть, что оператор B(φ) существует, но он отнюдь не единственный,
+его нужно как-то выбрать. Я буду налагать на него некоторые условия, которые
+позволят мне развить теорию рассеяния. В частности, я буду требовать, чтобы он
+был линеен по φ. Как я объяснял, каждому вектору в пространстве представления
+алгебры отвечает состояние. В данном случае это состояние записывается формулой
+(σ′(φ))(A) = ⟨Aσ(φ), σ(φ)⟩.
+Если вектор представлен в виде (47), справедлива следующая формула:
+σ′(φ) = L(φ)ω,
+(48)
+где
+L(φ) = ˜B(φ)B(φ).
+(49)
+Итак, в алгебраическом подходе у меня появилось, некоторое отображение σ′ : h → C,
+действующее в соответствии с формулой (48).
+В геометрическом подходе нужно забыть про алгебру, но при этом остается конус
+всех состояний C. По определению отображение σ′ элементарного пространства h в ко-
+нус C задает элементарное возбуждение если оно коммутирует с пространственными
+и временными трансляциями.
+64
+
+Мы постулируем, что так же, как и в алгебраическом случае, отображение σ′(φ)
+получается в соответствии с (48) действием некоторого оператора L(φ) на транс-
+ляционно инвариантное стационарное состояние ω. Изначально рассматриваемое в
+алгебраической ситуации отображение σ(φ) было линейным, а вот отображение σ′(φ)
+отнюдь не линейное. Действительно, из формулы (49) следует, что в алгебраическом
+подходе L(φ) — это квадратичное выражение, или, точнее говоря, эрмитово, потому
+что здесь по одной переменной имеет место линейность, по другой — антилиней-
+ность. (Отображение f называется эрмитовым, если его можно представить в виде
+f(x) = F(x, x∗), где F(x, y) линейно по первому аргументу и антилинейно по второ-
+му.) Естественно потребовать чтобы в геометрическом подходе L(φ) удовлетворяло
+тем же условиям. В дальнейшем я просто буду употреблять слово “квадратичное”
+вместо “эрмитово”, но следует понимать, что на самом деле оно не совсем квадратич-
+ное.
+Если кому-то больше нравится работать с линейными отображениями, то это мож-
+но сделать с помощью следующей общей алгебраической конструкции. Для каждо-
+го линейного комплексного пространства E в тензорном произведении E ⊗ ¯E этого
+пространства на комплексное сопряженное можно сконструировать конус C(E) как
+минимальный конус, содержащий все элементы вида e ⊗ ¯e. (Черта здесь обозначает
+комплексное сопряжение. В дальнейшем мы будем иметь дело только с комплексными
+пространствами.)
+Конус C(h) я называю элементарным конусом. Он отвечает элементарному про-
+странству h, при этом σ′ можно рассматривать как линейное отображение σ′ : C(h) →
+C элементарного конуса в конус состояний.
+Чтобы упростить обозначения, я сначала рассмотрю случай, когда элементарное
+пространство состоит из скалярных функций ( r = 1).
+Предположим, что носитель supp(φ) функции φ(p) в импульсном пространстве
+является компактным множеством. В таком случае можно найти ограниченное мно-
+жество Uφ для которого все точки вида ∇ǫ(p), где p принадлежит supp(φ), являются
+внутренними точками (функция ǫ(p) предполагается гладкой).
+Тогда при больших |τ| и x
+τ /∈ Uφ в координатном пространстве для функции
+(Tτφ)(x) =
+�
+dkeikx−iǫ(k)τφ(k)
+справедливо соотношение
+|(Tτφ)(x)| < Cn(1 + |x|2 + τ 2)−n,
+где n — некоторое целое число.
+Иными словами, с течением времени в координатном пространстве функция (Tτφ)(x)
+оказывается мала вне множества τUφ, которое я называю существенным носителем
+функции (Tτφ)(x).
+Вернемся теперь к общему случаю, когда элементарное пространство h состоит из
+векторнозначных функций. Мы говорим, что множество τUφ является существенным
+носителем функции
+(Tτφ)(x) =
+�
+dkeikx−iE(k)τφ(k)
+если
+||(Tτφ)(x)|| < Cn(1 + |x|2 + τ 2)−n,
+при больших |τ| и x
+τ /∈ Uφ.
+Мы говорим, что функции φ и φ′ не перекрываются, если расстояние между мно-
+жествами Uφ и Uφ′ положительно; тогда соответствующие существенные носители
+тоже не перекрываются, более того, при больших τ они далеки друг от друга. Мы
+65
+
+говорим, что φ1, ..., φn — это неперекрывающееся семейство функций если φi не пе-
+рекрывается с φj при i ̸= j. Мы всегда будем предполагать, что неперекрывающихся
+семейств функций достаточно много (точнее линейные комбинации неперекрываю-
+щихся семейств функций всюду плотны в интересующем нас пространстве семейств
+функций). При r = 1 это выполнено, например, когда функция ǫ(p) строго выпукла.
+Что нужно назвать двухчастичным состоянием в алгебраическом подходе? Я хо-
+чу заметить, что при определении одночастичного пространства, мне были нужны
+только пространственные и временные сдвиги, а сейчас нужно больше. Раньше я
+использовал представление в виде (49), чтобы описать одночастичное состояние с
+волновой функцией φ. Когда же есть две частицы, B нужно применить два раза:
+B(φ)B(φ′)θ. По крайней мере, когда φ и φ′ имеют носители в координатном про-
+странстве далеко отстоящие друг от друга, можно считать, что этот вектор описыва-
+ет состояние из двух отдаленных частиц. Нужно потребовать при этом, чтобы B(φ)
+и B(φ′) почти коммутировали друг с другом (тогда частицы будут бозонными) или
+почти антикоммутировали (и тогда частицы будут фермионными). Это определение
+дано в терминах состояний, описываемых векторами, но можно дать определение в
+терминах состояний, описываемых положительными функционалами. Для этого мы
+заметим,что состояние, которое отвечает вектору B(φ)B(φ′)θ, можно написать в виде
+L(φ)L(φ′)ω, где
+L(φ) = ˜B(φ)B(φ), L(φ′) = ˜B(φ′)B(φ′).
+При этом L(φ) во всех случаях почти коммутирует с L(φ′).
+В геометрическом подходе двухчастичное состояние записывается как L(φ)L(φ′)ω,
+где L(φ) почти коммутирует с L(φ′).
+Таким образом, при переходе к геометрическому подходу различие между бозо-
+нами и фермионами сглаживается.
+В дальнейшем я все время буду говорить о бозонах, но переход к фермионам
+тривиален: нужно только заменить коммутаторы на антикоммутаторы.
+7.2
+Рассеяние; in- и out-состояния
+Я хотел бы процитировать замечательное высказывание Бертрана Рассела:
+The axiomatic method has many advantages over honest work — Аксиоматический
+метод имеет много преимуществ перед честной работой.
+Дело в том, что все дальнейшее будет очень просто, но, к сожалению, эта простота
+достигнута за счет работы исключительно в аксиоматическом методе. Я хочу напом-
+нить, что аксиомы локальной квантовой теории поля были предложены Вайтманом
+в 50-х годах и до сих пор неизвестно ни одного примера нетривиальной теории, про
+который доказано, что он удовлетворяет всем аксиомам Вайтмана в нашем трехмер-
+ном пространстве. Большой сдвиг произошел, когда в одномерном пространстве было
+построено множество таких теорий в формализме конформной теории поля, но на се-
+годняшний день аксиомы Вайтмана в трехмерном пространстве проверены только в
+рамках теории возмущений. В моем подходе ситуация чуть лучше, но тем не менее
+проверка нужных аксиом остается большой проблемой. Это будет обсуждаться на
+следующей лекции. К счастью, по крайней мере в теории возмущений все прекрасно.
+Теперь будем рассматривать элементарные возбуждения в обоих подходах. В ал-
+гебраическом подходе я требую, чтобы определяющее частицу или квазичастицу отоб-
+ражение элементарного пространства в пространство состояний могло быть записано
+в форме
+σ(φ) = B(φ)θ.
+(50)
+В геометрическом подходе я требую, чтобы было отображение L : h → End(L), где
+σ′(φ) = L(φ)ω.
+(51)
+66
+
+В обоих подходах я определю состояния, описывающие процесс рассеяния и для этого
+введу новые операторы.
+В алгебраическом подходе возьмем оператор B(f) (где f стоит вместо использо-
+вавшегося ранее φ) и сначала сдвинем аргумент по времени в обратном направлении,
+а потом сдвинем получившийся элемент алгебры по времени в прямом направлении.
+Получим следующий оператор:
+B(f, τ) = TτB(T−τf))T−τ
+(Нужно помнить, что действие сдвига по времени на оператор — это сопряжение с
+оператором Tτ.)
+В геометрическом подходе запишем аналогичные формулы:
+L(f, τ) = TτL(T−τf)T−τ,
+после чего проведем цепочку выкладок:
+L(f, τ)ω = TτL(T−τf)ω = Tτσ′(T−τf) = σ′(f),
+где последовательно использованы свойство инвариантности ω по отношению к сдви-
+гам по времени, формула (51) и свойство σ′ коммутировать с временными транс-
+ляциями. В результате зависимость от времени пропадает — L(f, τ)ω не зависит от
+τ.
+Ровно такими же рассуждениями можно показать, что B(f, τ)θ не зависит от τ.
+Отсюда следуют соотношения:
+˙L(f, τ)ω = 0,
+˙B(f, τ)θ = 0,
+где точкой наверху обозначена производная по τ. Это будет мой основной инструмент.
+В случае многих частиц по аналогии с тем, как вводилось определение одноча-
+стичного состояния, я сначала много раз применяю B(fi, τ) с разными fi к θ:
+Ψ(f1, · · · , fn|τ) = B(f1, τ)...B(fn, τ)θ
+а потом у полученного выражения беру предел при τ → −∞:
+Ψ(f1, · · · , fn| − ∞) =
+lim
+τ→−∞ Ψ(f1, · · · , fn|τ)
+(52)
+Этот предел (который лежит в гильбертовом пространстве ¯H, пополнении простран-
+ства H) я буду называть in-состоянием. Чуть позже я объясню его физический смысл.
+В геометрическом подходе все то же самое, только вместо B(f, τ) стоит L(f, τ):
+Λ(f1, ..., fn|τ) = L(f1, τ), ...L(fn, τ)ω
+(53)
+Λ(f1, · · · , fn| − ∞) =
+lim
+τ→−∞ Λ(f1, · · · , fn|τ)
+(54)
+Получается in-состояние лежащее в L. В алгебраическом подходе оно соответствует
+вектору Ψ(f1, · · · , fn| − ∞).
+Применение оператора Tτ к L(f, τ ′) соответствует сдвигу по времени и в функции,
+и в аргументе:
+Tτ(L(f, τ ′)) = Tτ+τ ′L(T−τ ′f)T−τ−τ ′ = L(Tτf, τ + τ ′).
+(55)
+Это чисто формальное вычисление.
+67
+
+Применяя сдвиг по времени к in-состоянию и воспользовавшись формулой (55),
+получим, что такое преобразование сводится к действию оператора сдвига по времени
+на аргументы:
+TτΛ(f1, · · · , fn| − ∞) = Λ(Tτf1, · · · , Tτfn| − ∞).
+(56)
+Из этого соотношения следует, что если существенные носители функций Tτfi не пере-
+крываются (это происходит если функции не перекрываются), то в пределе τ → −∞
+получается несколько далеких друг от друга частиц. Другими словами, асимптотиче-
+ски по времени эволюция in-состояния Λ описывается просто сдвигами аргументов в
+функциях, которые далеки друг от друга. Это как раз то, что имеет место в процессе
+рассеяния.
+Обычно рассматривается рассеяние частиц с определенными импульсами. Но с
+определенными импульсами в моем подходе работать неудобно, потому что в таком
+случае волновая функция будет ненормированной, поэтому следует рассматривать
+ситуацию, когда импульс находится в каком-то узком диапазоне, то есть носитель
+волновой функции — это маленький кусок пространства импульсов. Если рассмат-
+ривать столкновение частиц с различными импульсами, то тогда множества Uφ, ко-
+торые я рассматриваю, не будут перекрываться. Это та ситуация, в которой я могу
+жить. Я говорю, что состояние TτΛ(f1, · · · , fn| − ∞) описывает столкновение частиц
+с волновыми функциями (f1, · · · , fn), если эти функции не перекрываются.
+Я предполагаю, что в случае когда функции f1, ..., fn не перекрываются соответ-
+ствующие операторы L(fi, τ) будут почти коммутировать, то есть их коммутатор в
+пределе τ → −∞ будет равен нулю:
+lim
+τ→−∞ ||[L(fi, τ), L(fj, τ)]|| = 0.
+(57)
+Почему я это предполагаю? В аксиоматическом подходе я могу предположить
+все, но желательно, чтобы мои предположения имели физический смысл. Когда τ →
+−∞, то в L(fi, τ) функции fi сдвигаются по времени, и тогда существенные носители
+соответствующих функций будут далеки друг от друга. В этом случае с точки зрения
+физики естественно думать, что соответствующие операторы почти коммутируют.
+Условие (57) можно вывести из требования, чтобы коммутаторы двух операторов
+L, зависящих от функций φa(x) и ψa(x), удовлетворяли неравенству
+||[L(φ), L(ψ)]|| ≤
+�
+dxdx′Dab(x − x′)|φa(x)| · |ψb(x′)|,
+(58)
+где Dab(x) стремится к нулю быстрее любой степени x → ∞. При выполнении этих
+условий, если множества Ufi для каждой пары функций не перекрываются, то ком-
+мутаторы, входящие в формулу (57) близки к нулю, и поэтому я в формуле (53) для
+in-состояния могу в пределе τ → −∞ переставлять операторы L. Отсюда следует, что
+in-состояния симметричны (не меняются при перестановке аргументов fi).
+Докажем теперь, что рассматриваемый предел существует. Для этого наложим
+дополнительно условие малости коммутатора [ ˙L(fi, τ), L(fj, τ)] при τ → −∞, где в
+качестве аргументов стоят функции fi, fj, i ̸= j. Это — опять же аксиома. Одного
+стремления к нулю мало — нужно, чтобы выполнялось условие
+||[ ˙L(fi, τ), L(fj, τ)]|| ≤ c(τ),
+где c(τ) стремится к нулю настолько быстро, чтобы был конечен интеграл
+�
+|c(τ)|dτ < ∞.
+Можно предположить, например, что c(τ) ∼ 1/τ a, где a > 1.
+68
+
+Теперь я очень просто докажу, что in-состояние действительно существует (выра-
+жение Λ(τ) = Λ(f1, · · · , fn|τ) имеет предел при τ → −∞). Для этого оценим произ-
+водную от Λ(τ) по τ и заметим, что имеет место равенство
+Λ(τ2) − Λ(τ1) =
+� τ2
+τ1
+˙Λ(τ)dτ
+(59)
+Я буду доказывать, что ˙Λ(τ) суммируема и, значит, интеграл в правой части стре-
+мится к нулю когда τ1, τ2 → −∞. Если выражение (59) стремится к нулю, то Λ(τ)
+имеет предел. Это следует из полноты пространства, в котором лежит этот вектор.
+Теперь нужно доказать, что ˙Λ(τ) мало. Напомню, что при определении Λ(τ) (фор-
+мула (53)) я несколько раз применял оператор L к состоянию ω. Продифференциру-
+ем это выражение по τ , и применим правило Лейбница. В результате получается
+несколько слагаемых, в каждом из которых продифференцирован один из множите-
+лей L. Теперь будем менять местами L и ˙L и потребуем, чтобы в пределе τ → −∞
+коммутаторы [L, ˙L] стремились к нулю достаточно быстро. Будем сдвигать ˙L напра-
+во, переместив в итоге на самое последнее место. Когда дойду до самого конца, вос-
+пользуюсь равенством ˙Lω = 0. В результате получится, что ˙Λ(τ) будет суммируемой
+функцией от τ, так как коммутаторы [ ˙L(fi, τ), L(fj, τ)] суммируемы.
+Таким образом доказано существование пределов. Это очень важная вещь. Это
+доказывает, что я могу рассматривать рассеяние частиц в моей картинке. Я потребо-
+вал очень мало, но мои аксиомы достаточны для того, чтобы доказать существование
+предела, доказать, что есть понятие рассеяния.
+Условия, которые я наложил на L в случае геометрического подхода — это про-
+сто аксиомы, а в случае алгебраического подхода аналогичные условия можно по-
+лучить как следствия более физических требований (например, из асимптотической
+коммутативности алгебры A). Но, так или иначе, они выполнены и тут, и там. Все
+приведенные рассуждения справедливы и в алгебраическом подходе. Напомню, что
+в алгебраическом подходе L(φ) = ˜B(φ)B(φ), поэтому я могу наложить условия
+||[ ˙B(fi, τ), B(fj, τ)]|| ≤ c(τ),
+где c(τ) -суммируемая функция. Следовательно, и коммутатор [ ˙L(fi, τ), L(fj, τ)] будет
+маленьким, и вектор
+Ψ(τ) = B(f1, τ)...B(fn, τ)θ
+будет иметь предел в гильбертовом пространстве ¯H при τ → −∞. (Пространство
+должно быть полным, чтобы применить условие сходимости.)
+Я хочу чуточку усилить сделанное утверждение. Раньше я применял операторы B
+к циклическому вектору θ в один и тот же момент времени τ, а теперь буду применять
+в разные времена, но все из них стремятся к бесконечности, возможно, по-разному.
+В таком случае я утверждаю, что для вектора
+Ψ(f1, τ1, ..., fn, τn) = B(f1, τ1)...B(fn, τn)θ
+(60)
+при τj → −∞ все равно существует предел в ¯H, обозначаемый как
+Ψ(f1, ..., fn| − ∞).
+При этом предполагаются выполненным условие на коммутаторы:
+||[ ˙B(φ), B(ψ)]|| ≤
+�
+dxdx′Dab(x − x′)|φa(x)| · |ψb(x′)|,
+где Dab(x) → 0 быстрее любой степени, когда x → ∞ (это условие аналогично усло-
+вию (58)).
+69
+
+Теперь я определю понятие асимптотического бозонного фоковского пространства
+Has, считая, что операторы B на больших расстояниях коммутируют. (Я могу рабо-
+тать как с коммутаторами, так и с антикоммутаторами, но для определенности я
+выберу коммутаторы). Я определю асимптотическое бозонное фоковское простран-
+ство Has как фоковское представление канонических коммутационных соотношений:
+[b(ρ), b(ρ′)] = [b+(ρ), b+(ρ′)] = 0, [b(ρ), b+(ρ′)] = ⟨ρ, ρ′⟩,
+где ρ, ρ′ ∈ h.
+В случае когда вместо коммутатора стоит антикоммутатор бозонное фоковское
+пространство нужно заменить на фермионное фоковское пространство. Все будет
+таким же, только будет иметь место антисимметрия.
+Пространственные и временные трансляции естественно действуют в этом фоков-
+ском пространстве потому что аргументы ρ принадлежат элементарному простран-
+ству h. В элементарном пространстве есть действие трансляций и оно распространя-
+ется на фоковское пространство.
+Теперь я определю матрицу Мёллера — половинку S-матрицы. Эта матрица
+переводит состояние из фоковского пространства b+(f1)...b+(fn)|0⟩ в in-состояние
+Ψ(f1, ..., fn| − ∞). При этом важно, что in-состояние симметрично. То, что можно
+переставлять fi существенно, потому что иначе это определение не имело бы смысла,
+так как n-частичное подпространство в фоковском пространстве представляет собой
+cимметрическое тензорное произведение одночастичных пространств.
+Матрицы Мёллера коммутируют с трансляциями. Мы на самом деле это дока-
+зали, и это я сейчас объясню. Вернемся к формуле (56), из которой следует, что
+действие временного сдвига на in-состояние отвечает временному сдвигу аргумен-
+тов. Временной сдвиг аргументов — это то, как действует временной сдвиг в фо-
+ковском пространстве, поэтому в формуле (56) написано то, что временной сдвиг в
+in-пространстве отвечает временному сдвигу в гильбертовом пространстве ¯H. (Берет-
+ся пополненное гильбертова пространства, так как только там лежит in-состояние.)
+Вот, эта очень важная вещь формально очень просто доказывается. То, что матрицы
+Мёллера коммутируют с пространственными трансляциями проверяется еще легче.
+Матрица Мёллера обычно обозначается как S−. На следующей лекции я буду
+доказывать, что при условии распадения корреляций S− является изометрическим
+вложением Has в ¯H. Вместо S− мы можем точно так же рассмотреть S+. Если обе
+эти матрицы Мёллера не только изометрические, но и унитарные, то есть, они яв-
+ляются сюръективными отображениями на все пространство ¯H, тогда мы говорим,
+что теория имеет интерпретацию в терминах частиц. Это означает, что почти всякое
+состояние является in-состоянием (множество in-состояний всюду плотно). Другими
+словами, почти всякое состояние в пределе τ → −∞ представляется совокупностью
+удаленных частиц.
+Аналогично можно время устремить к плюс бесконечности. Получится картинка,
+аналогичная той, что была описана в классической теории солитонов при обсуждении
+так называемой soliton resolution conjecture, когда почти любое начальное состояние
+распадется на солитоны и почти линейный хвостик. Здесь примерно то же самое: на
+плотном множестве состояние распадается на множество удаленных друг от друга
+частиц.
+Матрицу рассеяния можно определить формулой
+S = S−1
++ S−.
+Похожую формулу я писал в солитонной картинке. Это определение той самой мат-
+рицы рассеяния, которая является главным объектом в квантовой теории поля.
+70
+
+Теперь я определю in-операторы a+
+in с помощью предела операторов B(f, τ) при
+τ → −∞:
+a+
+in(f) =
+lim
+τ→−∞ B(f, τ).
+(61)
+Почему это законное определение? Обратимся к формуле (60), где операторы
+B(fi, τi) стоят с разными временами. Это значит, что по одному из аргументов я
+могу перейти к пределу раньше, чем по другим аргументам. Отсюда получается суще-
+ствование предела в формуле (61) для in-оператора. Можно сказать, что in-оператор
+хорошо определен именно потому, что есть это свойство. Следует сказать, что in-
+оператор не всегда определен, но, по крайней мере, если все функции f1, ..., fn не
+перекрываются, тогда все хорошо. В нашем определении in-операторы линейно зави-
+сят от функций f. Эти операторы можно рассматривать как обобщенные функции и
+ввести следующее обозначение:
+a+
+in(f) =
+�
+dpf k(p)a+
+in,k(p),
+где a+
+in,k(p) -обобщенная функция, а индекс k определяет тип частиц.
+Точно так же определяются out-операторы, но только τ нужно стремить к плюс
+бесконечности:
+a+
+out(f) =
+lim
+τ→+∞ B(f, τ).
+In-операторы связаны с операторами в асимптотическом пространстве с помощью
+формул:
+a+
+in(ρ)S− = S−b+(ρ),
+S−|0⟩ = θ.
+Эти формулы можно считать альтернативным определением in-операторов, и тогда
+точно так же можно определить операторы ain, связанные с операторами уничтоже-
+ния в фоковском пространстве:
+ain(ρ)S− = S−b(ρ),
+aout(ρ)S+ = S+b(ρ).
+Существует очевидная связь между определениями в геометрическом и алгебра-
+ическом подходах. Если геометрический подход строится в рамках алгебраического,
+то оператор L(f, τ) в пространстве состояний соответствует оператору B(f, τ) в ¯H в
+соответствии с формулой L(f, τ) = ˜B(f, τ)B(f, τ), и тогда состояние Λ(f1, . . . , fn|τ)
+соответствует вектору Ψ(f1, . . . , fn|τ), а состояние Λ(f1, . . . , fn| − ∞) (in-состояние)
+соответствует вектору Ψ(f1, . . . , fn| − ∞).
+Аналог матрицы Мёллера в геометрическом подходе обозначается как ˜S−. В то
+время как S−-матрица Мёллера — это линейный оператор, ˜S− — нелинейный опера-
+тор. Для теорий, которые могут быть сформулированы алгебраически S− отображает
+в ¯H симметричную степень h, рассматриваемую как подпространство пространства
+Фока. Взяв композицию этого отображения с естественным отображением ¯H в конус
+состояний C мы получаем ˜S−.
+Когда L является квадратичным или эрмитовым, он индуцирует мультилинейное
+отображение симметричной степени конуса C(h), соответствующего h в конус C.
+Матрица рассеяния описывает процесс рассеяния. Связь сечения рассеяния с мат-
+рицей рассеяния объясняется в общем курсе квантовой механики. Я объясню связь с
+инклюзивным сечением, и это проще.
+Сначала я должен объяснить, что такое инклюзивное сечение рассеяния. Сечение
+рассеяния связано с вероятностью перехода, скажем, от пары частиц к n частицам
+(M, N) → (Q1..., Qn). Мы же рассмотрим процесс, когда в конце получаются частицы
+(Q1..., Qn) плюс что-нибудь еще:
+(M, N) → (Q1, ..., Qm, R1, ..., Rn)
+71
+
+По частицам R1, ..., Rn проводится суммирование или, точнее, интегрирование. По-
+лучается инклюзивное сечение рассеяния. Все сказанное годится в теории, когда есть
+интерпретация в терминах частиц и когда все распадается на частицы, но я могу
+определить инклюзивное сечение, даже если нет такой интерпретации, когда просто
+происходит процесс (M, N) → (Q1, ..., Qn + something). Можно не знать, что пред-
+ставляет собой это “something”, но просуммировать по всему, что есть и получить
+инклюзивное сечение.
+В геометрическом подходе только инклюзивное сечение имеет смысл. В алгебраи-
+ческом подходе необязательно ограничиваться инклюзивным сечением, но все равно
+проще работать с инклюзивным сечением.
+Я рассматриваю произвольное состояние ν и пишу следующую формулу для плот-
+ности вероятности:
+ν(a+
+out,k1(p1)aout,k1(p1) . . . a+
+out,km(pm)aout,km(pm))
+(62)
+Отмечу, что здесь стоят out-операторы, то есть предел на + бесконечности. Входя-
+щие в формулу выражения вида a+
+out,ki(pi)aout,ki(pi) — это, по сути, числа частиц
+с импульсом pi, так что формула (62) представляет плотность вероятности в про-
+странстве импульсов нахождения m исходящих частиц типов k1, . . . , km с импульсами
+p1, . . . , pm. При этом на остальные частицы я не смотрю.
+До сих пор ν было любым состоянием, а теперь в качестве ν рассмотрим in-
+состояние
+ν = Λ(g1, ..., gn| − ∞) =
+lim
+τ→−∞ L(g1, τ)...L(gn, τ)ω
+In-состояние определяется входящими частицами. Например, когда я определял ин-
+клюзивное сечение, я сталкивал две частицы, но можно сталкивать несколько. Если
+в in-состоянии измерять входящее в формулу (62) произведение операторов с индек-
+сом out, как раз получается инклюзивное сечение рассеяния по определению. Значит,
+если посчитать выражение (62), когда ν будет in-состоянием, получится инклюзивное
+сечение. Мы сейчас произведем это вычисление.
+Рассмотрим следующее выражение:
+⟨1|L(g′
+1, τ ′)...L(g′
+n′, τ ′)L(g1, τ)...L(gn, τ)|ω⟩
+(63)
+в предположении, что g′
+i и gj не перекрываются, а времена устремляются к беско-
+нечности, причем τ ′ → +∞, τ → −∞. Полученное при действии операторов L на
+состояние ω — это линейный функционал на алгебре, поэтому мы можем посчитать,
+чему он будет равен на единичном элементе алгебры. Обращаю внимание, что я ис-
+пользовал то, что называется bra-ket notations, при этом в скобках слева и справа
+стоят элементы из сопряженных друг другу пространств. Итак, рассмотрим выраже-
+ние (63) и обозначим его как Q. Взяв предел τ → −∞, получим
+Q =
+lim
+τ ′→+∞⟨1|L(g′
+1, τ ′)...L(g′
+n′, τ ′)ν⟩,
+(64)
+где ν = Λ(g1, ..., gn| − ∞). Получается формула (64). Так как я предположил, что
+функции не перекрывается, всякие коммутаторы стремятся к нулю и Q не меняется
+при перестановке g′
+1, ..., g′
+n′.
+Теперь посмотрим на эти формулы в алгебраическом подходе. Тогда L(g, τ) =
+˜B(g, τ)B(g, τ), а кроме того у меня есть формула ( ˜
+MNν)(X) = ν(M∗XN). (Операто-
+ры с волной умножают с одной стороны, без волны — с другой стороны.) В формуле
+(64) эти операторы применены к ν. Они просто меняют аргумент у ν. Кроме того,
+⟨1|σ⟩ = σ(1). В результате получается выражение
+Q =
+lim
+τ ′→+∞ ν(B∗(g′
+n′, τ ′)...B∗(g′
+1, τ ′)B(g′
+1, τ ′)...B(g′
+n′, τ ′)).
+72
+
+В пределе операторы B переходят в in- и out-операторы :
+lim
+τ ′→+∞ B(g, τ ′) = a+
+out(g),
+lim
+τ ′→+∞ B∗(g, τ ′) = aout(g).
+Учитывая это, а также условие, что в пределе все операторы коммутируют, получаем
+следующее выражение для Q:
+Q = ν(aout(g′
+n′)...aout(g′
+1)a+
+out(g′
+1)...a+
+out(g′
+n′))).
+Я назову выражение
+Q = Q(g′
+1, ..., g′
+n′, g1, .., gn)
+инклюзивной матрицей рассеяния. Это выражение квадратично зависит от своих ар-
+гументов. Я могу перейти от квадратичных выражений к билинейным — тогда ко-
+личество аргументов удвоится. Получаемое выражение также будет называться ин-
+клюзивной матрицей рассеяния. Из него можно получить инклюзивное сечение. Это
+не вполне тривиальный процесс. Дело в том, что определении инклюзивной матри-
+цы рассеяния я рассматривал ее как функционал на неперекрывающихся семействах
+функциий. Этот функционал линеен или антилинеен, поэтому его можно рассматри-
+вать как обобщенную функцию, но но аргументы обобщенной функции (импульсы)
+следует считать различными. В выражении для инклюзивного сечения импульсы мо-
+гут совпадать, поэтому необходим предельный переход.
+В геометрическом подходе я могу определить инклюзивную матрицу рассеяния,
+взяв наряду с ω ∈ L какое-нибудь трансляционно инвариантное состояние α ∈ L∗:
+lim
+τ ′→+∞,τ→−∞⟨α|L(g1, τ ′)...L(gm, τ ′) × L(f1, τ)...L(fn, τ)|ω⟩.
+(65)
+Такая формула может быть применена и в алгебраической ситуации. В ней состояния
+α и ω входят абсолютно симметрично. Можно сформулировать это так: формулa (65)
+дает скалярное произведение out-состояния в L∗ и in-состояния в L. Иными словами,
+одна и та же формула будет давать инклюзивную матрицу рассеяния элементарных
+возбуждений состояния ω и инклюзивную матрицу рассеяния элементарных возбуж-
+дений состояния α. Это некая двойственность, на мой взгляд, абсолютно таинствен-
+ная. В алгебраическом подходе ее тоже можно рассматривать.
+73
+
+8
+Лекция 8
+8.1
+Связь с локальной квантовой теорией поля
+То, что я рассказываю в этом курсе, сильно отличается от того, что обычно расска-
+зывается в учебниках по релятивистской квантовой теории поля — в них рассматри-
+ваются локальные теории. Основная идея дальнейшего изложения — подчеркнуть,
+что локальность абсолютно не по существу в большинстве случаев, да и сами поля
+несущественны. Я не знаю, что нужно называть полями в том подходе, о котором я
+говорю, хотя вся квантовая теория поля здесь есть.
+Я хочу начать с установления связи того, что я рассказываю с тем, что обычно
+называется локальной релятивистской квантовой теорией поля.
+В аксиоматическом подходе к локальной теории есть разные системы аксиом на-
+чиная с аксиом Вайтмана, о которых вкратце можно сказать, что в них в качестве
+основного объекта рассматриваются поля, являющиеся обобщенными операторными
+функциями. Это не очень удобно: поля локальны, но это обобщенные функции. Если
+их проинтегрировать, то получаются обычные операторы. Они уже не локальны, но
+в каком-то смысле почти локальны. Их можно считать сосредоточенными в какой-то
+области.
+Я буду говорить про систему аксиом, которая принадлежит Араки, Хаагу и Ка-
+стлеру. В ней рассматриваются, строго говоря, не локальные поля, а поля, сосредо-
+точенные в каком-то открытом подмножестве пространства Минковского. В этой си-
+стеме аксиом считается, что такие поля образуют алгебру операторов, действующих
+в гильбертовом пространстве; эта алгебра замкнута относительно слабой сходимости
+(что не так существенно). Считается, что в том гильбертовом пространстве E, в кото-
+ром действуют эти операторы, действует унитарное представление группы Пуанкаре
+P.
+Предполагается, что каждой ограниченной области (ограниченному открытому
+подмножеству) O пространства Минковского сопоставлена алгебра операторов A(O)
+так, что
+• если область становится больше: O1 ⊂ O2, то и алгебра становится больше:
+A(O1) ⊂ A(O2);
+• действие группы Пуанкаре P на операторах согласованно с действием на обла-
+стях: A(gO) = gA(O)g−1, если g ∈ P;
+• если интервал между точками двух областей O1 и O2 пространственно-подобный,
+то соответствующие операторы A(O1) и A(O2) коммутируют (грубо говоря, это
+означает, что если интервал пространственно-подобный, то причинной связи не
+может быть — все абсолютно независимо);
+• основное состояние θ оператора энергии инвариантно относительно группы Пу-
+анкаре (имея представление группы Пуанкаре, можно рассмотреть оператор
+энергии (гамильтониан) и операторы импульса как инфинитезимальные гене-
+раторы, соответственно, временных и пространственных трансляций);
+• вектор, отвечающий основному состоянию, является циклическим по отношению
+к объединению A всех алгебр A(O).
+Это аксиоматика релятивистской локальной квантовой теории поля.
+В этой аксиоматике частица определяется как неприводимое подпредставление
+представления алгебры Пуанкаре в пространстве E.
+74
+
+Вернемся теперь к определению рассеяния в алгебраическом подходе. Прошлое
+мое рассмотрение базировалось на аксиомах, которые нелегко проверить. Сейчас же
+я наложу требования, которые много легче проверяются. В частности, будет видно,
+что они выполнены в релятивистской локальной теории.
+Мой исходный пункт, как и раньше — это ассоциативная алгебра A с инволюцией.
+Пространственно-временные трансляции — это автоморфизмы этой алгебры.
+У меня есть понятие состояния. Есть и понятие ненормализованного состояния.
+Такие состояния отвечают положительным линейным функционалам на алгебре A
+и образуют конус C. Я буду работать именно с ненормализованными состояниями.
+Трансляционно-инвариантное стационарное состояние всегда будет обозначаться как
+ω ∈ C. Возбуждения состояния ω — это элементы предгильбертова пространства H,
+которое построено по ω с помощью конструкции GNS.
+В ассоциативной алгебре A, с которой я начинал, нет никакой нормы, но поскольку
+она представлена в предгильбертовом пространстве H и его пополнении, гильберто-
+вом пространстве ¯H, можно рассматривать норму соответствующих операторов ˆA.
+Более того, я могу работать с пополненной по этой норме алгеброй A(ω), но это не
+обязательно.
+Пусть теперь у меня есть элемент A алгебры A который представлен ограни-
+ченным оператором ˆA в гильбертовом пространстве. Я могу этот оператор сдвигать
+по времени и по пространству. В результате получится некий оператор ˆA(x, τ). Бо-
+лее того, я могу усреднить такой оператор с какой-то гладкой и быстро убывающей
+функцией:
+B =
+�
+dτdxα(x, τ) ˆA(x, τ).
+Можно сдвинуть оператор B по времени и пространству:
+B(x, τ) =
+�
+dτ ′, dx′α(x − x′, τ − τ ′)A(x′, τ ′).
+Под знаком интеграла можно дифференцировать. Поскольку сама функция α(x, τ)
+предполагается гладкой, можно продифференцировать сколько угодно раз. Я всегда
+буду работать именно с такими операторами и буду называть их гладкими.
+Условие асимптотической коммутативности следует наложить таким способом,
+чтобы коммутатор сдвинутого оператора с другим оператором становился малым
+при больших пространственных сдвигах. Формализовать это можно по-разному. Я
+сделаю это так, чтобы было мгновенно понятно, что в аксиоматике Араки, Хаага и
+Кастлера мое условие выполнено. Именно я потребую, чтобы норма ||[B1(x, τ), B2]||
+коммутатора сдвинутого оператора с другим оператором, отвечающим элементу ал-
+гебры A убывала быстрее любой степени ||x||, когда x → ∞. То же условие я наложу
+на ||[ ˙B1(x, τ), B2]||, где точка обозначает производную по времени. Все операторы,
+напомню, у меня гладкие.
+В аксиоматике Араки, Хаага и Кастлера это всегда выполняется, потому что там
+при большом пространственном сдвиге пространственно-временной интервал меж-
+ду соответствующими областями, становится пространственно-подобным и поэтому
+можно сказать, что начиная с некоторого момента рассматриваемый мною коммута-
+тор просто равен нулю (и тем самым убывает быстрее любой степени).
+Другим определением асимптотической коммутативности является условие
+||[B1(x, τ), B2]|| ≤
+Cn(τ)
+1 + ||x||n ,
+где Cn(τ) имеет не более чем полиномиальный рост, а n -произвольно (сильная асимп-
+тотическая коммутативность). Это условие в аксиоматике Араки, Хаага и Кастлера
+выполнено, если спектр масс ограничен снизу положительным числом.
+75
+
+Кроме условия асимптотической коммутативности я хочу наложить условие кла-
+стеризации, о которых я говорил в прошлый раз. В простейшей форме это означает,
+что если один из операторов убегает на бесконечность, то все распадается с точностью
+до маленького слагаемого ρ(x, t):
+ω(A(x, t)B) = ω(A)ω(B) + ρ(x, t),
+где ω — это функционал на алгебре A, то самое состояние, возбуждение которого я
+рассматриваю. Можно наложить примерно такое же условие, когда операторов много.
+Это очень удобное условие, но формулируется оно довольно громоздко.
+Для формулировки условия кластеризации для многих операторов мне нужно вве-
+сти понятие корреляционной функции, являющейся обобщением функции Вайтмана
+из релятивистской квантовой теории поля. Я беру какие-то операторы A1 . . . Ar ∈ A,
+сдвигаю их и по пространству, и по времени. Перемножив их, получаю элемент ал-
+гебры и после этого применяю ω или, что то же самое, беру среднее по состоянию ω.
+В результате получается:
+wn(x1, t1, . . . xn, tn) = ω(A1(x1, t1) · · · An(xn, tn)) = ⟨A1(x1, t1) · · · An(xn, tn)⟩.
+Это корреляционная функция.
+Полезно определить понятие усеченной корреляционной функции
+wT
+n (x1, t1, . . . xn, tn) ≡ ⟨A1(x1, t1) · · · An(xn, tn)⟩T .
+Это делается несколько формально с помощью индуктивной формулы, связывающей
+усеченные корреляционные функции с обычными корреляционными функциями:
+wn(x1, τ1, k1 . . . , xn, τn, kn) =
+n
+�
+s=1
+�
+ρ∈Rs
+wT
+α1(π1) . . . wT
+αs(πs).
+Здесь Rs обозначает совокупность всех разбиений множества {1, ..., n} на подмноже-
+ства s, обозначаемые π1, ..., πs, количество элементов в подмножестве πi обозначается
+через αi, а wT
+αi(πi) обозначает усеченную корреляционную функцию с аргументами
+xa, τa, ka, где a ∈ πi. Эта формула выражает корреляционные функции через усечен-
+ные функции при всевозможных разбиениях множества индексов.
+В случае когда есть всего два оператора, усеченная корреляционная функция име-
+ет вид
+wT
+2 (x1, τ1, k1, x2, τ2, k2) = ω(A1(x1, t1)A2(x2, t2)) − ω(A1(x1, t1))ω(A2(x2, t2)).
+Так как ω трансляционно инвариантнo, как обычная, так и усеченная корреляци-
+онные функции зависят только от разностей xi−xj, ti−tj. Мы говорим, что выполнено
+условие кластеризации если усеченные корреляционные функции становятся малы-
+ми при xi − xj → ∞. Малость можно понимать в разных смыслах, но я имею в виду
+самое сильное условие: при фиксированных ti они стремятся к нулю быстрее, чем
+любая степень разности d = min ∥xi − xj∥. Точнее говоря, мы предполагаем,что
+|wT
+n (x1, t1, . . . xn, tn)| ≤ Cs(t)
+ds
+,
+где s любое натуральное число, а Cs(t)-полиномиальная функция от времен ti.
+Можно применить преобразование Фурье, и тогда факт зависимости только от
+разности будет означать появление δ-функции от суммы импульсов. Если эту функ-
+цию выделить, то условие кластеризации, означает, что усеченная корреляционная
+функция представляет собой гладкую функцию, умноженную на δ-функцию:
+νn(p2, . . . , pn, t1, . . . , tn)δ(p1 + . . . + pn).
+76
+
+Напомню, что если функция быстро убывает по x, то в импульсном представлении
+она является гладкой, и это здесь написано.
+В релятивистской квантовой теории условие кластеризации выполнено если массы
+частиц ограничены снизу положительным числом (mass gap).
+8.2
+Функции Грина. Связь с матрицей рассеяния.
+Функция Грина отличается от корреляционной функции тем, что операторы, кото-
+рые стоят под знаком ω, считаются упорядоченными по времени в порядке убывания.
+Это то, что называется хронологическим произведением. Если применить ω к хро-
+нологическому произведению (или взять среднее по вектору, который отвечает ω в
+конструкции GNS), получится функция
+Gn = ω(T(A1(x1, t1) . . . Ar(xr, tr))) = ⟨θ|T( ˆA1(x1, t1) . . . ˆAr(xr, tr))|θ⟩,
+которая называется функцией Грина, записанной в (x, t)-представлении (в коорди-
+натном представлении).
+Как всегда, можно переходить к импульсному представлению, взяв преобразова-
+ние Фурье по x. Это будет то, что называется (p, t)-представлением (импульсном и
+временном). Можно еще взять (обратное) преобразование Фурье по временной пере-
+менной и тогда функции Грина будут в (p, ǫ)-представлении, когда основные перемен-
+ные — это импульсы и энергии. Мне потребуются все эти представления. Отличаются
+они преобразованиями Фурье, всегда можно переходить от одного к другому.
+Благодаря трансляционной инвариантности функция Грина в (x, t)-представлении
+зависят от разностей xi − xj, ti − tj и, следовательно, в (p, t)-представлении выделя-
+ется множитель δ(p1 + . . . + pr), отвечающий закону сохранения импульса, а в (p, ǫ)-
+представлении —еще и множитель δ(ǫ1 + . . . + ǫr), отвечающий закону сохранения
+энергии.
+Переходя к вопросу о полюсах функции Грина, нужно заметить, что я всегда не
+обращаю внимания на δ-функции, говоря про полюса. В частности, когда функция
+Грина включает только два оператора, в (p, ǫ)-представлении мы имеем два импульса,
+две энергии и две δ-функции — одна от импульсов, другая от энергий:
+G(p1, ǫ1|A, A′)δ(p1 + p2)δ(ǫ1 + ǫ2).
+Остается функция от переменной p1 и о переменной ǫ1. Важно отметить, что полюса
+такой двухточечной функции Грина по энергии при фиксированном импульсе отве-
+чают частицам. Эти полюса зависят от импульса, а соответствующая функция ε(p)
+дает закон дисперсии для частиц (зависимость энергии от импульса). Эти хорошо
+известные факты легко вывести из рассуждений, которые будут использованы ниже.
+Я докажу, что для того, чтобы найти амплитуды рассеяния, нужно рассмот-
+реть асимптотическое поведение функции Грина в (p, t)-представлении, когда t →
+±∞. Это первое и основное замечание. А другое замечание состоит в том, что это
+асимптотическое поведение в (p, t)-представлении управляется полюсами в (p, ǫ)-
+представлении. Точнее говоря, асимптотика описывается вычетами в этих полюсах.
+Это называется “on-shell value of Green function”.
+Есть хорошо известный математический факт: если асимптотическое поведение
+какой-либо функции ρ(t) при t → ±∞ имеет вид e−itE±A± или, говоря по-другому,
+существует предел limt→±∞ eitE±ρ(t) = A±, тогда (обратное) преобразование Фурье
+ρ(ǫ) имеет полюса в точках E± ± i0 с вычетами ∓2πiA±. Иными словами предел
+отвечает вычетам, а показатели экспоненты отвечают полюсам; при этом полюс ока-
+зывается немножко сдвинутым в комплексной плоскости либо наверх, либо вниз. Это
+чрезвычайно важное замечание.
+77
+
+Можно либо смотреть на полюса в энергетическом представлении, либо смотреть
+на асимптотику во временном представлении. Это означает, что вычисление амплитуд
+рассеяния сводится к выяснению асимптотического поведения функций Грина (это
+мы сделаем). Переходя к энергетическому представлению, мы можем сказать, что
+амплитуды рассеяния выражаются через on-shell значения функций Грина. Это —
+формула Лемана, Симанчика и Циммермана (LSZ).
+Ниже я докажу формулу LSZ в случае, когда в теории есть интерпретация в терми-
+нах частиц. Это означает, что существуют половинки матрицы рассеяния — матрицы
+Мёллера S±, которые дают унитарную эквивалентность между свободным гамиль-
+тонианом в асимптотическом пространстве Has и полученым с помощью процедуры
+GNS гамильтонианом в пространстве H.
+Я хочу упростить обозначения, и поэтому я буду обсуждать случай, когда есть
+только один тип частиц. Напоминаю, что я рассматривал обобщенную функцию Φ(p),
+отвечающую состоянию частицы с заданным импульсом p, и это состояние — соб-
+ственное как для импульса, так и для оператора энергии. Действие гамильтониана
+на Φ(p) сводится к умножению на функцию ε(p) (закон дисперсии):
+ˆHΦ(p) = ε(p)Φ(p),
+ˆPΦ(p) = pΦ(p).
+Нужно помнить, что на самом деле Φ(p) не очень-то существует — это обобщенная
+функция. Для того, чтобы все это приобрело точный математический смысл, следует
+проинтегрировать с какой-то пробной функцией.
+Теперь я хочу сделать предположение, что одночастичный спектр не перекрыва-
+ется с многочастичным спектром. Как вычислить спектр гамильтониана ˆH? У меня
+есть интерпретация в терминах частиц, и поэтому матрица Мёллера связывает асимп-
+тотический гамильтониан с гамильтонианом в пространстве H.
+Асимптотический гамильтониан является свободным. У него (а, значит, и у ˆH)
+спектр полностью определяется одночастичными функциями ε(p). Энергии многоча-
+стичных возбуждений — это просто суммы ε(p1) + . . . + ε(pn). Если я хочу сказать,
+что одночастичный спектр не перекрывается с двухчастичным, а, следовательно, и
+многочастичным, нужно потребовать выполнения неравенства
+ε(p1 + p2) < ε(p1) + ε(p2).
+Это означает, что частицы с импульсом p1 + p2 не могут распасться на частицы с
+импульсом p1 и импульсом p2. Закон сохранения (в данном случае закон сохранения
+энергии) запрещает распад.
+Теперь я аккуратно сформулирую формулу Лемана, Симанчика и Циммермана.
+Для этого я зафиксирую какие-то элементы Ai ∈ A той алгебры, из которой я исхо-
+дил. (Напоминаю, что я работаю с гладкими элементами, но здесь это не так важно.)
+Также я требую, чтобы применив оператор ˆAi к вектору θ (который в релятивистской
+квантовой теории интерпретируется как физический вакуум) и осуществив проекцию
+на пространство, натянутое на одночастичные состояния (одночастичное простран-
+ство), я получу в результате отличную от нуля величину. Точнее говоря, я требую,
+чтобы проекция вектора ˆAiθ была одночастичным состоянием вида:
+Φ(φi) =
+�
+φi(p)Φ(p)dp,
+где φi(p) — это не обращающаяся нигде в ноль функция. Проекция этого вектора на
+вектор θ должна обращаться в ноль.
+Для простоты я еще веду обозначение Λi(p) = φi(p)−1. В этих обозначениях бу-
+дем рассматривать содержащую как сами операторы Ai, так и сопряженные им A∗
+i
+78
+
+функцию Грина в (p, t)-представлении:
+Gmn = ω(T(A∗
+1(x1, t1) . . . A∗
+m(xm, tm)Am+1(xm+1, tm+1) . . . Am+n(xm+n, tm+n))).
+Затем перейдем к (p, ǫ)-представлению, но при этом там, где стоят операторы A∗
+i
+при 1 ≤ i ≤ m удобно сменить знак у переменных pi и ǫi. Функцию Грина в новом
+представлении умножим на выражение:
+�
+1≤i≤m
+Λi(pi)(ǫi + ε(pi))
+�
+m<j≤m+n
+Λj(pj)(ǫj − ε(pj)).
+и перейдем к пределу ǫi → −ε(pi) для 1 ≤ i ≤ m и ǫj → ε(pj) для m < j ≤ m + n.
+Вклад в полученное выражение будут вносить только полюса, потому что введен-
+ный множитель стремится к нулю, и вне полюса все полученное выражение обраща-
+ется в ноль, а в полюсе получается вычет. Иными словами, данная операция сводится
+к взятию вычета в этом полюсе. Кроме того, я еще умножаю на функции Λi(p) и со-
+пряженные к ним. В физике это называется перенормировкой волновой функции. В
+случае когда эти множители не включены, я буду говорить про on-shell функции Гри-
+на, а если включены, я буду говорить, что рассматриваю нормализованные on-shell
+функции Грина.
+Основное высказывание в подходе Лемана, Симанчика и Циммермана состоит в
+том, что нормализованная on-shell функция Грина дает амплитуду рассеяния. Для
+его доказательства, прежде всего, я буду рассматривать случай, когда операторы ˆAi
+просто дают одночастичные состояния ˆAiθ = Φ(φi) (не нужно проецировать). Будем
+их называть хорошими операторами. В конце лекции я объясню, что к этому случаю
+все можно свести.
+До сих пор рассматривался случай, когда имеется всего один тип частиц. Давай-
+те рассмотрим случай, когда есть много типов частиц, иными словами, есть много
+функций Φk(p), являющиеся собственными как для импульса, так и для энергии:
+PΦk = pΦk(p),
+HΦk(p) = εk(p)Φk(p),
+но с разными законами дисперсии εk(p), задающимися гладкими функциями. Как
+всегда, Φk(p) — это обобщенные функции, то есть, нужны пробные функции. Я рас-
+сматриваю пробные функции из пространства S гладких быстро убывающих функ-
+ций. Чтобы гарантировать, что сдвиги по времени действуют в пространстве S, сле-
+дует предположить, что функции εk(p) растут не более чем полиномиально.
+Как уже было сказано, я буду работать с хорошими операторами Bk ∈ A, яв-
+ляющимися гладкими и переводящими вектор θ в одночастичные состояния ˆBkθ =
+Φk(φk). Теперь я определяю оператор ˆBk(f, t), зависящий от функции f = f(p) сле-
+дующим образом:
+ˆBk(f, t) =
+�
+˜f(x, t) ˆBk(x, t)dx,
+где функция ˜f(x, t) получена преобразованием Фурье от функции f(p)e−iεk(p)t по
+импульсной переменной.
+Аналогичные операторы рассматривались на прошлой лекции. Они обладают тем
+свойством, что если их применить к θ, получится независящее от t одночастичное
+состояние
+ˆBk(f, t)θ = Φk(fφk).
+(66)
+(На прошлой лекции функция φk равнялась единице). В целом же, все то, что го-
+ворилось на прошлой лекции можно повторить и здесь. Тот факт, что полученное
+состояние не зависит от времени — это результат формального счета. Вычисления
+79
+
+становятся совсем простыми если сделать преобразование Фурье по пространствен-
+ной переменной
+ˆBk(p, t) =
+�
+dxe−ipx ˆBk(x, t).
+В терминах этих операторов получается простое выражение
+ˆBk(f, t) =
+�
+dpf(p)eiεk(p)t ˆBk(p, t).
+(67)
+Если этот оператор применить к θ, получится одночастичное состояние, зависи-
+мость которого от времени определяется такой же экспонентой, что и стоит под зна-
+ком интеграла, только с минусом перед ее показателем. Эти две экспоненты сократят-
+ся и зависимость от времени пропадет. Это очень важная для меня вещь. В прошлый
+раз, я тоже рассматривал аналогичные операторы, и мне очень важно, чтобы я мог
+продифференцировать по t и получить ноль, действуя на θ.
+Теперь я делаю ровно то, что делал в прошлый раз, но обозначения изменились,
+потому что мне неудобно сейчас говорить про элементарное пространство. Поэтому я
+явно выписываю индексы. В элементарном пространстве у меня тоже были индексы,
+но я их мог спрятать, а тут я индексы пишу явно. Я рассматриваю вектор
+Ψ(k1, f1, . . . , kn, fn|t1, . . . , tn) = ˆBk1(f1, t1) · · · ˆBkn(fn, tn)θ,
+(68)
+где предполагается, что функции f1, . . . , fn имеют компактные носители. В прошлой
+лекции я в основном считал, что все времена, входящие в аналогичное выражение,
+одинаковы; случай, когда времена разные, но все стремятся к бесконечности немногим
+сложнее.
+Теперь я, как и в прошлый раз, рассматриваю градиенты от энергии vi(p) =
+∇εki(p), которые имеют смысл скорости. Теперь я обозначаю буквой Ui множество
+всех возможных скоростей vi(p), таких что fi(p) ̸= 0. Дальше я рассматриваю за-
+мыкание Ui. Я требую, чтобы все эти множества не перекрывались, тогда я буду
+называть эти функции неперекрывающимися. Это означает, что все классические
+скорости различны, и поэтому волновые пакеты расходятся в разных направлени-
+ях, тогда, как я объяснял в прошлый раз, в координатном представлении почти не
+перекрываются соответствующие волновые функции.
+Это то, что уже было сказано, а теперь я буду брать предел ti → ∞, опять для
+простоты буду рассматривать случай, когда все времена одинаковы. Я буду доказы-
+вать, что у вектора Ψ(k1, f1, . . . , kn, fn|t1, . . . , tn) есть предел, который обозначим как
+Ψ(k1, f1, . . . , kn, fn| ± ∞).
+Доказательство повторяет то, что было в прошлый раз. Я снова положу одинако-
+выми все ti = t и продифференцирую по t. Для того чтобы доказать, что есть предел,
+мне нужно знать, что производная по t мала (на самом деле достаточно, чтобы про-
+изводная была суммируемой функцией). Это условие выполняется. По определению
+вектор Ψ является результатом многократного применения к θ операторов ˆBki(fi, ti).
+Когда я дифференцирую это выражение по t, у меня появляется точка над одним
+из операторов ˆB, который я после этого могу сдвинуть направо, используя условие
+асимптотической коммутативности (но только если я работаю с неперекрывающими-
+ся функциями). Эта производная примененная к θ дает ноль, поэтому предел есть.
+Поскольку предел существует, я могу определить матрицу Мёллера. Для этого я
+введу асимптотическое пространство Has как фоковское представление операторов
+a+
+k (f), ak(f) и определю матрицы Мёллера S− и S+ как операторы, определенные в
+Has и принимающие значения в ¯H по формуле:
+Ψ(k1, f1, . . . , kn, fn| ± ∞) = S±(a+
+k1(f1φk1) . . . a+
+kn(fnφkn)|0⟩),
+(69)
+80
+
+где |0⟩ — фоковский вакуум. Это та же формула, которая была на прошлой лекции с
+той лишь разницей, что здесь появились множители φki, которые мне нужны для того,
+чтобы учесть то, что при действии хорошего оператора ˆBki на Φ мы получаем Φki(φki).
+Матрицы Мёллера определены на всюду плотном подпространстве асимптотического
+гильбертова пространства Has.
+Можно доказать и я сейчас это сделаю, что получилось изометрическое вложение
+асимптотического пространства в пространство H. Физический смысл полученного
+можно понять из формулы (которая в других обозначениях была на прошлой лекции):
+e−iHtΨ(k1, f1, . . . , kn, fn| ± ∞) = Ψ(k1, f1e−iεk1t, . . . , kn, fne−iεknt| ± ∞).
+Эта формула означает, что когда мы рассматриваем эволюцию в пространстве H,
+то в пределе t → ±∞ действию на вектор Ψ оператора эволюции в асимптотическом
+пространстве будет отвечать просто эволюция функций fi. Все происходит независи-
+мо. Иными словами, эволюция вектора Ψ сводится при больших t к эволюции системы
+из n далеких друг от друга частиц с неперекрывающимися волновыми функциями
+f1φ1e−iεk1t,. . . ,fnφne−iεknt.
+Данное нами определение S± может быть неоднозначным. Например, мы можем
+использовать разные хорошие операторы в построении, и неясно, получим ли мы
+один и тот же ответ. Однако можно доказать, что ответ не зависит от случайностей
+построения. Я докажу, что S± — изометрические операторы. Они сохраняют норму и
+сохраняют скалярное произведение. Такие операторы не могут быть многозначным.
+Заодно мы увидим, что вектор Ψ(k1, f1, ..., kn, fn|±∞) не меняется, при перестановках
+аргументов (ki, fi) и (kj, fj).
+Основная линия доказательства состоит в следующем. Чтобы получить матрицу
+Мёллера в формуле (68) положим для простоты все времена одинаковыми и рас-
+смотрим предел вектора Ψ(t) при t → ±∞. Заметим при этом, что согласно (68)
+такой вектор получается в результате многократного применения операторов B к ос-
+новному состоянию. Теперь легко увидеть, что, скалярное произведение двух таких
+векторов можно выразить через корреляционные функции, определяемые как сред-
+ние значения произведений рассматриваемых операторов по основному состоянию.
+Но корреляционные функции выражаются через усеченные корреляционные функ-
+ции. А в усеченных корреляционных функциях только двухточечные корреляционные
+функции будут выживать в пределе, t → ±∞, если я потребую кластеризацию.
+Теперь я могу сказать, что матрица Мёллера — это изометрическое отображение.
+Изометрическое отображение никак не может быть многозначным, потому что если
+два вектора совпадают, то расстояние между ними 0. Это условие должно сохранять-
+ся, и два совпадающих вектора должны переходить в совпадающие векторы.
+После того, как я ввел понятие матрицы Мёллера, я могу ввести понятия out-
+оператора и in-оператора:
+ain(f)S− = S−a(f), a+
+in(f)S− = S−a+(f),
+aout(f)S+ = S+a(f), a+
+out(f)S+ = S+a+(f).
+Грубо говоря, если S− и S+ унитарны, то это соответствие взаимно однозначно, и
+я могу все перекидывать из асимптотического пространства в основное пространство.
+(Здесь я опять не пишу индекс, описывающий тип частиц.)
+Справедливы также формулы
+a+
+in(fφ) = lim
+t→−∞
+ˆB(f, t), a+
+out(fφ) = lim
+t→∞
+ˆB(f, t).
+(70)
+То, что я рассказываю, немножко отличается от того, что было на прошлой лек-
+ции. Во-первых, операторы B(f, t) в прошлый раз я вводил аксиоматически, а в этот
+81
+
+раз построил с помощью явной формулы, а, во-вторых, мне пришлось заплатить за
+это появлением буквы φ, которая никому не мешает, но тем не менее она присутствует.
+Предел в (70) существует на множестве всех векторов вида Ψ(k1, f1, . . . , kn, fn|±)
+при условии, что f, f1, ..., fn-это неперекрывающееся семейство функций. Интересно,
+что когда размерность пространства d ≥ 3, то при некоторых условиях этот предел
+существует без условия неперекрывания. (Это для нас несущественно, потому что
+условие неперекрывания дает предел на всюду плотном множестве, и этого абсолютно
+достаточно.)
+Теперь, на основе сказанного, я могу выписать явно то, как действуют определен-
+ные нами операторы
+a+
+in(fφ)Ψ(f1, . . . , fn| − ∞) = Ψ(f, f1, . . . , fn| − ∞),
+a+
+out(fφ)Ψ(f1, . . . , fn|∞) = Ψ(f, f1, . . . , fn|∞).
+ain(f)Ψ(φ−1f, f1, . . . , fn| − ∞) = Ψ(f1, . . . , fn| − ∞),
+aout(f)Ψ(φ−1f, f1, . . . , fn|∞) = Ψ(f1, . . . , fn|∞).
+Эти формулы можно рассматривать как определения операторов ain и aout. Грубо
+говоря, операторы a+
+in/out добавляют одну функцию к Ψ(f1, . . . , fn|∓∞), а сопряжен-
+ные к ним операторы уничтожают одну из этих функций.
+Если операторы S+ и S− являются унитарными, мы говорим, что теория имеет
+интерпретацию в терминах частиц. В этом случае (а также в более общем случае,
+когда образ S− совпадает с образом S+), мы можем определить матрицу рассеяния
+S = S∗
++S−.
+как унитарный оператор в асимптотическом пространстве Has. Асимптотическое про-
+странство — это фоковское пространство. У него есть обобщенный базис |p1, . . . , pn⟩ =
+1
+n!a+(p1) . . . a+(pn)|0⟩.
+В этом базисе матричные элементы унитарного оператора S (амплитуды рассея-
+ния) могут быть выражены в терминах in- и out-операторов. Это те самые матричные
+элементы, квадраты которых дают эффективные сечения рассеяния. Получается сле-
+дующая формула:
+Smn(p1, . . . , pm|q1, . . . , qn) = ⟨a+
+in(q1) . . . a+
+in(qn)θ, a+
+out(p1) . . . a+
+out(pm)θ⟩
+(71)
+Это следует непосредственно из определения in- и out-операторов. В формуле (71) и
+в последующих я буду опускать числовые коэффициенты (m!)−1(n!)−1.
+Приведенная выше формула доказана только для случая, когда все значения
+импульсов pi, qj различны. (Точнее, мы должны предположить, что все векторы
+v(pi) = ∇ε(pi), v(qj) = ∇ε(qj) различны. В случае когда функция ε(p) является
+строго выпуклой, достаточно предположить, что pi, qj различны. Это несуществен-
+ное ограничение, но оно есть.) Формулу (71) следует понимать в смысле обобщенных
+функций. Это означает, что следует в качестве пробных функций брать множество
+функций fi(pi), gj(qj) с неперекрывающимися подмножествами U(fi), U(gj).
+Распишем теперь формулу (71) подробнее:
+Smn(f1, . . . , fm|g1, . . . , gn) =
+=
+�
+dmpdnq
+�
+fi(pi)
+�
+gj(qj)Smn(p1, . . . , pm|q1, . . . , qn) =
+= ⟨a+
+in(g1) . . . a+
+in(gn)θ, a+
+out( ¯f1) . . . a+
+out( ¯fm)θ⟩.
+82
+
+Вспомнив, что S-матрица рассеяния определена с помощью пределов при t → ±∞
+и воспользовавшись формулами (70), мы приходим к следующему представлению:
+Smn(f1, , fm|g1, . . . , gn) =
+lim
+t→∞,τ→−∞⟨θ| ˆB( ¯fmφ−1, t)∗ . . . ˆB( ¯f1φ−1, t)∗ ˆB(g1φ−1, τ) . . . ˆB(gnφ−1, τ))|θ⟩ =
+lim
+t→∞,τ→−∞ ω(B( ¯fm ¯φ−1, t)∗ . . . B( ¯f1 ¯φ−1, t)∗B(g1φ−1, τ) . . . B(gnφ−1, τ)),
+где B(f, t)∗ =
+�
+dxB∗(x, t) ˜f(x, t).
+Можно записать и более общую формулу
+Smn(f1, . . . , fm|g1, ..., gn) =
+lim
+ti→∞,
+τj→−∞
+ω(Bm( ¯fmφ−1
+m , tm)∗ . . . B1( ¯f1φ−1
+1 , t1)∗Bm+1(g1φ−1
+m+1, τ1) . . . Bm+n(gnφ−1
+m+n, τn)),
+справедливую, когда все Bi — разные хорошие операторы и Biθ = Φ(φi).
+Очень важное замечание состоит в том, что из условия неперекрывания, в пределе
+ti → ∞, τj → −∞ порядок времен несущественен как в группе со звездочкой, так и
+в группе без звездочки. Это значит, что на больших временах я могу эти операторы
+переставлять. В частности, я могу их считать упорядоченными по времени и получу
+то же, что появлялось у меня при введении функции Грина. Иными словами, я могу
+считать, что здесь стоит функция Грина. Это конец доказательства.
+Чтобы быть совершенно аккуратным, нужно все-таки сказать, какая получится
+функция Грина? Для того чтобы это установить, выразим согласно (67) операторы
+ˆBk(f, t) через ˆBk(p, t) и получим в результате:
+Smn(f1, . . . , fm|g1, . . . , gn) =
+�
+dm+np
+lim
+ti→∞,
+τj →−∞
+⟨θ|fm ¯φ−1
+m eiεm(pm)tm ˆBm(pm, tm)∗...f1 ¯φ−1
+1 eiε1(p1)t1 ˆB1(p1, t1)∗×
+g1φ−1
+m+1e−iεm+1(pm+1)τ1 ˆBm+1(pm+1, τn)...gnφ−1
+m+ne−iεm+n(pm+n)τn ˆBm+n(pm+n, τn)|θ⟩.
+Для матричных элементов матрицы рассеяния получается следующая формула:
+Smn(p1, . . . , pm|pm+1 . . . , pm+n) =
+lim
+t1,...,tm→∞,
+tm+1,...,tm+n→−∞
+⟨θ|¯φ−1
+m eiεm(pm)tm ˆBm(pm, tm)∗...¯φ−1
+1 eiε1(p1)t1 ˆB1(p1, t1)∗×
+φ−1
+m+1e−iεm+1(pm+1)tm+1 ˆBm+1(pm+1, tm+1)...×
+φ−1
+m+ne−iεm+n(pm+n),tm+n ˆBm+n(pm+n, tm+n)|θ⟩,
+где у меня стоят операторы ˆBm(pm, tm) и производится усреднение по состоянию θ.
+Это среднее совпадает с функцией Грина в (p, t)-представлении. Эта формула мне
+говорит, что если я исхожу из хороших операторов, матрица рассеяния выражается
+через функции Грина в (p, t)-представлении, точнее через их асимптотику приti → ∞
+и τj → −∞. У меня появились множители φ−1
+i . Это ровно те самые Λ, которые вво-
+дились для того, чтобы получить нормированные функции Грина. То, что я рассмат-
+риваю асимптотику, означает, что в энергетическом представлении я беру функции
+Грина on-shell. То, что у меня появились множители φ−1
+i
+означает, что я получаю
+нормированные функции Грина, и это конец истории.
+83
+
+Я дал доказательство формулы для хороших операторов. Из него, как я уже гово-
+рил, можно сделать вывод для много более широкого класса операторов. Я объясню
+это, вернувшись снова к ситуации, когда есть только один тип частиц.
+Дело в том, что в подходе Лемана, Симанчика и Циммермана операторы Ai почти
+произвольны. Нужно только чтобы проекция вектора ˆAiθ на одночастичные состоя-
+ния была ненулевой, а проекция этого вектора на вектор θ обращалась в ноль.
+Мне важно то, что в определении on-shell функции Грина эти операторы можно
+заменить на операторы A′
+i =
+�
+α(x, t)Ai(x, t)dxdt, которые осуществляют сглажива-
+ние. Легко проверить, что при этом не меняются нормированные on-shell функции
+Грина. Это первое замечание. А второе замечание состоит в том, что A′
+i можно счи-
+тать хорошими операторами. Дело в том, что я могу взять α(x, t) таким образом,
+чтобы носитель ее преобразования Фурье ˆα(p, ω) не пересекался с многочастичным
+спектром и не содержал нуля. (Я предполагаю, что одночастичный спектр не пере-
+секается с многочастичным спектром.) При этом я автоматически получу хороший
+оператор.
+Действительно, оператор ˆα осуществляет умножение на α в (p, ω)-представлении.
+(Я уже говорил, что переход к сглаженному оператору описывается сверткой. А в
+(p, ǫ)-представлении свертка превращается в умножение.) Если рассмотреть спектр
+операторов энергии и импульса, то умножение на функцию в (p, ǫ)-представлении
+убивает все точки спектра, где эта функция обращается в ноль. Функция, которую я
+рассматриваю, убивает многочастичный спектр, и у меня получается, хороший опе-
+ратор.
+84
+
+9
+Лекция 9
+9.1
+Обобщенные функции Грина
+В этой лекции я буду рассматривать обобщенные функции Грина. Эти функции есте-
+ственно появляются в формализме L-функционалов, в формализме Келдыша, а так-
+же и в других формализмах. Я хочу показать как инклюзивная матрица рассеяния
+выражается через обобщенные функции Грина. Для этого я заново докажу формулу
+Лемана, Симанчика и Циммермана (LSZ), но доказательство проведу по-другому. Я
+покажу как через обычные функции Грина выражается обычная матрица рассеяния,
+но доказательства будут построены так, чтобы было ясно, что их можно повторить
+и для обобщенных функций Грина.
+Мои рассуждения во многом повторяют уже приведенные на прошлых лекциях,
+но я постараюсь сделать так, чтобы эта лекция не зависела от двух предыдущих
+лекций.
+Я немного изменю обозначения. Временную переменную я буду обозначать буквой
+τ и писать B(τ, x) вместо B(x, τ) . Состояние ω, как и раньше, будет предполагаться
+трансляционно инвариантным и стационарным, соответствующий вектор в предгиль-
+бертовом пространстве H будет обозначаться θ.
+Обычная функция Грина в состоянии ω — это среднее значение произведения
+операторов Bi ∈ A :
+N = T(B1(τ1, x1) . . . Bn(τn, xn)),
+стоящих в порядке уменьшения времени (хронологического произведения). В обоб-
+щенной функции Грина
+Gn′n = ω(MN)
+есть как хронологическое произведение N, в котором времена уменьшаются, так и
+антихронологическое произведение
+M = T opp(B′∗
+1 (τ ′
+1, x′
+1) . . . B′∗
+n′(τ ′
+n′, x′
+n′)),
+в котором времена увеличиваются. Я докажу, что инклюзивная матрица рассеяния
+выражается через асимптотическое поведение обобщенных функций Грина в пред-
+ставлении, когда аргументами являются импульс и время. Я объяснял уже, что по
+общим свойствам преобразования Фурье это означает, что инклюзивная матрица рас-
+сеяния выражается через полюса обобщенных функций Грина, где аргументами явля-
+ются энергия и импульс. Точнее говоря, она совпадает с нормализованной обобщенной
+функцией Грина on-shell.
+9.2
+Матрица Мёллера
+Я ограничиваюсь случаем, когда есть только один тип частиц. Иными словами, рас-
+сматривается обобщенная векторная функция Φ(p), являющаяся собственной, как
+для оператора энергии, так и для оператора импульса:
+HΦ(p) = ǫ(p)Φ(p),
+PΦ(p) = pΦ(p).
+(72)
+(Здесь можно добавить отвечающий типу частиц индекс, и тогда это будет общий
+случай.)
+Еще одно определение, которое я буду использовать — это определение гладкого
+оператора. Я говорю, что оператор ˆB, который задается формулой
+ˆB =
+�
+dτdxα(τ, x) ˆA(τ, x),
+α ∈ S,
+85
+
+это гладкий оператор, если α принадлежит пространству быстро убывающих гладких
+функций S. Если произвести сдвиг в пространстве и времени,
+ˆB(τ, x) =
+�
+dτ ′dx′α(τ − τ ′, x − x′) ˆA(τ ′, x′),
+получится гладкая функция от τ и x, так как свертку можно дифференцировать под
+знаком интеграла. Это важно: мне придется дифференцировать, и поэтому я требую,
+чтобы все рассматриваемые операторы были гладкими.
+Для оператора ˆB(τ, x) можно взять преобразование Фурье:
+ˆB(τ, x) =
+�
+dpeipx ˆB(τ, p).
+Возникает вопрос о существовании преобразования Фурье, но всегда можно рассмат-
+ривать ˆB(τ, p) в смысле обобщенных функций. (Точнее, этот оператор нужно рас-
+сматривать как обычную функцию по τ и обобщенную по p.)
+Исходным пунктом теории было трансляционно инвариантное стационарное со-
+стояние ω, к которому может быть применена конструкцию Гельфанда, Наймарка
+и Сигала (GNS). Эта конструкция дает предгильбертово пространство H, в котором
+лежат одночастичные физические состояния
+�
+dpf(p)Φ(p). Я потребую, чтобы опера-
+тор ˆB переводил состояние θ (которое отвечает ω в пространстве H) в одночастичное
+пространство:
+ˆBθ =
+�
+dpφ(p)Φ(p).
+Я потребую также, чтобы волновая функция частицы φ(p) не обращалась в ноль.
+(Это не обязательно, но для упрощения я хочу это потребовать.) Тогда я буду гово-
+рить, что оператор ˆB хороший. (Мне важно, чтобы хороший оператор был гладким,
+но я уже сказал, что все операторы предполагаю гладкими.)
+Как уже было сказано, любой оператор, в том числе и хороший, можно сдвигать
+по x и τ. При сдвиге состояния по времени оно умножается на экспоненту от −iHτ,
+а по пространству — на экспоненту от iPx, поэтому, действуя оператором ˆB(τ, x) на
+θ, мы снова получим одночастичное состояние, только с некоторыми множителями:
+ˆB(τ, x)θ =
+�
+dpe−iǫ(p)τeixpφ(p)Φ(p).
+Если сделать преобразование Фурье только по p, это выражение преобразуется сле-
+дующим образом:
+ˆB(τ, p)θ = e−iǫ(p)τφ(p)Φ(p).
+Удобно ввести обозначение:
+B(p, τ) = eiǫ(p)τ ˆB(τ, p),
+где ǫ(p) — это закон дисперсии для рассматриваемой частицы. Здесь поменялись
+местами p и τ. Если рассматривать действие оператора B(p, τ) на θ, экспонента со-
+кратится и получится выражение независящее от τ:
+B(p, τ)θ = φ(p)Φ(p).
+Теперь все это можно усреднить с какой-то гладкой функцией f(p), имеющей
+компактный носитель: B(f) =
+�
+dpf(p) ˆB(0, p) =
+�
+dx ˜f(x) ˆB(0, x). После сдвига по τ
+это выражение преобразуется в следующее:
+B(f, τ) = TτB(T−τf)T−τ =
+�
+dpf(p)B(p, τ).
+86
+
+Очевидно, что B(f, τ)θ не зависит от τ. Иными словами, выполняется условие
+˙B(f, τ)θ = 0.
+Это – то, с чем я буду работать.
+Следующие выражения для вектора n-частичного состояния
+Ψ(τ) = Ψ(f1, .., fn|τ) = B1(f1, τ)...Bn(fn, τ)θ,
+(73)
+Ψ(f1, .., fn| ± ∞) =
+lim
+τ→±∞ Ψ(f1, .., fn|τ)
+(74)
+повторяют то, что было представлено на прошлой лекции, но здесь опущен индекс,
+описывающий тип частиц, и считается, что все операторы Bi хорошие. (Как мы уви-
+дим, последнее требование можно ослабить.)
+Теперь определим понятие in-состояния, устремляя τ → −∞, и аналогично опре-
+делим out-состояние, устремляя τ → +∞ (предел берется в гильбертовом простран-
+стве ¯H). Эти пределы описывают процесс рассеяния. Они существуют при некоторых
+условиях. Самое простое — потребовать, чтобы коммутаторы операторов Bk и ˙Bj (где
+точка обозначает дифференцирование по τ) вне зависимости от выбора операторов
+B1, ..., Bn стремились к нулю достаточно быстро при τ → ±∞ :
+||[ ˙Bk(fk, τ), Bl(fl, τ)]|| ≤
+Ca
+1 + |τ|a ,
+(75)
+где a > 1. (Достаточно потребовать, чтобы интеграл по τ от левой части сходился
+абсолютно.) Если это условие выполнено, то предел (74) существует.
+Вкратце я воспроизведу доказательство, изложенное на прошлой лекции. Доста-
+точно в выражении (73) продифференцировать Ψ(τ) по времени, при этом по правилу
+Лейбница образуется n слагаемых, каждое из которых содержит производную ˙Bi. Все
+замечательно в том случае, когда эта производная стоит на самом последнем месте:
+примененная к θ она дает ноль. Если производная стоит не на последнем месте, то
+благодаря условиям на коммутаторы (75) можно переставить ее на последнее место
+и получить ноль, но при этом нужно заплатить коммутаторами, которые становятся
+малыми при τ → ∞. В таком случае разность Ψ(τ1) − Ψ(τ2), представимая в виде
+интеграла от производной от ˙Ψ(τ), становится мала при τ1, τ2 → ∞; в силу полноты
+гильбертова пространства можно применить признак Коши для обоснования суще-
+ствования предела (74).
+Данное рассуждение во многом повторяет то, что уже было на прошлой лекции.
+Оно еще будет применяться неоднократно. Основное в нем — малость коммутатора.
+Для того чтобы проверить утверждение (75), говорящее, что встречающиеся в выра-
+жении (73) операторы почти коммутируют при больших τ, нужно, во-первых, пред-
+положить асимптотическую коммутативность (о которой я буду говорить немножко
+позже) и, во-вторых, предположить, что фигурирующие в формуле (75) функции
+fk(p) не перекрываются. Напомню, что когда мы рассматривали поведение волновой
+функции fk в зависимости от τ, мы видели, что при эволюции в x-пространстве важ-
+ную роль играет множество возможных скоростей Ufk. (Определяя множество Uf мы
+берем градиент от закона дисперсии v = ∇ǫ(p) в тех точках, где f(p) ̸= 0.) После
+этого берем замыкания ¯Ufk и требуем, чтобы эти множества не перекрывались. Грубо
+говоря, это означает, что частицы движутся в разных направлениях и существенные
+носители волновых функций в координатном пространстве далеко отстоят друг от
+друга. Это условие я буду все время накладывать. (Оно выполнено не всегда, но я
+требую, чтобы оно было выполнено для семейств функций f1, ..., fn, принадлежащих
+всюду плотному подмножеству пространства Sn.) Если наложить условие асимпто-
+тической коммутативности, из него вытекает малость коммутаторов.
+87
+
+Условие асимптотической коммутативности, прежде всего, означает, что комму-
+татор операторов Bk при одном и том же времени, но в далеко отстоящих простран-
+ственных точках будет мал. Условие малости можно варьировать, но как минимум
+мне нужно, чтобы этот коммутатор был мал в следующем смысле:
+∥[ ˆBk(τ, x), ˆBl(τ, x′)]∥ <
+Ca
+1 + |x − x′|a ,
+(76)
+где a > 1. Это условие должно быть выполнено не только для самих операторов,
+но и для их производных по времени и по пространству. Данный вариант несколько
+отличается от приведенного на прошлой лекции — мне так удобнее.
+Вспомним теперь, что все операторы ˆBk гладкие, то есть их можно получать пу-
+тем сглаживания некоторых операторов ˆBk =
+�
+gk(t, x) ˆAk(t, x)dxdt, где gk ∈ S. На
+операторы ˆAk можно наложить условие сильной асимптотической коммутативности:
+∥[ ˆAk(t, x), ˆAl(t′, x′)]∥ <
+Ca(t − t′)
+1 + |x − x′|a .
+(77)
+Это значит, что мы требуем, чтобы коммутаторы убывали быстрее любой степени,
+когда пространственное расстояние стремится к бесконечности. В числителе долж-
+на стоять полиномиальная функция Ca(t). В таком случае условия на производные
+накладывать не надо — они выводятся из приведенного условия.
+Для того чтобы вывести условие (76) из сильной асимптотической коммутативно-
+сти, достаточно заметить, что его можно свести к интегралу от выражения, включа-
+ющего произведение функций fk с различными индексами, у которых существенные
+носители далеки друг от друга. Эти интегралы будут малы потому, что при далеких
+операторах коммутаторы стремятся к нулю быстрее любой степени. Сделать оценку
+в этом случае очень просто.
+Можно ожидать, что сильная асимптотическая коммутативность имеет место в
+том случае, когда в спектре гамильтониана есть щель, то есть, спектр принадлежит
+лучу (ǫ, +∞), начинающемуся при каком-то положительном значении ǫ > 0. В случае
+релятивистской теории это отвечает случаю, когда все частицы имеют массу. Ко-
+гда же масса нулевая, сильной асимптотической коммутативности не будет, но более
+слабые условия (a > 1) выполняются в конформных теориях, где все аномальные
+размерности > 1
+2.
+Я определю понятие матрицы Мёллера следующим образом. Я введу понятие
+асимптотического пространства. Асимптотическое пространство Has определяется
+как гильбертово пространство, в котором действует фоковское представление кано-
+нических коммутационных соотношений:
+[a(p), a+(p′)] = δ(p, p′),
+[a(p), a(p′)] = [a+(p), a+(p′)] = 0.
+(Я работаю с импульсными переменными.) Вместо того, чтобы рассматривать опера-
+торные обобщенные функции a(p), a+(p′), можно рассматривать операторы a(f) =
+�
+dpf(p)a(p), a+(f) =
+�
+dpf(p)a+(p), где f ∈ h. В фоковском пространстве есть опе-
+раторы сдвига по пространству и сдвига по времени — это очевидно, если вспомнить,
+что это пространство может быть представлено как пополнение прямой суммы сим-
+метрических степеней элементарного пространства h. Матрицы Мёллера определим
+как отображения S± : Has → ¯H из асимптотического пространства Has в ¯H (в по-
+полнение пространства GNS). Здесь нельзя ограничиваться предгильбертовым про-
+странством — нужно рассматривать все гильбертово пространство. Это отображение
+определяется следующим образом:
+Ψ(f1, .., fn| ± ∞) = S±(a+(g1)...a+(gn)|0⟩).
+(78)
+88
+
+В левой части этой формулы стоит in-состояние — это асимптотическое состояние, за-
+висящее от функций fi. Я хочу интерпретировать их как волновые функции частиц,
+но для этого должно выполняться условие, чтобы одночастичное состояние в асимп-
+тотическом пространстве Has переходило в одночастичное состояние в пространстве
+¯H с той же волновой функцией. Чтобы это было выполнено, нужно наложить усло-
+вие, чтобы функции fi и gi асимптотически были связаны между собой соотношением
+fi = giφ−1
+i , где, напомню, φi ̸= 0 ни в одной точке.
+Из формулы (78) могут быть выведены различные свойства матрицы Мёллера.
+Одно из них заключается в том, что матрица Мёллера коммутирует с операторами
+пространственных и временных сдвигов. Это почти сразу следует из определений.
+Если рассматривать эволюцию во времени in-состояния, определяемого выражением
+(74), то ее можно свести к эволюции входящих в это выражение функций fi. Это в
+точности означает, что асимптотическая динамика in-состояния Ψ сводится к дина-
+мике функций fi.
+Покажем, что формула (78) определяет матрицу Мёллера как изометричное отоб-
+ражения S± : Has → ¯H.
+Прежде всего докажем, что матрицы Мёллера не зависят от выбора хороших опе-
+раторов B. Напомню, что операторы B в формуле (73) можно взять различными.
+Изменим один из этих операторов — тот оператор Bn, который стоит на последнем
+месте (заменим его другим хорошим оператором), оставляя функцию fn неизменной.
+Легко видеть, что при этом ничего не изменяется, потому что изменение последнего
+оператора сводится к изменению одночастичного состояния. То, как меняется одноча-
+стичное состояние, должно быть согласованно в асимптотическом пространстве Has
+и в пространстве ¯H, поэтому изменение последнего оператора ничего не меняет.
+Теперь изменим первый оператор. Поскольку по причине асимптотической комму-
+тативности операторы Bi имеют исчезающе малые коммутаторы, я могу оператор B1
+переставить на последнее место. За это нужно заплатить коммутаторами, но в пре-
+деле это ничего не меняет. После перестановки этого оператора, так же как и любого
+другого, на последнее место можно применить предыдущие рассуждения. Это дока-
+зывает, что матрица Мёллера не зависит от выбора операторов Bi, использовавшихся
+в определении (73).
+Для того, чтобы матрица Мёллера была хорошо определена, в формуле (78) долж-
+на быть симметрия по отношению к переменным gi. Эта симметрия присутствует,
+потому что коммутаторы малы и есть возможность переставлять операторы a+(gi)
+(если есть сильная асимптотическая коммутативность и мы имеем дело с неперекры-
+вающимися функциями).
+Если кроме асимптотической коммутативности наложить требование распадения
+корреляций (кластеризации), то можно доказать, что матрицы Мёллера изометрич-
+ны. Они, напомню, были определены только для набора неперекрывающихся функ-
+ций, но, раз они изометричны, то это ограниченные операторы. Их можно распро-
+странить на все гильбертово пространство и поэтому операторы S±, представляют
+собой изометрические вложения асимптотического пространства Has в пополненное
+пространство ¯H. Из изометричности следует, в частности, что матрицы Мёллера не
+зависят от случайностей построения (изометрический оператор не может быть мно-
+гозначным).
+Все перечисленные доказательства можно провести, не применяя свойства асимп-
+тотической коммутативности, а используя только свойство кластеризации, но с асимп-
+тотической коммутативностью все выглядит намного более понятно.
+89
+
+9.3
+Матрица рассеяния. Формула LSZ
+Теперь я хочу определить понятие матрицы рассеяния. Это разумно, если в теории
+есть интерпретация в терминах частиц, что подразумевает, что матрицы Мёллера яв-
+ляются не только изометричными, но и унитарными операторами. В этом случае они
+определяют изоморфизм асимптотического пространства Has и пространства ¯H. Это
+означает, грубо говоря, что любое (или почти любое) состояние с течением времени
+распадается на частицы. (Что-то похожее возникало при обсуждении солитонов.)
+Перейдем теперь к вычислению матричных элементов матрицы рассеяния. Пусть
+имеются какое-то начальное состояние и конечное состояние, которые обозначаются,
+соответственно, буквами i и f. Матрица рассеяния определяется формулой S = S∗
++S−.
+Обращаю внимание на то, что S− задает отображение Has → ¯H, а S∗
++ действует в про-
+тивоположном направлении, и поэтому матрица рассеяния — это оператор в асимпто-
+тическом пространстве. Я должен брать ее матричные элементы между состояниями
+в асимптотическом пространстве: ⟨f|S|i⟩ = ⟨f|S∗
++S−|i⟩.
+Рассмотрим состояния в асимптотическом пространстве, которые получаются из
+фоковского вакуума с помощью применения некоторого количества операторов a+:
+|i⟩ = a+(g1)...a+(gn)|0⟩,
+|f⟩ = a+(g′
+1)...a+(g′
+m)|0⟩.
+Выражая матричный элемент через операторы B, мы приходим к следующей фор-
+муле:
+⟨f|S|i⟩ =
+lim
+τ′
+k→+∞,
+τj →−∞
+⟨θ|B′∗
+m(f ′
+m, τ ′
+m)...B′∗
+1 (f ′
+1, τ ′
+1)B1(f1, τ1)...Bn(fn, τn)|θ⟩,
+(79)
+где fj = gjφ−1
+j , f ′
+k = g′
+kφ′−1
+k
+. Здесь использованы формулы (73), (74) с небольшой
+поправкой. Раньше при определении in-вектора считалось, что все времена в форму-
+ле (73) одинаковы. Этого достаточно для определения матрицы Мёллера, но здесь
+удобно (хотя и не обязательно) считать, что времена разные, но все стремятся, со-
+ответственно, к плюс или минус бесконечности — предел при этом не изменится.
+Доказательство этого может быть легко получено, но я его проводить не буду.
+Очень важное наблюдение заключается в том, что в формуле (79) можно все вре-
+мена упорядочить в порядке убывания. Ясно, что τ ′ я могу считать большим, чем τ,
+потому что τ ′ → +∞, а τ → −∞. Если же брать два времени τi1, τi2, то коммута-
+тор соответствующих операторов исчезающе мал, поскольку соответствующие функ-
+ции fi1, fi2 не перекрываются. В таком случае я могу поставить операторы в порядке
+убывания времен и считать, что здесь появляется хронологическое произведение или,
+иными словами, появляется функция Грина.
+Для того же, чтобы все было более аккуратно, я перейду к стандартному обоб-
+щенному базису |i⟩ = |p1, ..., pn⟩ , |f⟩ = |p′
+1, ..., p′
+m⟩. Собственно говоря, в физи-
+ке матричные элементы матрицы рассеяния нужно брать именно в таком базисе,
+в котором имеются частицы с заданными импульсами. Перепишем теперь форму-
+лу (79) в этом базисе. Стоящие в этой формуле операторы могут быть представ-
+лены в виде B(f, τ) =
+�
+dpf(p)B(p, τ). С другой стороны, напомню, что у опера-
+тора B(p, τ) можно поменять местами аргументы, и за это заплатить экспонентой
+B(p, τ) = eiǫ(p)τ ˆB(τ, p).
+В результате для матричного элемента в импульсном базисе получается следую-
+щее выражение:
+⟨f|S|i⟩ =
+lim
+τ ′
+k→+∞,τj→−∞
+ω(
+�
+e−iǫ(p′
+k)τ ′
+k( ¯φ′
+k)−1 ˆB′∗
+k (τ ′
+k, p′
+k)
+�
+eiǫ(pj)τj(φj)−1 ˆBj(τj, pj)).
+90
+
+Если в этой формуле вынести численные множители за знак (линейного) функци-
+онала ω, останется только ω( � ˆB′∗
+k (τ ′
+k, p′
+k) ˆBj(τj, pj)) и можно считать, что времена
+упорядочены, то есть, появляется функция Грина. Это — то, что мне было нужно.
+Полученное выражение будет иметь некоторый предел, и это означает, что мат-
+рица рассеяния выражена через асимптотику функции Грина в (τ, p)-представлении.
+Как уже объяснялось, асимптотика в (ǫ, p)-представлении определяется полюсами по
+энергетической переменной. Это (для хороших операторов) — в точности то, что есть
+в формуле LSZ.
+Матрицы Мёллера коммутируют со сдвигами по времени и пространству и, сле-
+довательно, с операторами энергии и импульса. Если теория имеет интерпретацию в
+терминах частиц, то в пространстве ¯H совместный спектр ˆH и ˆP совпадает со спек-
+тром свободного бозона, поскольку должны выполняться соотношения:
+a+
+out(f)S+ = S+a+(f),
+a+
+out(g) =
+lim
+τ→+∞ B(f, τ),
+где g = fφ, Bθ = Φ(φ).
+В доказательстве существования предела, определяющего in- (out)-состояние, пред-
+положение о том, что ˆBj являются хорошими операторами, использовалось только
+для вывода утверждения, что ˙ˆBj(f, τ)θ = 0. Если теория имеет интерпретацию в
+терминах частиц, можно доказать существование предела для любых гладких опера-
+торов ˆBj.
+Будем предполагать, что проекция ˆBjθ на одночастичное пространство имеет вид
+Φ(φj), где функция φj не обращающаяся в ноль. Асимптотическая коммутативность
+позволяет переместить множитель с производной по времени вправо. Докажем, что
+полученное выражение мало для большого τ (достаточно доказать, что это — сумми-
+руемая функция от τ).
+Я не предполагаю, что ˆBjθ, лежит в одночастичном пространстве, но мне важно,
+чтобы при проектировании на одночастичное пространство получалось Φ(φj), где φj
+— не обращающиеся в 0 функции. В то время как для хорошего оператора требова-
+лось, чтобы это выражение было равно одночастичному состоянию, здесь требуется
+только чтобы таковой была проекция. Все будет хорошо, если только выполняется
+условие ˙ˆBj(f, τ)θ → 0. Перейдем теперь к выводу этого условия.
+Если есть представление в терминах частиц, любой вектор можно разложить по
+векторам a+
+in(p1)...a+
+in(pr)θ. (Я мог бы это сделать в асимптотическом пространстве
+¯
+H⊣∫, и тогда у меня не было бы индекса in.) Cамо пространство ¯H изоморфно асимп-
+тотическому, если есть интерпретация в терминах частиц. В таком случае имеет место
+следующее разложение (справедливое для любого вектора):
+�
+r≥0
+�
+dp1...dprcr(p1, ..., pr)a+
+in(p1)...a+
+in(pr)θ.
+Если оператор ˆBj гладкий, то функции cr(p1, ..., pr) для ˆBjθ также будут гладкими
+(cr ∈ S). Это легко понять, так как гладкость оператора ˆBj подразумевает усреднение
+и, соответственно, функция cr представляется интегралом, позволяющим доказать
+гладкость. Для простоты формул предположим, что для среднего значения от глад-
+кого оператора ˆBj выполняется условие ⟨θ| ˆBj|θ⟩ = 0. Тогда мы можем представить
+ˆBj(f, τ)θ в виде
+ˆBj(f, τ)θ =
+�
+dpφj(p)f(p)a+
+in(p)θ+
+�
+r≥2
+�
+dpdp1...dpre−iτ(ε(p1)+...ε(pr)−ε(p))cr(p1, ..., pr)f(p)a+
+in(p1)...a+
+in(pr)θ.
+91
+
+В этой формуле суммирование должно быть по всем r, но при r = 1 в экспоненте
+происходит сокращение, и зависимость от времени исчезает. По этой причине выделе-
+но суммирование по r ≥ 2, а случай r = 1 соответствует одночастичному состоянию,
+для которого ответ известен, и он здесь представлен. Самое важное состоит в том, что
+при τ → ∞ у слагаемых, для которых r ≥ 2 в экспоненте стоит большой показатель,
+в то время как cr(p1, ..., pr) — хорошая гладкая функция.
+После дифференцирования по τ показатель в экспоненте останется и при этом
+возникнет некий множитель, который будет подавлен экспонентой, а так как cr —
+хорошая функция и показатель в экспоненте большой, то в пределе τ → ∞ вклад
+таких слагаемых исчезает.
+Таким образом, формула LSZ доказана не только для хороших операторов, но и
+для всех гладких операторов. (Практически все операторы, что встречается в физике,
+являются гладкими.) На этом и завершается доказательство формулы LSZ.
+9.4
+Инклюзивная матрица рассеяния
+Теперь я хочу применить полученные результаты для того, чтобы выразить инклю-
+зивную матрицу рассеяния в терминах обобщенных функций Грина. Для этого я буду
+рассматривать in-состояния. Ранее я их рассматривал как векторы в гильбертовом
+пространстве, но теперь я их буду рассматривать как состояния в смысле алгебраи-
+ческого подхода. Такие состояния — это положительные функционалы на алгебре, и
+у меня есть связь этих функционалов с векторами. Если я применяю некоторый опе-
+ратор, например B(f, τ) в гильбертовом пространстве, то в пространстве состояний я
+должен применять оператор вида L(g, τ) = ˜B(f, τ)B(f, τ), где f = gφ−1. (Напоминаю,
+что элемент алгебры определяет два оператора на функционалах: я могу либо умно-
+жать аргумент справа, либо умножать аргумент слева. В последнем случае берется
+сопряженный оператор.)
+В терминах таких операторов in-состояние, рассматриваемое как положительный
+функционал, может быть записано в следующем виде:
+ν =
+lim
+τ→−∞ L(g1, τ)...L(gn, τ)ω.
+Теперь рассмотрим следующее выражение:
+⟨1|L(g′
+1, τ ′)...L(g′
+n′, τ ′)L(g1, τ)...L(gn, τ)|ω⟩.
+В правой части формулы я, подействовав операторами L, получил in-состояние
+при τ → −∞, а в левой части вычисляю какие-то значения при τ → +∞. Это как раз
+то, что делается при рассмотрении инклюзивных матриц рассеяния. Я рассматриваю
+in-состояние и вычисляю что-то при больших временах. Можно доказать существо-
+вание предела Q этого выражения при τ ′ → +∞, τ → −∞ в предположении, что все
+функции g′
+i и gj не перекрываются:
+Q =
+lim
+τ ′→+∞⟨1|L(g′
+1, τ ′)...L(g′
+n′, τ ′)ν⟩.
+(Напомню, что асимптотическая коммутативность у меня всегда предполагается.)
+Используя соотношение ⟨1|σ⟩ = σ(1) и формулу limτ ′→+∞ B(f, τ ′) = a+
+out(g), полу-
+ченную на одной из предыдущих лекций, мы приходим к следующему выражению:
+Q = ν(aout(g′
+n′)...aout(g′
+1)a+
+out(g′
+1)...a+
+out(g′
+n′))).
+Через определенное таким образом выражение Q = Q(g′
+1, ..., g′
+n′, g1, .., gn) можно
+вычислить инклюзивные сечения. Я назову это выражение инклюзивной матрицей
+92
+
+рассеяния. Здесь есть проблема, связанная с тем, что это нелинейное выражение по g
+и g′ — это квадратичное (точнее, эрмитово) выражение. Всякое квадратичное выра-
+жение можно заменить билинейной формой, а эрмитово выражение можно заменить
+полуторалинейной формой — по одной переменной она будет линейной, по другой —
+антилинейной.
+Для того, чтобы получить выражение, которое будет линейным (или где-то анти-
+линейным), я введу обозначение L(˜g, g, τ) = ˜B( ˜f, τ)B(f, τ). Тут разведены перемен-
+ные f и ˜f, и теперь то, что раньше рассматривалось как эрмитово выражение, будет
+рассматриваться как полуторалинейное выражение от удвоенного числа переменных:
+ρ(˜g′
+1, g′
+1, ..., ˜g′
+n′, g′
+n′, ˜g1, g1, ..., ˜gn, gn) =
+lim
+τ′→+∞,
+τ→−∞
+⟨1|L(˜g′
+1, g′
+1, τ ′)...L(˜g′
+n′, g′
+n′, τ ′)L(˜g1, g1, τ)...L(˜gn, gn, τ)|ω⟩.
+Это выражение тоже будет называться инклюзивной матрицей рассеяния и его я
+буду выражать через обобщенные функции Грина. Расписав его, приходим к формуле
+ρ(˜g′
+1, g′
+1, ..., ˜g′
+n′, g′
+n′, ˜g1, g1, ..., ˜gn, gn) =
+lim
+τ′→+∞,
+τ→−∞
+⟨1|B(f ′
+1, τ ′)...B(f ′
+n′, τ ′) ˜B( ˜f ′
+n′, τ ′) ˜B( ˜f ′
+1, τ ′)×
+B(f1, τ)...B(fn, τ) ˜B( ˜fn, τ)... ˜B( ˜f1, τ)|ω⟩.
+Мне удобнее пользоваться несколько более общей формулой
+ρ(˜g′
+1, g′
+1, ..., ˜g′
+n′, g′
+n′, ˜g1, g1, ..., ˜gn, gn) =
+lim
+τ′
+j ,˜τ′
+j→+∞,
+τi,˜τi→−∞
+⟨1|B(f ′
+1, τ ′
+1)...B(f ′
+n′, τ ′
+n′) ˜B( ˜f ′
+n′, ˜τ ′
+n′) ˜B( ˜f ′
+1, ˜τ ′
+1)×
+B(f1, τ1)...B(fn, τn) ˜B( ˜fn, ˜τn)... ˜B( ˜f1, ˜τ1)|ω⟩.
+Мы можем считать, что в этом выражении времена упорядочены. (Часть времен
+стремится к +∞, другая часть к −∞. Внутри каждой группы в силу асимптотической
+коммутативности мы можем переставлять множители в любом порядке, в частности,
+в порядке убывания времен.) Применив формулу (36) мы видим, что то, что стоит
+под знаком предела, можно выразить через обобщенную функцию Грина. Инклю-
+зивная матрица рассеяния выражается через временную асимптотику этой функции.
+Такие же рассуждения были раньше для обычной матрицы рассеяния, только число
+аргументов удвоилось.
+93
+
+10
+Лекция 10
+10.1
+Удаление лишних состояний
+В этой лекции я собираюсь рассказать о том, как можно представить квантовую
+механику как классическую механику, в которой мы имеем возможность измерять
+только часть наблюдаемых. С точки зрения физики это вполне естественно. Наши
+приборы позволяют измерять какие-то наблюдаемые, но не исключено, что могут
+появиться какие-то более совершенные приборы, позволяющие измерять и другие
+вещи.
+Я буду исходить из геометрического подхода к квантовой теории. В геометриче-
+ском подходе, описывая физическую теорию, мы начинаем с множества состояний,
+представляющего собой ограниченное выпуклое множество C0-подмножество тополо-
+гического векторного пространства L. Операторы эволюции должны принадлежать
+некоторой группе V, которая состоит из автоморфизмов пространства состояний C0
+(либо все автоморфизмы, либо часть их). Оператор эволюции σA(t) удовлетворяет
+уравнению
+dσA(t)
+dt
+= AσA(t),
+(80)
+где A ∈ Lie(V) — это элемент алгебры Ли группы V (“гамильтониан”). Это уравнение
+должно иметь решение (то есть “гамильтониан” должен порождать однопараметриче-
+скую подгруппу σA(t) группы V). Хотелось бы сказать, что “гамильтонианы” — это и
+есть наблюдаемые в геометрическом подходе, но наблюдаемые должны давать какие-
+то числа, и поэтому я буду говорить, что наблюдаемая- это пара (A, a) , где A ∈ Lie(V)
+— это “гамильтониан”, а a -это линейный функционал, инвариантный относительно
+группы σA(t), порожденной оператором A (это означает, что удовлетворяется условие
+a(Az) = 0).
+В обычной квантовой механике наблюдаемая определяется самосопряженным опе-
+ратором ˆA, “гамильтониан” действует на матрицы плотности как коммутатор (с точ-
+ностью до множителя i), функционал a определяется формулой a(K) = Tr ˆAK.
+Группа V естественным образом действует на наблюдаемые: A преобразуется по
+присоединенному представлению, a — как функция на L.
+Теперь исследуем, нет ли в нашей теории лишних состояний? Когда есть два состо-
+яния x, y ∈ C0, такие, что a(x) = a(y) для всякой наблюдаемой (A, a), то мы говорим,
+что из этих двух состояний можно оставить только одно. Если отождествить те состо-
+яния, которые дают один и тот же ответ для всех наблюдаемых, то в результате будет
+получена новая теория без лишних состояний эквивалентная, по существу, исходной
+теории.
+10.2
+Квантовая механика из классической механики
+Применим эти соображения к случаю, когда в качестве исходной берется класси-
+ческая теория. В этой теории чистые состояния — это точки фазового пространства
+(симплектического многообразия) M. Смешанные состояния — это распределения ве-
+роятностей на M; каждое смешанное состояние может быть однозначно представлено
+как смесь чистых состояний. Физические наблюдаемые — это вещественные функ-
+ции на M. Наблюдаемая a задает векторное поле A на M как гамильтоново вектор-
+ное поле с гамильтонианом a. (Идентифицируя векторное поле с дифференциальным
+оператором первого порядка, мы можем выразить A в терминах скобки Пуассона:
+Af = {a, f}.) Мы предполагаем, что, интегрируя это векторное поле, получаем од-
+нопараметрическую группу σA(t) канонических преобразований (симплектоморфиз-
+94
+
+мы) многообразия M. Эта группа действует также и на смешанные состояния, и на
+наблюдаемые, описывающие временную эволюцию этих объектов. Уравнение дви-
+жения для плотности распределения вероятностей (уравнение Лиувилля) имеет вид
+dρ
+dt = −{a, ρ}, а уравнение движения для наблюдаемых имеет вид df
+dt = {a, f}.
+Предположим, что наши устройства способны видеть только часть наблюдаемых
+объектов и что множество Λ “наблюдаемых наблюдаемых” представляет собой ли-
+нейное пространство, замкнутое относительно скобки Пуассона. Будем индексиро-
+вать это множество элементами алгебры Ли, обозначаемой g. (Отображение γ → aγ,
+переводящее γ ∈ g в aγ ∈ Λ является изоморфизмом алгебр Ли g и Λ.)
+Гамильтоновы векторные поля Aγ с гамильтонианами aγ определяют действие
+алгебры Ли g на M. Предположение о том, что векторные поля Aγ генерируют одно-
+мерные подгруппы, означает, что это действие индуцируется действием односвязной
+группы Ли G, имеющей g в качестве алгебры Ли.
+Приведенные выше соображения могут быть применены также и к бесконечномер-
+ным симплектическим многообразиям и к бесконечномерным алгебрам и группам Ли.
+Однако в бесконечномерном случае эти соображения не являются строгими.
+Определим отображение моментов µ из M в g∨ как отображение x → µx, где
+µx(γ) = aγ(x). (Здесь x ∈ M, γ ∈ g, а g∨ обозначает пространство линейных функцио-
+налов на g.) Это отображение является G-эквивариантным относительно коприсоеди-
+ненного действия G на g∨. (Иными словами оно коммутирует с преобразованиями из
+группы G.) Для каждого состояния классической системы (для каждого распределе-
+ния вероятностей ρ на M) определим точку ν(ρ) ∈ g∨ как интеграл от µx по x ∈ M с
+мерой ρ:
+ν(ρ) =
+�
+x∈M
+µxdρ.
+Точка ν(ρ) принадлежит выпуклой оболочке N из µ(M). (Выпуклая оболочка под-
+множества E топологического векторного пространства определяется как наимень-
+шее выпуклое замкнутое множество, содержащее E.)
+Группа G естественным образом действует на пространстве классических состоя-
+ний. Из G-эквивариантности отображения моментов следует, что отображение ν явля-
+ется G-эквивариантным отображением этого пространства в g∨ с коприсоединенным
+действием G.
+Будем говорить, что два классических состояния (два распределения вероятностей
+ρ и ρ′) эквивалентны, если
+�
+x∈M
+aγ(x)dρ =
+�
+x∈M
+aγ(x)dρ′
+(81)
+для каждого γ ∈ g. Другими словами, мы говорим, что два состояния эквивалентны,
+если вычисления с этими состояниями дают одинаковые результаты для каждого
+гамильтониана aγ. (Наши устройства не могут различать эти два состояния.)
+Докажем теперь следующее утверждение: два состояния ρ и ρ′ эквивалентны
+тогда и только тогда, когда ν(ρ) = ν(ρ′).
+Сначала заметим, что для каждого γ ∈ g
+ν(ρ)(γ) =
+�
+x∈M
+µx(γ)dρ =
+�
+x∈M
+aγ(x)dρ
+и точно так же
+ν(ρ′)(γ) =
+�
+x∈M
+µx(γ)dρ′ =
+�
+x∈M
+aγ(x)dρ′.
+В классической теории где допустимы только гамильтонианы из множества Λ = {aγ},
+где γ ∈ g, должны быть отождествлены эквивалентные состояния (мы исключаем
+95
+
+избыточные состояния). Отображение ν индуцирует биективное отображение про-
+странства классов эквивалентности на множество N, полученное в виде выпуклой
+оболочки множества µ(M) (“квантовые состояния”). G-эквивариантность отображе-
+ния ν означает, что эволюция классических состояний согласуется с эволюцией кван-
+товых состояний.
+Применим теперь наши конструкции к комплексному проективному пространству
+CP. Мы определяем это пространство как сферу ||x|| = 1 в комплексном гильбертовом
+пространстве H с отождествлениями x ∼ λx, где λ ∈ C, |λ| = 1. Группа U унитарных
+операторов действует транзитивно на CP. Существует единственная (с точностью до
+постоянного множителя) U-инвариантная симплектическая структура на этом про-
+странстве; это позволяет рассматривать комплексное проективное пространство как
+однородное симплектическое многообразие.
+Предположим, что мы можем наблюдать только гамильтонианы вида aC(x) =
+⟨x, Cx⟩ где C — самосопряженный оператор. Множество Λ таких гамильтонианов
+является алгеброй Ли относительно скобки Пуассона. Эта алгебра Ли изоморфна ал-
+гебре Ли g самосопряженных операторов, где операция определяется как коммутатор,
+умноженный на i. Однопараметрическая группа из унитарных операторов, соответ-
+ствующих гамильтониану aC, задается формулой σ(t) = e−iCt.
+Отображение моментов преобразует точку x в линейный функционал на про-
+странстве самосопряженных операторов, отображающий оператор C в TrKxC, где
+Kx является проекцией H на вектор x, т.е. Kx(z) = ⟨z, x⟩x. (Напомню, что в наших
+обозначениях точки комплексного проективного пространства представлены норми-
+рованными векторами.) Выпуклая оболочка образа отображения моментов состоит
+из положительно определенных самосопряженных операторов с единичным следом
+(т.е. она состоит из матриц плотности).
+Мы видим, что, применяя нашу общую конструкцию к комплексному проектив-
+ному пространству, мы получаем обычную квантовую механику. В этом случае наши
+соображения близки к “нелинейной квантовой механике” Вайнберга, предложившего
+рассматривать классическую теорию на CP как деформацию квантовой механики.
+Богатый источник примеров описанной ранее конструкции дают орбиты группы
+G в коприсоединенном представлении (в пространстве g∨ двойственном к алгебре Ли
+g группы G). Эти орбиты играют важную роль в теории представлений. Они явля-
+ются однородными симплектическими многообразиями. Квантуя их, можно получать
+унитарные представления группы G.
+Для того чтобы ввести на орбите структуру симплектического многообразия, нуж-
+но заметить, что для функций на g∨ определена скобка Пуассона. (Элементы алгебры
+Ли g можно рассматривать как линейные функции на g∨ ; для них скобка Пуассо-
+на определяется как операция в алгебре Ли. Пользуясь свойствами скобки Пуассо-
+на, можно определить скобку для произвольных гладких функций.) Пространство
+g∨ получает структуру Пуассонова многоообразия, но это многообразие не являет-
+ся симплектическим, потому что скобка Пуассона вырождена. Ограничивая скобку
+Пуассона на орбиту, мы получаем невырожденную скобку, определяющую симплек-
+тическую структуру.
+Элементы алгебры Ли g определяют семейство Λ функций на орбите; мы при-
+меним к нему описанную выше конструкцию. Она сильно упрощается, потому что
+в рассматриваемом случае отображение моментов - это просто вложение орбиты в
+g∨. Мы видим, что считая в классической теории допустимыми только гамильтониа-
+ны из семейства Λ мы получаем теорию, в которой множеством состояний является
+выпуклая оболочка орбиты. Группу V можно считать изоморфной группе G.
+Проиллюстрируем приведенные выше конструкции в случае, когда G является
+группой U унитарных преобразований гильбертова пространства H. В этом случае
+96
+
+мы можем идентифицировать элементы алгебры Ли g с ограниченными самосопря-
+женными операторами. При подходящем выборе топологии в g можно отождествить
+двойственное пространство g∨ с линейным пространством самосопряженных опера-
+торов имеющих след. Для упрощения обозначений предположим, что гильбертово
+пространство конечномерно, однако наши соображения могут быть применены и в
+бесконечномерном случае.
+Рассмотрим орбиты U в пространстве g∨ (в коприсоединенном представлении).
+Если dim H = n, орбита индексируется различными вещественными числами λ1, ..., λr
+(собственными значениями операторов, принадлежащих орбите) и неотрицательными
+целыми числами k1, ..., kr (кратностями собственных значений). (Кратности долж-
+ны удовлетворять условию k1 + ...kr = n.) Стационарная группа U(n) точки ор-
+биты изоморфна прямому произведению групп U(ki), поэтому орбита гомеоморфна
+U(n)/U(k1) × ... × U(kr) (многообразию флагов).
+Если r = 2, то орбита гомеоморфна грассманиану. Если r = 2, k1 = n − 1, k2 = 1,
+мы получаем комплексное проективное пространство. (Грассманиан Gk(H) опреде-
+ляется как пространство всех k-мерных подпространств пространсва H. Его можно
+рассматривать как симплектическое U-многообразие. Ортонормированный базис k-
+мерного подпространства определен с точностью до преобразования из унитарной
+группы U(k). Это означает, что точки Gk(H) описываются ортонормированными си-
+стемами векторов φ1, ..., φk с идентификацией φ′
+i ∼ ul
+iφl, где ul
+i — унитарная матрица.)
+Some problems (Mostly mathematical problems related to the material of my lectures)
+0. A complex vector space E is equipped with non-negative scalar product. Prove that
+we can obtain pre Hilbert space factorizing E with respect to vectors with⟨x, x⟩ = 0.
+Hint. Check that these vectors constitute a linear subspace of E.
+Density matrices are defined as positive-definite self-adjoint operators having unit trace
+and acting in complex Hilbert space.
+1. Prove that the set of density matrices is convex. Check that extreme points of this set
+are one-dimensional projectors KΨ(x) = ⟨x, Ψ⟩Ψ where ||Ψ|| = 1 (they are in one-to-one
+correspondence with non-zero vectors of Hilbert space with identification Ψ ∼ λΨ).
+2. Prove that the set of density matrices in two-dimensional Hilbert space is a three-
+dimensional ball and set of of its extreme points is a two-dimensional sphere.
+The linear envelope T of the set of density matrices is the space of all self-adjoint
+operators belonging to trace class. (A self-adjoint operator belongs to trace class if it
+has discrete spectrum and the the series of its eigenvalues is absolutely convergent.) We
+consider T as a normed space with the norm ||T|| = � |λk| where λk are eigenvalues of
+T. By definition an automorphism of the set of density matrices is a bicontinuous linear
+operator in T generating a bicontinuous map of the set density matrices. (One says that a
+map is bicontinuous if it is continuous and has a continuous inverse.) It is obvious that a
+unitary operator specifies an automorphism of the set of density matrices by the formula
+T → UTU −1.
+3.Prove that automorphisms of the set of density matrices are in one-to-one correspondence
+with unitary operators.
+I do not know how to solve this problem (this does not mean that it is difficult, I did
+not try). Maybe this fact is proved somewhere, but I do not know any reference.
+In the next problems the term "operator"means "linear operator". It is convenient to
+define etA where A is an operator as a solution of the equation
+dU(t)
+dt
+= A · U(t)
+97
+
+with initial condition U(0) = 1. If (A1, ..., An) is a family of commuting operators we
+define e
+� tiAi as a product et1A1...etnAn.
+4. Prove that eiaP = Tℏa, where P denotes the momentum operator P = ℏ
+i ∇ and Ta
+stands for translation operator transforming the function f(x) into a function f(x + a).
+5. Let us define an operator CA acting in the space of operators by the formula CA(X) =
+[A, X]. (For definiteness one can assume that A and X are bounded operators in Hilbert
+space, but this assumption is not important.) Prove that
+etCA(X) = etAXe−tA,
+or equivalently
+etAXe−tA = X + t[A, X] + t2
+2![A, [A, X]] + t3
+3![A, [A, [A, X]]] + ...
+Hint. Differentiate these equalities.
+6. Let us assume that the commutator of operators X and Y is a number C (or, more
+generally, an operator C, commuting with X and Y ). Prove that
+eXeY = eX+Y e
+1
+2C,
+eXeY = eCeY eX.
+7. Let us assume that for an operator A acting in Banach space the norms of operators
+etA where t ∈ R are uniformly bounded (i.e. sup−∞<t<+∞ ||etA|| is finite). Prove that all
+eigenvalues ofA are purely imaginary.
+8. Let A denote an operator acting in finite-dimensional complex vector space. Assume
+that the norms of operators etA where t ∈ R are uniformly bounded. Prove that the
+operator A is diagonalizable (i.e. there exists a basis consisting of eigenvectors of A).
+Hint. Use Jordan normal form. Prove that all Jordan cells are one-dimensional.
+9. Let us consider Grassmann algebra with generators ǫ1, ǫ2, ǫ3, ǫ4. Calculate
+(ǫ1 + ǫ2)(ǫ2 + ǫ3)(ǫ3 + ǫ4),
+�
+(ǫ1 + ǫ2)(ǫ2 + ǫ3)(ǫ3 + ǫ4)(ǫ4 + ǫ1)d4ǫ,
+∂
+∂ǫ2
+(ǫ1ǫ2ǫ3),
+cos(ǫ1ǫ2 + ǫ3ǫ4).
+10.Let us consider a unital associative algebra with commuting generators x1, ..., xn and
+anticommuting generators ξ1, ..., ξn (tensor product of polynomial algebra and Grassmann
+algebra). Prove that d2 = 0 and ∆2 = 0 where
+d =
+�
+ξi
+∂
+∂xi
+,
+∆ =
+� ∂
+∂xi
+∂
+∂ξi
+.
+Weyl algebra is defined as unital associative algebra with generators a∗
+k, ak obeying
+CCR( [ak, a∗
+l ] = δk,l, [ak, al] = [a∗
+k, a∗
+l ] = 0). An involution ∗ in Weyl algebra transforms ak
+into a∗
+k. Fock representation of Weyl algebra=representation with cyclic vector |0⟩ obeying
+ˆak|0⟩ = 0. Scalar product is defined by the condition that ˆa∗
+k is adjoint to ˆak
+98
+
+Fock space= completion of the space of Fock representation.
+11. Poisson vector is defined by the formulaΨλ = eλˆa∗|0⟩ where λˆa∗ = � λkˆa∗
+k.
+a) Check that Poisson vector is an eigenvector of the operator ˆak
+b) Find scalar product of two Poisson vectors.
+12. Let us consider the Hamiltonian ˆH(t) = ω(t)ˆa∗ˆa (harmonic oscillator with time-
+dependent frequency).
+a) Let us suppose that ω(t) = ω0 + ω exp(−αt) for t ≥ 0. (Here α is a small positive
+number.) Calculate the evolution of matrix entries of the density matrix in ˆH-representation
+(i.e. express matrix entries of K(t) in terms of matrix entries of K(0)). Here ˆH = ˆH(0).
+b) Assuming that ω is a random variable with given mean value and dispersion uniformly
+distributed on some interval calculate the average of matrix entries of the density matrix
+K(t) for t = const
+α .
+13. A molecule is placed near a microwave oven. Give a rough estimate of decoherence
+time for this molecule.
+Hint. Decoherence appears if the phase factors entering the expressions for non-diagonal
+matrix entries of density matrix are changed significantly by the electric field of the oven.
+You can calculate this change in perturbation theory; the first order contribution comes
+from dipole momentum. You can find the information about dipole momenta of ground
+state and excited states on the web; you need only the order of magnitude of these momenta
+and of electric field.
+14. Let us define the L-functional corresponding to the density matrix K by the the
+formula
+LK(α) = tre−αˆa∗eα∗ˆaK.
+(We consider the case when there is only one degree of freedom, [ˆa, ˆa∗] = 1.)
+Calculate the L-functional corresponding to the coherent state (to the normalized
+Poisson vector).
+Reminder. Every normalized vector Φ defines a density matrix KΦ and trAKΦ =
+⟨AΦ, Φ⟩. Coherent state is a normalized eigenvector of ˆa.
+15. Let us assume that supp(φ) is a compact set. Then for large |τ| we have
+|(Tτφ)(x)| < Cn(1 + |x|2 + τ 2)−n
+where x
+τ /∈ Uφ, the initial data φ = φ(x) is the Fourier transform of φ(k), and n is an
+arbitrary integer.
+Here supp(φ) is the closure of the set of points were φ(k) ̸= 0, Uφ is a set of all points
+of the form ∇ǫ(k) where k belongs to a neighborhood of supp(φ),
+Tτφ)(x) =
+�
+dkeikx−iǫ(k)τφ(k),
+the function ǫ(k) is smooth.
+16. A generalized function ρ(ǫ) is a sum of of the function
+A
+ǫ−E−i0 and a square integrable
+function. Find asymptotic behavior of its Fourier transform ρ(t).
+17. Let us consider a system with the Hamiltonian
+ˆH =
+�
+dpǫ(p)ˆa∗(p)ˆa(p) and
+momentum operator ˆP =
+�
+dpp)ˆa∗(p)ˆa(p) where ˆa, ˆa∗ obey CCR.
+a) Calculate time and space translations in the formalism of L-functionals
+b) Find the operators ˆa(τ, p), ˆa∗(τ, p) and corresponding operators acing on L-functionals.
+c) Prove that the L-functional ω = exp(−
+�
+dpα∗(p)n(p)α(p)) specifies a stationary
+translation-invariant state. (Here n(p) is an arbitrary function.)
+d) Calculate two-point GGreen functions in (τ, p)- and (ε, p)-representations for the
+state ω.
+99
+
+Список литературы
+[1] Schwarz A. Geometric approach to quantum theory. SIGMA. Symmetry, Integrability
+and Geometry: Methods and Applications. 2020 Apr 1;16:020.
+[2] Schwarz, A., 2021. Geometric and algebraic approaches to quantum theory. arXiv
+preprint arXiv:2102.09176.
+[3] Schwarz A. Scattering theory in algebraic approach to quantum theory. Associative
+algebras, arXiv preprint arXiv:2107.08553, 2021
+[4] Schwarz A. Scattering theory in geometric approach to quantum theory. arXiv
+preprint arXiv:2107.08557, 2021
+[5] Schwarz A. Scattering theory in algebraic approach to quantum theory. Jordan
+algebras (in preparation)
+[6] A.S. Shvarts, New formulation of quantum theory, Dokl. Akad. Nauk SSSR, 173, 793
+(1967).
+[7] Schwarz A. Inclusive scattering matrix and scattering of quasiparticles. Nuclear
+Physics B. 2020 Jan 1;950:114869.
+100
+
diff --git a/G9E2T4oBgHgl3EQfTge-/content/tmp_files/load_file.txt b/G9E2T4oBgHgl3EQfTge-/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..244e4c79534beb1a25ccffe1f3090a7317386276
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf,len=3850
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='03804v1  [quant-ph]  10 Jan 2023  Quantum mechanics and quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Algebraic and geometric approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Квантовая механика и квантовая теория поля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Алгебраический и геометрический подходы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Schwarz  Department of Mathematics  University of California  Davis, CA 95616, USA,  schwarz @math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='edu  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Frolov  Department of Mathematics,  National Research Nuclear University MEPhI  (Moscow Engineering Physics Institute)  115409, Kashirskoe shosse 31, Moscow, Russia  frolovi55@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ru  Аннотация  This is a non-standard exposition of main notions of quantum mechanics and quantum  field theory that also includes some recent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' It is based on algebraic approach where  the starting point is an associative algebra with involution and states are defined as positive  linear functionals on this algebra and on geometric approach where the starting point is a  set of states considered as a convex subset of linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' The exposition does not depend  on textbooks in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Standard formulas for quantum probabilities are derived from decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' This derivation  allows us to go beyond quantum theory in geometric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Particles are defined as  elementary excitations of ground state (and quasiparticles as elementary excitations of  any translation invariant state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' It follows from this definition that the notion of identical  particles is very natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' The scattering of particles is analyzed in the framework of  generalization of Haag-Ruelle theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' The conventional scattering matrix does not work for  quasiparticles (and even for particles if the theory does not have particle interpretation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  The analysis of scattering in these cases is based on the notion of inclusive scattering  matrix, closely related to inclusive cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' It is proven that the conventional scattering  matrix can be expressed in terms of Green functions (LSZ formula) anf inclusive scattering  matrix can be expressed in terms of generalized Green functions that appear in the  Keldysh formalism of non-equilibrium statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' It is shown that generalized  Green functions and inclusive scattering matrices appear also in the formalism of L-  functionals that can be identified with positive functionals on Weyl or Clifford algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  The derivation of the expression of the evolution operator and other physical quantities  in terms of functional integrals is based on the notion of symbol of operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' these arguments  1  can be applied also in geometric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' This result can be used, in particular, to give  a simple derivation of diagram technique for generalized Green functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  The notion of inclusive scattering matrix makes sense in geometric approach (but it  seems that one cannot give a definition of conventional scattering matrix in this situation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  The geometric approach is used to show that quantum mechanics and its generalizations  can be considered as classical theories where our devices are able to measure only a part  of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  This text is based on first ten lectures of the course taught by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Schwarz in the Spring  of 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' see www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='mathnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ru for lectures (in Russian) and slides (in English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Keywords Inclusive scattering matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' generalized Green function, geometric approach  2  Содержание  1  Лекция 1  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Введение .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
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+page_content='2  Представления алгебры Клиффорда .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
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+page_content='  32  4  Лекция 4  36  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Адиабатическое приближение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Декогерентность .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  92  10 Лекция 10  94  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1 Удаление лишних состояний .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
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+page_content='  94  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2 Квантовая механика из классической механики .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  94  3  1  Лекция 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Введение  В обычном изложении квантовой механики мы живем в гильбертовом пространстве и  рассматриваем операторы в этом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Самосопряженные операторы отвеча-  ют наблюдаемым.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это подход, которым пользуются физики почти всегда, но он имеет  свои недостатки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду говорить про другие подходы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это прежде всего, алгебраиче-  ский подход, где исходным пунктом является алгебра наблюдаемых — ассоциативная  алгебра с инволюцией, в которой самосопряженные элементы является наблюдаемы-  ми.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот подход, по-моему, почти столь же стар, как и сама квантовая механика.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Кроме того, я буду говорить про геометрический подход, в котором исходная точка  — это множество состояний [1-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот подход я предложил пару лет тому назад, и  он много более общий чем алгебраический подход.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Основная вещь, которая, по-моему, недостаточно подчеркивается в обычном изло-  жении квантовой механики (чуть больше говорится в квантовой теории поля) это то,  что понятие частицы не является основным в квантовой механике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это понятие произ-  водное.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Частицы — это элементарные возбуждения основного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Квазичасти-  цы (тоже важное понятие) — это элементарные возбуждения любого трансляционно  инвариантного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Основное понятие, которое есть в физике элементарных  частиц — это понятие матрицы рассеяния и о нем я буду больше всего говорить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  понятие тесно связано с понятием сечения рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Кроме того, я буду говорить о понятии инклюзивной матрицы рассеяния, тесно  связанном с понятием инклюзивного сечения рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Матрицы рассеяния выража-  ются через функции Грина известной формулой, принадлежащей Леману, Симанчику  и Циммерману, а инклюзивные матрицы рассеяния выражаются через обобщённые  функции Грина, которые появились впервые в неравновесной статистической физике  в формализме Келдыша.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Выпуклые множества  Перед тем как переходить к физике, я хочу сказать несколько слов про выпуклые  множества, которые у меня будут появляться многократно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Выпуклое множество C — это подмножество векторного пространства, которое  вместе с каждыми двумя точками содержит отрезок, соединяющий эти точки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Важ-  но то, что в выпуклом множестве можно рассматривать смесь точек этого множества.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если взять несколько точек множества и каждой точке приписать неотрицательное  число так, чтобы сумма чисел равнялась единице, то тогда сумма точек ei ∈ C с коэф-  фициентами pi ≥ 0, � pi = 1 также будет принадлежать выпуклому множеству.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта  сумма � piei ∈ C называется смесью точек множества с вероятностями pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно  рассматривать числа pi как веса, и тогда эта сумма будет представлять центр тя-  жести.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Другое важное понятие — это понятие крайней точки выпуклого множества.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Крайняя точка — это такая точка, которая не лежит внутри никакого отрезка с кон-  цами, принадлежащими множеству.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У многогранника крайние точки — это вершины.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  У шара крайние точки — это сфера, ограничивающая этот шар.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я всегда буду считать, что в том векторном пространстве, которое я рассматри-  ваю, есть какая-то топология, есть понятие непрерывности и, тем самым, понятие  замкнутого множества.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду предполагать, что все выпуклые множества, которые  я рассматриваю, замкнуты, и тогда можно рассматривать смесь не только конечно-  го числа точек, но и смесь счетного числа точек.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В последнем случае сумму нужно  считать бесконечной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  4  Также можно рассматривать смесь точек любого подмножества выпуклого мно-  жества, если на подмножестве задано распределение вероятностей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если, скажем, на  сфере, ограничивающей шар, задано распределение вероятности с плотностью ве-  роятности ρ(λ), то можно рассматривать смесь исходя из этого распределения ве-  роятности: просто вместо суммы нужно брать интеграл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если λ — это параметры,  описывающие точки на сфере, ω(λ) — это точки сферы, тогда можно рассматривать  такую смесь, беря вместо суммы интеграл  �  ω(λ)ρ(λ)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если выпуклое множество  компактно, то каждая точка является смесью его крайних точек.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Квантовая теория.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Геометрический подход  В двух словах я повторю то, что мне нужно из квантовой механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Начну с того, что  для меня понятие состояния в квантовой механике — это понятие матрицы плотно-  сти.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Матрица плотности — это самосопряженный оператор K, который положительно  определен и у которого след равен единице: trK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Множество матриц плотности  выпукло, а его крайние точки называются чистыми состояниями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти чистые состо-  яния отвечают векторам в гильбертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Каждому нормированному  вектору Ψ соответствует матрица плотности, которая определяется как ортогональ-  ная проекция на этот вектор:  KΨ(x) = ⟨x, Ψ⟩Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Нужно заметить что если два вектора пропорциональны (Ψ′ = λΨ), то соответ-  ствующие матрицы плотности совпадают: KΨ′ = KΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  У матрицы плотности K существует базис из собственных векторов ei с неотри-  цательными собственными значениями pi сумма которых равна 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно считать,  что каждому из этих векторов соответствует чистое состояние, а матрица плотности  — это смесь этих чистых состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В том представлении, в котором матрица диаго-  нальна, ее диагональные элементы обозначим как pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку след равен единице,  их сумма равна единице: � pi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку матрица положительно определена,  эти диагональные элементы неотрицательны: pi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Следовательно, можно сказать,  что матрица плотности — это смесь чистых состояний с вероятностями pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В обычных курсах все это рассказывается в обратном порядке начиная с чистых  состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Матрицы плотности определяются как смешанные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я обсудил один способ представления матрицы плотности как смеси чистых состо-  яний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом деле, это можно делать бесконечным множеством разных способов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе исходной точкой является множество состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пред-  полагается, что это ограниченное замкнутое выпуклое подмножество топологического  линейного пространства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оказывается, что этих предположений достаточно для того,  чтобы построить содержательную теорию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='3  Алгебраический подход к квантовой теории  В то время как в геометрическом подходе основное — это пространство состояний,  в алгебраическом подходе основное — это алгебра наблюдаемых A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что  алгебра — это векторное пространство, в котором можно умножать элементы с дис-  трибутивным законом, но еще я потребую, чтобы в этой алгебре была инволюция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Типичный пример алгебры (для меня основной) — это алгебра ограниченных опера-  торов в гильбертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этой алгебре есть инволюция A → A∗, которая  отвечает переходу к сопряженному оператору.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Она обладает тем свойством, что, пе-  рейдя дважды к сопряженному оператору, мы возвращаемся к исходному: A∗∗ = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если берать сопряженный к произведению оператор, это будет опять же произведе-  ние, но в обратном порядке: (AB)∗ = B∗A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кроме того, инволюция антилинейна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В  5  алгебре операторов это простые свойства, а для произвольных ассоциативных алгебр  с инволюцией (∗-алгебр) это — аксиомы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я всегда предполагаю, что в алгебре есть какая-то топология, в которой все опе-  рации непрерывны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я обычно про эти топологии говорить не буду, во-первых, по-  тому, что это требует времени, а, во-вторых, потому, что разные топологии могут  быть одинаково разумны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В некоторых ситуациях более удобны одни, в других си-  туациях — другие.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Того, что есть топология обычно бывает мало.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иногда требуется,  чтобы была норма, в которой алгебра является банаховым пространством, тогда тре-  буется, чтобы норма произведения была меньше или равна чем произведение норм  (||AB|| ≤ ||A|| · ||B||) (это определение банаховой алгебры).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иногда требуется нечто  большее, например, чтобы алгебра была C∗ -алгеброй.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Это значит, что норма произ-  ведения A∗A равна квадрату нормы оператора A, то есть ||A∗A|| = ||A||2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я обычно  не буду уточнять, какая именно топология выбрана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хочу подчеркнуть, что при  рассмотрении гомоморфизма или автоморфизма алгебры я буду всегда считать, что  он согласован с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если есть алгебра с инволюцией, то самосопряженные элементы (A = A∗) отве-  чают физическим величинам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сами по себе самосопряженные элементы алгебры не  образуют.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Произведение самосопряженных элементов — это не обязательно самосо-  пряженный элемент.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот недостаток алгебраического подхода было замечен еще в  30-е годы, что привело к понятию йордановой алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Йордан заметил, что хотя  произведение самосопряженных элементов не является самосопряженным , но если  взять антикоммутатор A ◦ B = AB + BA, где A и B — самосопряженные, этот ан-  тикоммутатор снова будет самосопряжённым.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Он аксиоматизировал эту операцию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теория йордановых алгебр была построена в 30-е годы, в основном, в знаменитой  работе Йордана, Вигнера и фон Нойманна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но хотя йорданова алгебра — очень кра-  сивый и действительно полезный во многих частях математики объект, в физике до  сих пор он не получил большого применения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сейчас он естественно появился в гео-  метрическом подходе и, может быть, снова вернется в физику.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если начинать со ∗-алгебры (с алгебры с инволюцией), можно определить понятие  состояния: состояние — это линейный функционал ω на алгебре A, удовлетворяющий  условию неотрицательности на элементах вида A∗A:  ω(A∗A) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Состояния, отличающиеся численным множителем, отождествляются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Часто удобно  рассматривать только состояния, удовлетворяющие условию ω(1) = 1 (нормирован-  ные состояния).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь можно определить понятие математического ожидания в данном состоя-  нии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для функции f(A) от A ∈ A математическое ожидание задается формулой:  ⟨f(A)⟩ω = ω(f(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для C∗-алгебры можно определить f(A) для любой непрерывной функции от само-  сопряженного элемента A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' тогда, зная математическое ожидание для всех непрерыв-  ных функций, можно определить понятие распределения вероятностей физической  наблюдаемой в нормированном состоянии (состоянии, для которого ω(1) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Одного понятия состояния мало: нужно еще понятие эволюции состояния, потому  что целью физики, как и любой науки — делать предсказания.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Физик прежде всего  рассматривает задачу: если есть какое-то начальное состояние, нужно предсказать,  что будет потом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В алгебраическом подходе для того, чтобы определить эволюцию, нужно прежде  всего рассмотреть группу Aut автоморфизмов алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомним, что автоморфиз-  мы A всегда должны коммутировать с инволюцией, и тогда эта группа автоморфиз-  6  мов, естественно действующая на линейных функционалах, переводит положитель-  ные функционалы в положительные (состояния в состояния).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В любом подходе к  квантовой теории состояния должны зависеть от времени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Должен быть оператор  эволюции U(t), который переводит состояние в начальный момент в состояние ω(t) в  какой-то другой момент времени t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда ω(t) = U(t)ω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В алгебраическом подходе можно считать, что операторы U(t) являются автомор-  физмами алгебры;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' они порождают обозначаемые точно так же операторы, действую-  щие на состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Как и в обычной квантовой механике, есть картина Шредингера,  где эволюционирует состояние, и есть картина Гейзенберга, где эволюционирует опе-  ратор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти две картины эквивалентны:  ω(t)(A) = ω(A(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Наблюдать динамику состояния по времени при неподвижном элементе алгебры, это  то же самое, что наблюдать динамику элемента алгебры A при неизменном состоянии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В физике оператор эволюции обычно вычисляется исходя из уравнения движе-  ния, описывающих тот же оператор эволюции, но за бесконечно малое время.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если  есть инвариантность относительно сдвига времени, то можно утверждать, что опе-  ратор, описывающий изменение за бесконечно малый промежуток времени, сам от  времени не зависит.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Уже было сказано, что изменение за конечное время должно  быть автоморфизмом алгебры, значит, изменения за бесконечно малые промежутки  времени — это инфинитезимальные автоморфизмы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Зная инфинитезимальный авто-  морфизм A, будем решать уравнение движения dV/dt = AV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Решение выражается  через экспоненту от инфинитезимального автоморфизма eAt ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате по-  лучаем однопараметрическую группу автоморфизмов, состоящую из преобразований  вида eAt (операторов эволюции).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Физики в курсах квантовой механики пишут обыч-  но мнимую единицу в экспоненте — я ее не пишу, но, конечно, никакой разницы в  этом нет).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Решение уравнения движения имеет вид V (t) = eAtV (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я не дал формального определения инфинитезимального автоморфизма.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Одно  из возможных формальных определений: инфинитезимальный автоморфизм — это  касательный вектор к кривой в группе автоморфизмов в единичном элементе этой  группы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно потребовать несколько больше: чтобы эта кривая была однопарамет-  рической подгруппой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Важно отметить, что инфинитезимальный автоморфизм является дифференци-  рованием алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что он должен удовлетворять правилу Лейбница:  применяя его к произведению xy нужно сначала применить его к первому сомно-  жителю, оставляя второй неизменным, потом — ко второму сомножителю, первый  оставив неизменным A(xy) = (Ax)y + x(Ay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это следует мгновенно из самого опре-  деления автоморфизма и самого определения инфинитезимального автоморфизма.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если есть инфинитезимальный автоморфизм, то 1 + tA при малом t — это уже авто-  морфизм.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Точнее, 1+tA плюс нечто высшего порядка по t — это есть автоморфизм).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если применить определение автоморфизма, как раз получится правило Лейбница.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обратно, если есть дифференцирование, то есть если выполнено правило Лейб-  ница, и еще к тому же оно согласованно с инволюцией, то есть, выполняется условие  (Ax)∗ = A(x∗), то можно надеяться, что это инфинитезимальный автоморфизм.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для  того чтобы проверить, что это так, нужно написать уравнение движения  dU/dt = AU,  где U(t) — это элемент алгебры A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если это уравнение имеет решение с начальным  условием U(0) = 1, то A является инфинитезимальным автоморфизмом — анало-  гом гамильтониана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду говорить, что A это “гамильтониан”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если алгебра конеч-  номерна, то можно применить теорему существования решений дифференциальных  7  уравнений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае дифференцирование и инфинитезимальный автоморфизм  — это одно и то же.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку алгебра в физике бесконечномерна, то это в общем  случае не так.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Не всякое дифференцирование определяет инфинитезимальный ав-  томорфизм.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно, чтобы уравнение движения имело решение, то есть,чтобы от-  вечающая этому уравнению эволюция существовала.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поэтому есть различие между  дифференцированием и инфинитезимальным автоморфизмом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко проверить, что  дифференцирования образуют алгебру Ли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это же верно для дифференцирований,  которые согласованы с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно считать, что дифференцирования согла-  сованные с инволюцией образуют алгебру Ли группы автоморфизмов Aut(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для  случая бесконечномерных групп понятие алгебры Ли не очень хорошо определено, но,  тем не менее, это важное понятие, которое очень хорошо работает во многих случаях.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я рассматривал случай, когда уравнение движения не зависит от времени, но,  вообще говоря, это не обязательно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' “Гамильтониан” может зависеть от времени, и  тогда уравнение движения для операторов эволюции имеет вид:  dU  dt = A(t)U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если оператор A(t) не зависит от t, то операторы эволюции образуют однопарамет-  рическую группу:  U(t + τ) = U(t)U(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матрице плотности K отвечает линейный функционал ω(A) = trKA на алгеб-  ре ограниченных операторов;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' этот функционал удовлетворяет условию ω(A∗A) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Легко понять, что это условие вытекает из положительной определенности оператора  K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эволюция матрицы плотности, как это рассказывается в курсе квантовой механи-  ки, описывается уравнением, в котором в правой части стоит коммутатор с самосопря-  женным оператором.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это уравнение имеет вид dK/dt = H(K), где H(K) = [ ˆH, K]/iℏ,  ˆH — это обычный гамильтониан, а H — это “гамильтониан”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь введены следующие обозначения: операторы в гильбертовом пространстве  — операторы с крышечкой, а операторы на матрицах — без крышечки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По теореме  Стоуна самосопряженные операторы в гильбертовом пространстве (не обязательно  ограниченные) отвечают однопараметрическим подгруппам группы унитарных опе-  раторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Унитарные операторы можно считать действующими на матрицах плот-  ности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В теореме Стоуна требуется, чтобы подгруппы были непрерывны в сильном  смысле.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Я не буду пояснять, что это такое — мне это не понадобится).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если са-  мосопряженный оператор ограничен, то соответствующая однопараметрическая под-  группа дифференцируема в смысле сходимости по норме.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В дальнейшем я не буду  обращать внимания на эти тонкости.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я хочу еще пару слов сказать про связь алгебраического подхода со стандартным  подходом, основанным на гильбертовых пространствах, и объяснить, почему алгебра-  ический подход лучше.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пусть у нас имеется сохраняющее инволюцию представление  алгебры A операторами в гильбертовом пространстве H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, рассмотрим  сохраняющий инволюцию гомоморфизм алгебры A в алгебру операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обозначим  оператор, отвечающий элементу A алгебры A через ˆA, тогда каждому нормирован-  ному вектору Φ ∈ H отвечает нормированное же состояние ω алгебры A по формуле  ω(A) = ⟨ ˆAΦ, Φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Более того, каждой матрице плотности K отвечает состояние по формуле ω(A) =  tr(K ˆA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, можно получать состояния алгебры из векторов в гиль-  бертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возникает естественный вопрос: все ли состояния алгебры  можно так получить?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ответ на этот вопрос положительный.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Всякое состояние может  8  быть представлено вектором в гильбертовом пространстве, и это причина того, что  физики в состоянии работать все время в гильбертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Почему это во многих случаях неудобно?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что исходя из алгебры наблю-  даемых невозможно ограничиться одним гильбертовым пространством — в разных  ситуациях придется рассматривать разные гильбертовы пространства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например, в  статистической физике рассматриваются равновесные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Каждое равновес-  ное состояние лежит в своем гильбертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это не всегда удобно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Одним гильбертовым пространством, как правило, можно обойтись в квантовой  теории поля, потому что там обычно рассматривается гильбертово пространство, в  котором лежит основное состояние.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Его элементы отвечают возбуждениям основного  состояния, которые только и нужны в квантовой теории поля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По этой причине в  квантовой теории поля обычно можно обойтись одним гильбертовым пространством,  но это не всегда — в квантовой электродинамике этого сделать нельзя: там электрон  одевается облаком мягких фотонов, которые выводят из изначального гильбертова  пространства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Сейчас я собираюсь доказать, что каждое состояние алгебры с инволюцией пред-  ставляется вектором из гильбертова пространства, которое я собираюсь построить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Построение такого гильбертова пространства осуществляется конструкцией Гельфанда  – Наймарка – Сигала (GNS-конструкция), которую я сейчас объясню.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для каждой  алгебры A я построю предгильбертово пространство E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ( Здесь мне будет удобно  работать с предгильбертовыми пространствами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Я построю представление алгебры  операторами в этом предгильбертовом пространстве таким образом, что некоторому  циклическому вектору, который я обозначаю буквой θ ∈ E, будет отвечать состояние  ω(A) = ⟨ ˆAθ, θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что вектор циклический означает, что всякий другой вектор мож-  но из него получить с помощью операторов из алгебры (все векторы имеют вид ˆAθ,  где A ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Конструкция, которую я буду объяснять, однозначна с точностью до эквивалент-  ности, как будет видно из построения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для ее построения я предположу, что такое  представление у меня уже есть, и определю в алгебре скалярное произведение по  формуле  ⟨A, B⟩ = ω(B∗A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Зная это скалярное произведение в алгебре, я могу вычислить произведение векторов  ˆAθ и ˆBθ просто перекидывая ˆB на первый сомножитель, а дальше пользуюсь тем,  что гомоморфизм, сохраняет инволюцию, я могу с этой пары операторов перекинуть  крышечку на B∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это как раз и есть ω(B∗A):  ⟨ ˆAθ, ˆBθ⟩ = ⟨ ˆB∗ ˆAθ, θ⟩ = ⟨ �  (B∗A)θ, θ⟩ = ω(B∗A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В результате получилось так, что если я знаю ω(B∗A), то могу вычислить скаляр-  ное произведение ˆAθ и ˆBθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Так как вектор θ циклический, каждый вектор простран-  ства имеет вид ˆAθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это условие цикличности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь я могу сказать, что имеется  отображение ν : A → E, которое переводит A в ˆAθ, и это отображение сюрьективное  (то есть, на).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отсюда следует, что E получается из A факторизацией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно профак-  торизовать по всем векторам, которые дают 0 в скалярном произведении с любыми  другими векторами (нулевым векторам).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Чтобы дать теперь окончательный ответ на вопрос: как построить E исходя из  алгебры A и ω, нужно просто взять алгебру A, ввести в ней скалярное произведе-  ние по формуле ω(B∗A) и профакторизовать по нулевым векторам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получится пред-  гильбертово пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Факторизуя по нулевым векторам, можно снова опре-  делить скалярное произведение в фактор-пространстве).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате получилась  эта конструкция, которую я, по существу, вывел, а не придумал.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я увидел, что она  9  обязательно должна быть такой, потому что исходя из того, что вектор θ цикли-  ческий, я сделал две вещи: во-первых, построил предгильбертово пространство, а,  во-вторых, доказал, что моя конструкция по существу единственна, ничего другого  сделать нельзя.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я ее вывел из цикличности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это рассуждение (конструкция Гель-  фанда–Наймарка–Сигала) — самый важный элемент из того, что я в этой лекции  собираюсь рассказывать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этой конструкцией я буду пользоваться много раз.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вместо  предгильбертова пространства я могу рассматривать его пополнение ¯E — гильберто-  во пространство (тогда вектор θ будет циклическим в более слабом смысле: векторы  вида Aθ будут плотны в ¯E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для иллюстрации возьмем какое-то стационарное состояние (состояние, которое  не меняется при эволюции) и применим к нему GNS- конструкцию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда получится  некоторое гильбертово пространство и циклический вектор в нем, который тоже будет  стационарным, не будет зависеть от времени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Утверждение: если исходить из стационарного состояния, то в новом гильбертовом  пространстве, полученном с помощью данной конструкции будет действовать группа  унитарных операторов, отвечающих операторам эволюции U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это очень просто понять.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что GNS-конструкция использовала в каче-  стве скалярного произведения ω(B∗A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но это скалярное произведение инвариантно  относительно операторов U(t) потому что ω инвариантно (то есть, не изменяется  оператором эволюции).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раз скалярное произведение инвариантно, то операторы U(t)  спускаются в унитарные операторы ˆU(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Операторы ˆU(t) образуют однопараметри-  ческую группу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У нее есть генератор (инфинитезимальный автоморфизм) ˆH, и это  то, что в физике называется гамильтонианом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (На самом деле это не совсем так, пото-  му что в физике гамильтониан считается самосопряженным оператором, и для этого  нужно поставить мнимую единицу в определении: ˆU(t) = e−i ˆ  Ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я говорю, что состояние ω — это основное состояние, если спектр оператора ˆH  неотрицателен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обращаю внимание, что основное состояние будет иметь нулевую  энергию при таком определении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это оказывается согласованным со стандартным  определением.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если применить GNS-конструкцию к алгебре ограниченных опера-  торов и состоянию, отвечающему собственному вектору гамильтониана с собствен-  ным значением E, то тогда генератором группы ˆU(t) построенной с помощью GNS-  конструкции будет ˆH − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В квантовой теории поля всегда говорят, что мы будем  отсчитывать энергию от основного состояния, и не будем обращать внимание на тот  бесконечный вклад, который присутствует в буквальном подходе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Алгебраическая  GNS-конструкция делает это сама по себе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я все время говорил, что рассматриваю алгебраический подход в квантовой ме-  ханике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот подход прекрасно работает и в классической механике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чтобы пока-  зать это, будем работать в гамильтоновом формализме и тогда можно повторить  те же самые рассуждения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чистое состояние описывается обобщенными импульсами  p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', pn) и обобщенными координатами q = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', qn), представляющими точки  в 2n-мерном пространстве, которое называется фазовым пространством.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — чи-  стое состояние, но, точно так же как в квантовой механике, можно рассматривать и  смешанные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Смешанное состояние — это распределение вероятностей на фазовом простран-  стве или положительная мера на фазовом пространстве (считается, что мера все-  го пространства равна единице).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все эти распределения вероятностей образуют вы-  пуклое множество D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чистые состояния — это крайние точки этого множества.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чи-  стое состояние — это распределение вероятностей, которое сосредоточено ровно в  одной точке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ему отвечает плотность распределения вероятностей, которая является  дельта-функцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Любая функция является суперпозицией дельта-функций (иными  словами, любую функцию можно представить как интеграл от дельта-функций).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  10  означает, что всякому распределению вероятностей на фазовом пространстве отве-  чает распределение вероятностей на чистых состояниях, а чистые состояния можно  отождествить с крайними точками пространства всех состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В рассматриваемом случае, соответствующем в классической механике, всякое со-  стояние может быть представлено единственным образом как смесь чистых состо-  яний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это отличает классическую механику от квантовой механики, где состояние  может быть представлено как смесь чистых состояний множеством разных спосо-  бов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Позже я расскажу, как квантовую механику можно получить из классической  механики, если дозволить себе рассматривать не все наблюдаемые.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Естественно с фи-  зической точки зрения предположить, что наши приборы не могут измерить всё, что  есть на свете.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мое утверждение: если есть такая ситуация, то классическая механика  может превратиться в квантовую механику.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дальше я напомню хорошо известное уравнение движения Гамильтона  dq  dt = ∂H  ∂p , dp  dt = −∂H  ∂q  и уравнение Лиувилля для плотности распределения вероятностей, которое записы-  вается в терминах скобок Пуассона как:  d  dtρ(p, q, t) = {H, ρ(p, q, t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Скобки Пуассона при этом определяются формулой  {f, H} = −∂f  ∂p  ∂H  ∂q + ∂f  ∂q  ∂H  ∂p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того чтобы определить эволюцию U(t) нужно будет решить уравнение d  dtU(t) =  LU(t), где Lρ = {H, ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это уравнение эквивалентно уравнению Лиувилля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того,  чтобы в этом убедиться, нужно просто проверить, что на чистых состояниях оно сво-  дится к гамильтоновым уравнениям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В классической механике, точно так же как в  квантовой, можно следить как развиваются наблюдаемые (а не состояния).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Наблюда-  емые - это действительные функции f(p, q) на фазовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко можно  вывести из уравнений Гамильтона, что это развитие происходит в соответствии с  уравнением  d  dtf(p(t), q(t)) = −∂f  ∂p  ∂H  ∂q + ∂f  ∂q  ∂H  ∂p  или  d  dtf(p(t), q(t)) = {f, H},  где в правой части стоит скобка Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь мы видим, что находимся в той же  ситуации как квантовая механика или, наоборот, квантовая механика находится в той  же ситуации, как и классическая механика.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У нас есть наблюдаемые.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они образуют  алгебру A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти наблюдаемые вы можете перемножать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть понятие инволюции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Инволюция — это просто комплексное сопряжение).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Каждому состоянию ω отвечает  линейный функционал на наблюдаемых A: нужно взять интеграл от функции по  распределению вероятностей  ω(f) =  �  fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Распределение вероятностей — это мера, по которой можно проинтегрировать).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот  функционал удовлетворяет условию положительности : ω(f) ≥ 0 при f ≥ 0 потому  что для каждой положительной функции будет неотрицательный ответ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Функции ти-  па A∗A, несомненно, положительны (просто квадрат модуля), так что классическая  механика входит как маленький кусочек в алгебраический подход к квантовой меха-  нике, но есть разница: в классической механике алгебра наблюдаемых коммутативна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  11  2  Лекция 2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Квантовая механика как деформация классической  механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Алгебра Вейля  Теперь я попытаюсь объяснить как математик мог бы вывести квантовую механику из  классической.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Он знает, что квантовая механика в пределе малой постоянной Планка  сводится к классической.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что квантовая механика получается дефор-  мацией — маленьким изменением классической механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Должно быть семейство  алгебр Aℏ, которое зависит от постоянной Планка ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Зависимость эта непрерывная и  дифференцируемая.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При равной нулю постоянной Планка ℏ = 0 должна получаться  классическая механика, то есть, должна получаться коммутативная алгебра с произ-  ведением A · B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать (это не обязательно, как мы увидим вскоре), что все  эти алгебры определены в том же самом векторном пространстве, то есть, сложение  и умножение на число независимы от постоянной Планка, а умножение элементов  алгебры зависит от постоянной Планка A·ℏ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь рассмотрим коммутатор в этой  алгебре, зависящей от постоянной Планка:  [A, B]ℏ = A ·ℏ B − B ·ℏ A  Наше основное требование — чтобы этот коммутатор обращался в ноль при обраще-  нии в ноль постоянной Планка : ℏ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать, что зависимость от постоянной  Планка является гладкой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что коммутатор представляется как линей-  ная по ℏ часть и нечто более высокого порядка:  [A, B]ℏ = i{A, B}ℏ + O(ℏ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  То, что линейно по ℏ обозначено как фигурная скобка, умноженная на мнимую еди-  ницу;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' я докажу, что фигурная скобка — это и есть скобка Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее, она имеет  те же свойства, которые есть у скобки Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что новая операция  есть дифференцирование (удовлетворяет правилу Лейбница):  {A · B, C} = {A, C} · B + A · {B, C}  и, кроме того, удовлетворяет аксиомам алгебры Ли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нетрудно доказать, что это так.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Cвойства обычного коммутатора в ассоциативной алгебре:  [A, B] = −[B, A],  [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0,  [AB, C] = [A, C]B + A[B, C]  должны удовлетворяться при каждой постоянной Планка ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Разложим все эти ра-  венства по постоянной Планка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Во втором равенстве нужно разлагать до второго  порядка, а в остальных — до первого достаточно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если приравнять члены первого  порядка по ℏ, то как раз получаются нужные свойства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть, мы убедились, что  в пределе из квантовой механики получается классическая механика.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Коммутатор  переходит в скобку Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Давайте посмотрим: можно ли пойти наоборот.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы видели как из квантовой меха-  ники получается классическая.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' А теперь мы интересуемся: можно ли из классической  механики получить квантовую?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого я сначала опишу все возможные скобки  Пуассона в том случае, когда алгебра A является алгеброй полиномиальных функций  на каком-то векторном пространстве с координатами (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если она является  12  алгеброй полиномов, то для того, чтобы вычислить, как работает скобка Пуассона,  нужно знать только какова скобка Пуассона у этих координат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это обусловлено тем,  что любая другая функция — это полином, а у меня есть свойство, которое позволяет  вычислять скобку Пуассона от произведения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Полином — это линейная комбинация  произведений координат, и поэтому скобку двух полиномов можно вычислить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ре-  зультат вычисления таков:  {A, B} = 1  2σkl(u) ∂A  ∂uk  ∂B  ∂ul,  (1)  где σkl(u) обозначает скобку Пуассона координат uk, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Наоборот, можно проверить, что если σ является антисимметрической и незави-  сящей от u, то это выражение будет удовлетворять условиям, наложенным на скобку  Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это как раз ситуация, которая возникает, когда мы имеем дело со стан-  дартной скобкой Пуассона в фазовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я могу задать вопрос: как продеформировать скобку Пуассона, чтобы по-  лучилось то, что мне нужно в квантовой механике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта задача непростая.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Она была  решена сравнительно недавно Концевичем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но в том случае, когда матрицы σ не  зависят от u, то есть, когда скобка Пуассона двух координат не зависит от u, то сде-  лать это очень легко.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Именно, я определю алгебру Aℏ как ассоциативную алгебру с  генераторами ˆuk, связанными соотношением:  ˆukˆul − ˆulˆuk = iℏσkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (2)  Я просто переписал то условие, которое у меня было наложено на скобку Пуас-  сона координат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это, конечно, можно было сделать и в том случае, когда σ зависит  от u, но тогда неизвестно: получилась бы ассоциативная алгебра или нет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же σ  не зависит от u, то получается ассоциативная алгебра, которая называется алгеброй  Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если мы исходим из полиномов, то это — единственный (с точностью до вы-  бора генераторов) способ продеформировать скобку Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я ввожу инволюцию  в алгебре Вейля считая генераторы ˆuk самосопряженными.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мы получили коммутационные соотношения, которые несколько в другом виде  хорошо известны из квантовой механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чтобы это показать, я потребую, чтобы  матрица σ была обратима.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда эту матрицу можно упростить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это антисимметрич-  ная матрица и, в буквальном смысле, ее диагонализовать нельзя, однако ее можно  записать в подходящем базисе как блочно-диагональную матрицу, состоящую из дву-  мерных блоков.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если этим воспользоваться, то заменив систему генераторов, можно  свести коммутационные соотношения в алгебре Вейля к хорошо известным в кванто-  вой механике коммутационным соотношениям:  ˆpkˆpl = ˆplˆpk,  ˆqkˆql = ˆqlˆqk,  ˆpkˆql − ˆqlˆpk = ℏ  i δl  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это — коммутационные соотношения для координат и импульсов в стандартной кван-  товой механике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они называются каноническими коммутационными соотношениями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вместо самосопряженных генераторов можно взять другие генераторы, которые  не самосопряжены, но сопряжены друг с другом и удовлетворяют соотношениям  ˆakˆal = ˆalˆak,  ˆa∗  kˆa∗  l = ˆa∗  l ˆa∗  k,  ˆakˆa∗  l − ˆa∗  l ˆak = ℏδkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь ˆak =  1  √  2(ˆqk + iˆpk), ˆa∗  k =  1  √  2(ˆqk − iˆpk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Так обозначаются операторы рождения и уничтожения, но, пока что, это у меня  — формальный математический объект.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я их ввел формально и написал для них  коммутационные соотношения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти коммутационные соотношения тоже называются  каноническими коммутационными соотношениями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  13  Могу ли я теперь сказать, что полученная алгебра Aℏ — это деформация ком-  мутативной алгебры?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Формально — не могу, потому что, когда я определял понятие  деформации, то потребовал, чтобы все эти алгебры были определены на одном и том  же пространстве, иначе было бы трудно рассматривать все одновременно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Моя ком-  мутативная алгебра состояла из полиномов, а эта алгебра состоит неизвестно из чего.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Но ее тоже можно сделать состоящей из полиномов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это делается очень просто.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Построенная алгебра порождена элементами ˆqk и ˆpk, то есть, ее элементы явля-  ются суммами мономов, составленных из генераторов ˆqk и ˆpk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Благодаря наличию  коммутационного соотношения я могу сдвинуть все генераторы ˆqk налево а ˆpk на-  право (можно и наоборот) и потом снять крышечки с них.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда получится обычный  полином и я могу сказать, что элемент из моей алгебры представлен полиномом, кото-  рый называется q-p-символом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда алгебра определена на пространстве полиномов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это не очень хорошее представление, потому что оно не согласовано с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Тем не менее оно тоже очень полезно во многих случаях.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если исходить из операторов ˆak, ˆa∗  k, то можно использовать ту же самую идею:  сдвинуть ˆa∗  k налево, ˆak — направо и получится то, что называется нормальной фор-  мой элемента вейлевской алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь опять можно снять крышечки и получить  полином, который называется символом Вика.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Физики термин “символ Вика” не ис-  пользуют, но слово “нормальная форма” они используют все время.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Как легко видеть,  переход к нормальной форме согласован с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мы можем рассматривать алгебру Вейля с бесконечным числом образующих.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' До  сих пор параметр k у нас считался дискретным (хотя количество uk или ak с ин-  дексом k могло быть бесконечным), но можно считать этот параметр непрерывным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Например, рассматривать алгебру с генераторами ˆa(k), ˆa∗(l) и соотношениями  ˆa(k)ˆa(l) = ˆa(l)ˆa(k),  ˆa∗(k)ˆa∗(l) = ˆa∗(l)ˆa∗(k),  ˆa(k)ˆa∗(l) − ˆa∗(l)ˆa(k) = ℏδ(k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В этом случае вместо символа Кронекера возникает непрерывный его аналог: δ−функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Так как функция δ(k, l) — это обобщенная функция, генераторы алгебры ˆa(k), ˆa∗(l)  нужно также рассматривать как обобщенные функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обобщенная функция — это  такая функция, которая имеет смысл только под знаком интеграла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Имеют смысл  только элементы ˆa(f) =  �  f(k)ˆa(k)dk и ˆa∗(g) =  �  g(l)ˆa∗(l)dl, представляющие собой  формальные интегралы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти элементы линейно зависят соответственно от f и g и  удовлетворяют коммутационным соотношениям:  ˆa(f)ˆa(g) = ˆa(g)ˆa(f),  ˆa∗(f)ˆa∗(g) = ˆa∗(g)ˆa∗(f),  ˆa(f)ˆa∗(g) − ˆa∗(g)ˆa(f) = ℏ⟨f, ¯g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того, чтобы эти соотношения имели смысл, нужно чтобы было определено ска-  лярное произведение ⟨f, ¯g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку скалярное произведение зависит от g анти-  линейно, то в нем стоит комплексное сопряжение от g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я обычно буду считать, что  индекс k дискретен, но следует понимать, что все можно перенести на случай непре-  рывного индекса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я уже говорил, что элементы алгебры Вейля можно представлять полиномами,  например, с помощью понятия виковского символа, который определяется с помощью  сдвига всех генераторов ˆa∗  k налево и снятием крышечек.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Виковские символы согласованы с определением инволюции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Инволюции в ал-  гебре отвечает комплексное сопряжение полиномов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Переход к символам — это операция, которая тесно связана с операцией квантова-  ния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Что такое квантование?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пусть имеется классический гамильтониан и мы хотим  14  получить его квантовый аналог.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если гамильтониан зависит от uk, то при простой за-  мене u на ˆu возникает проблема, в каком порядке ставить эти генераторы?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В классике  не важен порядок: u1u2 и u2u1 — это одно и то же, но если переходить к квантовой  механике, к алгебре Вейля, то все зависит от порядка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это – то, что по-английски  называется «ordering ambiguity» и означает, что нет однозначной процедуры кванто-  вания.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ее можно сделать однозначной выбрав понятие символа, но символов разных  много.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть такие случаи, когда квантовый гамильтониан имеет естественное определе-  ние.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это, например, стандартная в классической механике ситуация, когда гамиль-  тониан делится на кинетическую и потенциальную энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кинетическая энергия  зависит только от импульсов, а потенциальная энергия — только от координат, и  тогда можно ставить шапочки и все будет абсолютно однозначно потому что кван-  товые импульсы между собой коммутируют и квантовые координаты между собой  коммутируют.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того, чтобы написать уравнение движения есть стандартный способ: в класси-  ческих уравнениях движения на месте скобок Пуассона нужно писать коммутаторы:  ∂ˆuk  dt = i[ ˆH, ˆuk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В этой формуле я считаю, что ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В дальнейшем постоянная Планка ℏ всегда  будет приниматься равной единице, если не сказано обратное.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Уравнение движения осмыслено, если гамильтониан ˆH является элементом ал-  гебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я уже объяснял, что в уравнении движения должна стоять операция,  которая удовлетворяет правилу Лейбница.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такая операция называется дифференци-  рованием.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Коммутатор вида Dh(a) = [h, a] как раз удовлетворяет правилу Лейбница  [h, ab] = [h, a]b + a[h, b] для любой алгебры и поэтому пока ˆH является элементом  алгебры Вейля — все замечательно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Единственное, что нехорошо – это то, что он  очень часто не является элементом алгебры Вейля в случае бесконечного количества  степеней свободы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Типичный пример — гамильтониан вида  ˆH =  �  ǫkˆa∗  kˆak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Когда число индексов бесконечно, это бесконечная сумма.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В алгебре Вейля ниче-  го похожего нет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот гамильтониан не принадлежит алгебре Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тем не менее,  можно формально взять коммутатор гамильтониана ˆH и ˆak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получается уравнение  движения, которое еще встретится неоднократно:  dˆak  dt = −iǫkˆak,  dˆa∗  k  dt = iǫkˆa∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, в случае бесконечного числа степеней свободы гамильтониан — это  не элемент алгебры Вейля, а просто формальное выражение вида  ˆH =  �  Γm,n(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', km, l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ln)ˆa∗  k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆa∗  kmˆal1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆaln,  которое само по себе смысла оператора и смысла элемента алгебры Вейля не имеет,  но, тем не менее, имеет смысл под знаком коммутатора в уравнениях движения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  происходит не всегда, но есть очень простые условия, когда оно точно имеет смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Когда берется коммутатор операторов ˆak или ˆa∗  k с произведением, следует прокомму-  тировать с каждым элементом этого произведения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При коммутировании возникнет  символ Кронекера, а в коммутатор войдут только те коэффициенты, у которых ин-  декс совпадает с индексом у операторов, с которыми происходит коммутирование.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если у всякого индекса в гамильтониане есть только конечное число ненулевых ко-  эффициентов содержащих этот индекс, то уравнение движения имеет смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Квадратичные гамильтонианы  Перейдем к рассмотрению квадратичных гамильтонианов вида  H(u) = 1  2Hklukul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь упорядочение не играет никакой роли, потому что при смене порядка появит-  ся константа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта константа несущественна, поскольку гамильтониан используется  только под знаком коммутатора, где константы исчезают.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Классические уравнения  движения и квантовые уравнения движения совершенно одинаковы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Более того, ес-  ли мы умеем решать классические уравнения движения, то сразу же умеем решать  и квантовые уравнения движения, потому что все уравнения движения линейны, а  разница между классической и квантовой механикой возникает только тогда, когда  перемножаются операторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тут же перемножать ничего не нужно, так что все очень  просто.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Эту же задачу можно сделать еще проще, а именно, можно гамильтониан упро-  стить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если предположить, что гамильтониан положительно определен и матрица  Hkl невырождена, то можно представить гамильтониан как сумму квадратов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Остается еще проблема с матрицей σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ее тоже можно сделать проще, сохраняя при  этом представление гамильтониана как суммы квадратов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того чтобы сохранить  это свойство, следует брать только ортогональные преобразования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Заметим, что σ  — антисмметричная матрица (так как она появилась из алгебры Вейля).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если анти-  симметричную матрицу умножить на мнимую единицу получится матрица, которая  отвечает самосопряженному оператору и ее можно диагонализовать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта диагонали-  зация происходит в комплексной области, но можно сказать, что вместе с каждым  собственным вектором будет и комплексно сопряженный собственный вектор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мож-  но рассмотреть картинку, которая содержит два комплексно сопряженных вектора,  взять в ней действительную и мнимую части и получить представление матрицы σkl  в блочно-диагональном виде с двумерными блоками.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти двумерные блоки будут яв-  ляться антисимметричными матрицами, поэтому гамильтониан примет форму суммы  гамильтонианов вида  ˆH = 1  2(ˆp2 + ǫ2ˆq2),  где [ˆp, ˆq] = 1  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это чрезвычайно важное упрощение, которое всегда можно сделать  для положительного квадратичного гамильтониана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я это рассказывал для случая  конечного числа степеней свободы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Важно заметить, что для случая бесконечного  числа степеней свободы это все тоже правильно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Единственная проблема заключает-  ся в том, что при диагонализации самосопряженного оператора кроме дискретного  спектра может появиться непрерывный спектр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я к этому еще вернусь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь рас-  смотрим гамильтониан  ˆH =  � 1  2(ˆp2  k + ǫ2  k(ˆqk)2)  и будем решать соответствующие ему уравнения движения  dˆpk  dt = −ǫ2  kˆqk,  dˆqk  dt = ˆpk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это стандартные уравнения движения осциллятора точно такие же, как в классике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Их можно решить десятками способов, но самый простой способ — это ввести новые  переменные  ˆak =  1  √  2(√ǫkˆqk + iˆpk  √ǫk  ),  ˆa∗  k =  1  √  2(√ǫkˆqk − iˆpk  √ǫk  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  16  В таком случае уравнения движения  dˆak  dt = −iǫkˆak,  dˆa∗  k  dt = iǫkˆa∗  k,  которые отвечают гамильтониану  ˆH =  �  ǫkˆa∗  kˆak,  (3)  имеют очень простое решение:  ˆak(t) = e−itǫkˆak(0),  ˆa∗  k(t) = eitǫkˆa∗  k(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В случае бесконечного числа степеней свободы в силу спектральной теоремы тоже  можно считать, что все диагонально, но вместо суммы возникнет интеграл и тогда  гамильтониан будет иметь вид:  ˆH =  �  dλǫ(λ)ˆa∗(λ)ˆa(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В физике λ — это обычно совокупность непрерывных и дискретных индексов, а ин-  теграл включает как обычное интегрирование, так и суммирование по дискретному  индексу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В частном случае когда теория трансляционно инвариантна, мы считаем, что  операторы ˆa∗(x) и ˆa(x) зависят от координат x, которые можно сдвигать не меняя  гамильтониана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это значит, что гамильтониан имеет вид  ˆH =  �  dxdyǫ(x − y)ˆa∗(x)ˆa(y),  где коэффициент зависит только от разности x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (В рассматриваемом выражении  могут быть также дискретные индексы — по ним нужно проводить суммирование.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Можно перейти к импульсному представлению (взять преобразование Фурье).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То-  гда гамильтониан примет вид:  ˆH =  �  dkǫ(k)ˆa∗(k)ˆa(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мы будем все время рассматривать трансляционно инвариантные гамильтонианы, и  эта формула будет существенно использоваться.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='3  Стационарные состояния  Теперь кратко обсудим стационарные (не зависящие от времени) состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если  операторы эволюции обозначить U(t), то для стационарного состояния выполняется  условие U(t)ω = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если работать в формализме матриц плотности K, то, учитывая что уравнения  движения для матрицы плотности записывается как коммутатор с гамильтонианом  ˆH, можно сделать вывод что матрица плотности представляет стационарное состоя-  ние, если она коммутирует с гамильтонианом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В частности, если матрица плотности  является функцией оператора ˆH, то состояние стационарно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если говорить не о матрице плотности, а о векторе состояния, то стационарное со-  стояние удовлетворяет условию ˆHΨ = EΨ, то есть стационарное состояние является  собственной функцией гамильтониана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Гамильтониан имеет смысл оператора энер-  гии, а E — это уровень энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вектор Ψ с течением времени просто умножается на  множитель:  Ψ(t) = ˆU(t)Ψ = e−itEΨ ∼ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  17  Вектор меняется — состояние не меняется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В алгебраическом подходе гамильтониан может быть формальным выражением,  но можно применить GNS-конструкцию к какому-то стационарному состоянию ω и  получить гильбертово пространство, в котором есть унитарные операторы, описыва-  ющие эволюцию (сдвиг по времени) и есть генератор группы временных трансляций,  который имеет смысл гамильтониана — оператора энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Его собственные значе-  ния имеют смысл уровней энергии возбуждений состояния ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее говоря, такая  интерпретация будет совершенно правильной в том случае, когда само ω стационар-  но и трансляционно инвариантно (инвариантно как относительно пространственных,  так и временных трансляций).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Проиллюстрировать это можно следующим образом: трансляционно инвариант-  ное состояние можно представить в виде горизонтальной прямой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У него обычно бес-  конечная энергия, но все отсчитывается от этой бесконечной энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (В алгебраи-  ческой квантовой механике никакой бесконечной энергии нет — просто надо отсчи-  тывать энергию от энергии этого состояния, считая ее равной нулю).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возбуждение  нужно воспринимать как сосредоточенный в конечной области горбик на этой гори-  зонтальной прямой, и тогда имеет смысл понятие энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта энергия отсчитывается  уже от энергии трансляционно инвариантного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Важное и простое замечание, которое в курсе квантовой механики объясняется  в несколько менее общей ситуации, заключается в следующем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Рассмотрим класси-  ческий гамильтониан H(u), у которого есть минимум в невырожденной критической  точке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что квадратичная часть в разложении Тейлора положительно  определена;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' тогда нет нулевых мод, квантовый гамильтониан в первом приближении  квадратичен;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' в соответствующих координатах он будет иметь вид:  ˆH =  �  dλǫ(λ)ˆa∗(λ)ˆa(λ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=',  где обозначенные как .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' члены начинаются с кубических по ˆa∗, ˆa (линейных членов  быть не может потому что мы находимся в критической точке).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Члены высшего по-  рядка по ˆa∗, ˆa также являются членами высшего порядка по постоянной Планка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По  крайней мере, в квазиклассике можно этими членами пренебречь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же мы работа-  ем не в квазиклассическом приближении, здесь можно поставить какой-то множитель  и рассматривать теорию возмущений по этому множителю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='4  Пространство Фока  Перейдем теперь к рассмотрению представлений алгебры Вейля (или, что то же —  представлений канонических коммутационных соотношений).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Среди этих представ-  лений есть одно замечательное, самое простое, которое называется представлением  Фока, а пространство, в котором оно живет, называется пространством Фока.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Опре-  деляется фоковское представление просто: в нем существует циклический вектор |0⟩,  который уничтожается всеми операторами ˆak:  ˆak|0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это условие с точностью до эквивалентности однозначно определяет представление.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Попробуем сейчас выписать в конкретной форме то как оно устроено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вектор |0⟩ — циклический.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Определение цикличности зависит от того, живем ли  мы в предгильбертовом или в гильбертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если мы живем в пред-  гильбертовом пространстве, то, применяя элементы алгебры к циклическому вектору,  можно получить все векторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если мы живем в гильбертовом пространстве это не со-  всем так: нужно разрешить взятие предела.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть, взять еще замыкание множества  18  тех векторов, которые получаются с помощью алгебраических операций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Давайте по-  смотрим, что будет здесь?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оператор ˆak уничтожает |0⟩, поэтому действовать им на  |0⟩ не надо — ничего нового не получится.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно действовать только операторами  рождения ˆa∗  k:  (ˆa∗  k1)n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='(ˆa∗  ks)ns|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (4)  Здесь применены операторы рождения много раз.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Конечное число раз, так как в  алгебре Вейля существуют только конечные комбинации этих операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Теперь спрашивается: можем ли мы получить что-нибудь новое, если применим к  этому выражению оператор ˆak?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ответ: нет, не получим, потому что можно пользуясь  коммутационными соотношениями переставить оператор ˆak так, чтобы он действо-  вал на вакуум.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда оператор ˆak исчезнет в силу наложенного условия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поэтому  только выражения вида (4) и их линейные комбинации принадлежат пространству  Фока, если пользоваться буквальным определением цикличности без взятия предела.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дальше следует заметить, что все выписанные состояния (4) являются собственными  векторами любого гамильтониана формы ˆH = � ǫkˆa∗  kˆak с собственными значениями,  равными � nkǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как это проверить?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко посчитать коммутатор гамильтониана (3), с которым  мы сейчас работаем, с оператором ˆa∗  l :  ˆHˆa∗  l = ˆa∗  l ˆH + ǫlˆa∗  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Чтобы вычислить действие гамильтониана ˆH на (4) достаточно, пользуясь этим  соотношением, передвинуть гамильтониан на |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Несколько более простое формаль-  ное рассуждение таково: известно, как операторы ˆa∗  k изменяются с течением времени  — просто умножаются на численную экспоненту.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По этой причине динамика вектора  состояния сводится к умножению на экспоненту с некоторым показателем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Динамика  здесь ровно такая, какая требуется для стационарного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мы получили ортогональный (но не ортонормированный) базис фоковского про-  странства, который состоит из собственных векторов (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Все полученные формулы можно применить для случая многомерного гармони-  ческого осциллятора.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Там операторы ˆak, ˆa∗  k называются операторами рождения и  уничтожения квантов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В кристалле атомы как-то между собой взаимодействуют, но  кристалл находится в стационарном состоянии близком к основному состоянию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По  крайней мере, в первом приближении кристалл описывается квадратичным гамильто-  нианом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для квантов в этой ситуации есть другое название: фононы — кванты звука.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В общем случае мы имеем дело с системой невзаимодействующих бозонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Опера-  торы ˆak, ˆa∗  k называются операторами рождения и уничтожения частиц, а числа nk в  формуле для уровней энергии � nkǫk называются числами заполнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если мы хотим, чтобы фоковское пространство было гильбертовым простран-  ством, нужно взять пополнение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть свои преимущества в том и в другом подходе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В предгильбертовом простран-  стве рассматриваемые операторы определены везде.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В гильбертовом пространстве —  это неограниченные операторы, которые не всюду определены, но есть возможность  рассматривать пределы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предгильбертово фоковское пространство может быть представлено как простран-  ство полиномов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В самом деле, есть следующая формула для базиса:  (ˆa∗  k1)n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='(ˆa∗  ks)ns|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Чтобы получить полином, я предлагаю вычеркнуть |0⟩ и снять крышечку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получится  некоторый моном.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Линейная комбинация таких мономов — это полином.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть, каждому элементу  фоковского пространства (которое до пополнения мы считаем предгильбертовым)  19  можно сопоставить полином.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко посчитать, что скалярное произведение в таком  представлении в виде полиномов будет даваться формулой:  ⟨F, G⟩ =  �  da∗daF(a∗)G(a∗)∗e−a∗a,  (5)  где F(a∗) и G(a∗) — это полиномы, отвечающие разным векторам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обращаю внима-  ние, что G(a∗) стоит со звездочкой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Звездочка, примененная к a дважды — это опять  a, и поэтому здесь уже будет полином от a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Проверим, что скалярное произведение записывается в виде (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это можно посчи-  тать, но этого даже не нужно делать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что фоковское пространство одно-  значно определяется наличием циклического вектора, который уничтожается всеми  операторами уничтожения при том, что операторы рождения и уничтожения удовле-  творяют нужным коммутационным соотношениям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В пространстве полиномов от a∗  k  я определяю оператор ˆa∗  k просто как умножение на a∗  k, а оператор ˆak — как диф-  ференцирование по a∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко понять, что умножение и дифференцирование как раз  удовлетворяют необходимым коммутационным соотношениям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если умножить на a∗  k,  а потом продифференцировать и применить правило Лейбница, то как раз получится  то что нужно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, мы имеем коммутационные соотношения, есть и циклический век-  тор, который просто равен единице.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Полином равный единице обнуляется введенными  операторами уничтожения и является правильным кандидатом на |0⟩, цикличность  тоже имеет место.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Осталось проверить, что ˆak и ˆa∗  k сопряжены друг к другу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Это  нужно, чтобы правильно работала инволюция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Действительно, в формуле (5) можно  применить интегрирование по частям и убедиться, что действительно для этого ска-  лярного произведения умножения и дифференцирования сопряжены друг другу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В  итоге все свойства фоковского представления выполнены, и поэтому не нужно срав-  нивать исходное скалярное произведение с новым: оно обязательно будет таким же  (с точностью до численного множителя).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  До сих пор мы имели дело с полиномами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но я хочу все-таки иметь возможность  работать в гильбертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого нужно пополнить пространство  полиномов, и тогда кроме полиномов появятся и другие функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они будут голо-  морфными функциями от a∗  k, но это будут не все голоморфные функции, а только  те, что имеют конечную норму в нашем скалярном произведении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Важно отметить, что все эти рассуждения применимы и к бесконечному числу  степеней свободы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Хотя там интеграл бесконечномерен, но, все равно, он оказывается  хорошо определенным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть еще один способ описания фоковского пространства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что по-  линомы связаны с симметрическими функциями от дискретного аргумента.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Хорошо  известно, что если имеется квадратичная форма, то всегда можно считать, что коэф-  фициенты этой квадратичной формы симметричны по индексу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если есть кубичный  полином, можно считать, что коэффициент зависит от трех индексов, но, опять же,  можно наложить условие симметрии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Его нужно наложить, если вы хотите иметь  однозначное представление.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для полиномов высших степеней ситуация аналогична.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Поэтому можно считать что есть единственное представление для всякого элемента  предгильбертова фоковского пространства в форме:  �  n  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=',kn  fn(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', kn)ˆa∗  k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆa∗  kn|0⟩,  где n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', а коэффициенты симметричны по индексам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это значит, что фоков-  ское пространство представимо как последовательность симметрических функций от  20  растущего числа переменных:  f0, f1(k), f2(k1, k2), f3(k1, k2, k3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Пока мы работаем с полиномами, должно быть только конечное количество сим-  метричных функций от k1, k2, k3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='. В этом пространстве можно взять пополнение,  и тогда придется рассматривать последовательность функций f0, f1(k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', которые  должны удовлетворять единственному условию: норма этой бесконечной последова-  тельности конечна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обычно эта последовательность записывается как столбик (фо-  ковский столбик).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта запись имеет смысл и в том случае, когда k — непрерывный  параметр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В лекциях по квантовой механике фоковское пространство обычно опреде-  ляется именно как пространство состоящее из столбиков симметрических функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='5  Гамильтонианы сохраняющие число частиц  В квантовой теории важную роль играет оператор  ˆN =  �  ˆa∗  kˆak =  �  dλˆa∗(λ)ˆa(λ),  который представляет собой сумму по всем k (или интеграл, если есть непрерывный  индекс).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот оператор имеет физический смысл числа частиц или числа квантов,  когда имеем дело с осцилляторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Название не важно, а важно, что если есть соб-  ственный вектор X оператора числа частиц с каким-то собственным значением N,  то операторы a∗  k, действуя на X, увеличивают число частиц на единицу, а ak, на-  оборот, уменьшают на единицу (рождают частицу или уничтожают частицу).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  значит, что оператор, сохраняющий число частиц, должен содержать одинаковое ко-  личество операторов рождения и уничтожения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем рассматривать квадратичные  гамильтонианы обладающие этим свойством.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Квадратичные гамильтонианы сохраняющие число частиц очень важны потому,  что вблизи состояния с минимальной энергией (основного состояния) они играют  основную роль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возьмем квадратичный гамильтониан, который сохраняет число ча-  стиц:  ˆH =  �  dxdyA(x, y)ˆa∗(x)ˆa(y) = ⟨a∗, Aa⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В нем есть произведения типа a∗a, а произведений типаa∗a∗, aa, — нет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если у опе-  ратора A есть только дискретный спектр, то можно записать гамильтониан в виде:  ˆH =  �  ǫkˆa∗  kˆak,  где ˆak =  �  dxφk(x)ˆa(x) отвечает собственной функции φk(x) оператора A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Давайте теперь возьмем в качестве x и y векторы в евклидовом пространстве и  будем считать что A записан в виде: A = − 1  2m∆ + ˆU, который хорошо известен из  обычной квантовой механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это тот оператор, который получается из квантования  классического гамильтониана вида p2  2m +U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае гамильтониан ˆH описы-  вает систему невзаимодействующих нерелятивистских тождественных бозонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если  добавить сохраняющие число частиц неквадратичные члены, получится гамильтони-  ан системы взаимодействующих нерелятивистских тождественных бозонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Хочу обратить внимание что содержание этой лекции — это, в каком-то смыс-  ле, объяснение того, как математик мог бы угадать квантовую механику, но не уга-  дал.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Угадали, конечно, физики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Смотрите, какова была логика.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Математик знает,  что наблюдаемые в классической теории — это просто алгебра функций на фазовом  21  пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Он знает, что квантовая механика — это несколько продеформирован-  ная классическая механика: все подчиняется классическим законам, но в некоторых  ситуациях должны быть поправки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Исходя из этого он говорит: давайте я эту ком-  мутативную алгебру продеформирую: ассоциативность оставлю, но введу некомму-  тативность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Простейшая деформация, которая есть — это алгебра Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Математик  угадал, что нужно жить с алгеброй Вейля, а после этого он понимает, что в алгебре  Вейля нужно рассматривать простейший гамильтониан, который описывает жизнь  около основного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — квадратичный гамильтониан.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Уровни энергии это-  го гамильтониана даются формулой:  �  ǫkˆnk,  где ˆnk = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' — числа заполнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта формула описывает уровни энергии си-  стемы невзаимодействующих тождественных частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть, в появлении тожде-  ственных частиц никакой тайны нет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' С точки зрения математика, именно они долж-  ны возникнуть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Наоборот, нетождественных частиц может не быть, а тождественные  всегда есть, потому что простейший квадратичный гамильтониан уже описывает тож-  дественные частицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В нерелятивистском случае одночастичные уровни энергии ǫk должны получаться  при квантовании классического одночастичного гамильтониана  p2  2m + U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Добавив  неквадратичные члены, мы приходим к гамильтониану системы нерелятивиских тож-  дественных бозонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='6  Представления алгебры Вейля  Теперь перейдем к вопросу, как на самом деле устроены другие представления алгеб-  ры Вейля и есть ли они вообще?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Прежде всего отметим, что другие представления,  несомненно, есть: можно взять прямую сумму двух представлений, и это будет новое  представление.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Так как фоковское представление, как нетрудно видеть, неприводимо,  (внутри него нет никакого другого представления), то основной вопрос заключает-  ся в том, какие есть неприводимые представления?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ответ на этот вопрос такой: в  случае когда рассматриваются представления алгебры Вейля с конечным числом ге-  нераторов (случай коечного числа степеней свободы), то единственное (с точностью  до эквивалентности) неприводимое представление — это фоковское.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Его можно запи-  сывать в разных видах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть, например, представление, в котором оператор импульса  реализуется как дифференцирование, а оператор координаты — как умножение, но  оно эквивалентно фоковскому.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Там тоже можно ввести операторы ˆa∗, ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В общем случае ответ на этот вопрос трудно дать, потому что вопрос плохо сфор-  мулирован.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что в гильбертовом пространстве операторы ˆa∗  k, ˆak имеют  какую-то область определения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Она — не все гильбертово пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно об-  ласть определения изменить немножко, а оператор оставить тем же самым и спросить  себя: это то же самое представление или другое?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' С формально математической точки  зрения — другое.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом деле, конечно, то же самое.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При работе с неограниченны-  ми операторами эта проблема всегда возникает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ее можно решить, но лучше иметь  дело с ограниченными операторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это можно сделать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно рассмотреть экспоненты:  Vα = eiαk ˆuk,  в показателе которых стоит линейная комбинация самосопряженных операторов ˆuk,  удовлетворяющих коммутационным соотношениям:  ˆukˆul − ˆulˆuk = iσk,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  22  Коэффициенты в экспоненте Vα выбраны тдействительными, тогда в показателе сто-  ит оператор вида мнимая единица, умноженная на самосопряженный оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда  то, что получилось — это оператор унитарный, а унитарный оператор ограничен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' С  такими операторами удобно жить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Имеет место соотношение:  VαVβ = e−i 1  2 ασβVα+β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (6)  В этом легко убедиться, если использовать формулу:  eXeY = eX+Y e  1  2C,  которая верна, если коммутатор операторов X и Y представляет собой число, которое  здесь обозначено буквой C:  [X, Y ] = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В рассматриваемом нами случае ровно такая ситуация, и поэтому математики,  если они не хотят иметь дело с неограниченными операторами, работают с этими  унитарными операторами и с экспоненциальной формой коммутационных соотноше-  ний (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — экспоненциальная форма алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' С точки зрения физика это  — та же самая алгебра, с которой мы имели дело, с точки зрения математика, это не  совсем так.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Существует единственное неприводимое представление алгебры Вейля в случае  конечного числа степеней свободы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я дам доказательство этого факта, не пользуясь  экспоненциальной формой алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оно не вполне строгое, но, с точки зрения  физика, оно вполне приемлемо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Рассуждение такое: рассмотрим уже упомянутый оператор числа частиц ˆN =  � ˆa∗  kˆak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае конечного числа генераторов это хороший оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возьмем соб-  ственный вектор этого оператора и начнем применять к нему операторы уничтоже-  ния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Математик спросит: откуда вы знаете, что есть такой собственный вектор, но  физик, наверное, это делать не будет).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если все операторы уничтожения дают ноль,  я скажу, что это как раз тот самый фоковский вакуум, который нам нужен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если  есть такой оператор, который не дает ноль, тогда я применю оператор уничтожения  еще раз и еще раз и еще раз и буду делать это до тех пор, пока у меня не получится  такой вектор |0⟩, который обнуляется всеми операторами уничтожения: ak|0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Такой вектор обязательно есть, потому что оператор числа частиц положительно  определен, а операторы уничтожения уменьшают число частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' После этого я возьму  то подпредставление, в котором он лежит.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я применю к вектору |0⟩ все операторы  ˆa∗  k много раз и, когда я возьму линейные комбинации получившихся выражений, у  меня получится пространство, которое инвариантно относительно всех операторов  рождения и уничтожения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — подпредставление моего представления и оно будет,  несомненно, фоковским представлением потому, что там есть вектор, который уни-  чтожается всеми ˆak и этот вектор циклический (я все могу получить из него).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отсюда  все получается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В случае бесконечного числа генераторов эти рассуждения неприменимы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я сей-  час объясню, как построить пример представления, которое не является фоковским,  но, тем не менее является хорошим представлением.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я возьму фоковское простран-  ство и возьму новые операторы, которые я буду обозначать буквой ˆAk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это старые  операторы минус число:  ˆAk = ˆak − fk,  ˆA∗  k = ˆa∗  k − ¯fk  (для каждого ˆak я вычитаю свое число).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Рассматривая коммутационные соотношения,  которые мне нужны для алгебры Вейля, я сразу увижу, что все они удовлетворяют-  ся.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, я снова получил представление канонических коммутационных  соотношений, представление алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  23  Теперь я попытаюсь решить уравнение ˆAkΘ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Его решения — это собственные  функции операторов ˆak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы с этими векторами встретимся много раз.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они называ-  ются иногда пуассоновскими векторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко понять, что ответ будет представлен  экспонентой  Θ = efˆa∗|0⟩  где fˆa∗ = � fkˆa∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если рассматривать представление векторов из фоковского про-  странства с помощью функций, легко увидеть, что это действительно собственный  вектор всех операторов ˆak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Легко вычислить норму пуассоновского вектора и скалярное произведение двух  пуассоновских векторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Скалярное произведение задается интегралом;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' этот интеграл гауссов и легко бе-  рется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Норма Θ будет конечной, если сумма � |fk|2 конечна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если норма конечна, то-  гда вектор Θ принадлежит фоковскому пространству и представление эквивалентно  фоковскому представлению, но если норма бесконечна, то это не так и в представле-  нии нет вектора, который уничтожается операторами ˆak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оно не является фоковским  раз нет такого вектора и следовательно это представление — пример того, что есть  неприводимые представления, которые неэквивалентны фоковскому представлению.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В таком представлении нет даже подпредставления эквивалентного фоковскому.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В  случае конечного числа степеней свободы � |fk|2 всегда конечна, и там эти рассуж-  дения ничего не дают.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я хочу обобщить эту конструкцию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я рассмотрю оператор ˆAk, определен-  ный несколько по-другому.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раньше я просто добавлял fk, а тут я еще возьму линейные  преобразования:  ˆAk = Φl  kˆal + Ψl  kˆa∗  l + fk  ˆA∗  k = ¯Ψl  kˆal + ¯Φl  kˆa∗  l + ¯fk  и потребую, чтобы новые операторы ˆAk, ˆA∗  k тоже удовлетворяли каноническим ком-  мутационным соотношением.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это накладывает некоторые условия на коэффициенты.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Такое преобразование называется линейным каноническим преобразованием.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Опера-  торы ˆAk, ˆA∗  k определяют новое представление алгебры и, опять же, если вектор Θ,  являющийся решением уравнения ˆAkΘ = 0, принадлежит фоковскому пространству,  то тогда линейное каноническое преобразование эквивалентно фоковскому представ-  лению.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такое каноническое преобразование называется собственным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же эти  условия не выполнены, то это преобразование не эквивалентно фоковскому представ-  лению.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Линейные канонические преобразования часто бывают полезны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Их аналоги  в фермионном случае называются боголюбовскими преобразованиями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они важны в  теории сверхпроводимости.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  24  3  Лекция 3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Алгебра Клиффорда и алгебра Грассманна  У вейлевской алгебры есть близкие родственники, которые определяются теми же  самыми формулами, как и вейлевская алгебра, только вместо коммутаторов стоят  антикоммутаторы:  ukul + uluk = 2hkl,  где hkl — обратимая матрица.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта формула описывает алгебру Клиффорда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть еще другое понятие — алгебра Грассмана, которая получится, если hkl счи-  тать равным нулю:  ukul + uluk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В клиффордовской алгебре требуется, чтобы справа была обратимая матрица.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Грас-  сманова алгебра не является частным случаем клиффордовской — здесь в правой  части стоит ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Элементы грассмановой алгебры — это тоже полиномы, но только,  как говорят, от антикоммутирующих переменных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для клиффордовой алгебры верно  все то же, что было сказано для алгебры Вейля почти буквально.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Просто всюду нуж-  но заменять коммутаторы на антикоммутаторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Единственное отличие состоит в том,  что в то время как все сказанное в прошлой лекции о невзаимодействующих бозонах,  в алгебре Клиффорда будет относиться к невзаимодействующим фермионам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Начнем с грассмановой алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это ассоциативная алгебра с единицей и с анти-  коммутирующими генераторами ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ǫn:  ǫiǫj = −ǫjǫi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (7)  Она обозначается Λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Каждый элемент грассмановой алгебры можно однозначным образом записать  как сумму мономов по ǫi так, чтобы в каждом мономе индексы возрастали: ω = α +  �  i αiǫi+�  i<j αijǫiǫj+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='+�  i1<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='<ik αi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=',ikǫi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ǫik+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='+α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=',nǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это очевидно: мы  можем воспользоваться отношениями антикоммутации, чтобы переставить меньшие  индексы налево, а двух совпадающих индексов не может быть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Если положить i = j,  получится, что квадрат равен нулю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я напоминаю, что все алгебры рассматриваются  над комплексными числами и на 2 можно делить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Грассманова алгебра, так же, как и обычная алгебра полиномов, Z-градуированная.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это значит, что есть понятие степени: степень каждого монома — это число генерато-  ров в этом мономе;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ясно, что при умножении степени складываются, так что в этом  смысле все так же как у обычных полиномов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Следующее замечание: можно получить  другое однозначное представление элементов грассманновой алгебры не упорядочи-  вая индексы, а наложив требование антисимметричности коэффициентов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это точно  так же как в обычных полиномах — генераторы можно переставлять и поэтому мож-  но считать, что коэффициенты симметричны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь же можно переставлять, но со  знаком минус и поэтому можно считать, что коэффициенты антисимметричны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я уже говорил, что есть понятие степени (Z-градуировки).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Важнее всего то, что из  этой Z -градуировки можно получить Z2-градуировку сказав, что существуют четная  и нечетная части, четные и нечетные элементы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть то, что натянуто на четные  мономы Λeven  n  = �  k≥0 Λ2k  n и то, что натянуто на нечетные мономы Λodd  n  = �  k≥0 Λ2k+1  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это разбиение на четную и нечетную части здесь более важно, чем степень, потому  что именно оно заведует умножением.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Четный элемент коммутирует с чем угодно, и это понятно, потому что при пе-  рестановке двух генераторов возникает знак минус, но если это проделать дважды,  25  то минус на минус даст плюс и ничто не изменится.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Два нечетных элемента между  собой антикоммутируют.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно сказать, что элемент грассмановой алгебры — это полином от антиком-  мутирующих элементов ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ǫn или, иными словами, это функция антикоммутиру-  ющих переменных, но она автоматически полиномиальна, потому что есть только  конечное число мономов (в случае, когда есть конечное число антикоммутирующих  переменных).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Аналогия с полиномами — это важная идея, потому что она подсказы-  вает, что должен быть анализ в грассмановой алгебре, и, действительно, такой анализ  есть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно определить дифференцирование по переменной ∂i = ∂/∂ǫi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Просто надо  вычеркнуть эту переменную.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если переменной ǫi в мономе нет, то эта производная  равна нулю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вычеркивая переменную ǫi нужно учесть, что переменная может сто-  ять где угодно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для случая антикоммутирующих переменных есть понятия левой  производной и правой производной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем для определенности рассматривать левую  производную;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' это означает, что перед тем как вычеркивать следует соответствующий  элемент переместить налево — поставить на первое место:  ∂i(ǫiǫi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ǫin) = ǫi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ǫin  (8)  ∂i(ǫi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ǫin) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (9)  Понятие производной связано с правилом Лейбница.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь тоже есть правило  Лейбница, но только с той разницей, что появляется знак, который связан с переста-  новками, о которых уже было сказано:  ∂i(ωρ) = (∂iω) · ρ + (−1)¯ω · ω · ∂iρ,  (10)  где ω, ρ ∈ Λn, а ω обладает определенной четностью ¯ω, то есть ω является либо  четной (¯ω = 0) либо нечетной (¯ω = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это называется градуированным правилом  Лейбница.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если выполнено это правило, говорят про нечетное дифференцирование,  если выполнено обычное правило Лейбница, говорят про четное дифференцирование.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть и понятие интегрирования  �  : Λn → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Интеграл любого монома не макси-  мальной степени дает ноль, а интеграл монома максимальной степени — плюс или  минус единицу:  �  ǫi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ǫikdnǫ = 0  if k < n,  �  ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ǫndnǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я выбрал обозначение, при котором плюс единица получается тогда, когда эти гене-  раторы упорядочены в возрастающем порядке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко понять, что интеграл от произ-  водной всегда равен нулю:  �  (∂iω)dnǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (11)  Это происходит потому, что в производной не может быть члена максимальной сте-  пени — при дифференцировании уменьшается количество генераторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из этого и из  правила Лейбница можно вывести правило интегрирования по частям:  �  (∂iω) · ρdnǫ = −(−1)¯ω  �  ω · ∂iρdnǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Хочу заметить, что, рассматривая грассманову алгебру Λn, мы фиксировали си-  стему образующих ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Естественным образом, можно перейти к другим образу-  ющим (это — аналог замены переменных) и тогда изменится все — изменится понятие  дифференцирования, изменится понятие интегрирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для таких преобразований  26  существует аналог теоремы о дифференцировании сложной функции, аналог якоби-  ана, о которых я много говорить не буду.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я хочу рассмотреть только частный случай, когда замена переменных линейна:  ˜ǫi =  �  Ai  jǫj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (12)  Легко понять, что в интеграле появится детерминант при замене переменных точно  так же, как при обычной замене переменных с той только разницей, что там, где  стоял детерминант, здесь стоит детерминант в минус первой степени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Следующее замечание состоит в том, что в гладкую функцию f, которая зависит  от действительной переменной x ∈ R, можно подставить четный элемент грассма-  новой алгебры ω, и выражению f(ω) можно придать смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В грассмановой алгебре  всякий элемент может быть представлен в виде суммы ω = a+ν, где a — число, а ν —  нильпотентный элемент (т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' νk = 0 для некоторого k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возьмем разложение функции  f(a + ν) в ряд Тейлора по нильпотентной части:  f(ω) = f(a) + f ′(a)  1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  ν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + f l(a)  l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  νl + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (13)  и заметим, что из-за нильпотентности этот ряд Тейлора имеет конечное число членов  — остается только полином, и это имеет четкий смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В частности, можно рассмотреть экспоненту.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно заметить, что для экспоненты  имеются обычные правила.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Четные элементы между собой коммутируют, и поэтому  все формальные рассуждения можно проводить для экспоненты четного элемента  грассмановой алгебры так же, как и для числа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') В качестве примера рассмотрим  экспоненту квадратичного выражения:  eλ1ǫ1ǫ2+λ2ǫ3ǫ4+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='+λkǫ2k−1ǫ2k = eλ1ǫ1ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='eλkǫ2k−1ǫ2k =  (1 + λ1ǫ1ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='(1 + λkǫ2k−1ǫ2k),  отсюда следует, что  �  eλ1ǫ1ǫ2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='+λkǫ2k−1ǫ2kd2kǫ = λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (14)  В более общем случае, когда ω = 1  2  � aijǫiǫj, можно привести антисимметричную  невырожденную матрицу a к блочно-диагональному виду путем замены переменных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Используя правило замены переменных, можно вычислить гауссов интеграл:  �  eωdnǫ = (det a)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (15)  Ответ такой же, как и в обычном случае, но только там, где был детерминант  в степени −1/2 здесь будет детерминант в степени +1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее говоря, совпаде-  ние имеет место с точностью до множителя, который может быть убран с помощью  соответствующей нормировки меры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Следующее замечание заключается в том, что можно рассматривать функции од-  новременно от коммутирующих и от антикоммутирующих переменных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это просто  означает, что в разложениях по антикоммутирующим переменным коэффициенты  можно считать функциями от коммутирующих переменных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно рассматривать  либо полиномиальные функции, либо гладкие функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы полугаем функции на  суперпространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это просто слова — вы говорите, что суперпространство это та-  кое пространство, на которой заданы функции (эти функции я определил), а точек  у суперпространства в таком формальном определении нет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом деле, мож-  но определить понятие точки суперпространства — это важно и удобно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Об этом я  несколько слов скажу, но не буду здесь давать никаких точных определений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  27  Есть понятие супермногообразия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы можем рассматривать полиномиальные или  гладкие функции коммутирующих переменных x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', xm и антикоммутирующих пе-  ременных ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='., ǫn — элементов алгебры C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', xm] ⊗ Λn или алгебры C∞(Rm) ⊗ Λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Кроме того, могут быть некоторые уравнения: на коммутирующие и антикоммутиру-  ющие переменные можно наложить условия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Важно чтобы в условии все слагаемые  были одинаковой четности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например, может быть такое условие: x1 + ǫ1ǫ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По-  лучаем алгебраические супермногообразия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обычные алгебраические многообразия  задаются полиномиальными уравнениями в обычном линейном пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь  есть коммутирующие переменные и полиномиальные уравнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Представим себе, что коэффициенты этих уравнений являются целыми числами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если есть такие уравнения, то можно считать переменные xi какими угодно: действи-  тельными, комплексными, элементами какого-то кольца, элементами поля вычетов по  модулю n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это все имеет смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно рассматривать алгебраическое многообразие над разными полями или над  разными кольцами, то есть, можно говорить, что есть точки алгебраического много-  образия над каким-то полем, каким-то кольцом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например, если рассмотреть, ска-  жем, окружность мнимого радиуса x2 + y2 + 1 = 0, то действительных точек у такого  объекта нет, а над комплексными числами точки есть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда рассматриваются су-  пермногообразия, возникает такая же ситуация.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно рассматривать переменные  в уравнении принадлежащими некоторой грассмановой алгебре (любой).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' И тогда по-  явится понятие Λ-точки супермногообразия, где Λ — любая грассманнова алгебра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Нужно только позаботиться о том, чтобы вместо четной переменной подставлялся  четный элемент грассманновой алгебры, а вместо нечетной переменной нечетный.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Иными словами, нужно чтобы четность сохранялась при введении Λ-точек.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Понятие Λ-точки очень удобно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оно, например, позволяет дать очень простое  определение понятия супералгебры Ли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Супералгебра Ли — это Z2-градуированная  алгебра с некоторым обобщением правила Якоби.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Запоминать это обобщение не на-  до.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Просто нужно сказать, что это обобщение должно быть таким, чтобы Λ-точки  супералгебры Ли образовывали обыкновенные алгебры Ли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Подробно я об этом го-  ворить не буду.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что я рассказывал — это маленький кусочек того, что называется  суперматематикой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Представления алгебры Клиффорда  Следующие рассуждения, как я уже обещал, будут в точности повторять рассуж-  дения предыдущей лекции, только вместо “Вейль” я буду говорить “Клиффорд” и  вместо слова “коммутатор” я буду говорить “антикоммутатор”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Согласно определению алгебры Клиффорда, антикоммутатор генераторов явля-  ется симметричной невырожденной матрицей hkl = hlk:  ˆukˆul + ˆulˆuk = 2hkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Эти же соотношения встречаются при рассмотрении уравнений Дирака: гамма-матрицы  являются генераторами алгебры Клиффорда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Будем рассматривать алгебру Клиффорда, в которой есть инволюция, предпола-  гая при этом, что генераторы ˆak, ˆa∗  k получаются друг из друга с помощью инволюции  и удовлетворяют равенствам:  ˆakˆal + ˆalˆak = 0, ˆa∗  kˆa∗  l + ˆa∗  l ˆa∗  k = 0, ˆakˆa∗  l + ˆa∗  l ˆak = δkl,  которые называются каноническими антикоммутационными соотношениями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они по-  лучаются из канонических коммутационных соотношений заменой коммутаторов на  28  антикоммутаторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отмечу, что, представленные здесь антикоммутационные соотно-  шения — это в точности антикоммутационные соотношения, которые получаются,  если в алгебре Грассмана взять дифференцирования и умножения на генераторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно рассматривать алгебры Клиффорда с бесконечным числом генераторов  или генераторы ˆa(k), ˆa(k)∗, зависящие от непрерывного параметра и удовлетворяю-  щие соотношениям:  ˆa(k)ˆa(l) = −ˆa(l)ˆa(k),  ˆa∗(k)ˆa∗(l) = −ˆa∗(l)ˆa∗(k),  ˆa(k)ˆa∗(l) + ˆa∗(l)ˆa(k) = δ(k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  — все то же самое, что и для алгебры Вейля, не стоит повторяться.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В алгебре Клиффорда, определяемой с помощью генераторов ˆak, ˆa∗  k, есть понятие  нормальной формы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точно так же как и в случае алгебры Вейля генераторы ˆa∗  k уво-  дятся налево, ˆak уводятся направо, но есть небольшая особенность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раньше, получая  нормальную форму, я мог определить виковский символ, просто снимая крышечки  и рассматривая ak, a∗  k как комплексные переменные.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь этого нельзя сделать —  получится ноль, но я могу сделать то же самое, говоря при этом, что, снимая кры-  шечки, я получаю антикоммутирующие переменные.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, из элементов  алгебры Клиффорда я могу получить полином от антикоммутирующих переменных,  произнося все те же слова, только вставляя при этом приставку “анти”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Рассмотрим формальный гамильтониан  ˆH =  �  Γm,n(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', km, l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ln)ˆa∗  k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆa∗  kmˆal1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆaln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Гамильтониан всегда нужно считать четным элементом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда количество перемен-  ных бесконечно, гамильтониан обычно не является элементом алгебры Клиффорда,  но коммутаторы, тем не менее имеют смысл, если наложить те же условия, что и для  алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Определение фоковского представления алгебры Клиффорда в точности повто-  ряет данное для алгебры Вейля: требуется, чтобы существовал циклический вектор  |0⟩, для которого выполнено условие ˆak|0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В силу условия цикличности можно  получить все что угодно, применяя к вектору |0⟩ операторы ˆa∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все состояния будут  линейными комбинациями мономов вида Π(ˆa∗  k)nk|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Единственная разница заключается в том, что показатели nk в этих мономах могут  быть только ноль или единица (nk = 0, 1) потому что ˆa∗2  k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — то, что назы-  вается принципом Паули.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Опять же, элементы этого базиса являются собственными  векторами любого гамильтониана вида � ǫkˆa∗  kˆak и собственные значения даются точ-  но той же формулой, как и в бозонном случае: � nkǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Показатели nk называются числами заполнения;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' операторы ˆa∗  k является опера-  торами рождения, а ˆak — операторами уничтожения по тем же причинам, что и в  бозонном случае.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно сказать, что гамильтониан � ǫkˆa∗  kˆak описывает невзаимо-  действующие фермионы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь вспомним, что в бозонном случае для получения представления элемен-  тов фоковского пространства полиномами я в мономах снимал крышечки и получал  полином от комплексных переменных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сейчас я хочу сделать то же самое: снять кры-  шечки и получить полином от антикоммутирующих переменных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае все  будет абсолютно то же.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Форма скалярного произведения абсолютно такая же, только  интегрирование будет вестись по антикоммутирующим переменным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Как и в случае  алгебры Вейля, легко проверить, что в этом представлении оператор ˆa∗  k действует  как умножение, ˆak действует как дифференцирование, и это дает правильные анти-  коммутационные соотношения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  29  Единственное, что осталось проверить, это то, что в скалярном произведении  ⟨F, G⟩ =  �  da∗daF(a∗)G(a∗)∗e−a∗a  умножения и дифференцирования являются сопряженными операторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это можно  сделать, применяя интегрирование по частям, с той лишь разницей, что при вычис-  лениях следует применять градуированное правило Лейбница.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если рассматривать одни полиномы мы получим предгильбертово пространство,  но можно взять пополнение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае конечного числа степеней свободы брать по-  полнение не надо — там только полиномы и есть, а в случае бесконечного числа  степеней свободы, если мы хотим работать в гильбертовом пространстве, фоковское  пространство нужно пополнить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В фоковском пространстве векторы можно представить в виде суммы мономов с  антисимметрическими коэффициентами:  �  n  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=',kn  fn(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', kn)ˆa∗  k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆa∗  kn|0⟩  в то время как для алгебры Вейля коэффициенты симметрические.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В пополненном  фоковском пространстве эти суммы бесконечны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Другими словами, точка фоковско-  го пространства представляется последовательностью антисимметрических функций  тогда как в алгебре Вейля она была последовательностью симметрических функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это стандартное представление из учебников квантовой механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такое представ-  ление работает также в случае, когда k является непрерывным параметром.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно, опять же, рассмотреть оператор числа частиц:  ˆN =  �  ˆa∗  kˆak =  �  dλˆa∗(λ)ˆa(λ)  и снова ˆa∗  k увеличивает число частиц на единицу, ˆak — уменьшает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Перейдем теперь к рассмотрению операторов, которые сохраняют число частиц,  как и в нерелятивистской квантовой механике, где гамильтониан сохраняет число  частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Самый простой гамильтониан — квадратичный гамильтониан.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  ˆH =  �  dxdyA(x, y)ˆa∗(x)ˆa(y) = ⟨a∗, Aa⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если оператор A имеет дискретный спектр, то, диагонализуя его, мы получим опе-  ратор  ˆH =  �  ǫkˆa∗  kˆak,  где ǫk, φk(x) — собственные значения и собственные функции оператора A:  ˆak =  �  dxφk(x)ˆa(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В прошлой лекции было сказано, что, взяв A = − 1  2m∆ + ˆU, мы получаем си-  стему невзаимодействующих нерелятивистских бозонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Имея тот же оператор, но  используя канонические антикоммутационные соотношения, мы получаем систему  невзаимодействующих нерелятивистских фермионов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Единственная разница заклю-  чается в том, что нужно считать, что этот оператор действует на многокомпонентные  функции от x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Он имеет точно такой же вид, но есть еще и дискретный индекс.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Какие канонические соотношения нужно брать — зависит от того, как действу-  ет на волновые функции ортогональная группа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Действие ортогональной группы на  дискретные индексы определяет спин частицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае полуцелого спина нужно  30  квантовать с помощью алгебры Клиффорда, а в случае целого спина — с помощью  алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По-другому можно сказать, что если представление неприводимое, то  все определяется числом индексов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если четное число, то мы имеем дело с алгеброй  Вейля, если нечетное — с алгеброй Клиффорда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того чтобы описывать взаимодействующие частицы в нерелятивистской кван-  товой механике, нужно добавить члены более высокого порядка с равным количе-  ством операторов рождения и уничтожения;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' тогда эти члены сохраняют число ча-  стиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В случае конечного числа степеней свободы существует единственное неприводи-  мое представление алгебры Клиффорда, и оно изоморфно фоковскому представле-  нию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Доказательство неприводимости фоковского представления и его единственно-  сти такое же, как и в случае алгебры Вейля с той лишь разницей, что представленное  доказательство будет строгим.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В предыдущем случае оно было нестрогое, потому что  операторы ˆak и ˆa∗  k в случае алгебры Вейля являются неограниченными операторами,  а здесь все эти операторы ограничены.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это легко понять, потому что выполняется  соотношение ˆakˆa∗  k + ˆa∗  kˆak = 1, в котором оба слагаемые положительно определены, их  сумма равна единице и, следовательно, каждый из операторов ограничен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если мы имеем бесконечное число степеней свободы, то число операторов ˆak и  ˆa∗  k бесконечно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно рассмотреть каноническое преобразование, введя новые гене-  раторы, которые по-прежнему удовлетворяют антикоммутационным соотношениям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь очень легко построить пример канонического преобразования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что  условия антикоммутативности симметричны относительно ˆak и ˆa∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если поменять  местами операторы рождения и уничтожения, то в коммутаторе изменится знак, а  в антикоммутаторе ничего не изменится и, поэтому, если просто поменять местами  операторы рождения и уничтожения, появится снова представление алгебры Клиф-  форда, и нужно задать вопрос: будет ли это представление эквивалентно исходному?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Ответ: не всегда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дальнейшие рассуждения точно такие же, как и в бозонном случае.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возьмем в  пространстве Фока операторы, определенные формулой ˆAk = ˆak для k ∈ I, ˆAk = ˆa∗  k  для k /∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти операторы подчиняются каноническим антикоммутационным соот-  ношениям, следовательно, они задают представление алгебры Клиффорда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того  чтобы это представление было фоковским, у него должен быть циклический вектор  Θ который обнуляется всеми новыми операторами уничтожения: ˆAkΘ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Очень легко найти решение этого уравнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно подействовать на |0⟩ теми  операторами ˆa∗  k, которые изменилиcь (k /∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, изменившиеся операторы  ˆa∗  k стали операторами уничтожения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь если взять моном с операторами ˆa∗  k, то  действуя на этот моном любым оператором ˆa∗  k из этого множества, получим ноль  потому что операторы ˆa∗  k будут появляться дважды с одним и тем же индексом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тех  же операторов ˆak, которые не менялись в этом произведении нет — они тоже дают  ноль, потому что их можно перенести на |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, у нас получился моном  Θ, который удовлетворяет соотношению ˆAkΘ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дальше все зависит от того, сколько операторов было изменено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если их чис-  ло конечно — все в порядке, будет получен конечный моном.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же изменилось  бесконечное число операторов, будет получено что-то, что фоковскому пространству  совершенно не принадлежит — моном бесконечной степени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Новое представление не  будет эквивалентно фоковскому, так как у него нет циклического вектора, который  нам нужен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, мы получили пример неэквивалентных представлений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это рассуждение восходит к Дираку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дираковское море подразумевает заполнение  бесконечного количества состояний, в результате чего получается неэквивалентное  представление антикоммутационных соотношений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  31  Перейдем к рассмотрению линейных канонических преобразований вида:  ˆAk = Φl  kˆal + Ψl  kˆa∗  l ,  ˆA∗  k = ¯Ψl  kˆal + ¯Φl  kˆa∗  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В отличие от вейлевского случая, здесь нельзя добавлять численные слагаемые, но  все остальное то же самое.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если потребовать, чтобы новые операторы удовлетворяли  каноническим антикоммутационным соотношениям, мы получим новое представле-  ние алгебры Клиффорда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — то что называется боголюбовским преобразованием.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вот, более или менее все, что я хотел рассказать про алгебру Клиффорда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В заклю-  чение я еще хотел добавить, что алгебра Клиффорда тесно связана с представлениями  ортогональной группы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из нее мгновенно получается то, что называется спинорным  представлением ортогональной группы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='3  Статистическая физика  Переходя к новой теме, напомню некоторые положения из статистической физики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как в классической, так и в квантовой механике есть понятие равновесного состо-  яния и в обоих случаях оно определяется как состояние с максимальной энтропией  при заданных условиях.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Выражение “заданные условия” может иметь разные смыслы  в разных ситуациях, но условие максимума энтропиии при заданных условиях долж-  но быть выполнено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если состояние представлено матрицей плотности, то энтропия  состояния записывается формулой  S = −TrK log K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если матрица плотности диагональна, то диагональные элементы матрицы pi ин-  терпретируются как вероятности, и эта формула дает обычное классическое выраже-  ние для энтропии:  S = −  �  pi log pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если задан гамильтониан ˆH, то мы можем зафиксировать среднюю энергию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мож-  но зафиксировать допустимый интервал энергии — тогда это будет микроканониче-  ское распределение, но давайте зафиксируем среднюю энергию E = Tr ˆHK (тогда  мы получим каноническое распределение).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Максимизируя энтропию при заданной  средней энергии, увидим, что матрица плотности, на которой достигается максимум  энтропии, дается формулой:  e−β ˆ  H  Z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (16)  Константу Z мы можем вычислить, потому что матрица плотности по определению  должна иметь след равный единице и поэтому знаменатель должен быть равен следу  оператора e−β ˆ  H:  Z = Tre−β ˆ  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это выражение носит название статистической суммы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если спектр ˆH дискретен с уровнями Ei, то  Z =  �  e−βEi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Физический смысл β — это обратная температура: β =  1  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из выражения для ста-  тистической суммы видно, что при β → ∞ или, что то же самое, T → 0, вклад в  это выражение вносит только член с минимальной энергией, то есть, член отвечаю-  щий основному состоянию, которое мы считаем невырожденным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, в  32  случае нулевой температуры получается чистое состояние — единственное основное  состояние.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вывод выписанной формулы (16) для состояния с максимальной энтропией ис-  пользует метод Лагранжа для случая когда зафиксирована средняя энергия Tr ˆHK =  E, а кроме того, как мы знаем, что след матрицы плотности равен единице TrK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Все это нужно включить в число условий.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно стандартным способом посчитать  вариацию выражения  L = −TrK log K − βTr( ˆHK − E) − ζ(TrK − 1)  и найти стационарные точки:  − log K − 1 − β ˆH − ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Единственная тонкость заключается в том, что нужно использовать формулу:  δTrφ(K) = Trφ′(K)δK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (17)  Достаточно проверить эту формулу для случая φ(K) = Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда рассматривается  вариация функции Kn, то в силу некоммутативности будет n членов, но когда вы  берете след, то все эти члены становятся одинаковыми и получается ответ, соответ-  ствующий (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Следующее замечание заключаются в том, что хотя статистическая сумма Z сама  по себе не имеет физического смысла, но через нее более или менее все выражается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Тесно связаны с логарифмом статсуммы выражения для средней энергии  E = ¯H = − 1  β  ∂ log Z  ∂β  и энтропии  S = βE + log Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть еще важная характеристика, которая называется свободной энергией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Она  определяется формулой F = E − TS и выражается через статсумму так:  F = −T log Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Свободная энергия удобна тем, что вместо нахождения максимума энтропии можно  искать минимум свободной энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это сразу следует из метода множителей Лагран-  жа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как правило физические величины можно получать так: берем какой-то гамиль-  тониан, к нему что-то прибавляем и смотрим что получится в пределе, когда то, что  было прибавлено, стремится к нулю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этом следует заметить, что когда гамиль-  тониан ˆH чуточку меняется:  ˆH(λ) = ˆH + λA + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ,  то изменение статистической суммы  Z(λ) = Z + (−β)λTrAe−β ˆ  H + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  управляется средним значением этой добавки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Оператор A — это тоже физическая величина, и она заведует изменением стат-  суммы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Изменение статсуммы выражается в терминах среднего значения (математи-  ческого ожидания) этой физической величины:  ¯A = TrAe−β ˆH  Z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  33  (Для этого выражения я буду использовать еще альтернативное обозначение ⟨A⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Перейдем теперь к основной нашей цели – к тому, что называется корреляцион-  ной функцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Корреляционная функция — это просто среднее значение произведе-  ния некоторых физических величин ⟨A1 · · · An⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно считать также, что эти фи-  зические величины зависят от времени — являются гейзенберговскими операторами  и удовлетворяют гейзенберговским уравнениям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда выражение ⟨A1(t1) · · · An(tn)⟩  тоже называется корреляционной функцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно рассматривать корреляционные  функции по любому состоянию — не обязательно состоянию равновесия, но если речь  идет о состоянии равновесия, я пишу в качестве индекса значение обратной темпера-  туры: ⟨A1(t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' An(tn)⟩β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь я хочу заметить, что все сказанное про статсумму и связанные с ней вещи  применимо очень часто в случае конечного числа степеней свободы, а в случае бес-  конечного числа степеней свободы применимо достаточно редко.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В нерелятивистской  квантовой механике применимо, когда мы находимся в конечном объеме.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В беско-  нечном объеме это не работает даже в нерелятивистской квантовой механике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Очень  важно понимать, что на самом деле даже в нерелятивистской квантовой механике, но  в бесконечном объеме следует рассматривать статсумму сначала в конечном объеме, а  после этого — перейти к переделу бесконечного объема в корреляционных функциях.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Нельзя работать сразу в бесконечном объеме.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Еще я должен заметить, что при переходе к бесконечному объему обычно берется  предел, при котором сохраняется не число частиц, а плотность числа частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными  словами, при переходе к пределу число частиц меняется пропорционально объему, и  тогда плотность числа частиц остается постоянной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предположим теперь, что набор корреляционных функций в бесконечном объеме  получен — что с ним делать?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нет никакого гильбертова пространства, в котором опре-  делены эти операторы A в бесконечном объеме, но есть корреляционные функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обычно в таком случае по этим корреляционным функциям строится гильбертово  пространство, применяя некий аналог конструкции GNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Физики это обычно делают  неявно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они просто говорят “теперь у нас есть равновесное состояние — его можно  представить вектором в гильбертовом пространстве или матрицей плотности в гиль-  бертовом пространстве и там действуют эти операторы A”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом деле, нужно  применять конструкцию, которая в аксиоматической квантовой теории поля называ-  ется “reconstruction theorem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Там роль корреляционных функций играют функции  Вайтмана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Перейду теперь к вопросу: как можно поступать в алгебраическом подходе с рав-  новесными состояниями в ситуации, когда невозможно использовать принцип макси-  мума энтропии?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь можно применить то, что называется условием Кубо — Мар-  тина — Швингера (KMS):  ⟨A(t)B⟩β = ⟨BA(t + iβ)⟩β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (18)  Это условие на корреляционные функции наблюдаемых A и B в равновесном состо-  янии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Его легко вывести в случае конечномерного гильбертова пространства, потому  что в этом случае все хорошо определено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае есть оператор Гейзенберга, в  определении которого участвует eiHt, есть матрица плотности, в которой фигурирует  e−βH, но время можно считать комплексным и тогда, когда комплексное число явля-  ется действительным, получается операторы эволюции, а в случае когда оно является  чисто мнимым и Im(t) > 0, то тогда будет получена матрица плотности равновесного  состояния (с точностью до множителя).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это важное замечание, которое заключается  в том, что можно пытаться жить в комплексном времени, заменяя время веществен-  ное временем комплексным или чисто мнимым.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это – то, что называется словом  “виков поворот”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Не любое комплексное время годится, потому что если в показателе  34  степени будет стоять положительное выражение, то это будет вещь в конечномерном  случае вполне хорошая, а в бесконечномерном случае — плохая.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Однако корреляци-  онную функцию ⟨BA(t)⟩β можно продолжить аналитически в полосу 0 ≤ Im(t) ≤ β  и за счет этого условие KMS будет иметь смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Условие KMS не использует понятие энтропии — нужно знать только корреляци-  онные функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оно работает также в бесконечном объеме.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно рассматривать  условие KMS как определение равновесного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Равновесное состояние, как я  его определял раньше, практически почти всегда является единственным, а здесь по-  является простор для того, чтобы равновесное состояние было неединственным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта  неединственность равновесного состояния связана с наличием фазовых переходов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Условие KMS является заменой условия максимума энтропии в случае алгебраиче-  ской квантовой теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Разберем теперь примеры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Самый простой пример — квадратичный гамильтониан.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Положительно определенный квадратичный гамильтониан можно привести к ви-  ду:  ˆH =  �  ǫiˆa∗  i ˆai,  описывающему невзаимодействующие бозоны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Невзаимодействующие бозоны с точки  зрения формальной — это то же самое, что многомерный гармонический осциллятор,  для которого легко можно посчитать матрицу плотности равновесного состояния, ста-  тистическую сумму, среднее значение энергии и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='. Статсумма просто распадается  на произведение статсумм для разных значений k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для каждого k можно просумми-  ровать геометрическую прогрессию, а потом взять произведение  Z = Π  1  1 − eβǫi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Среднее значение энергии вычисляется по формуле:  E = ¯H =  �  ǫi¯ni,  где ¯ni = (eβǫi − 1)−1 — средние числа заполнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для случая фермионов существенной разницы нет:  Z = Π(1 + eβǫi),  ¯H =  �  ǫi¯ni,  где ¯ni = (eβǫi +1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Основная разница в том, что в выражении для чисел заполнения  стоит не минус, а плюс и поэтому средние числа заполнения всегда не превышают  единицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В заключение добавлю, что, как я говорил, все корреляционные функции можно  получать с помощью дифференцирования статистической суммы, но удобнее диф-  ференцировать ее логарифм, который совпадает с точностью до множителя со сво-  бодной энергией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этом дифференцировании получаются не сами корреляционные  функции, а то, что называется усеченными корреляционными функциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Что это та-  кое?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если рассматривать, например, корреляционные функции двух операторов, это  среднее значение ⟨AB⟩ , а если дифференцировать свободную энергию, то получится  ⟨AB⟩ − ⟨A⟩⟨B⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Грубо говоря, вычитается часть, отвечающая меньшему количеству  операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такие функции будут мне важны позже.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  35  4  Лекция 4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Адиабатическое приближение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Декогерентность  Эту лекцию я начну с объяснения того, что происходит, когда гамильтониан ˆH(t)  зависит от времени, но меняется медленно (адиабатически).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я предполагаю, что все  уровни энергии En(t) различны и зависят от t непрерывно и даже дифференцируе-  мо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Соответствующие собственные векторы я обозначаю φn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я предполагаю, что  зависящий от времени вектор φn(t) меняется медленно и его производной по t мож-  но пренебречь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы покажем, что если начать с собственного вектора гамильтониана  ˆH(t = 0), то во время эволюции, управляемой медленно меняющимся гамильтони-  аном ˆH(t) он остается собственным, но это будет собственный вектор переменного  гамильтониана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При эволюции будет возникать фазовый множитель αn(t):  ˆU(t)φn(0) = e−iαn(t)φn(t),  (19)  который определяется уравнением:  dαn(t)  dt  = En(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (20)  Это легко получить — нужно просто продифференцировать предыдущее равенство с  учетом правила Лейбница, пренебрегая при этом производной от φn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это рассуж-  дение чуточку неаккуратно, потому что я предположил, что φn(t) с течением времени  меняется медленно, что не очевидно, потому что собственный вектор определен неод-  нозначно (только с точностью до множителя).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно ли подобрать множитель так,  чтобы это условие выполнялось — это не так ясно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Проведем более аккуратное рассмотрение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать, что гамильтониан ˆH(g)  зависят от какого-то параметра или многих параметров, которые я обозначил буквой  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Предположим, что собственные векторы φn(g) и собственные значения En(g) глад-  ко зависят от g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать, что параметр g зависит от времени таким образом,  что производной от g по t можно пренебречь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Стандартный выбор таков: фиксиру-  ется g(t) и строится семейство ga(t) = g(at), где a → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Очевидным образом, на  таких функциях условие малости производной по t выполняется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае все  приведенные выше рассуждения становится вполне аккуратным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того чтобы обобщить эти рассуждения на матрицы плотности, следует просто  заметить, что зависимость матрицы плотности K(t) от времени определяется урав-  нением  dK  dt = H(t)K(t) = 1  iℏ[ ˆH(t), K(t)],  где стоит коммутатор с гамильтонианом ˆH(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот коммутатор и есть оператор H(t)  (без крышечки), действующий на K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь нужно рассмотреть собственные векторы ψmn(t) оператора H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Их лег-  ко найти.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это будут просто проекции на собственные векторы гамильтониана ˆH(t),  которые я обозначил как φn(t) :  ψmn(t)x = ⟨x, φn(t)⟩φm(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В представлении где оператор ˆH(t) диагонален это будут матрицы, у которых стоит  единица в позиции (m, n), а остальные элементы нулевые.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае в точности  такое же рассуждение, которое было приведено выше, определяет эволюцию U(t)  этих собственных векторов:  U(t)ψmn(0) = e−iβmn(t)ψmn(t),  dβmn(t)  dt  = Em(t) − En(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (21)  36  Это легко проверяется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно продифференцировать по t и пренебречь производной  вектора ψmn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Очень важное замечание состоит в том, что когда m = n можно  считать, что фаза обращается в ноль: βmm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Представим то же самое в несколько других обозначения, а именно, возьмем мат-  рицу плотности и запишем ее как сумму по собственным векторам с какими-то коэф-  фициентами kmn:  K =  �  kmnψmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вместо того, чтобы рассматривать эволюцию собственных векторов ψmn рассмотрим  эволюцию коэффициентов kmn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Формулы те же самые — коэффициенты kmn(t)  получают фазовые множители, а в показателе стоит βmn(t):  kmn(t) = e−iβmn(t)kmn,  βmn(t) =  � t  0  (Em(τ) − En(τ))dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если адиабатический гамильтониан ˆH(t) таков, что в момент T он возвращается к  тому, что был в нулевой момент времени: ˆH(T) = ˆH(0), то диагональные элементы  матрицы K не меняются, но меняются недиагональные элементы — они умножаются  на фазовый множитель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Фиксируем теперь гамильтониан ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно себе представить молекулу или атом  или что-нибудь побольше — любую квантовую систему и эта система живет в каком-то  внешнем мире.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать, что внешнее окружение просто меняет гамильтониан,  который управляет молекулой;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' новый гамильтониан ˆH(t) может зависеть от време-  ни, но мы считаем, что он меняется медленно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно себе представить, что около  молекулы довольно далеко от нее пролетает какая-то космическая частица.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком  случае есть электрическое поле, которое возникает от этой частицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает,  что гамильтониан из-за космической частицы изменился.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если эта частица пролетает  достаточно далеко, то можно считать, что изменение адиабатично.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мой любимый пример: вы проводите эксперимент, а в соседней комнате кто-то  решил подогреть свой завтрак и включил микроволновку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда у вас тоже возникает  какое-то электрическое поле, которое, несомненно, можно считать адиабатическим.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Другой пример: мы знаем, что живем в мире, где есть микроволновое космиче-  ское излучение, которое тоже порождает некоторое электромагнитное поле.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Очень  маленькое, но тем не менее оно есть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этих адиабатических возмущений мы не знаем,  но мы знаем, что на диагональные элементы матрицы плотности адиабатическое воз-  мущение не оказывает влияния, а у недиагональных элементов матрицы плотности  появляются фазовые множители, которые мы, естественно, не знаем, потому что не  знаем возмущения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  То, что я сказал, можно интерпретировать другим способом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно рассмотреть  линейные комбинации вида α0φ0 + α1φ1 двух или большего количества собственных  векторов гамильтониана ˆH = ˆH(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При рассмотрении эволюции этого состояния бу-  дут появляться фазовые множители: αk(t) = e−iEktαk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти фазовые множители по-  являются всегда, они предсказуемы, но если наложено адиабатическое возмущение,  то тогда фазовые множители становятся непредсказуемыми, а абсолютные значения  коэффициентов αk(t) остаются постоянными.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раньше два собственных вектора коге-  рентно менялись с течением времени (пока не было адиабатического возмущения), а  сейчас эта когерентность исчезла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — то, что называется декогерентностью.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я хочу объяснить как из этих очень простых соображений появляется  стандартный рецепт теории измерений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать, что та же самая молекула  взаимодействует с окружающей средой и есть случайное адиабатическое возмуще-  ние ˆH(t) гамильтониана рассматриваемой системы ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает что есть гамиль-  тониан, который зависит от каких-то параметров λ ∈ Λ, по которым есть некото-  рое распределение вероятности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать, что это адиабатическое возмущение  37  действует в период времени от 0 до T и тогда, как я уже говорил, коэффициенты  kmn матрицы плотности Kλ(T) в представлении ˆH получают фазовые множители  Kλ(T) = � Cmn(λ, T)kmnψmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Фазовые множители Cmn(λ, T) равны единице для  диагональных элементов и не равны единице для остальных элементов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку  гамильтониан случайный, матрицу плотности следует усреднять по этому возмуще-  нию, то есть, у недиагональных элементов нужно усреднить этот фазовый множи-  тель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Совершенно ясно, что при усреднении этих фазовых множителей получается  нечто по модулю меньшее единицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Наложив некоторые условия, легко проверить,  что среднее от недиагональных матричных элементов будет равно нулю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Формальное доказательство заключается в следующем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я уже говорил, что мы  можем включить наш гамильтониан в некоторое семейство гамильтонианов ˆH(g), где  g принадлежит некоторому множеству параметров обозначенному через Λ (g ∈ Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Будем считать, что все эти возмущения таковы, что g(0) = 0, g(1) = 0 а зависимость  гамильтониана от времени определяется формулой ˆH(g(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я определяю адиабати-  ческий гамильтониан следующим образом:  ˆHα(t) = ˆH(g(αt)),  где α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этом растянется время.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если раньше оно изменялось от нуля до  единицы, то сейчас — от нуля до T = α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если обозначить через En(g) собственные  значения гамильтониана ˆH(g), то значения фазовых множителей вида e−iβmn(t) при  t = T будут определяться следующей формулой:  βmn =  � T  0  dτ(Em(g(ατ)) − En(g(ατ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Проведя замену переменных ατ = τ ′, можно 1/α вынести за знак интеграла:  βmn = 1  α  � 1  0  dτ ′(Em(g(τ ′)) − En(g(τ ′))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Воспользуемся теперь леммой Римана – Лебега, которая утверждает, что если  ищется среднее значение экспоненты eikx с большим, стремящемся к бесконечности  показателем и среднее берется с какой-то сколько-нибудь приличной функцией (точ-  ное высказывание — она должна быть абсолютно интегрируемой), то тогда искомое  среднее в пределе будет стремиться к нулю:  �  eikxρ(x)dx → 0  при k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если распределение вероятностей по λ не слишком плохое (с приличной плотно-  стью распределения вероятности), то коэффициенты kmn матрицы плотности исчез-  нут, если m ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате матрица плотности K становится диагональной после  введения случайных адиабатических возмущений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обозначим теперь диагональные матричные элементы буквами pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В обычном  подходе pn — это вероятности различных состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У нас же так и получилось:  усредненная матрица плотности — это смесь чистых состояний с вероятностями pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, получилась обычная формула для вероятностей разных чистых со-  стояний в данном смешанном состоянии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из этих соображений получаются обычные  формулы теории измерений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Отмечу, что обычно говорится, что если имеет место взаимодействие с классиче-  ской системой, то матрица плотности становится диагональной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В терминах волновых  функций это называется коллапсом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В приведенных рассуждениях нет классической  системы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  38  Таким образом коллапс волновой функции можно вывести из случайного адиаба-  тического взаимодействия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь нет постоянной Планка — есть только адиабатика.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  До сих пор мы рассматривали обычную квантовую механику, а теперь рассмот-  рим геометрический подход к квантовой теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я уже говорил, что есть алгебраиче-  ский подход, где исходная точка это алгебра наблюдаемых — ассоциативная алгебра  с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Собственно наблюдаемыми являются самосопряженные элементы ал-  гебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Состояния определяются как положительные линейные функционалы на этой  алгебре, то есть, понятие состояния вторично.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Давайте будем считать, что понятие  состояния первично, что есть некоторое множество состояний и подумаем, что от него  нужно потребовать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Первым делом я хочу потребовать, чтобы у меня было понятие смеси состояний  — не смешанного состояния, а смеси состояний, чтобы любые состояния можно бы-  ло смешивать с некоторыми вероятностями, с некоторыми весами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это совершенно  необходимое требование и, в общем-то, оно не имеет отношения к физике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно, на-  пример, в экономике или в теории игр рассматривать смешанные стратегии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того  чтобы можно было смешивать, множество должно быть выпуклым.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать,  что выпуклое множество — это подмножество некоторого векторного пространства  и тогда понятие смешивания будет определено очевидным образом: если xi — точки  выпуклого множества, pi — неотрицательные числа, сумма которых равна единице,  то смесь этих точек дается формулой ¯x = � pixi (числаpi рассматриваются как веса  или вероятности).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Что нужно еще потребовать?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если мы смешиваем конечное количество состояний,  то ничего больше не надо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но если мы хотим смешивать бесконечное количество со-  стояний, то необходимо понятие предела, а для этого нужны две вещи.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Во-первых,  нужна топология в векторном пространстве L, в котором лежит выпуклое множе-  ство C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оно должно быть топологическим векторным пространством.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для простоты  предположу, что это банахово пространство (нормированное полное пространство).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Во-вторых нужно потребовать, чтобы пределы принадлежали тому же самому мно-  жеству — чтобы множество было замкнутым.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При выполнении этих условий можно  смешивать любое бесконечное множество состояний и поэтому мы требуем, чтобы  множество состояний было замкнутым выпуклым множеством.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Еще потребуем, чтобы множество было ограниченным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это, по существу, един-  ственное требование, которое нужно для того чтобы развивать теорию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Также мне нужно иметь понятие оператора эволюции σ(t), который переводит  состояние в момент времени ноль в состоянии какой-то момент времени t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Ведь что  самое главное не просто в физике — в науке?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно уметь предсказывать будущее.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Что мне нужно потребовать от этого оператора эволюции?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для каждого состояния я  должен получить другое состояние, то есть, оператор эволюции должен отображать  множество состояний в себя.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кроме того, я считаю, что t может быть отрицательным;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  тогда оператор эволюции будет обратимым пробразованием множества состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Будем считать, что оператор эволюции — это линейное преобразование.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее,  мы считаем, что он может быть продолжен как линейный оператор на векторное  пространство L, включающее множество C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Введем понятие группы автоморфизмов  множества состояний C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это группа линейных операторов в объемлющем простран-  стве L, которые обратимы и отображают множество C0 на себя.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Естественно считать,  что оператор эволюции будет автоморфизмом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иногда удобно наложить на оператор  эволюции дополнительные условия, например, потребовать, чтобы операторы эволю-  ции принадлежали некоторой подгруппе V группы автоморфизмов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это необходимо,  например, в классической механике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дальше нужно написать уравнение движения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обычно мы считаем, что знаем  изменение системы за бесконечно малое время.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это называется уравнением движения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  39  В самом общем виде уравнение движения можно записать так:  dσ  dt = H(t)σ(t),  (22)  где H(t) это линейный оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Формально можно сказать, что H(t) определяется из этого уравнения, но в фи-  зике мы обычно считаем, что оператор H(t) известен и нужно найти оператор эво-  люции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Назовем оператор H(t) “гамильтонианом” (в кавычках).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если “гамильтониан”  не зависит от времени, то оператор эволюции — это экспонента от “гамильтониана” :  σ(t) = exp(Ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Всё как в обычной квантовой механике, я лишь не поставил мнимую  единицу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно считать это просто определением экспоненты: для независящего от  времени оператора H в банаховом пространстве экспоненту можно определить как  решение уравнения (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Из уравнения (22) следует, что “гамильтониан” принадлежит алгебре Ли группы V  (он является касательным вектором группы V в единичном элементе группы).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Группа  V, вообще говоря, бесконечномерна, поэтому понятие касательного вектора зависит от  выбора топологии в группе, но я не буду обращать внимание на эти тонкости.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кроме  того, разумно потребовать, чтобы для не зависящего от времени “гамильтониана”  уравнение (22) имело решение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь у меня есть понятие состояния, у меня есть уравнения движения, но еще  мне нужно ввести понятие наблюдаемой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раньше я шел от наблюдаемых к состоя-  ниям, а сейчас хочу перейти от состояний к наблюдаемым.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Что такое наблюдаемая?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Прежде всего, это должен быть оператор A на который мы наложим те же условия,  как и на “гамильтониан”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, мы требуем, чтобы была определена экс-  понента σA(t) = exp(At), которую можно рассматривать как однопараметрическую  подгруппу в V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Зафиксируем еще функционал a, который инвариантен относительно операторов  σA(t) = exp(At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Это условие эквивалентно условию a(Ax) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Функционал a опре-  деляет значения наблюдаемой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В частности, в случае обычной квантовой механики V — это группа унитарных  операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Она действует на матрицы плотности по формуле U(K) = ˆUK ˆU −1, где ˆU  — унитарный оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если ˆA — самосопряженный оператор (не обязательно огра-  ниченный), то экспоненту ei ˆ  At можно рассматривать как однопараметрическую под-  группу группы V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Так получаются все однопараметрические подгруппы непрерывные  в сильной топологии (теорема Стоуна).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это позволяет отождествить самосопряжен-  ные операторы с элементами алгебры Ли группы унитарных операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь есть  маленькие трудности, которые заключаются в том, что коммутатор неограниченных  самосопряженных операторов не всегда определен и поэтому, строго говоря, струк-  тура алгебры Ли нарушается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти трудности возникают всегда и к рассматриваему  подходу не имеют никакого отношения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  “Гамильтониан” выражается через самосопряженный оператор ˆH по формуле  H(K) = 1  i [ ˆH, K],  где K — матрица плотности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Мы считаем, что ℏ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Аналогично, любому самосопряженному оператору ˆA мы сопоставляем оператор  на матрицах плотности по формуле:  A(K) = 1  i [ ˆA, K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (23)  Наблюдаемая задается парой ( ˆA, a), где ˆA — самосопряженный оператор, действую-  щий на матрицах плотности по формуле (23) и функционал на матрицах плотности  a задается формулой a(K) = tr( ˆAK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  40  В алгебраическом подходе группа V состоит из автоморфизмов ассоциативной  алгебры с инволюцией A (∗-алгебры);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' в частности, самосопряженный элемент A ал-  гебры A определяет инфинитезимальный автоморфизм (элемент алгебры Ли группы  V) по формуле αA(X) = i[A, X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Линейный функционал ω на алгебре называется по-  ложительным, если ω(X∗X) ≥ 0 для всех X ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' множество всех положительных  функционалов я обозначаю буквой C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Множество состояний C0 состоит из нормированных положительных функциона-  лов (функционалов, удовлетворяющих условию ω(1) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — ограниченное вы-  пуклое множество;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' на нем естественным образом действует группа V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Наблюдаемая  — это пара (A, a), где A — это самосопряженный элемент алгебры A, рассматрива-  емый как инфинитезимальный автоморфизм, а a-это линейный функционал на C0  сопоставляющий состоянию ω число a(ω) = ω(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе точно так же есть декогерентность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Доказательство  этого, практически такое же, как и раньше.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я начну с независящего от времени “га-  мильтониана”, обозначаемого буквой H (без крышечки).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда оператор эволюции —  это просто экспонента σ(t) = etH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я предположу, что оператор H диагонализуем, то  есть, существует базис (ψj), состоящий из собственных векторов оператора H:  Hψj = ǫjψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я предположил, что оператор эволюции отображает множество состояний в себя и что  множество состояний является ограниченным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этих условиях следует ожидать,  что этот оператор диагонализуем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это легко доказать в конечномерном случае.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если σ(t) = etH — это семейство  ограниченных операторов в конечномерном пространстве и эти операторы не просто  ограничены — они ограничены вместе, то есть, норма всех этих операторов ограни-  чена одним и тем же не зависящим от t числом, то из этого сразу следует две вещи.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Во-первых, собственные значения этих операторов чисто мнимые, потому что функ-  ция вида eǫt ограничена только в том случае, если ǫ чисто мнимое.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Асимптотика  определяется вещественной частью, а мнимая часть дает фазовый множитель;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' если ǫ  — действительное число, ограниченность будет только в том случае, когда ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Во-  вторых, в случае наличия жордановой клетки размера больше единицы, экспонента  σ(t) = etH тоже не будет ограниченной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, в конечномерном случае во-  первых все собственные значения чисто мнимые, а во-вторых все жордановы клетки  одномерны, то есть H диагонализуем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В бесконечномерном случае этого недостаточно для диагонализуемости, но тем не  менее жордановых клеток быть не может и в этом случае.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поэтому нужно ожидать,  что оператор H диагонализуем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я это и буду предполагать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я повторяю свои рассуждения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я включаю мой “гамильтониан” в семейство  H(g) “гамильтонианов”, с собственными значениями (ǫj(g)) и собственными функци-  ями (ψj(g)):  H(g)ψj(g) = ǫj(g)ψj(g),  удовлетворяющими прежним условиям — базис гладко зависит от g и для значения  g = 0 получается исходный “гамильтониан”: H(0) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дальше я говорю, что ψj — это робастная нулевая мода, если ǫj(g) ≡ 0, то есть,  собственное значение равно нулю при любом g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно было бы сказать, что это —  устойчивая нулевая мода (она была у H и остается нулевой модой после введения  H(g)), но слово “устойчивость” занято для других целей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я хочу сделать все то же, что я делал раньше — буду считать, что взаимо-  действие со средой определяется случайным “гамильтонианом” H(g(t)), где g зависит  от t, и буду считать, что эти случайные “гамильтонианы” адиабатичны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они медлен-  но меняются так, что производной от g по t можно пренебречь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно сказать, что  41  рассматривается адиабатическая эволюция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Собственный вектор при этом остается  собственным вектором меняющегося “гамильтониана”, но в нем появляется фазовый  множитель:  σ(t)ψj = eρj(t)ψj(g(t)),  где dρj  dt = ǫj(g(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тут нет мнимой единицы, но ǫj(g) само чисто мнимое.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для дока-  зательства просто нужно продифференцировать правую часть этого выражения по t  применяя уравнения движения и пренебрегая производной ˙g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Фазовый множитель не появляется для робастных нулевых мод, а для неробаст-  ных появляется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же адиабатическое возмущение действовало некоторое конеч-  ное время, то в конце для робастных нулевых мод просто ничего не изменится.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В  середине эти моды менялись, они становились нулевыми модами другого гамильто-  ниана, а в конце они приходят к тому, что было: σ(T)ψj = ψj, если g(T) = g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У  всех остальных мод есть фазовые множители.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дальше все так же, как в обычной квантовой механике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для неробастных нулевых  мод все случайные фазовые факторы после усреднения по случайному возмущению  обнуляются, а для робастных нулевых мод ничего не меняется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В обычной квантовой механике робастные нулевые моды — это диагональные эле-  менты матрицы плотности в ˆH — представлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они, как мы знаем, являются ну-  левыми модами, потому что все диагональные матрицы коммутируют между собой,  поэтому здесь применимы предыдущие высказывания.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Интерпретировать это нужно так: произвольное состояние x ∈ C0 можно разло-  жить по собственным функциям оператора H, и при этом взаимодействие с окру-  жающей средой убьет все моды, кроме робастных нулевых мод.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я обозначу через P ′  тот оператор, который убивает все, кроме робастных нулевых мод, и теперь можно  сказать, что вся наблюдаемая физика лежит в проекции на робастные нулевые моды  P ′x ∈ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Далее нужно разложить состояния P ′x ∈ C0 по чистым робастным нулевым  модам P ′x = � piui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Коэффициенты pi в этом разложении должны быть интерпрети-  рованы как вероятности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В обычной квантовой механике таким образом получаются  обычные вероятности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В общем случае коэффициент pi — это вероятность чистой  робастной нулевой моды ui в состоянии x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Гамильтониану H отвечает физическая величина (H, h), где функционал h нужно  отождествить с энергией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Коэффициент pi следует интерпретировать как вероятность  найти энергию h(ui) в состоянии x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Предполагается, что все числа h(ui) различны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если некоторые из них совпадают, то чтобы получить вероятность того, что энергия  равна h, нужно просуммировать все коэффициенты pi, для которых h(ui) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Если все нулевые моды робастные, то можно написать простую формулу для опе-  ратора: P ′ = P, где P — оператор, убивающий все ненулевые моды:  P = lim  T→∞  1  T  � T  0  σ(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если в этом интеграле разлагать σ(t) по собственным векторам, то для ненулевых мод  появляются ненулевые фразовые множители;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' усредняя их по времени, в результате  получаем ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Сейчас я объяснил то, как появляются вероятности для “гамильтонианов”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они  интерпретируются как вероятности различных уровней энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Повторить те же  самые рассуждения можно рассматривая и другие наблюдаемые, представляемые в  виде (A, a), где A ∈ V, a — функционал, для которого a(Ax) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно определить  понятие робастной нулевой моды для любой наблюдаемой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я дам несколько другое, более общее, определение робастной нулевой моды.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда  говорится, что x — это робастная нулевая мода, прежде всего, нужно потребовать,  42  чтобы это была нулевая мода: Ax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если A слегка меняется (заменяется на близ-  кий элемент A′ группы V) , то нужно потребовать, чтобы у A′ была нулевая мода  x′, близкая к исходной нулевой моде x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дальше все то же самое, что было сказано  для энергии, можно повторить для произвольной наблюдаемой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно рассмотреть  проекцию PA на пространство нулевых мод: CA = (KerA) ∩ C = Im(PA), где  PA = lim  T→∞  1  T  � T  0  dteAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Я предположил, что все нулевые моды робастные).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' После этого нужно разложить  проекцию на нулевые моды по крайним точкам:  PA(x) =  �  piui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Коэффициенты pi будут интерпретироваться как вероятности значений a(ui) в состо-  янии x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  L-функционалы  Я сейчас объяснил, что есть формализм, в котором начальный объект — это мно-  жество состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возникает вопрос, который должен задать физик: удобен ли этот  формализм, удобно ли в этом формализме считать?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На этот вопрос я собираюсь сей-  час ответить, вернувшись уже к обычному формализму квантовой механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду  работать в рамках алгебраического подхода, в котором алгеброй является алгебра  Вейля с генераторами ˆui и соотношениями  ˆukˆul − ˆulˆuk = iℏσk,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я хочу ввести понятие L-функционала, определяемого для всякой матрицы плот-  ности K в любом представлении алгебры Вейля по формуле:  LK(α) = treiαk ˆukK = trVαK,  где Vα = eiαk ˆuk = eiαˆu — операторы, которые мы уже рассматривали, αk — это набор  чисел, отвечающих генераторам алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что алгебру Вейля можно  задавать генераторами Vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если uk — самосопряженные, а αk -действительные, то  эти генераторы унитарны и удовлетворяют соотношениям  VαVβ = e−i ℏ  2ασβVα+β,  где ασβ = αkσk,lβl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это так называемая экспоненциальная форма алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Важное свойство L-функционалов заключается в том, что они объединяют вме-  сте все представления алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Проблема неэквивалентности представлений  алгебры Вейля в формализме L-функционалов полностью исчезает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В формуле для  L-функционала унитарный оператор умножается на оператор из класса операторов  имеющих след, и поэтому след хорошо определен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я могу ввести пространство L всех линейных функционалов на алгебре Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно считать, что L-функционал определяет линейный функционал на алгебре  Вейля: LK ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ( Это вытекает из замечания, что всякий элемент, выраженный через  генераторы Vα с помощью конечного числа операций сложения и умножения явля-  ется линейной комбинацией генераторов Vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Как легко проверить, функционал LK  положителен и, кроме того, нормирован (на единичном элементе алгебры он равен  единице).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В дальнейшем я буду отождествлять L-функционалы с положительными  функционалами на алгебре Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  43  Определим операции в рассматриваемом пространстве L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти операции определе-  ны для любой алгебры с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если имеется алгебра с инволюцией, то на линейных функционалах есть две опе-  рации, отвечающие элементу A этой алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно определить операцию на функ-  ционале ω(x), где x — элемент алгебры, умножая x справа на A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Другая операция  получается если умножать слева на A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Первую из них я обозначил той же самой  буквой A, а другую — ˜A :  (Aω)(x) = ω(xA), ( ˜Aω)(x) = ω(A∗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Зададимся вопросом: остается ли функционал положительным при действии эти-  ми операторами?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ответ: нет, но если применить комбинацию ˜AA, тогда положитель-  ные функционалы будут переходить в положительные функционалы (которые не обя-  зательно нормированы).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что положительные функционалы должны быть  неотрицательны на элементах вида x = B∗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко проверить, что при действии опе-  рации ˜AA происходит преобразование x → A∗xA = (BA)∗xBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отсюда видно, что  положительность сохраняется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это важное высказывание.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы обозначили через C пространство всех положитель-  ных функционалов (ненормированных состояний).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В нем действует оператор ˜AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Следующее замечание состоит в том, что операторы вида ˜A всегда коммутируют  с операторами вида A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это происходит потому, что один умножает слева, а другой —  справа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Далее, если оператор имеет вид A = eitH, тогда ˜A = e−it ˜  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отсюда легко вывести,  что если уравнения движения записать как  idσ/dt = ( ˜H − H)σ,  (24)  то оператор эволюции будет иметь вид e−it( ˜  H−H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Согласно предыдущему замеча-  нию, это выражение может быть представлено в виде ˜AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть если уравнения  движения записать в виде (24), то σ не выводит за пределы конуса положительных  функционалов (конуса ненормированных состояний).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это я буду использовать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Перейдем теперь к вопросу о виде уравнений движения в случае алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Легко вычислить операторы Vβ и ˜Vβ :  (VβL)(α) = e+i ℏ  2 ασβL(α + β), ( ˜VβL)(α) = e+i ℏ  2ασβL(α − β),  Запишем ˆH как интеграл  ˆH =  �  dβh(β)Vβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Этот оператор будет самосопряженным если h(−β) = h(β)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Введем постоянную  Планка и сделаем замену 1  ℏ ˆH → ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Уравнение движения, в котором ˆH играет роль  гамильтониана примет вид:  iℏdσ  dt = ( ˜ˆH − ˆH)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это уравнение в L-функционалах можно записать в виде:  iℏdL  dt =  �  dβh(−β)e+i ℏ  2 ασβL(α − β) −  �  dβh(β)e+i ℏ  2ασβL(α + β) =  = −  �  dβh(β)(e+i ℏ  2ασβ − e−i ℏ  2ασβ)L(α + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В результате приходим к формуле:  dL  dt = −  �  dβh(β)2 sin(ℏ  2ασβ)  ℏ  L(α + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Из этой формулы ясно, что уравнение движения для L-функционалов имеет пре-  дел когда ℏ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  44  5  Лекция 5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Функциональные интегралы  Настоящая лекция посвящена, прежде всего, широко применяемым в квантовой ме-  ханике функциональным интегралам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду рассказывать об основанном на идее  Березина подходе, который позволяет применять их и в обычном подходе к кванто-  вой механике, и в геометрическом подходе, и в формализме L-функционалов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В квантовой механике физическая величина представляется в виде функциональ-  ного интеграла — бесконечномерного интеграла, где подынтегральное выражение  включает экспоненту от действия, умноженную на что-то.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Действие — это функ-  ционал от кривой (точнее, от функции q(τ), график которой мы рассматриваем как  кривую):  S[q(τ)] =  � t  0  dτL(q(τ), ˙q(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В данном выражении q(τ) стоит в квадратных скобках, чтобы подчеркнуть, что S  — это не функция, а функционал, который зависит не от точки кривой, а от кривой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матричный элемент ⟨q2| ˆU(t)|q1⟩ оператора эволюции в координатном представле-  нии: ˆU(t) = e− it ˆ  H  ℏ , выражается через функциональный интеграл с подынтегралным  выражением:  e  i  ℏ S[q(τ)],  для которого область интегрирования — это кривые q(τ) с заданными начальными и  конечными значениями q(0) = q1, q(t) = q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Что же означает слово “функциональный интеграл”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Интеграл описанного мной  типа можно аппроксимировать конечномерными интегралами: под знаком функци-  онального интеграла заменить обычный интеграл, скажем, на интегральные суммы  и взять предел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Проблема в том, что в отличие от теории из курса обычного инте-  грального исчисления, когда способ приближения несущественен, для функциональ-  ных интегралов это вовсе не так.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кроме того, предел, как правило, не существует: он  обычно или бесконечный, или нулевой, и нужно предпринимать какие-то действия  для того, чтобы извлечь из всего этого конечный ответ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  То, что я собираюсь обсуждать, может быть сделано строгим до некоторого момен-  та.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Строгие вещи можно сделать для гауссовых интегралов, представляющих собой  интегралы от экспонент квадратичных выражений, возможно, умноженных на поли-  ном.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для вычисления таких интегралов в конечномерном случае можно воспользо-  ваться формулой:  �  exp( i  2⟨Ax, x⟩ + i⟨b, x⟩)dx = (det A)− 1  2 exp(− i  2⟨A−1b, b⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (В этой формуле должен стоять постоянный множитель, который мы не пишем, вклю-  чив его в определение интеграла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оператор A может быть комплексным, но нужно  наложить условия, гарантирующие, что интеграл имеет смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Дифференцируя это  выражение по b, мы придем к выводу, что интеграл вида  �  P(x) exp( i  2⟨Ax, x⟩)dx,  где P(x) — некоторый полином может быть выражен через детерминанты, а, по край-  ней мере, для случая эллиптических операторов существует вполне хорошая теория  45  вычисления таких детерминантов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Опять же, такой детерминант нужно чем-то при-  ближать и после этого выбрасывать бесконечные члены разложения для логарифма  детерминанта.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Тем не менее если имеется подынтегральное выражение вида W(x) = Q(x)+gV (x),  где Q — квадратичное выражение по x, а константу g, мы считаем малой, то тогда  нормированный функциональный интеграл вида:  �  eW (x)dx  �  eQ(x)dx  можно представить, в виде суммы фейнмановских диаграмм.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В более общем случае,  когда W(x) представляется как квадратичное по x − x0 выражение плюс кусочек,  который содержит только мономы большей степени по x − x0, теория возмущений  тоже хорошо определена.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  После этих предварительных слов перейдем к вопросу, как можно построить эти  функциональные интегралы?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Их можно строить для более или менее любой физи-  ческой величины.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду строить функциональный интеграл для экспоненты опера-  тора, действующего в банаховом (не обязательно в гильбертовом) пространстве или,  что почти то же, я буду решать уравнение движения:  dσ  dt = H(t)σ(t)  (25)  и искать оператор эволюции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Оператор H(t) в уравнении движения может быть функцией от t, но, для того  чтобы упростить формулы, я буду считать, что H не зависит от t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' это совершенно  несущественно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Давайте теперь то пространство, в котором действует оператор H, обозначим бук-  вой L и определим понятие символа оператора.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы встречались с этим понятием.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я  хочу дать очень общее определение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Символ оператора — это функция, определенная  на каком-то пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду считать, что на пространстве с мерой, потому что  хочу интегрировать, но мне ничего не нужно из теории пространств с мерой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Доста-  точно понимать, что символ оператора — это функция, определенная где-то, где есть  понятие интегрирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Символы будем выделять подчеркиванием снизу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Основные свойства символов  следующие.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Символ равен тождественно единице, если оператор равен единице.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сим-  вол A должен зависеть линейно от оператора A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Произведению операторов C = AB  должна отвечать некоторая операция на символах, которую я обозначу символом ∗,  то есть C = A ∗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я проведу совершенно тривиальные рассуждения, которые сразу приво-  дят к функциональному интегралу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду использовать стандартную формулу для  экспоненты:  σ(t) = etH = lim  N→∞(1 + tH  N )N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если я буду рассматривать экспоненту от символа exp t  N H, то при больших N, в  первом приближении будет справедлива формула:  1 + tH  N = e  t  N H + O(N −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Ошибка будет иметь порядок  1  N2 , но когда N → ∞, то этой поправкой можно прене-  бречь, и в результате получается выражение:  σ(t) = lim  N→∞ IN(t),  46  IN(t) = e  t  N H ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ∗ e  t  N H  (N сомножителей).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мое утверждение заключается в том, что я вывел представление оператора эволю-  ции в качестве функционального интеграла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно только дать примеры символов  и расшифровать, что такое “звездочка”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я хочу подчеркнуть одну вещь, на мой взгляд, важную и физиками недооценен-  ную.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что до сих пор я приводил тривиальные рассуждения, которые могут  быть сделаны строгими.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом деле, не нужно говорить про функциональные ин-  тегралы — просто нужно исследовать функции IN(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно, например, применять  метод стационарной фазы и получать более или менее те же результаты, что и на  языке функциональных интегралов, проводя вполне строгие рассуждения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Позже я введу большой класс операторов, для которых операция ∗ записывается  в простом виде.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот класс операторов включает в себя q-p-символы и виковские  символы, про которые я уже рассказывал.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сейчас я буду обсуждать их в несколько  ином виде.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я считаю, что символ — это функция от двух переменных A(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для  этого большого класса символов выражение для символа произведения выглядит так:  C(α, β) =  �  dγdγ′A(α, γ)B(γ′, β)ec(α,γ)+c(γ′,β)−c(α,β)−r(γ′,γ),  (26)  где c(α, β) и r(α, β) — некоторые функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для q-p-символов это просто скалярные  произведения с точностью до знака:  c(q, p) = r(q, p) = −ipq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (27)  Позже я вернусь к вопросу как получаются такого типа формулы, а пока я считаю,  что есть такая формула для символа произведения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Зная формулу для произведения  двух операторов, можно записать формулу для произведения n операторов A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', AN:  C(α, β) =  �  dγ1dγ′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='dγN−1dγ′  N−1A1(α, γ1)A2(γ′  1, γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='An(γ′  N−1, β)eρN ,  где  ρN = c(α, γ1) + c(γ′  1, γ2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + c(γ′  N−1, β) − c(α, β)) − r(γ′  1, γ1) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' − r(γ′  N−1, γN−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В результате получаем произведение символов, умноженное на экспоненту от неко-  торого выражения, которое обозначено символом ρN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возвращаясь к выражению  IN(t), приближающему оператор эволюции, мы можем представить его в виде:  IN(t) =  �  dγ1dγ′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='dγN−1dγ′  N−1eρN exp( t  N (H(α, γ1) + H(γ′  1, γ2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + H(γ′  N−1, β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  То, что я рассказал — это очень широкая схема.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть один совершенно конкретный  пример — это q-p-символ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хочу заметить, что если рассматривать ядро единичного  оператора в смысле математики (или матрицу единичного оператора, как говорят  физики), то это — δ-функция: в координатном представлении ⟨q2|1|q1⟩ = δ(q1 − q2),  а в импульсном представлении ⟨p2|1|p1⟩ = δ(p1 − p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы же хотим, чтобы символ  единичного оператора был равен единице, а не δ-функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это очень просто сде-  лать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что преобразование Фурье от δ-функции — это константа, поэтому  для того, чтобы получить символ, дающий для единичного оператора единицу, мы  должны просто взять преобразование Фурье и помножить на постоянный множитель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Раз матрица — это δ-функция от аргумента q1 − q2, значит, возьмем преобразование  Фурье матричного элемента ⟨q2|A|q1⟩ по этому аргументу:  Aq−p(q, p) =  �  dy⟨y|A|q⟩eip(q−y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  47  Здесь написано, что q-p-символ (я так назвал этот символ) — это просто преобра-  зование Фурье матричного элемента по разности между аргументами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это действи-  тельно символ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для единицы это единица.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Очевидно, можно взять обратное преоб-  разование Фурье и выразить матричные элементы ⟨p2|A|p1⟩ через q-p-символ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По-  скольку мы знаем, как выразить матричный элемент произведения через матричные  элементы сомножителей, мы можем вычислить, чему равен q-p-символ от произведе-  ния двух операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ответ дается формулой (26), где функции c(α, β) и r(α, β) —  просто скалярные произведения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Определение q-p-символа, которое было только что дано, отличается от опреде-  ления, данного раньше.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это определение применимо к любому оператору, лишь бы  интеграл сходился.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же A — это дифференциальный оператор, то легко понять,  что определение q-p-символа, которое я сейчас дал с помощью интеграла, соответ-  ствует предыдущему: если имеется дифференциальный оператор с полиномиальными  коэффициентами, то его можно записать как полином от операторов координат qj и  операторов импульсов ˆpj = 1  i  ∂  ∂qj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' После этого операторы координат нужно сдвинуть  налево, а операторы импульсов — направо, после чего нужно снять шляпочки с опе-  раторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получится из операторного выражения полиномиальная функция, которую  я называл q-p-символом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Отмечу, что я считал, что постоянная Планка ℏ = 1, но иногда удобно ее оставить  в формулах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Перейдя к обычной квантовой механике, мы обнаружим, что формула для IN  принимает вид:  IN(q, p, t) =  �  N−1  �  1  dqαdpα exp(i  N  �  1  pα(qα − qα−1) − it  N  N  �  1  H(pα, qα−1)),  (28)  где pN = p, q0 = qN = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, мы получили оператор эволюции как предел конечнократных ин-  тегралов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  На этом вполне разумно было бы остановиться и просто изучать это представле-  ние, но также можно сказать слова, которые содержат выражение “функциональный  интеграл”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого следует обратить внимание на то, что выражение, которое стоит  в экспоненте — это в точности интегральная сумма для некоторого интеграла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот  интеграл хорошо известен физикам — это функционал действия  S[p(τ), q(τ)] =  � t  0  (p(τ) d  dtq(τ) − H(p(τ), q(τ))dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Все уже очень близко к тому, что я хочу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Давайте подойдем еще ближе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что  было написано – это q-p-символ для оператора эволюции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы уже знаем, что q-p-  символ — это просто Фурье преобразование от матричного элемента.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате  мы можем сказать, что q-p-символ — это функциональный интеграл от экспоненты  eiS[p(τ),q(τ)],  где стоит функционал S от пары функций p(τ), q(τ), удовлетворяющих граничным  условиям  p(0) = p(t) = p, q(0) = q(t) = q,  когда τ изменяется от 0 до t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Далее можно перейти к матричным элементам, сделав  Фурье — преобразование.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате получится такой же интеграл, но с другими  граничными условиями: q(0) = q1, q(t) = q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Чтобы прийти к той формуле, про которую я уже говорил, рассмотрим частный  случай, когда символ H(p, q) — это сумма квадратичной функции от p (кинетиче-  ской энергии) и некоторой функции V (q) (потенциальной энергии).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда ничего не  48  стоит проинтегрировать по p — это гауссов интеграл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате матричный эле-  мент оператора эволюции может быть представлен как функциональный интеграл с  подынтегральным выражением вида экспоненты от функционала действия  eiS[q(τ)] = ei(  � t  0 dτ(T( ˙q(τ))−V (q(τ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Функционал действия при этом представляет собой интеграл от разности кинети-  ческой и потенциальной энергий.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По p(τ) интегрирование уже проведено, и остаётся  только интеграл по функциям q(τ), удовлетворяющим условиям q(0) = q1, q(t) = q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, я вывел функциональный интеграл, с которого я начал и даже  в более общем виде.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда кинетическая энергия квадратична по импульсам, это  вычисление всегда работает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В приведенных рассуждениях допущена определенная недосказанность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом  деле, в выражении (28) для IN должен стоять постоянный множитель, стремящийся к  нулю при N → ∞ (этот множитель происходит от константы, которую мы выбросили,  когда писали выражение для гауссова интеграла).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть, здесь выброшен нулевой  множитель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это все время делается в функциональных интервалах, потому что на  самом деле хороший объект — это частное от деления функциональных интегралов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я хочу построить большое количество примеров.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти примеры обобщают то, что  Березин называл ковариантными символами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я возьму два банаховых пространства L и L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно взять два сопряженных  банаховых пространства, но это не обязательно — важно только чтобы между ними  было невырожденное скалярное произведение (спаривание).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку я хочу, чтобы  частным случаем было гильбертово пространство, где скалярное произведение ⟨l, l′⟩  линейно по одному аргументу l ∈ L и антилинейно по другому аргументу l′ ∈ L′, я и  здесь такое потребую.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Итак, у меня есть два банаховых пространства, которые почти двойственны друг  другу в том смысле, что между ними есть невырожденное скалярное произведение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  После этого я возьму две системы векторов eα ∈ L, где α ∈ M, и e′  β ∈ L′, где β ∈ M′, в  этих пространствах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они не должны быть линейно независимы — это не базисы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это,  как говорят, переполненные системы векторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мне важно только то, чтобы любой  вектор можно было выразить через эти векторы как предел линейных комбинаций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Примером такой переполненной системы являются пуассоновы векторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Теперь я потребую, чтобы, как говорят физики, можно было вставить единицу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это означает, что если я хочу рассмотреть скалярное произведение ⟨l, l′⟩, то нужно  взять скалярные произведения ⟨l, e′  µ⟩ и ⟨eλ, l′⟩ и из этих скалярных произведений  с помощью некоторого интегрирования создать общее скалярное произведение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  всегда можно сделать и даже разными способами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду считать, что такой способ  зафиксирован:  ⟨l, l′⟩ =  �  ⟨l, e′  µ⟩⟨eλ, l′⟩e−r(λ,µ)dλdµ,  (29)  где r(λ, µ) — функция на пространстве с мерой M × M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  После этого я хочу определить ковариантный символ A(α, β) оператора A, дей-  ствующего в L (можно рассмотреть L′ с тем же успехом).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот символ определяется  простой формулой:  A(α, β) =  ⟨Aeα, e′  β⟩  ⟨eα, e′  β⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мое основное условие выполнено — символ единичного оператора равен едини-  це.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко посчитать символ C произведения операторов C = AB, воспользовавшись  49  соотношением: ⟨ABeα, e′  β⟩ = ⟨Beα, A∗e′  β⟩, и формулой (29) для ⟨l, l′⟩, где l = Beα,  l′ = A∗e′  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Введя обозначения ⟨eα, e′  β⟩ = ec(α,β), получаем следующее выражение:  C(α, β) =  �  dλdµB(α, µ)A(λ, β) exp(−r(λ, µ) − c(α, β) + c(α, µ) + c(λ, β));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  оно совпадает с формулой (26) для символа произведения, которую я постулировал  ранее.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обращаю внимание на то, что моя конструкция невероятно общая.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Векторы eα  и e′  β могли быть выбраны практически произвольным образом, лишь бы их было  достаточно много для того, чтобы это была переполненная система.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом можно построить громадное количество конкретных примеров.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Важно лишь, чтобы были достаточно простые выражения для скалярного произве-  дения ⟨eα, e′  β⟩ и для функции r(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если L = L′ — гильбертово пространство фоковского представления алгебры Вей-  ля, то можно взять в качестве eα пуассоновы векторы eα = eαˆa∗|0⟩, которые мы уже  рассматривали.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае c(α, β) = r(α, β) = ⟨α, β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это просто вычислить, по-  тому что все интегралы гауссовы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Замечу, что помимо гильбертова пространства, соответствующего случаю обыч-  ной квантовой механики, можно рассматривать и банаховы пространства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например,  взять в качестве L и L′ двойственные друг к другу пространства Lp и Lq = (Lp)∗, где  1  p + 1  q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  L-функционалы и функциональные интегралы  Перейдем теперь к L-функционалам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для L-функционала можно дать два определения, потому что для алгебры Вей-  ля было дано два определения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Одно из определений алгебры Вейля состояло в том,  что есть операторы ak и a+  k , которые подчиняются каноническим коммутационным  соотношениям, и это была алгебра полиномов этих операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В другом определе-  нии я рассматривал в качестве основных элементов унитарные операторы, которые  выражены как экспоненты линейных выражений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В этом определении красивые формулы будут получаться, если взять:  в качестве L′ — алгебру Вейля с операторами e′  α = Vα, представляющими собой  экспоненты линейных выражений;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  в качестве L — линейные функционалы на алгебре Вейля, а eβ — экспоненты  линейных выражений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Другой вариант определения L-функционала:  LK(α∗, α) = Tre−αa+eα∗aK,  (30)  где αa+ = � αka+  k и α∗a = � α∗  kak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В первом определении был след от экспоненты линейного выражения, а здесь —  след от произведения экспонент.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта разница невелика: они отличаются численным  множителем, но при таком определении понятие L-функционала я могу сказать, что  L-функционал — это просто производящий функционал для корреляционных функ-  ций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно разложить по α и α∗, после чего в пределе α = α∗ = 0 будет получен  след от произведения некоторого полинома по a и a+ на матрицу плотности K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  как раз и есть корреляционная функция (по определению).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Что можно еще сделать?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно считать индекс k не дискретным, а непрерывным  параметром, когда канонические коммутационные соотношения принимают вид:  [a(k), a+(k′)] = ℏδ(k, k′),  50  [a(k), a(k′)] = [a+(k), a+(k′)] = 0,  где k, k′ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того чтобы объединить дискретный случай с непрерывным в од-  ном выражении, будем считать, что M — пространство с мерой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В дискретном случае  здесь стоят сумма и δ-символ Кронекера, а в случае, когда индексы k, k′ непрерыв-  ны — интеграл и δ-функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае когда α — это квадратично интегрируемая  функция, выражение для LK хорошо определено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если бы была экспонента от ли-  нейного выражения (не произведение экспонент, а экспонента от суммы), то тогда все  будет конечно, потому что оператор унитарен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если стоит произведение экспонент, но  функция α интегрируема с квадратом, то будет тоже все хорошо, потому что просто  появится дополнительный конечный множитель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Перейдем к вопросу о действии алгебры Вейля A на пространстве L-функционалов  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Зададимся вопросом, что такое L-функционал?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это, по существу, положительный  линейный функционал на алгебре Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Как я уже говорил, если рассматривать  просто произвольный линейный функционал на любой алгебре, то всякий элемент  алгебры порождает два оператора на пространстве функционалов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Один из них по-  лучается, если аргумент умножить слева на что-то, а другой получится если умно-  жить справа на что-то.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда множитель ставится слева, берется еще и операция  сопряжения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, из каждого элемента алгебры получается два операто-  ра на функционалах: первый из них был обозначен тем же символом, как и элемент  алгебры, а другой — этим же символом, но с тильдой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Действие алгебры Вейля A на пространстве L-функционалов L реализуется опера-  торами b и b+, действие которых на функционалы LK сводится к умножению матрицы  плотности на операторы a+ и a справа:  b(k)LK = LKa+(k),  b+(k)LK = LKa(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Нетрудно проверить, что эти операторы удовлетворяют каноническим коммутацион-  ным соотношениям и могут быть представлены в следующем виде  b+(k) = −ℏc+  2 (k) + c1(k),  b(k) = −c2(k),  где c+  i (k) — операторы умножения на α∗  k для i = 1 и на αk для i = 2, а ci(k) —  производные, взятые, соответственно, по α∗  k и αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Альтернативное действие A на L реализуют операторы с тильдой, действие кото-  рых на функционалы LK сводится к умножению матрицы плотности на операторы a  и a+ слева:  ˜b(k)LK = La(k)K,  ˜b+(k)LK = La+(k)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Эти операторы также удовлетворяют каноническим коммутационным соотношениям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Они могут быть представлены в виде:  ˜b+(k) = ℏc+  1 (k) − c2(k),  ˜b(k) = c1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, на L-функционалах действует два экземпляра алгебры Вейля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Происходит то, что физики называют удвоением полей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Рассмотрим теперь формальный гамильтониан ˆH  ˆH =  �  m,n  �  ki,lj  Hm,n(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='km|l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ln)a+  k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+  kmal1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='aln,  (31)  выраженный через операторы рождения и уничтожения и приведенный к нормаль-  ной форме (то есть, все операторы рождения перемещены налево).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что в  теории, которую мы рассматриваем в алгебраическом подходе, формальные гамиль-  тонианы могут не иметь смысла как операторы, но соответствующие уравнения дви-  жения могут иметь смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Гамильтониану ˆH отвечают два формальных оператора,  51  действующего на L-функционалах (точнее говоря, на всех линейных функционалах  на алгебре Вейля):  ˆH =  �  m,n  �  ki,lj  Hm,n(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='km|l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ln)b+  k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='b+  kmbl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='bln,  (32)  ˜H =  �  m,n  �  ki,lj  Hm,n(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='km|l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ln)˜b+  k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='˜b+  km˜bl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='˜bln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (33)  Один из них обозначен тем же символом, другой — символом с тильдой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' И теперь  можно записать уравнение движения для L-функционала: L(α∗, α):  iℏdL  dt = HL = ˜HL − ˆHL,  (34)  где введено обозначение H = ˜H − ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Еще следует заметить, что если рассматривать трансляционно инвариантный га-  мильтониан, то в импульсном представлении коэффициенты Hm,n будут содержать  δ-функции δ(k1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + km − l1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' − ln), задающие закон сохранения импульса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Уравнения для L-функционала имеют то преимущество, что они имеют смысл да-  же в том случае, когда уравнение движения в фоковском пространстве плохо опреде-  лено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что в фоковском пространстве обычно все хорошо, когда мы живем  в конечном объеме, а когда мы живем в бесконечном объеме, то все плохо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тот эф-  фект неэквивалентности различных представлений канонических коммутационных  соотношений, который обсуждался, всегда проявляется, а для L-функционалов, если  нет ультрафиолетовых расходимостей, то все в порядке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Возвращаясь к выписанным ранее выражениям для ˆH и ˜H, можно сказать, что  мы оказываемся в абсолютно привычной обстановке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Действительно, имеется пред-  ставление двух алгебр Вейля (можно считать, что это одна алгебра Вейля, только  большая).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В курсе квантовой теории поля обычно все делается по теории возмуще-  ний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Берется оператор эволюции в представлении взаимодействия и выписывается  T-экспонента в качестве оператора эволюции в представлении взаимодействия и из  этого выводится диаграммная техника.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае L-функционалов ничего принципи-  ально не изменилось.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Существуют те же самые коммутационные соотношения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно  применять ровно ту же технику.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Единственное, что изменилось — удвоилось количество полей и не стало гиль-  бертова пространства, но то, что мы жили в гильбертовом пространстве, нигде не  использовалось и поэтому в формализме L-функционалов вся стандартная техника  из курса квантовой теории поля работает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Та техника функциональных интегралов,  о которой я рассказал, тоже работает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, формализм L-функционалов  с точки зрения вычисления ничем не хуже, чем обычный.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом деле, он лучше.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как я объяснял, он уничтожает проблемы с тривиальными объемными расходимостя-  ми.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Также он существенно лучше, если рассматривать адиабатическое приближение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Сейчас мы в этом убедимся.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Давайте рассмотрим семейство “гамильтонианов”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что я говорю, относится не  только к L-функционалам, а вообще к любой ситуации в геометрическом подходе к  квантовой теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например, рассмотрим семейство, отвечающее теории возмущений  H(g) = H0+gV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я утверждаю, что если рассматривать выражение ω(g(t)), являюще-  еся стационарным состоянием для “гамильтониана” H(g(t)), то тогда ω(g(t)) является  решением уравнения движения для нестационарного “гамильтониана”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Действительно, при рассмотрении адиабатического приближения в случае обыч-  ной квантовой механики стационарное состояние оставалось стационарным с тече-  нием времени, но поскольку мы жили в гильбертовом пространстве, состояние было  52  определено только до численного множителя, и на собственном векторе нарастал  численный (фазовый) множитель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь же этого нет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этому есть совершенно триви-  альная причина — просто можно пренебречь производной по времени ˙g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это дает  возможность написать следующую формулу:  ω(g) = lim  α→0 σα(0, −∞)ω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (35)  Чтобы ее получить я рассматриваю гамильтониан H0 + ge−α|t|V , в котором введен  адиабатический множитель, исчезающий на бесконечности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда я могу сказать, что  если у меня есть какой-то начальный “гамильтониан” H0 (обычно берется свободный,  но это не обязательно) и имеется стационарное состояние для этого “гамильтониана”,  то формула (35) описывает стационарное состояние “гамильтониана” с константой g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Такая же формула есть в обычной квантовой механике, только там стоит фазовый  множитель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В обычной квантовой механике данная формула обычно применяется ко-  гда ω(0) является основным состоянием и тогда ω(g) будет основным состоянием для  константы связи g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В нашем подходе можно применять эту формулу в несравненно  более общей ситуации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например, если гамильтониан трансляционно инвариантен,  можно применять эту формулу к любому трансляционно инвариантному стационар-  ному состоянию свободного гамильтониана H0, для которого все такие состояния лег-  ко вычислить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В результате мы получаем стационарное трансляционно инвариантное состояние  в случае, когда константа связи равна g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такой подход очень естественен как в равно-  весной, так и в неравновесной статистической физике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно взять в качестве ω(g),  скажем, равновесное состояние для какой-то температуры и, применив этот процесс,  получить равновесное состояние, правда, при другой температуре, но с той же эн-  тропией, потому что адиабатический процесс не меняет энтропии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В неравновесной  статистической физике есть формализм Келдыша, тесно связанный с формализмом  L-функционалов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В нем, по существу, применяется эта же формула, но, может быть,  с несколько меньшим основанием.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Наряду с оператором эволюции σα(0, −∞) можно при заданном α рассматривать  оператор эволюции σα(+∞, −∞), представляющий собой адиабатическую S-матрицу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если адиабатический параметр α стремиться к нулю, в формализме L-функционалов  адиабатическая S-матрица, умноженная на некоторые множители (которые я не хочу  здесь описывать, но это множители, которые связаны с уровнями энергии) стремится  к инклюзивной матрице рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Инклюзивная матрица рассеяния — это основа  для расчета инклюзивного сечения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот вопрос мы обсудим позже.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я только хо-  чу сказать, что обычная матрица рассеяния в квантовой механике тоже может быть  получена из адиабатического оператора эволюции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Голые частицы при этом превра-  щается в одетые, происходит перенормировка волновой функции, но не происходит  перенормировки константы связи.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Сейчас я хочу объяснить некие вещи, к которым я потом вернусь с другой точки  зрения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Хорошо известно (я буду доказывать это через некоторое время), что матрица  рассеяния выражается через функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это то, что называется формулой Ле-  мана — Симанчика — Циммермана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но если я хочу рассмотреть инклюзивную матри-  цу рассеяния, я должен рассмотреть то, что можно называть обобщенной функцией  Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обычная функция Грина определяется как хронологическое произведение  M = T(B∗  1(x1, t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' B∗  n(xn, tn)) = T(B∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  усредненное по какому-то состоянию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В хронологическом произведении все времена  53  идут в порядке убывания слева направо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно точно так же рассмотреть и анти-  хронологическое произведение (времена возрастают)  N = T opp(B1(x′  1, t′  1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Bn(x′  n, t′  n)) = T opp(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В обобщенных функциях Грина берется хронологическое произведение операто-  ров, умножается на антихронологическое произведение операторов и затем берется  среднее по какому-то состоянию:  Это обобщенная функция Грина в данном состоянии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Именно эти функции Грина  появляются в формализме Келдыша, и, как сейчас будет объяснено, они же появля-  ются в формализме L-функционалов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Напомню, что в L-функционалах есть операторы без тильды и операторы с тиль-  дой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Справедлива формула  (T( ˜BB)ω)(x) = ω(T(B∗)xT opp(B)) = ω(MxN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (36)  В левой части здесь стоит обычное хронологическое произведение, однако операторы  без тильды действуют, умножая справа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этом меняется порядок и хронологиче-  ское произведение превращается в антихронологическое.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Заметим, что формула (36) справедлива не только для L-функционалов (линей-  ных функционалов на алгебре Вейля), но и для линейных функционалов на любой  ассоциативной алгебре с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если в выписанном выражении положить x = 1, то в точности получится ω(MN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь M определено как хронологическое произведение, а N — как антихронологи-  ческое произведение, так что T-произведение на L-функционалах переходит в хро-  нологическое и антихронологическое произведения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Рассматривая эту картинку в  L-функционалах, я могу применять для вычисления обобщенных функций Грина  технику вычисления обычных функций Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это дает диаграммную технику для  обобщенных функций Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  54  6  Лекция 6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Солитоны как аналоги частиц  Я собираюсь дать определение квантовых частиц и квазичастиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Основное утвержде-  ние состоит в том, что понятие частицы вторично.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я определяю частицу как элемен-  тарное возбуждение основного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно также рассматривать элементарное  возбуждение любого стационарного трансляционно инвариантного состояния, тогда  элементарное возбуждение — это квазичастица.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Прежде всего, я хочу рассказать о классическом аналоге всего этого.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это понятия  солитона и обобщенного солитона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Рассмотрим трансляционно инвариантный гамиль-  тониан в бесконечномерном фазовом пространстве, которое состоит из векторнознач-  ных функций f(x), где x ∈ Rd — это пространственные координаты.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я считаю, что  пространственные трансляции действуют как сдвиги этих координат, а временные  определяются гамильтонианом, который коммутирует с пространственными транс-  ляциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Предположим, что соответствующее уравнение движения имеет вид  ∂f  ∂t = Af + B(f),  где A — линейный оператор и B — нелинейная часть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Считаем, что нелинейная часть  как минимум квадратична;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' тогда для малых f доминирует линейная часть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В част-  ности, мы можем сказать, что f ≡ 0 — это решение, и в его окрестности можно  пренебречь нелинейной частью.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь мы определяем солитон как решение, которое имеет вид s(x − vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Соли-  тон — это такой горбик.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы можем изображать решение f ≡ 0 как горизонтальную  прямую, тогда солитон (уединенная волна) — это решение, где этот горбик двигает-  ся равномерно без изменения формы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть еще понятие обобщенного солитона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  горбик, который двигается, с поостоянной средней скоростью но одновременно может  пульсировать, менять свою форму.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я не буду говорить об этом понятии подробно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я требую, чтобы солитон имел конечную энергию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для трансляционно инвариант-  ного решения понятие энергии бессмысленно — для него можно говорить о плотности  энергии, но я буду считать его энергию равной нулю и отсчитывать энергию солитона  от энергии трансляционно инвариантного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что энергия конечна, озна-  чает, грубо говоря, что солитон более или менее сосредоточен в какой-то конечной  области.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть гипотеза, которая была высказана в моей работе с Фатеевым и Тюпкиным  лет сорок пять тому назад.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Она состоит в том, что для очень многих систем и по-  чти для всех начальных условий с конечной энергией при временах стремящихся к  плюс или минус бесконечности мы получаем несколько солитонов и еще нечто, что  удовлетворяет линейному уравнению, по крайней мере, приблизительно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это хорошо  известно для интегрируемых систем в случае d = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' мы предположили, что это вер-  но без предположения интегрируемости в любой размерности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Не думаю, что нашу  работу прочел кто-нибудь из математиков, но эта гипотеза также была высказана в  других работах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В работе Соффера это называется “grand conjecture”, в работе Тао –  “soliton resolution conjecture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тем не менее гипотеза так и остается гипотезой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пока  что это остается вне пределов существующей математики за исключением случая,  когда солитоны отсутствуют изначально.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае можно предполагать, что в  конце мы асимптотически получим решение линейного уравнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это утверждение можно обосновать следующим образом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пусть имеются какие-  то отличные от нуля начальные условия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае происходит обычно то, что в  старых учебниках квантовой механики называлось расплыванием волнового пакета.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  55  То есть если начальные данные были где-то сосредоточены, то дальше они расплы-  ваются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этом энергия все-таки сохраняется, поэтому это расплывание приводит  к тому, что амплитуда волны уменьшается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же она действительно все время  уменьшается, то, как я говорил, в случае, малых амплитуд нелинейной частью можно  пренебречь и решение нелинейного уравнения приближается к решению линейного.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Конечно, это не происходит, если в теории есть солитон.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Амплитуда солитона по  определению не меняется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Какой высоты был горбик, таким он и остается, но можно  думать, что в конце концов останутся какие-то солитоны или обобщенные солито-  ны, а также хвостик, который приблизительно удовлетворяет линейному уравнению.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Конечно, никаким доказательством эти рассуждения не являются, но это очень ве-  роятная гипотеза.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нет сомнения, что это верно не всегда, но естественно думать, что  это верно в очень многих случаях.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Раз есть такая картинка, которую я описал, нужно думать, что есть понятие рассе-  яния солитонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть на свете точно решаемые модели размерности 1+1 (одномерное  пространство и время);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' для таких моделей мое утверждение — это теорема.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Что про-  исходит?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сталкиваются два таких горбика, образуется некая каша, а потом снова воз-  никают те же самые солитоны: это специфика интегрируемых случаев.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Стандартная  ситуация в неинтегрируемых случаях чуточку другая: после столкновения возника-  ют, возможно, другие солитоны и еще, может быть, то, что я назвал словом хвостик.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Хвостик — это образование, которое асимптотически стремится к решению линейного  уравнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я попытаюсь облечь эти общие рассуждения в некую форму.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обозначим  пространство возможных начальных данных буквой R, и тогда наложенное условие  означает для общего случая, что, задав какие-то начальные условия, в конце мы полу-  чим солитоны и хвостик.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Солитоны характеризуются какими-то данными, а хвостик  — решением линейного уравнения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Формально на математическом языке сформули-  рованная гипотеза означает, что на плотном множестве начальных данных можно  определить отображение D+(t) : R → Ras начальных данных в асимптотические  данные на плюс бесконечности (t → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я мог бы взять и минус бесконечность:  D−(t) : R → Ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Единственное отличие в том, что уже нельзя будет говорить слово  “начальные данные”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я предполагаю, что есть и обратное отображение, то есть исходя из асимп-  тотики можно найти начальные условия или, по крайней мере, доказать, что заданная  асимптотика получается из каких-то начальных условий.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть, я хочу рассмотреть  обратные операторы S(t, +∞) = (D+(t))−1 и S(t, −∞) = (D−(t))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь возникает задача : определить из асимптотических данных решения уравне-  ния и тем самым начальные данные.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это, по-видимому, нетрудно, но никто этого не  сделал.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я спросил у Тао, известно ли это?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Он сказал, что да, наверное, это нетрудно  сделать, но на тему о том, что кто-нибудь это сделал, он мне ничего не написал.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я  думаю, что это интересная и не очень трудная задача: по асимптотическим данным  построить решение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В квантовом случае решение этой задачи хорошо известно – это  то, что называется теорией рассеяния Хаага — Рюэля;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' я буду рассказывать обобщение  этой теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я могу определить то, что следует назвать нелинейной матрицей рассеяния:  S = S(0, +∞)−1S(0, −∞) : Ras → Ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Грубо говоря, мы задаем начальные условия на минус бесконечности и смотрим  асимптотику на плюс бесконечности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно надеяться, что есть предельный переход  от квантовой матрицы рассеяния к этой нелинейной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отмечу еще раз, что приведен-  ные рассуждения — это вещи гипотетические.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Еще одно маленькое замечание заключается в том, что если теория лоренц-инвариантна,  то можно применить преобразование Лоренца к солитону и снова получится солитон.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  56  Солитоны ходят семействами — солитоны с разными скоростями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То же верно и для  преобразований Галилея.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Завершая обсуждение, хочу сказать, что классический со-  литон нужно рассматривать как модель квантовой частицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В квантовой механике  понятие частицы — это асимптотическое понятие: если две частицы сталкиваются,  образуется некая каша, которая потом распадается на частицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Следующие рассуждения еще сильнее подчеркивают аналогию солитонов с кван-  товыми частицами в том определении, которое я буду давать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я рассматривал про-  странство всех начальных данных R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это симплектическое многообразие.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Рассмот-  рим ситуацию, когда имеются фазовое пространство и гамильтониан, то есть, име-  ется симплектическое многообразие и оператор эволюции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Также предположим, что  есть пространственные трансляции, а временные трансляции описываются трансля-  ционно инвариантным гамильтонианом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Формально математически это означает, что  на этом симплектическом многообразии M действует коммутативная группа T про-  странственных и временных трансляций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь возьмем стационарную трансляцион-  но инвариантную точку m ∈ M этого симплектического многообразия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В предыдущей  картинке такой точкой было решение f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это решение трансляционно инвариантно  и стационарно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Определим возбуждение трансляционно инвариантного стационарного состояния  как состояние с конечной энергией (напомню, что мы считаем энергию трансляционно  инвариантного состояния равной нулю).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я определю элементарное симплектическое многообразие как такое мно-  гообразие, в котором в координатах Дарбу p, x пространственные сдвиги действует  просто как сдвиги x → x + a, при этом p остается неизменным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы считаем, что  гамильтониан инвариантен относительно этих трансляций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что он за-  висит только от p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обозначим его как ǫ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда временные сдвиги будут просто  сдвигами с определенной скоростью x → x+v(p)t, где скорость v рассчитывается по  формуле v(p) = ∇ǫ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предположим теперь, что M представляет собой пространство вектор-функций  f(x), где x ∈ Rd, а пространственные трансляции действуют как сдвиги x → x+a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ес-  ли взять симплектическое вложение элементарного симплектического пространства  во множество возбуждений в M и потребовать, чтобы вложение коммутировало с  пространственно-временными трансляциями, то возникнет семейство солитонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  очень просто объяснить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При вложении точка (p, 0) переходит в некоторую функцию  sp(x), зависящую от p, а поскольку вложение E → M коммутирует с трансляциями,  то точка (p, a) будет переходить в сдвинутую функцию sp(x + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Условие того, что  отображение E → M коммутирует с временными сдвигами, как раз и будет означать,  что функция sp(x − v(p)t) удовлетворяет уравнению движения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Частицы и квазичастицы  Теперь введем понятие частицы и более общее понятие квазичастицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отличие только  в том, что частица — это возбуждение основного состояния, а квазичастица — это  возбуждение любого трансляционно инвариантного стационарного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для  того, чтобы определить понятие частицы, мне нужны понятия пространственных и  временных сдвигов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В обычной квантовой механике если мы рассматриваем эволюцию, нужно иметь  понятие временного сдвига Tτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хочу, чтобы были еще пространственные сдвиги  Ta, которые действовали бы на состояния и коммутировали с временными сдвигами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе пространство состояний это основная вещь, но здесь мне  удобно рассматривать ненормированные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что в алгебраическом  подходе состояниями были положительные функционалы, нормированные условием,  57  что функционал от единицы равен единице.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это условие нормировки я отброшу, и  в таком случае получается конус, которые я обозначаю буквой C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Состояние теперь  определено только с точностью до численного множителя.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я все время буду говорить  про этот конус ненормированных состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пространственно-временные трансляции  должны действовать на этом конусе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обозначим теперь группу пространственно-временных трансляций как T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В алгеб-  раическом подходе эта группа должна действовать на самой алгебре A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Симметрия в  алгебраическом подходе — это автоморфизм алгебры, то есть имеется гомоморфизм  группы T в группу автоморфизмов алгебры A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Группа автоморфизмов алгебры (и,  следовательно, группа T ) действует на C (напомню, что мы всегда считаем, что ав-  томорфизмы согласованы с инволюцией).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я ввожу стандартное обозначение, A(τ, x) = TτTxA для элемента A ∈ A сдвинуто-  го по времени и пространству.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Трансляционно-инвариантное стационарное состояние  ω в алгебраическом подходе удовлетворяет условию ω(A(τ, x)) = ω(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В частности, давайте рассмотрим алгебру Вейля A в координатном представ-  лении, когда заданы генераторы ˆa∗(x), ˆa(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем считать, что пространственные  сдвиги просто сдвигают аргумент, а временные сдвиги определяются формальным  гамильтонианом, который выражен через ˆa∗(x), ˆa(x) с какими-то коэффициентными  функциями, зависящими только от разностей xi−xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тем самым обеспечена трансля-  ционная инвариантность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Еще я потребую, чтобы коэффициентные функции быстро  убывали.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда уравнение движения будет иметь смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я могу сделать преобразование Фурье и перейти к импульсному представлению.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Тогда аргументом будет k и пространственные сдвиги сведутся к умножению на экс-  поненты вида exp(±ika), а временные сдвиги будут опять определяться гамильто-  нианом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Условие, что функции в координатном представлении зависят от разности,  приводит к появлению δ-функций, отвечающих сохранению импульса, а требование,  чтобы коэффициентные функции быстро убывали, означает, что функции в импульс-  ном представлении будут гладкими (после выделения δ-функции).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В общем случае, когда задан конус состояний, я могу определить понятие трансля-  ционно инвариантного стационарного состояния как состояния, которое не меняется  ни при пространственных, ни при временных сдвигах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это будет для меня основной  объект.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Стандартные примеры такого состояния — основное состояние и равновесное  состояние.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я хочу определить понятие возбуждения трансляционно инвариантного  стационарного состояния как аналог введенного ранее понятия состояния с конечной  энергией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда солитон уходит на бесконечность, мы его перестаем видеть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Говоря  формально, если имеется возбуждение σ трансляционно инвариантного состояния  ω ∈ C и к нему применяется пространственный сдвиг, то в пределе a → ∞ остается  одно ω ∈ C:  (Taσ)(A) → const · ω(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Тут еще написана константа, потому, что состояние в конусе определено только с точ-  ностью до численного множителя — там ненормированные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такое понятие  возбуждения — это общее понятие, которое может быть применено как в геометри-  ческом, так и в алгебраическом подходах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае солитонов аналогом этого было  то, что солитон имеет конечную энергию и тем самым более или менее сосредоточен  в конечной области.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В алгебраическом подходе по трансляционно инвариантному стационарному со-  стоянию можно построить с помощью конструкции GNS предгильбертово простран-  ство H (я хочу жить в предгильбертовом пространстве).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом пространстве есть  циклический вектор θ, соответствующий состоянию ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, это означает, что ω  58  — это положительный функционал  ω(A) = ⟨ ˆAθ, θ⟩,  где ˆA — оператор, который отвечает A в H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Пространственные и временные трансляции Ta и Tτ спускаются на предгильберто-  во пространство H как унитарные операторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Трансляции действуют в самой алгебре  как автоморфизмы алгебры, а мы строили предгильбертово пространство факторизуя  алгебру неким образом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это позволяет спустить эти операторы на H как унитарные  операторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дальше мы определяем операторы энергии и импульса как операторы  инфинитезимальных трансляций по времени и по пространству:  Ta = ei ˆPa,  Tτ = e−i ˆ  Hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В алгебраическом подходе элементы предгильбертова пространства H можно отож-  дествить с возбуждениями состояния ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Физический смысл GNS – конструкции состо-  ит в том, что она по некоторому трансляционно инвариантному состоянию позволяет  построить пространство H, в котором живут возбуждения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это, собственно говоря,  объяснение, почему эта конструкция так важна в физике.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  То, что я утверждал, не всегда верно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мне нужно потребовать, чтобы было то,  что по-английски называется cluster property, а по-русски — распадение корреляций  или кластеризация.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Представим себе ферромагнетик.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если спин имеет какое-то направление в начале  координат, то такое же направление спина будет везде, по крайней мере, статистиче-  ски.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это тот случай, когда нет распадения корреляций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В более стандартной ситуации  на больших расстояниях спин уже не помнит, каков был спин в начале координат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вот,  это и есть то, что называется кластеризацией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Формально математически это можно сформулировать следующим образом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если  взять ω(A(τ, x)B), где A, B — два элемента алгебры, и один из этих элементов сдви-  гать по пространству, оставляя время постоянным, то тогда произойдет распадение  этого среднего: limx→∞ ω(A(τ, x)B) = ω(A)ω(B)  Это простейшая форма распадения корреляций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Позже я сформулирую это более  формально и в более общем виде.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для меня важно в данный момент еще то, что когда  есть три оператора B′, A и B и один из этих операторов сдвигается на бесконечность,  то произойдет распадение:  lim  x→∞ ω(B′A(τ, x)B) = ω(A)ω(B′B)  Эти равенства можно продифференцировать:  lim  x→∞  d  dτ ω(A(τ, x)B) = d  dτ (ω(A))ω(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Любой вектор из H может быть представлен в виде Bθ ∈ H и тогда для соответ-  ствующего этому вектору состояния σ(A) справедлива формула:  σ(A) = ⟨ ˆA ˆBθ, ˆBθ⟩ = ω(B∗AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (37)  Теперь я буду сдвигать аргумент у A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда A будет уходить на бесконечность,  произойдет кластеризация, и в пределе x → ∞ получится следующее выражение:  (Txσ)(A) = σ(A(0, x) = ω(B∗A(0, x)B) → ω(A)ω(B∗B),  При этом ω(B∗B) можно считать константой, то есть, σ(A) является возбуждени-  ем в том смысле, в котором я его определил, и поэтому в алгебраическом подходе  59  все элементы предгильбертова пространства H дают возбуждения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В алгебраическом  подходе я только такие возбуждения и буду рассматривать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На самом деле, я мог  бы начать с этого момента — определить понятие возбуждения таким образом: взять  ω, применить конструкцию GNS и взять элементы предгильбертова пространства H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это будут возбуждения в алгебраическом подходе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я сейчас определю понятие элементарного возбуждения трансляционно инвари-  антного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Элементарное возбуждение — это то, что называется частицей  в квантовой теории поля, потому что там рассматривается возбуждение основного  состояния, а квазичастицы — это элементарные возбуждения любого трансляцион-  но инвариантного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку я буду рассматривать оба случая, то я буду  говорить про элементарные возбуждение, но я могу также употреблять термины “ча-  стица” или “квазичастица”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как понять, что такое элементарное возбуждение в алгебраическом подходе?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В  алгебраическом подходе мы живем в гильбертовом пространстве, которое получается  с помощью GNS-конструкции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хочу понять, что такое частица в этой ситуации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  У частицы есть импульс.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Бывают и другие квантовые числа — они просто будут  появляться как дискретные индексы, которые мне никак не мешают.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я скажу, что у  частицы есть состояние с определенным импульсом и обозначу его как Φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Импульс  этого состояния равняется p:  ˆPΦ(p) = pΦ(p)  (38)  а энергия этого состояния – некоторая функция ǫ(p), которая называется законом  дисперсии  ˆHΦ(p) = ǫ(p)Φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (39)  Заметим, что (38), (39) можно переписать в виде  TaΦ(p) = eipaΦ(p),  (40)  TτΦ(p) = e−iǫ(p)τΦ(p)  (41)  Важно отметить, что Φ(p) — это не вектор, а обобщенная векторная функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При  заданном значении импульса состояние является ненормированным, поэтому нужно  рассматривать интеграл этой функции с какой-то пробной функцией φ(p)  Φ(φ) =  �  dpφ(p)Φ(p),  (42)  Это будет хорошо определенный вектор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кроме того, обычно накладывается усло-  вие нормировки на δ-функцию — я его тоже наложу:  ⟨Φ(p), Φ(p′)⟩ = δ(p − p′)  (43)  Если рассматривать настоящие векторы, где аргументом является пробная функция,  то нормировка на δ-функцию будет означать, что скалярное произведение векторов  равняется скалярному произведению соответствующих пробных функций  ⟨Φ(φ), Φ(φ′)⟩ = ⟨φ, φ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (44)  Теперь я хочу определить понятие элементарного пространства h как простран-  ства пробных функций φa(x), принимающих значения в пространстве Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Действие  пространственных трансляций на пробные функции в x-представлении сводится к  простому сдвигу аргумента, а в импульсном представлении — к умножению на экс-  поненту eika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вывести то, как будут вести себя временные сдвиги, можно исходя из  60  требования, чтобы временной сдвиг коммутировал с пространственным сдвигом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  будет просто умножение на экспоненту e−iE(k)τ от некоторой эрмитовой матрицы  E(k) размерности r × r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я могу диагонализовать эту матрицу, тогда будет просто  умножение на скалярные фазовые множители.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это значит, что всегда можно ограни-  читься случаем r = 1,  Элементарное возбуждение трансляционно инвариантного стационарного состо-  яния ω можно определить как коммутирующее с пространственными и временны-  ми трансляциями изометричное отображение σ элементарного пространства h в про-  странство возбуждений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Важно отметить, что для скалярного случая r = 1 не сказано ничего нового.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Действительно, элементарные возбуждения были определены как функции Φ(p), яв-  ляющиеся собственными для импульса и энергии (38, 39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть еще условия норма-  лизации (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я также говорил, что нужно рассматривать пробные функции φ(p),  тогда Φ(φ) — это отображение пространства пробных функций в пространство воз-  буждений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Изометричность этого отображения вытекает из условия нормализации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' А  формулы (38, 39) гарантируют, что трансляции вектора Φ(p) отвечают трансляциям  функции φ(p) и по пространству, и по времени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Таким образом, в алгебраическом  подходе при r = 1 можно просто положить σ(φ) = Φ(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе я должен рассматривать в качестве пространств состо-  яний конусы — там никакой надежды на линейность быть не может.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если мы исходим  из теории в алгебраическом подходе, то функции φ из элементарного пространства  можно сопоставить состояние  (σ′(φ))(A) = ⟨AΦ(φ), Φ(φ)⟩,  и тогда оказывается, что σ′ — это квадратичное (точнее, эрмитово) отображение эле-  ментарного пространства в конус, которое коммутирует со всеми трансляциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это замечание подсказывает, что в геометрическом подходе нужно определять  элементарное возбуждение как коммутирующее с трансляциями отображение элемен-  тарного пространства в конус (условие эрмитовости необязательно, но часто удобно).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если Φ(φ) в алгебраическом подходе принадлежало пространству H, а θ — цик-  лический вектор в этом пространстве, то Φ(φ) получается применением какого-то  элемента из алгебры к циклическому вектору: Φ(φ) = B(φ)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда можно легко  убедиться в справедливости формулы  σ′(φ) = L(φ)ω,  (45)  где L(φ) = ˜B(φ)B(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Напомню, что в алгебраическом подходе на пространстве функционалов были  определены два вида операторов: один отвечает умножению аргумента слева, дру-  гой отвечает умножению аргумента справа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На один из них я поставил волну.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Существование оператора L(φ), удовлетворяющего соотношению (45), удобно вклю-  чить в определение элементарного возбуждения в геометрическом подходе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я уже говорил, что при построении теории рассеяния важна только трансляцион-  ная инвариантность, но если, скажем, мы имеем дело с лоренц-инвариантной теорией,  естественно думать, что, по крайней мере, вакуумный вектор лоренц-инвариантен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Тогда в пространстве возбуждений действует вся группа Пуанкаре, и элементарное  пространство будет являться представлением этой группы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отмечу, что в локальной  квантовой теории поля лоренц-инвариантные частицы определяются как неприводи-  мые представления группы Пуанкаре.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь тоже нужно требовать, чтобы представ-  ление было неприводимо или, по крайней мере, было суммой неприводимых.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Сделаем теперь следующее замечание.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пусть в нерелятивистской квантовой ме-  ханике задан трансляционно инвариантный гамильтониан.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае нет по-  61  тенциального поля, гамильтониан инвариантен относительно преобразований Гали-  лея, а энергия элементарного возбуждения задается привычной формулой: ǫ(p) =  p2/2m + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если же взять оператор ˆa∗(p) и применить к трансляционно инвариантному фо-  ковскому вакууму |0⟩, то получится элементарное возбуждение фоковского вакуума  Φ(p) = ˆa∗(p)|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это частица, но кроме такой частицы есть еще другие частицы, которые тоже удо-  влетворяют наложенным условиям, например, связанные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Что такое связанное состояние?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Гамильтониан действует на состояния с любым  числом частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возьмем n частиц и отделим движение центра инерции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если рас-  смотреть спектр гамильтониана в этом пространстве, то там могут оказаться норми-  рованные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они называются связанными состояниями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Говоря формально, если не отделять центр инерции и суммарный импульс равен  нулю, в выражении для элементарного возбуждения будет присутствовать δ-функция  от суммы импульсов:  �  dp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='dpnΨ(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='pn)δ(p1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + pn)ˆa∗(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆa∗(pn)|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Пусть теперь это состояние может двигаться.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У него может быть импульс;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' легко  понять, что такое движущееся связанное состояние будет описываться обобщенной  функцией  Φ(p) =  �  dp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='dpnΨ(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='pn)δ(p − p1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' − pn)ˆa∗(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ˆa∗(pn)|0⟩,  которая удовлетворяет всем выписанным условиям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То есть, это элементарное воз-  мущение в моем смысле, и для меня такие связанные состояния ничем не хуже, чем  элементарные возбуждения с n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кстати говоря, в свое время была такая про-  блема: как рассматривать рассеяние составных частиц?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В общую теорию, которую я  буду излагать, это все вполне укладывается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обращаю внимание, что в определении элементарного возбуждения — одноча-  стичного состояния Φ(φ) = B(φ)θ я использовал только понятие пространственных и  временных трансляций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же я хочу определить понятие рассеяния, то мне нужно  определить двухчастичные состояния, а также многочастичные состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того чтобы понять, как разумно определить понятие двухчастичного состо-  яния, я хочу рассмотреть то, как изменяется одночастичное состояние со временем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я рассматриваю одночастичное состояние и смотрю за динамикой пробной функ-  ции из h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напоминаю, что у меня действие оператора эволюции на пробные функции  соответствуют действию оператора эволюции на векторы:  TτΦ(φ) = Φ(Tτφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом при эволюции одночастичного состояния происходит умножение проб-  ной функции из h на матричную экспоненту  (Tτφ)(k) = e−iE(k)τφ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если я считаю для простоты, что пробные функции принимают значения в комплекс-  ных числах, то пробная функция просто умножается на фазовый множитель:  (Tτφ)(k) = e−iǫ(k)τφ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  62  (В более общем случае я просто могу разложить на одномерные собственные про-  странства — все будет то же самое, так что это — не ограничение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Переходя от  импульсного пространства к координатному, я должен сделать преобразование Фу-  рье.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате получается, что сдвиг по времени в координатном пространстве  дается интегралом  (Tτφ)(x) =  �  dkeixk−iǫ(k)τφ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  При больших |τ| фазовый множитель велик.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я могу применить метод стационарной  фазы, и тогда у меня возникнут уравнения:  x  τ = ∇ǫ(k)  (46)  Ясно, что нужно рассматривать только ситуацию, когда φ(k) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я определю множество U как окрестность множества точек x, где выпол-  нено условие (46) с τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' После этого я замечу, что вне множества τU уравнение (46)  не имеет решения, поэтому функция (Tτφ)(x очень мала при x /∈ τU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='Я говорю, что  τU — это существенный носитель волновой функции рассматриваемого состояния в  координатном представлении для больших |τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Более аккуратно я расскажу об этом  в следующей лекции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  63  7  Лекция 7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Одночастичные и многочастичные состояния  В прошлой лекции мы определили элементарное пространство h как пространство  пробных функций φa(x), где пространственные перемещения действуют, сдвигая ар-  гумент.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пробные функции принимают значения в Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для определенности мы пред-  полагаем, что пробные функции принадлежат пространству S гладких быстро убы-  вающих функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В импульсном представлении пространственные сдвиги действуют как умножение  на eika, а временные сдвиги как умножение на e−iE(k)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Это следует из предположе-  ния, что временные сдвиги являются унитарными операторами, коммутирующими с  пространственными перемещениями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Здесь E(k) обозначает эрмитову r ×r матрицу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Диагонализуя матрицу E(k), мы можем свести общий случай к случаю r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Напомню теперь, что такое элементарное возбуждение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Элементарное возбужде-  ние трансляционно инвариантного стационарного состояния ω (квазичастица) зада-  ется отображением σ из h в пространство возбуждений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это отображение должно  коммутировать с трансляциями (и пространственными, и временными).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Нужно еще сказать, что такое пространство возбуждений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В алгебраическом под-  ходе — это предгильбертово пространство H, которое получается с помощью кон-  струкции Гельфанда, Наймарка и Сигала (GNS), где мы исходим из некоторого со-  стояния, являющегося трансляционно инвариантным и по пространственным, и по  временным переменным (то есть, из стационарного трансляционно инвариантного  состояния).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оно представляется циклическим вектором, обозначенным буквой θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обсуждаемое отображение σ дает некоторый вектор σ(φ),который в прошлой лек-  ции был обозначен Φ(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оно является отображением в пространство H, и это значит,  что существует элемент B(φ) из алгебры, применив который к циклическому вектору,  получаем наш вектор  σ(φ) = B(φ)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (47)  Я предполагаю еще, что отображение σ изометрично.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Хочу подчеркнуть, что оператор B(φ) существует, но он отнюдь не единственный,  его нужно как-то выбрать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду налагать на него некоторые условия, которые  позволят мне развить теорию рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В частности, я буду требовать, чтобы он  был линеен по φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Как я объяснял, каждому вектору в пространстве представления  алгебры отвечает состояние.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В данном случае это состояние записывается формулой  (σ′(φ))(A) = ⟨Aσ(φ), σ(φ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если вектор представлен в виде (47), справедлива следующая формула:  σ′(φ) = L(φ)ω,  (48)  где  L(φ) = ˜B(φ)B(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (49)  Итак, в алгебраическом подходе у меня появилось, некоторое отображение σ′ : h → C,  действующее в соответствии с формулой (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе нужно забыть про алгебру, но при этом остается конус  всех состояний C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По определению отображение σ′ элементарного пространства h в ко-  нус C задает элементарное возбуждение если оно коммутирует с пространственными  и временными трансляциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  64  Мы постулируем, что так же, как и в алгебраическом случае, отображение σ′(φ)  получается в соответствии с (48) действием некоторого оператора L(φ) на транс-  ляционно инвариантное стационарное состояние ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Изначально рассматриваемое в  алгебраической ситуации отображение σ(φ) было линейным, а вот отображение σ′(φ)  отнюдь не линейное.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Действительно, из формулы (49) следует, что в алгебраическом  подходе L(φ) — это квадратичное выражение, или, точнее говоря, эрмитово, потому  что здесь по одной переменной имеет место линейность, по другой — антилиней-  ность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Отображение f называется эрмитовым, если его можно представить в виде  f(x) = F(x, x∗), где F(x, y) линейно по первому аргументу и антилинейно по второ-  му.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Естественно потребовать чтобы в геометрическом подходе L(φ) удовлетворяло  тем же условиям.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В дальнейшем я просто буду употреблять слово “квадратичное”  вместо “эрмитово”, но следует понимать, что на самом деле оно не совсем квадратич-  ное.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если кому-то больше нравится работать с линейными отображениями, то это мож-  но сделать с помощью следующей общей алгебраической конструкции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для каждо-  го линейного комплексного пространства E в тензорном произведении E ⊗ ¯E этого  пространства на комплексное сопряженное можно сконструировать конус C(E) как  минимальный конус, содержащий все элементы вида e ⊗ ¯e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Черта здесь обозначает  комплексное сопряжение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В дальнейшем мы будем иметь дело только с комплексными  пространствами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Конус C(h) я называю элементарным конусом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Он отвечает элементарному про-  странству h, при этом σ′ можно рассматривать как линейное отображение σ′ : C(h) →  C элементарного конуса в конус состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Чтобы упростить обозначения, я сначала рассмотрю случай, когда элементарное  пространство состоит из скалярных функций ( r = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предположим, что носитель supp(φ) функции φ(p) в импульсном пространстве  является компактным множеством.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае можно найти ограниченное мно-  жество Uφ для которого все точки вида ∇ǫ(p), где p принадлежит supp(φ), являются  внутренними точками (функция ǫ(p) предполагается гладкой).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Тогда при больших |τ| и x  τ /∈ Uφ в координатном пространстве для функции  (Tτφ)(x) =  �  dkeikx−iǫ(k)τφ(k)  справедливо соотношение  |(Tτφ)(x)| < Cn(1 + |x|2 + τ 2)−n,  где n — некоторое целое число.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Иными словами, с течением времени в координатном пространстве функция (Tτφ)(x)  оказывается мала вне множества τUφ, которое я называю существенным носителем  функции (Tτφ)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вернемся теперь к общему случаю, когда элементарное пространство h состоит из  векторнозначных функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы говорим, что множество τUφ является существенным  носителем функции  (Tτφ)(x) =  �  dkeikx−iE(k)τφ(k)  если  ||(Tτφ)(x)|| < Cn(1 + |x|2 + τ 2)−n,  при больших |τ| и x  τ /∈ Uφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мы говорим, что функции φ и φ′ не перекрываются, если расстояние между мно-  жествами Uφ и Uφ′ положительно;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' тогда соответствующие существенные носители  тоже не перекрываются, более того, при больших τ они далеки друг от друга.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы  65  говорим, что φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', φn — это неперекрывающееся семейство функций если φi не пе-  рекрывается с φj при i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы всегда будем предполагать, что неперекрывающихся  семейств функций достаточно много (точнее линейные комбинации неперекрываю-  щихся семейств функций всюду плотны в интересующем нас пространстве семейств  функций).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При r = 1 это выполнено, например, когда функция ǫ(p) строго выпукла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Что нужно назвать двухчастичным состоянием в алгебраическом подходе?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хо-  чу заметить, что при определении одночастичного пространства, мне были нужны  только пространственные и временные сдвиги, а сейчас нужно больше.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раньше я  использовал представление в виде (49), чтобы описать одночастичное состояние с  волновой функцией φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда же есть две частицы, B нужно применить два раза:  B(φ)B(φ′)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По крайней мере, когда φ и φ′ имеют носители в координатном про-  странстве далеко отстоящие друг от друга, можно считать, что этот вектор описыва-  ет состояние из двух отдаленных частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно потребовать при этом, чтобы B(φ)  и B(φ′) почти коммутировали друг с другом (тогда частицы будут бозонными) или  почти антикоммутировали (и тогда частицы будут фермионными).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это определение  дано в терминах состояний, описываемых векторами, но можно дать определение в  терминах состояний, описываемых положительными функционалами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого мы  заметим,что состояние, которое отвечает вектору B(φ)B(φ′)θ, можно написать в виде  L(φ)L(φ′)ω, где  L(φ) = ˜B(φ)B(φ), L(φ′) = ˜B(φ′)B(φ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  При этом L(φ) во всех случаях почти коммутирует с L(φ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе двухчастичное состояние записывается как L(φ)L(φ′)ω,  где L(φ) почти коммутирует с L(φ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, при переходе к геометрическому подходу различие между бозо-  нами и фермионами сглаживается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В дальнейшем я все время буду говорить о бозонах, но переход к фермионам  тривиален: нужно только заменить коммутаторы на антикоммутаторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Рассеяние;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' in- и out-состояния  Я хотел бы процитировать замечательное высказывание Бертрана Рассела:  The axiomatic method has many advantages over honest work — Аксиоматический  метод имеет много преимуществ перед честной работой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дело в том, что все дальнейшее будет очень просто, но, к сожалению, эта простота  достигнута за счет работы исключительно в аксиоматическом методе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хочу напом-  нить, что аксиомы локальной квантовой теории поля были предложены Вайтманом  в 50-х годах и до сих пор неизвестно ни одного примера нетривиальной теории, про  который доказано, что он удовлетворяет всем аксиомам Вайтмана в нашем трехмер-  ном пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Большой сдвиг произошел, когда в одномерном пространстве было  построено множество таких теорий в формализме конформной теории поля, но на се-  годняшний день аксиомы Вайтмана в трехмерном пространстве проверены только в  рамках теории возмущений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В моем подходе ситуация чуть лучше, но тем не менее  проверка нужных аксиом остается большой проблемой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это будет обсуждаться на  следующей лекции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' К счастью, по крайней мере в теории возмущений все прекрасно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь будем рассматривать элементарные возбуждения в обоих подходах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В ал-  гебраическом подходе я требую, чтобы определяющее частицу или квазичастицу отоб-  ражение элементарного пространства в пространство состояний могло быть записано  в форме  σ(φ) = B(φ)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (50)  В геометрическом подходе я требую, чтобы было отображение L : h → End(L), где  σ′(φ) = L(φ)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (51)  66  В обоих подходах я определю состояния, описывающие процесс рассеяния и для этого  введу новые операторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В алгебраическом подходе возьмем оператор B(f) (где f стоит вместо использо-  вавшегося ранее φ) и сначала сдвинем аргумент по времени в обратном направлении,  а потом сдвинем получившийся элемент алгебры по времени в прямом направлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Получим следующий оператор:  B(f, τ) = TτB(T−τf))T−τ  (Нужно помнить, что действие сдвига по времени на оператор — это сопряжение с  оператором Tτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  В геометрическом подходе запишем аналогичные формулы:  L(f, τ) = TτL(T−τf)T−τ,  после чего проведем цепочку выкладок:  L(f, τ)ω = TτL(T−τf)ω = Tτσ′(T−τf) = σ′(f),  где последовательно использованы свойство инвариантности ω по отношению к сдви-  гам по времени, формула (51) и свойство σ′ коммутировать с временными транс-  ляциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате зависимость от времени пропадает — L(f, τ)ω не зависит от  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Ровно такими же рассуждениями можно показать, что B(f, τ)θ не зависит от τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Отсюда следуют соотношения:  ˙L(f, τ)ω = 0,  ˙B(f, τ)θ = 0,  где точкой наверху обозначена производная по τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это будет мой основной инструмент.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В случае многих частиц по аналогии с тем, как вводилось определение одноча-  стичного состояния, я сначала много раз применяю B(fi, τ) с разными fi к θ:  Ψ(f1, · · · , fn|τ) = B(f1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(fn, τ)θ  а потом у полученного выражения беру предел при τ → −∞:  Ψ(f1, · · · , fn| − ∞) =  lim  τ→−∞ Ψ(f1, · · · , fn|τ)  (52)  Этот предел (который лежит в гильбертовом пространстве ¯H, пополнении простран-  ства H) я буду называть in-состоянием.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чуть позже я объясню его физический смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе все то же самое, только вместо B(f, τ) стоит L(f, τ):  Λ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn|τ) = L(f1, τ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(fn, τ)ω  (53)  Λ(f1, · · · , fn| − ∞) =  lim  τ→−∞ Λ(f1, · · · , fn|τ)  (54)  Получается in-состояние лежащее в L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В алгебраическом подходе оно соответствует  вектору Ψ(f1, · · · , fn| − ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Применение оператора Tτ к L(f, τ ′) соответствует сдвигу по времени и в функции,  и в аргументе:  Tτ(L(f, τ ′)) = Tτ+τ ′L(T−τ ′f)T−τ−τ ′ = L(Tτf, τ + τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (55)  Это чисто формальное вычисление.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  67  Применяя сдвиг по времени к in-состоянию и воспользовавшись формулой (55),  получим, что такое преобразование сводится к действию оператора сдвига по времени  на аргументы:  TτΛ(f1, · · · , fn| − ∞) = Λ(Tτf1, · · · , Tτfn| − ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (56)  Из этого соотношения следует, что если существенные носители функций Tτfi не пере-  крываются (это происходит если функции не перекрываются), то в пределе τ → −∞  получается несколько далеких друг от друга частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Другими словами, асимптотиче-  ски по времени эволюция in-состояния Λ описывается просто сдвигами аргументов в  функциях, которые далеки друг от друга.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это как раз то, что имеет место в процессе  рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обычно рассматривается рассеяние частиц с определенными импульсами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но с  определенными импульсами в моем подходе работать неудобно, потому что в таком  случае волновая функция будет ненормированной, поэтому следует рассматривать  ситуацию, когда импульс находится в каком-то узком диапазоне, то есть носитель  волновой функции — это маленький кусок пространства импульсов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если рассмат-  ривать столкновение частиц с различными импульсами, то тогда множества Uφ, ко-  торые я рассматриваю, не будут перекрываться.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это та ситуация, в которой я могу  жить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я говорю, что состояние TτΛ(f1, · · · , fn| − ∞) описывает столкновение частиц  с волновыми функциями (f1, · · · , fn), если эти функции не перекрываются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я предполагаю, что в случае когда функции f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn не перекрываются соответ-  ствующие операторы L(fi, τ) будут почти коммутировать, то есть их коммутатор в  пределе τ → −∞ будет равен нулю:  lim  τ→−∞ ||[L(fi, τ), L(fj, τ)]|| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (57)  Почему я это предполагаю?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В аксиоматическом подходе я могу предположить  все, но желательно, чтобы мои предположения имели физический смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда τ →  −∞, то в L(fi, τ) функции fi сдвигаются по времени, и тогда существенные носители  соответствующих функций будут далеки друг от друга.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае с точки зрения  физики естественно думать, что соответствующие операторы почти коммутируют.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Условие (57) можно вывести из требования, чтобы коммутаторы двух операторов  L, зависящих от функций φa(x) и ψa(x), удовлетворяли неравенству  ||[L(φ), L(ψ)]|| ≤  �  dxdx′Dab(x − x′)|φa(x)| · |ψb(x′)|,  (58)  где Dab(x) стремится к нулю быстрее любой степени x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При выполнении этих  условий, если множества Ufi для каждой пары функций не перекрываются, то ком-  мутаторы, входящие в формулу (57) близки к нулю, и поэтому я в формуле (53) для  in-состояния могу в пределе τ → −∞ переставлять операторы L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отсюда следует, что  in-состояния симметричны (не меняются при перестановке аргументов fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Докажем теперь, что рассматриваемый предел существует.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого наложим  дополнительно условие малости коммутатора [ ˙L(fi, τ), L(fj, τ)] при τ → −∞, где в  качестве аргументов стоят функции fi, fj, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — опять же аксиома.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Одного  стремления к нулю мало — нужно, чтобы выполнялось условие  ||[ ˙L(fi, τ), L(fj, τ)]|| ≤ c(τ),  где c(τ) стремится к нулю настолько быстро, чтобы был конечен интеграл  �  |c(τ)|dτ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно предположить, например, что c(τ) ∼ 1/τ a, где a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  68  Теперь я очень просто докажу, что in-состояние действительно существует (выра-  жение Λ(τ) = Λ(f1, · · · , fn|τ) имеет предел при τ → −∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого оценим произ-  водную от Λ(τ) по τ и заметим, что имеет место равенство  Λ(τ2) − Λ(τ1) =  � τ2  τ1  ˙Λ(τ)dτ  (59)  Я буду доказывать, что ˙Λ(τ) суммируема и, значит, интеграл в правой части стре-  мится к нулю когда τ1, τ2 → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если выражение (59) стремится к нулю, то Λ(τ)  имеет предел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это следует из полноты пространства, в котором лежит этот вектор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь нужно доказать, что ˙Λ(τ) мало.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что при определении Λ(τ) (фор-  мула (53)) я несколько раз применял оператор L к состоянию ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Продифференциру-  ем это выражение по τ , и применим правило Лейбница.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате получается  несколько слагаемых, в каждом из которых продифференцирован один из множите-  лей L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь будем менять местами L и ˙L и потребуем, чтобы в пределе τ → −∞  коммутаторы [L, ˙L] стремились к нулю достаточно быстро.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем сдвигать ˙L напра-  во, переместив в итоге на самое последнее место.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда дойду до самого конца, вос-  пользуюсь равенством ˙Lω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате получится, что ˙Λ(τ) будет суммируемой  функцией от τ, так как коммутаторы [ ˙L(fi, τ), L(fj, τ)] суммируемы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом доказано существование пределов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это очень важная вещь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  доказывает, что я могу рассматривать рассеяние частиц в моей картинке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я потребо-  вал очень мало, но мои аксиомы достаточны для того, чтобы доказать существование  предела, доказать, что есть понятие рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Условия, которые я наложил на L в случае геометрического подхода — это про-  сто аксиомы, а в случае алгебраического подхода аналогичные условия можно по-  лучить как следствия более физических требований (например, из асимптотической  коммутативности алгебры A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Но, так или иначе, они выполнены и тут, и там.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все  приведенные рассуждения справедливы и в алгебраическом подходе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что  в алгебраическом подходе L(φ) = ˜B(φ)B(φ), поэтому я могу наложить условия  ||[ ˙B(fi, τ), B(fj, τ)]|| ≤ c(τ),  где c(τ) -суммируемая функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Следовательно, и коммутатор [ ˙L(fi, τ), L(fj, τ)] будет  маленьким, и вектор  Ψ(τ) = B(f1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(fn, τ)θ  будет иметь предел в гильбертовом пространстве ¯H при τ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Пространство  должно быть полным, чтобы применить условие сходимости.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Я хочу чуточку усилить сделанное утверждение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раньше я применял операторы B  к циклическому вектору θ в один и тот же момент времени τ, а теперь буду применять  в разные времена, но все из них стремятся к бесконечности, возможно, по-разному.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В таком случае я утверждаю, что для вектора  Ψ(f1, τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn, τn) = B(f1, τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(fn, τn)θ  (60)  при τj → −∞ все равно существует предел в ¯H, обозначаемый как  Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn| − ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  При этом предполагаются выполненным условие на коммутаторы:  ||[ ˙B(φ), B(ψ)]|| ≤  �  dxdx′Dab(x − x′)|φa(x)| · |ψb(x′)|,  где Dab(x) → 0 быстрее любой степени, когда x → ∞ (это условие аналогично усло-  вию (58)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  69  Теперь я определю понятие асимптотического бозонного фоковского пространства  Has, считая, что операторы B на больших расстояниях коммутируют.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Я могу рабо-  тать как с коммутаторами, так и с антикоммутаторами, но для определенности я  выберу коммутаторы).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я определю асимптотическое бозонное фоковское простран-  ство Has как фоковское представление канонических коммутационных соотношений:  [b(ρ), b(ρ′)] = [b+(ρ), b+(ρ′)] = 0, [b(ρ), b+(ρ′)] = ⟨ρ, ρ′⟩,  где ρ, ρ′ ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В случае когда вместо коммутатора стоит антикоммутатор бозонное фоковское  пространство нужно заменить на фермионное фоковское пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все будет  таким же, только будет иметь место антисимметрия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Пространственные и временные трансляции естественно действуют в этом фоков-  ском пространстве потому что аргументы ρ принадлежат элементарному простран-  ству h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В элементарном пространстве есть действие трансляций и оно распространя-  ется на фоковское пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я определю матрицу Мёллера — половинку S-матрицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта матрица  переводит состояние из фоковского пространства b+(f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='b+(fn)|0⟩ в in-состояние  Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn| − ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этом важно, что in-состояние симметрично.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что можно  переставлять fi существенно, потому что иначе это определение не имело бы смысла,  так как n-частичное подпространство в фоковском пространстве представляет собой  cимметрическое тензорное произведение одночастичных пространств.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матрицы Мёллера коммутируют с трансляциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы на самом деле это дока-  зали, и это я сейчас объясню.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вернемся к формуле (56), из которой следует, что  действие временного сдвига на in-состояние отвечает временному сдвигу аргумен-  тов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Временной сдвиг аргументов — это то, как действует временной сдвиг в фо-  ковском пространстве, поэтому в формуле (56) написано то, что временной сдвиг в  in-пространстве отвечает временному сдвигу в гильбертовом пространстве ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Берет-  ся пополненное гильбертова пространства, так как только там лежит in-состояние.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Вот, эта очень важная вещь формально очень просто доказывается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что матрицы  Мёллера коммутируют с пространственными трансляциями проверяется еще легче.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матрица Мёллера обычно обозначается как S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На следующей лекции я буду  доказывать, что при условии распадения корреляций S− является изометрическим  вложением Has в ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вместо S− мы можем точно так же рассмотреть S+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если обе  эти матрицы Мёллера не только изометрические, но и унитарные, то есть, они яв-  ляются сюръективными отображениями на все пространство ¯H, тогда мы говорим,  что теория имеет интерпретацию в терминах частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что почти всякое  состояние является in-состоянием (множество in-состояний всюду плотно).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Другими  словами, почти всякое состояние в пределе τ → −∞ представляется совокупностью  удаленных частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Аналогично можно время устремить к плюс бесконечности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получится картинка,  аналогичная той, что была описана в классической теории солитонов при обсуждении  так называемой soliton resolution conjecture, когда почти любое начальное состояние  распадется на солитоны и почти линейный хвостик.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь примерно то же самое: на  плотном множестве состояние распадается на множество удаленных друг от друга  частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матрицу рассеяния можно определить формулой  S = S−1  + S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Похожую формулу я писал в солитонной картинке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это определение той самой мат-  рицы рассеяния, которая является главным объектом в квантовой теории поля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  70  Теперь я определю in-операторы a+  in с помощью предела операторов B(f, τ) при  τ → −∞:  a+  in(f) =  lim  τ→−∞ B(f, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (61)  Почему это законное определение?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обратимся к формуле (60), где операторы  B(fi, τi) стоят с разными временами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это значит, что по одному из аргументов я  могу перейти к пределу раньше, чем по другим аргументам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отсюда получается суще-  ствование предела в формуле (61) для in-оператора.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно сказать, что in-оператор  хорошо определен именно потому, что есть это свойство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Следует сказать, что in-  оператор не всегда определен, но, по крайней мере, если все функции f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn не  перекрываются, тогда все хорошо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В нашем определении in-операторы линейно зави-  сят от функций f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти операторы можно рассматривать как обобщенные функции и  ввести следующее обозначение:  a+  in(f) =  �  dpf k(p)a+  in,k(p),  где a+  in,k(p) -обобщенная функция, а индекс k определяет тип частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Точно так же определяются out-операторы, но только τ нужно стремить к плюс  бесконечности:  a+  out(f) =  lim  τ→+∞ B(f, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  In-операторы связаны с операторами в асимптотическом пространстве с помощью  формул:  a+  in(ρ)S− = S−b+(ρ),  S−|0⟩ = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Эти формулы можно считать альтернативным определением in-операторов, и тогда  точно так же можно определить операторы ain, связанные с операторами уничтоже-  ния в фоковском пространстве:  ain(ρ)S− = S−b(ρ),  aout(ρ)S+ = S+b(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Существует очевидная связь между определениями в геометрическом и алгебра-  ическом подходах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если геометрический подход строится в рамках алгебраического,  то оператор L(f, τ) в пространстве состояний соответствует оператору B(f, τ) в ¯H в  соответствии с формулой L(f, τ) = ˜B(f, τ)B(f, τ), и тогда состояние Λ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn|τ)  соответствует вектору Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn|τ), а состояние Λ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn| − ∞) (in-состояние)  соответствует вектору Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn| − ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Аналог матрицы Мёллера в геометрическом подходе обозначается как ˜S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В то  время как S−-матрица Мёллера — это линейный оператор, ˜S− — нелинейный опера-  тор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для теорий, которые могут быть сформулированы алгебраически S− отображает  в ¯H симметричную степень h, рассматриваемую как подпространство пространства  Фока.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Взяв композицию этого отображения с естественным отображением ¯H в конус  состояний C мы получаем ˜S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Когда L является квадратичным или эрмитовым, он индуцирует мультилинейное  отображение симметричной степени конуса C(h), соответствующего h в конус C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матрица рассеяния описывает процесс рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Связь сечения рассеяния с мат-  рицей рассеяния объясняется в общем курсе квантовой механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я объясню связь с  инклюзивным сечением, и это проще.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Сначала я должен объяснить, что такое инклюзивное сечение рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сечение  рассеяния связано с вероятностью перехода, скажем, от пары частиц к n частицам  (M, N) → (Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', Qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы же рассмотрим процесс, когда в конце получаются частицы  (Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', Qn) плюс что-нибудь еще:  (M, N) → (Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', Qm, R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', Rn)  71  По частицам R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', Rn проводится суммирование или, точнее, интегрирование.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По-  лучается инклюзивное сечение рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все сказанное годится в теории, когда есть  интерпретация в терминах частиц и когда все распадается на частицы, но я могу  определить инклюзивное сечение, даже если нет такой интерпретации, когда просто  происходит процесс (M, N) → (Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', Qn + something).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно не знать, что пред-  ставляет собой это “something”, но просуммировать по всему, что есть и получить  инклюзивное сечение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе только инклюзивное сечение имеет смысл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В алгебраи-  ческом подходе необязательно ограничиваться инклюзивным сечением, но все равно  проще работать с инклюзивным сечением.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я рассматриваю произвольное состояние ν и пишу следующую формулу для плот-  ности вероятности:  ν(a+  out,k1(p1)aout,k1(p1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' a+  out,km(pm)aout,km(pm))  (62)  Отмечу, что здесь стоят out-операторы, то есть предел на + бесконечности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Входя-  щие в формулу выражения вида a+  out,ki(pi)aout,ki(pi) — это, по сути, числа частиц  с импульсом pi, так что формула (62) представляет плотность вероятности в про-  странстве импульсов нахождения m исходящих частиц типов k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , km с импульсами  p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При этом на остальные частицы я не смотрю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  До сих пор ν было любым состоянием, а теперь в качестве ν рассмотрим in-  состояние  ν = Λ(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', gn| − ∞) =  lim  τ→−∞ L(g1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(gn, τ)ω  In-состояние определяется входящими частицами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например, когда я определял ин-  клюзивное сечение, я сталкивал две частицы, но можно сталкивать несколько.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если  в in-состоянии измерять входящее в формулу (62) произведение операторов с индек-  сом out, как раз получается инклюзивное сечение рассеяния по определению.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Значит,  если посчитать выражение (62), когда ν будет in-состоянием, получится инклюзивное  сечение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы сейчас произведем это вычисление.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Рассмотрим следующее выражение:  ⟨1|L(g′  1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(g′  n′, τ ′)L(g1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(gn, τ)|ω⟩  (63)  в предположении, что g′  i и gj не перекрываются, а времена устремляются к беско-  нечности, причем τ ′ → +∞, τ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Полученное при действии операторов L на  состояние ω — это линейный функционал на алгебре, поэтому мы можем посчитать,  чему он будет равен на единичном элементе алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Обращаю внимание, что я ис-  пользовал то, что называется bra-ket notations, при этом в скобках слева и справа  стоят элементы из сопряженных друг другу пространств.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Итак, рассмотрим выраже-  ние (63) и обозначим его как Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Взяв предел τ → −∞, получим  Q =  lim  τ ′→+∞⟨1|L(g′  1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(g′  n′, τ ′)ν⟩,  (64)  где ν = Λ(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', gn| − ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получается формула (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Так как я предположил, что  функции не перекрывается, всякие коммутаторы стремятся к нулю и Q не меняется  при перестановке g′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', g′  n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь посмотрим на эти формулы в алгебраическом подходе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда L(g, τ) =  ˜B(g, τ)B(g, τ), а кроме того у меня есть формула ( ˜  MNν)(X) = ν(M∗XN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Операто-  ры с волной умножают с одной стороны, без волны — с другой стороны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') В формуле  (64) эти операторы применены к ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они просто меняют аргумент у ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кроме того,  ⟨1|σ⟩ = σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате получается выражение  Q =  lim  τ ′→+∞ ν(B∗(g′  n′, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B∗(g′  1, τ ′)B(g′  1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(g′  n′, τ ′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  72  В пределе операторы B переходят в in- и out-операторы :  lim  τ ′→+∞ B(g, τ ′) = a+  out(g),  lim  τ ′→+∞ B∗(g, τ ′) = aout(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Учитывая это, а также условие, что в пределе все операторы коммутируют, получаем  следующее выражение для Q:  Q = ν(aout(g′  n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='aout(g′  1)a+  out(g′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+  out(g′  n′))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я назову выражение  Q = Q(g′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', g′  n′, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='., gn)  инклюзивной матрицей рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это выражение квадратично зависит от своих ар-  гументов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я могу перейти от квадратичных выражений к билинейным — тогда ко-  личество аргументов удвоится.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получаемое выражение также будет называться ин-  клюзивной матрицей рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из него можно получить инклюзивное сечение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  не вполне тривиальный процесс.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что определении инклюзивной матри-  цы рассеяния я рассматривал ее как функционал на неперекрывающихся семействах  функциий.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этот функционал линеен или антилинеен, поэтому его можно рассматри-  вать как обобщенную функцию, но но аргументы обобщенной функции (импульсы)  следует считать различными.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В выражении для инклюзивного сечения импульсы мо-  гут совпадать, поэтому необходим предельный переход.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В геометрическом подходе я могу определить инклюзивную матрицу рассеяния,  взяв наряду с ω ∈ L какое-нибудь трансляционно инвариантное состояние α ∈ L∗:  lim  τ ′→+∞,τ→−∞⟨α|L(g1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(gm, τ ′) × L(f1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(fn, τ)|ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (65)  Такая формула может быть применена и в алгебраической ситуации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В ней состояния  α и ω входят абсолютно симметрично.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно сформулировать это так: формулa (65)  дает скалярное произведение out-состояния в L∗ и in-состояния в L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами,  одна и та же формула будет давать инклюзивную матрицу рассеяния элементарных  возбуждений состояния ω и инклюзивную матрицу рассеяния элементарных возбуж-  дений состояния α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это некая двойственность, на мой взгляд, абсолютно таинствен-  ная.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В алгебраическом подходе ее тоже можно рассматривать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  73  8  Лекция 8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Связь с локальной квантовой теорией поля  То, что я рассказываю в этом курсе, сильно отличается от того, что обычно расска-  зывается в учебниках по релятивистской квантовой теории поля — в них рассматри-  ваются локальные теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Основная идея дальнейшего изложения — подчеркнуть,  что локальность абсолютно не по существу в большинстве случаев, да и сами поля  несущественны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я не знаю, что нужно называть полями в том подходе, о котором я  говорю, хотя вся квантовая теория поля здесь есть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я хочу начать с установления связи того, что я рассказываю с тем, что обычно  называется локальной релятивистской квантовой теорией поля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В аксиоматическом подходе к локальной теории есть разные системы аксиом на-  чиная с аксиом Вайтмана, о которых вкратце можно сказать, что в них в качестве  основного объекта рассматриваются поля, являющиеся обобщенными операторными  функциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это не очень удобно: поля локальны, но это обобщенные функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если  их проинтегрировать, то получаются обычные операторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они уже не локальны, но  в каком-то смысле почти локальны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Их можно считать сосредоточенными в какой-то  области.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я буду говорить про систему аксиом, которая принадлежит Араки, Хаагу и Ка-  стлеру.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В ней рассматриваются, строго говоря, не локальные поля, а поля, сосредо-  точенные в каком-то открытом подмножестве пространства Минковского.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этой си-  стеме аксиом считается, что такие поля образуют алгебру операторов, действующих  в гильбертовом пространстве;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' эта алгебра замкнута относительно слабой сходимости  (что не так существенно).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Считается, что в том гильбертовом пространстве E, в кото-  ром действуют эти операторы, действует унитарное представление группы Пуанкаре  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предполагается, что каждой ограниченной области (ограниченному открытому  подмножеству) O пространства Минковского сопоставлена алгебра операторов A(O)  так, что  если область становится больше: O1 ⊂ O2, то и алгебра становится больше:  A(O1) ⊂ A(O2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  действие группы Пуанкаре P на операторах согласованно с действием на обла-  стях: A(gO) = gA(O)g−1, если g ∈ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  если интервал между точками двух областей O1 и O2 пространственно-подобный,  то соответствующие операторы A(O1) и A(O2) коммутируют (грубо говоря, это  означает, что если интервал пространственно-подобный, то причинной связи не  может быть — все абсолютно независимо);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  основное состояние θ оператора энергии инвариантно относительно группы Пу-  анкаре (имея представление группы Пуанкаре, можно рассмотреть оператор  энергии (гамильтониан) и операторы импульса как инфинитезимальные гене-  раторы, соответственно, временных и пространственных трансляций);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  вектор, отвечающий основному состоянию, является циклическим по отношению  к объединению A всех алгебр A(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это аксиоматика релятивистской локальной квантовой теории поля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В этой аксиоматике частица определяется как неприводимое подпредставление  представления алгебры Пуанкаре в пространстве E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  74  Вернемся теперь к определению рассеяния в алгебраическом подходе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Прошлое  мое рассмотрение базировалось на аксиомах, которые нелегко проверить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сейчас же  я наложу требования, которые много легче проверяются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В частности, будет видно,  что они выполнены в релятивистской локальной теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мой исходный пункт, как и раньше — это ассоциативная алгебра A с инволюцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Пространственно-временные трансляции — это автоморфизмы этой алгебры.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  У меня есть понятие состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Есть и понятие ненормализованного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Такие состояния отвечают положительным линейным функционалам на алгебре A  и образуют конус C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду работать именно с ненормализованными состояниями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Трансляционно-инвариантное стационарное состояние всегда будет обозначаться как  ω ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Возбуждения состояния ω — это элементы предгильбертова пространства H,  которое построено по ω с помощью конструкции GNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В ассоциативной алгебре A, с которой я начинал, нет никакой нормы, но поскольку  она представлена в предгильбертовом пространстве H и его пополнении, гильберто-  вом пространстве ¯H, можно рассматривать норму соответствующих операторов ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Более того, я могу работать с пополненной по этой норме алгеброй A(ω), но это не  обязательно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Пусть теперь у меня есть элемент A алгебры A который представлен ограни-  ченным оператором ˆA в гильбертовом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я могу этот оператор сдвигать  по времени и по пространству.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В результате получится некий оператор ˆA(x, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Бо-  лее того, я могу усреднить такой оператор с какой-то гладкой и быстро убывающей  функцией:  B =  �  dτdxα(x, τ) ˆA(x, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно сдвинуть оператор B по времени и пространству:  B(x, τ) =  �  dτ ′, dx′α(x − x′, τ − τ ′)A(x′, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Под знаком интеграла можно дифференцировать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку сама функция α(x, τ)  предполагается гладкой, можно продифференцировать сколько угодно раз.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я всегда  буду работать именно с такими операторами и буду называть их гладкими.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Условие асимптотической коммутативности следует наложить таким способом,  чтобы коммутатор сдвинутого оператора с другим оператором становился малым  при больших пространственных сдвигах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Формализовать это можно по-разному.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я  сделаю это так, чтобы было мгновенно понятно, что в аксиоматике Араки, Хаага и  Кастлера мое условие выполнено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Именно я потребую, чтобы норма ||[B1(x, τ), B2]||  коммутатора сдвинутого оператора с другим оператором, отвечающим элементу ал-  гебры A убывала быстрее любой степени ||x||, когда x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То же условие я наложу  на ||[ ˙B1(x, τ), B2]||, где точка обозначает производную по времени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все операторы,  напомню, у меня гладкие.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В аксиоматике Араки, Хаага и Кастлера это всегда выполняется, потому что там  при большом пространственном сдвиге пространственно-временной интервал меж-  ду соответствующими областями, становится пространственно-подобным и поэтому  можно сказать, что начиная с некоторого момента рассматриваемый мною коммута-  тор просто равен нулю (и тем самым убывает быстрее любой степени).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Другим определением асимптотической коммутативности является условие  ||[B1(x, τ), B2]|| ≤  Cn(τ)  1 + ||x||n ,  где Cn(τ) имеет не более чем полиномиальный рост, а n -произвольно (сильная асимп-  тотическая коммутативность).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это условие в аксиоматике Араки, Хаага и Кастлера  выполнено, если спектр масс ограничен снизу положительным числом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  75  Кроме условия асимптотической коммутативности я хочу наложить условие кла-  стеризации, о которых я говорил в прошлый раз.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В простейшей форме это означает,  что если один из операторов убегает на бесконечность, то все распадается с точностью  до маленького слагаемого ρ(x, t):  ω(A(x, t)B) = ω(A)ω(B) + ρ(x, t),  где ω — это функционал на алгебре A, то самое состояние, возбуждение которого я  рассматриваю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно наложить примерно такое же условие, когда операторов много.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это очень удобное условие, но формулируется оно довольно громоздко.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для формулировки условия кластеризации для многих операторов мне нужно вве-  сти понятие корреляционной функции, являющейся обобщением функции Вайтмана  из релятивистской квантовой теории поля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я беру какие-то операторы A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ar ∈ A,  сдвигаю их и по пространству, и по времени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Перемножив их, получаю элемент ал-  гебры и после этого применяю ω или, что то же самое, беру среднее по состоянию ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В результате получается:  wn(x1, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' xn, tn) = ω(A1(x1, t1) · · · An(xn, tn)) = ⟨A1(x1, t1) · · · An(xn, tn)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это корреляционная функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Полезно определить понятие усеченной корреляционной функции  wT  n (x1, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' xn, tn) ≡ ⟨A1(x1, t1) · · · An(xn, tn)⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это делается несколько формально с помощью индуктивной формулы, связывающей  усеченные корреляционные функции с обычными корреляционными функциями:  wn(x1, τ1, k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , xn, τn, kn) =  n  �  s=1  �  ρ∈Rs  wT  α1(π1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' wT  αs(πs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Здесь Rs обозначает совокупность всех разбиений множества {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', n} на подмноже-  ства s, обозначаемые π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', πs, количество элементов в подмножестве πi обозначается  через αi, а wT  αi(πi) обозначает усеченную корреляционную функцию с аргументами  xa, τa, ka, где a ∈ πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта формула выражает корреляционные функции через усечен-  ные функции при всевозможных разбиениях множества индексов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В случае когда есть всего два оператора, усеченная корреляционная функция име-  ет вид  wT  2 (x1, τ1, k1, x2, τ2, k2) = ω(A1(x1, t1)A2(x2, t2)) − ω(A1(x1, t1))ω(A2(x2, t2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Так как ω трансляционно инвариантнo, как обычная, так и усеченная корреляци-  онные функции зависят только от разностей xi−xj, ti−tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы говорим, что выполнено  условие кластеризации если усеченные корреляционные функции становятся малы-  ми при xi − xj → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Малость можно понимать в разных смыслах, но я имею в виду  самое сильное условие: при фиксированных ti они стремятся к нулю быстрее, чем  любая степень разности d = min ∥xi − xj∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее говоря, мы предполагаем,что  |wT  n (x1, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' xn, tn)| ≤ Cs(t)  ds  ,  где s любое натуральное число, а Cs(t)-полиномиальная функция от времен ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно применить преобразование Фурье, и тогда факт зависимости только от  разности будет означать появление δ-функции от суммы импульсов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если эту функ-  цию выделить, то условие кластеризации, означает, что усеченная корреляционная  функция представляет собой гладкую функцию, умноженную на δ-функцию:  νn(p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , pn, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , tn)δ(p1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  76  Напомню, что если функция быстро убывает по x, то в импульсном представлении  она является гладкой, и это здесь написано.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В релятивистской квантовой теории условие кластеризации выполнено если массы  частиц ограничены снизу положительным числом (mass gap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Связь с матрицей рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Функция Грина отличается от корреляционной функции тем, что операторы, кото-  рые стоят под знаком ω, считаются упорядоченными по времени в порядке убывания.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это то, что называется хронологическим произведением.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если применить ω к хро-  нологическому произведению (или взять среднее по вектору, который отвечает ω в  конструкции GNS), получится функция  Gn = ω(T(A1(x1, t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ar(xr, tr))) = ⟨θ|T( ˆA1(x1, t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ˆAr(xr, tr))|θ⟩,  которая называется функцией Грина, записанной в (x, t)-представлении (в коорди-  натном представлении).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как всегда, можно переходить к импульсному представлению, взяв преобразова-  ние Фурье по x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это будет то, что называется (p, t)-представлением (импульсном и  временном).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно еще взять (обратное) преобразование Фурье по временной пере-  менной и тогда функции Грина будут в (p, ǫ)-представлении, когда основные перемен-  ные — это импульсы и энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мне потребуются все эти представления.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отличаются  они преобразованиями Фурье, всегда можно переходить от одного к другому.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Благодаря трансляционной инвариантности функция Грина в (x, t)-представлении  зависят от разностей xi − xj, ti − tj и, следовательно, в (p, t)-представлении выделя-  ется множитель δ(p1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + pr), отвечающий закону сохранения импульса, а в (p, ǫ)-  представлении —еще и множитель δ(ǫ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + ǫr), отвечающий закону сохранения  энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Переходя к вопросу о полюсах функции Грина, нужно заметить, что я всегда не  обращаю внимания на δ-функции, говоря про полюса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В частности, когда функция  Грина включает только два оператора, в (p, ǫ)-представлении мы имеем два импульса,  две энергии и две δ-функции — одна от импульсов, другая от энергий:  G(p1, ǫ1|A, A′)δ(p1 + p2)δ(ǫ1 + ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Остается функция от переменной p1 и о переменной ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Важно отметить, что полюса  такой двухточечной функции Грина по энергии при фиксированном импульсе отве-  чают частицам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти полюса зависят от импульса, а соответствующая функция ε(p)  дает закон дисперсии для частиц (зависимость энергии от импульса).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти хорошо  известные факты легко вывести из рассуждений, которые будут использованы ниже.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я докажу, что для того, чтобы найти амплитуды рассеяния, нужно рассмот-  реть асимптотическое поведение функции Грина в (p, t)-представлении, когда t →  ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это первое и основное замечание.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' А другое замечание состоит в том, что это  асимптотическое поведение в (p, t)-представлении управляется полюсами в (p, ǫ)-  представлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее говоря, асимптотика описывается вычетами в этих полюсах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это называется “on-shell value of Green function”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Есть хорошо известный математический факт: если асимптотическое поведение  какой-либо функции ρ(t) при t → ±∞ имеет вид e−itE±A± или, говоря по-другому,  существует предел limt→±∞ eitE±ρ(t) = A±, тогда (обратное) преобразование Фурье  ρ(ǫ) имеет полюса в точках E± ± i0 с вычетами ∓2πiA±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами предел  отвечает вычетам, а показатели экспоненты отвечают полюсам;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' при этом полюс ока-  зывается немножко сдвинутым в комплексной плоскости либо наверх, либо вниз.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  чрезвычайно важное замечание.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  77  Можно либо смотреть на полюса в энергетическом представлении, либо смотреть  на асимптотику во временном представлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что вычисление амплитуд  рассеяния сводится к выяснению асимптотического поведения функций Грина (это  мы сделаем).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Переходя к энергетическому представлению, мы можем сказать, что  амплитуды рассеяния выражаются через on-shell значения функций Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это —  формула Лемана, Симанчика и Циммермана (LSZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Ниже я докажу формулу LSZ в случае, когда в теории есть интерпретация в терми-  нах частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что существуют половинки матрицы рассеяния — матрицы  Мёллера S±, которые дают унитарную эквивалентность между свободным гамиль-  тонианом в асимптотическом пространстве Has и полученым с помощью процедуры  GNS гамильтонианом в пространстве H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я хочу упростить обозначения, и поэтому я буду обсуждать случай, когда есть  только один тип частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напоминаю, что я рассматривал обобщенную функцию Φ(p),  отвечающую состоянию частицы с заданным импульсом p, и это состояние — соб-  ственное как для импульса, так и для оператора энергии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Действие гамильтониана  на Φ(p) сводится к умножению на функцию ε(p) (закон дисперсии):  ˆHΦ(p) = ε(p)Φ(p),  ˆPΦ(p) = pΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Нужно помнить, что на самом деле Φ(p) не очень-то существует — это обобщенная  функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того, чтобы все это приобрело точный математический смысл, следует  проинтегрировать с какой-то пробной функцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я хочу сделать предположение, что одночастичный спектр не перекрыва-  ется с многочастичным спектром.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Как вычислить спектр гамильтониана ˆH?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У меня  есть интерпретация в терминах частиц, и поэтому матрица Мёллера связывает асимп-  тотический гамильтониан с гамильтонианом в пространстве H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Асимптотический гамильтониан является свободным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У него (а, значит, и у ˆH)  спектр полностью определяется одночастичными функциями ε(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Энергии многоча-  стичных возбуждений — это просто суммы ε(p1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' + ε(pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если я хочу сказать,  что одночастичный спектр не перекрывается с двухчастичным, а, следовательно, и  многочастичным, нужно потребовать выполнения неравенства  ε(p1 + p2) < ε(p1) + ε(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это означает, что частицы с импульсом p1 + p2 не могут распасться на частицы с  импульсом p1 и импульсом p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Закон сохранения (в данном случае закон сохранения  энергии) запрещает распад.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я аккуратно сформулирую формулу Лемана, Симанчика и Циммермана.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для этого я зафиксирую какие-то элементы Ai ∈ A той алгебры, из которой я исхо-  дил.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Напоминаю, что я работаю с гладкими элементами, но здесь это не так важно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Также я требую, чтобы применив оператор ˆAi к вектору θ (который в релятивистской  квантовой теории интерпретируется как физический вакуум) и осуществив проекцию  на пространство, натянутое на одночастичные состояния (одночастичное простран-  ство), я получу в результате отличную от нуля величину.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее говоря, я требую,  чтобы проекция вектора ˆAiθ была одночастичным состоянием вида:  Φ(φi) =  �  φi(p)Φ(p)dp,  где φi(p) — это не обращающаяся нигде в ноль функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Проекция этого вектора на  вектор θ должна обращаться в ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для простоты я еще веду обозначение Λi(p) = φi(p)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этих обозначениях бу-  дем рассматривать содержащую как сами операторы Ai, так и сопряженные им A∗  i  78  функцию Грина в (p, t)-представлении:  Gmn = ω(T(A∗  1(x1, t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' A∗  m(xm, tm)Am+1(xm+1, tm+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Am+n(xm+n, tm+n))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Затем перейдем к (p, ǫ)-представлению, но при этом там, где стоят операторы A∗  i  при 1 ≤ i ≤ m удобно сменить знак у переменных pi и ǫi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Функцию Грина в новом  представлении умножим на выражение:  �  1≤i≤m  Λi(pi)(ǫi + ε(pi))  �  m<j≤m+n  Λj(pj)(ǫj − ε(pj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  и перейдем к пределу ǫi → −ε(pi) для 1 ≤ i ≤ m и ǫj → ε(pj) для m < j ≤ m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вклад в полученное выражение будут вносить только полюса, потому что введен-  ный множитель стремится к нулю, и вне полюса все полученное выражение обраща-  ется в ноль, а в полюсе получается вычет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, данная операция сводится  к взятию вычета в этом полюсе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Кроме того, я еще умножаю на функции Λi(p) и со-  пряженные к ним.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В физике это называется перенормировкой волновой функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В  случае когда эти множители не включены, я буду говорить про on-shell функции Гри-  на, а если включены, я буду говорить, что рассматриваю нормализованные on-shell  функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Основное высказывание в подходе Лемана, Симанчика и Циммермана состоит в  том, что нормализованная on-shell функция Грина дает амплитуду рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для  его доказательства, прежде всего, я буду рассматривать случай, когда операторы ˆAi  просто дают одночастичные состояния ˆAiθ = Φ(φi) (не нужно проецировать).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем  их называть хорошими операторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В конце лекции я объясню, что к этому случаю  все можно свести.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  До сих пор рассматривался случай, когда имеется всего один тип частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Давай-  те рассмотрим случай, когда есть много типов частиц, иными словами, есть много  функций Φk(p), являющиеся собственными как для импульса, так и для энергии:  PΦk = pΦk(p),  HΦk(p) = εk(p)Φk(p),  но с разными законами дисперсии εk(p), задающимися гладкими функциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Как  всегда, Φk(p) — это обобщенные функции, то есть, нужны пробные функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я рас-  сматриваю пробные функции из пространства S гладких быстро убывающих функ-  ций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чтобы гарантировать, что сдвиги по времени действуют в пространстве S, сле-  дует предположить, что функции εk(p) растут не более чем полиномиально.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как уже было сказано, я буду работать с хорошими операторами Bk ∈ A, яв-  ляющимися гладкими и переводящими вектор θ в одночастичные состояния ˆBkθ =  Φk(φk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь я определяю оператор ˆBk(f, t), зависящий от функции f = f(p) сле-  дующим образом:  ˆBk(f, t) =  �  ˜f(x, t) ˆBk(x, t)dx,  где функция ˜f(x, t) получена преобразованием Фурье от функции f(p)e−iεk(p)t по  импульсной переменной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Аналогичные операторы рассматривались на прошлой лекции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они обладают тем  свойством, что если их применить к θ, получится независящее от t одночастичное  состояние  ˆBk(f, t)θ = Φk(fφk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (66)  (На прошлой лекции функция φk равнялась единице).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В целом же, все то, что го-  ворилось на прошлой лекции можно повторить и здесь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тот факт, что полученное  состояние не зависит от времени — это результат формального счета.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Вычисления  79  становятся совсем простыми если сделать преобразование Фурье по пространствен-  ной переменной  ˆBk(p, t) =  �  dxe−ipx ˆBk(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В терминах этих операторов получается простое выражение  ˆBk(f, t) =  �  dpf(p)eiεk(p)t ˆBk(p, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (67)  Если этот оператор применить к θ, получится одночастичное состояние, зависи-  мость которого от времени определяется такой же экспонентой, что и стоит под зна-  ком интеграла, только с минусом перед ее показателем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти две экспоненты сократят-  ся и зависимость от времени пропадет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это очень важная для меня вещь.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В прошлый  раз, я тоже рассматривал аналогичные операторы, и мне очень важно, чтобы я мог  продифференцировать по t и получить ноль, действуя на θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я делаю ровно то, что делал в прошлый раз, но обозначения изменились,  потому что мне неудобно сейчас говорить про элементарное пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поэтому я  явно выписываю индексы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В элементарном пространстве у меня тоже были индексы,  но я их мог спрятать, а тут я индексы пишу явно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я рассматриваю вектор  Ψ(k1, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , kn, fn|t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , tn) = ˆBk1(f1, t1) · · · ˆBkn(fn, tn)θ,  (68)  где предполагается, что функции f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn имеют компактные носители.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В прошлой  лекции я в основном считал, что все времена, входящие в аналогичное выражение,  одинаковы;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' случай, когда времена разные, но все стремятся к бесконечности немногим  сложнее.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я, как и в прошлый раз, рассматриваю градиенты от энергии vi(p) =  ∇εki(p), которые имеют смысл скорости.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь я обозначаю буквой Ui множество  всех возможных скоростей vi(p), таких что fi(p) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дальше я рассматриваю за-  мыкание Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я требую, чтобы все эти множества не перекрывались, тогда я буду  называть эти функции неперекрывающимися.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что все классические  скорости различны, и поэтому волновые пакеты расходятся в разных направлени-  ях, тогда, как я объяснял в прошлый раз, в координатном представлении почти не  перекрываются соответствующие волновые функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это то, что уже было сказано, а теперь я буду брать предел ti → ∞, опять для  простоты буду рассматривать случай, когда все времена одинаковы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я буду доказы-  вать, что у вектора Ψ(k1, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , kn, fn|t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , tn) есть предел, который обозначим как  Ψ(k1, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , kn, fn| ± ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Доказательство повторяет то, что было в прошлый раз.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я снова положу одинако-  выми все ti = t и продифференцирую по t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того чтобы доказать, что есть предел,  мне нужно знать, что производная по t мала (на самом деле достаточно, чтобы про-  изводная была суммируемой функцией).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это условие выполняется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По определению  вектор Ψ является результатом многократного применения к θ операторов ˆBki(fi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Когда я дифференцирую это выражение по t, у меня появляется точка над одним  из операторов ˆB, который я после этого могу сдвинуть направо, используя условие  асимптотической коммутативности (но только если я работаю с неперекрывающими-  ся функциями).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта производная примененная к θ дает ноль, поэтому предел есть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Поскольку предел существует, я могу определить матрицу Мёллера.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого я  введу асимптотическое пространство Has как фоковское представление операторов  a+  k (f), ak(f) и определю матрицы Мёллера S− и S+ как операторы, определенные в  Has и принимающие значения в ¯H по формуле:  Ψ(k1, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , kn, fn| ± ∞) = S±(a+  k1(f1φk1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' a+  kn(fnφkn)|0⟩),  (69)  80  где |0⟩ — фоковский вакуум.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это та же формула, которая была на прошлой лекции с  той лишь разницей, что здесь появились множители φki, которые мне нужны для того,  чтобы учесть то, что при действии хорошего оператора ˆBki на Φ мы получаем Φki(φki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матрицы Мёллера определены на всюду плотном подпространстве асимптотического  гильбертова пространства Has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно доказать и я сейчас это сделаю, что получилось изометрическое вложение  асимптотического пространства в пространство H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Физический смысл полученного  можно понять из формулы (которая в других обозначениях была на прошлой лекции):  e−iHtΨ(k1, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , kn, fn| ± ∞) = Ψ(k1, f1e−iεk1t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , kn, fne−iεknt| ± ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Эта формула означает, что когда мы рассматриваем эволюцию в пространстве H,  то в пределе t → ±∞ действию на вектор Ψ оператора эволюции в асимптотическом  пространстве будет отвечать просто эволюция функций fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все происходит независи-  мо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, эволюция вектора Ψ сводится при больших t к эволюции системы  из n далеких друг от друга частиц с неперекрывающимися волновыми функциями  f1φ1e−iεk1t,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ,fnφne−iεknt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Данное нами определение S± может быть неоднозначным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Например, мы можем  использовать разные хорошие операторы в построении, и неясно, получим ли мы  один и тот же ответ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Однако можно доказать, что ответ не зависит от случайностей  построения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я докажу, что S± — изометрические операторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они сохраняют норму и  сохраняют скалярное произведение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такие операторы не могут быть многозначным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Заодно мы увидим, что вектор Ψ(k1, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', kn, fn|±∞) не меняется, при перестановках  аргументов (ki, fi) и (kj, fj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Основная линия доказательства состоит в следующем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чтобы получить матрицу  Мёллера в формуле (68) положим для простоты все времена одинаковыми и рас-  смотрим предел вектора Ψ(t) при t → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Заметим при этом, что согласно (68)  такой вектор получается в результате многократного применения операторов B к ос-  новному состоянию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Теперь легко увидеть, что, скалярное произведение двух таких  векторов можно выразить через корреляционные функции, определяемые как сред-  ние значения произведений рассматриваемых операторов по основному состоянию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Но корреляционные функции выражаются через усеченные корреляционные функ-  ции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' А в усеченных корреляционных функциях только двухточечные корреляционные  функции будут выживать в пределе, t → ±∞, если я потребую кластеризацию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь я могу сказать, что матрица Мёллера — это изометрическое отображение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Изометрическое отображение никак не может быть многозначным, потому что если  два вектора совпадают, то расстояние между ними 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это условие должно сохранять-  ся, и два совпадающих вектора должны переходить в совпадающие векторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  После того, как я ввел понятие матрицы Мёллера, я могу ввести понятия out-  оператора и in-оператора:  ain(f)S− = S−a(f), a+  in(f)S− = S−a+(f),  aout(f)S+ = S+a(f), a+  out(f)S+ = S+a+(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Грубо говоря, если S− и S+ унитарны, то это соответствие взаимно однозначно, и  я могу все перекидывать из асимптотического пространства в основное пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Здесь я опять не пишу индекс, описывающий тип частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Справедливы также формулы  a+  in(fφ) = lim  t→−∞  ˆB(f, t), a+  out(fφ) = lim  t→∞  ˆB(f, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (70)  То, что я рассказываю, немножко отличается от того, что было на прошлой лек-  ции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Во-первых, операторы B(f, t) в прошлый раз я вводил аксиоматически, а в этот  81  раз построил с помощью явной формулы, а, во-вторых, мне пришлось заплатить за  это появлением буквы φ, которая никому не мешает, но тем не менее она присутствует.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предел в (70) существует на множестве всех векторов вида Ψ(k1, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , kn, fn|±)  при условии, что f, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn-это неперекрывающееся семейство функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Интересно,  что когда размерность пространства d ≥ 3, то при некоторых условиях этот предел  существует без условия неперекрывания.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Это для нас несущественно, потому что  условие неперекрывания дает предел на всюду плотном множестве, и этого абсолютно  достаточно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Теперь, на основе сказанного, я могу выписать явно то, как действуют определен-  ные нами операторы  a+  in(fφ)Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn| − ∞) = Ψ(f, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn| − ∞),  a+  out(fφ)Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn|∞) = Ψ(f, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn|∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  ain(f)Ψ(φ−1f, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn| − ∞) = Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn| − ∞),  aout(f)Ψ(φ−1f, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn|∞) = Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn|∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Эти формулы можно рассматривать как определения операторов ain и aout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Грубо  говоря, операторы a+  in/out добавляют одну функцию к Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fn|∓∞), а сопряжен-  ные к ним операторы уничтожают одну из этих функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если операторы S+ и S− являются унитарными, мы говорим, что теория имеет  интерпретацию в терминах частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае (а также в более общем случае,  когда образ S− совпадает с образом S+), мы можем определить матрицу рассеяния  S = S∗  +S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  как унитарный оператор в асимптотическом пространстве Has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Асимптотическое про-  странство — это фоковское пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У него есть обобщенный базис |p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , pn⟩ =  1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+(p1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' a+(pn)|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В этом базисе матричные элементы унитарного оператора S (амплитуды рассея-  ния) могут быть выражены в терминах in- и out-операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это те самые матричные  элементы, квадраты которых дают эффективные сечения рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Получается сле-  дующая формула:  Smn(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , pm|q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , qn) = ⟨a+  in(q1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' a+  in(qn)θ, a+  out(p1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' a+  out(pm)θ⟩  (71)  Это следует непосредственно из определения in- и out-операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В формуле (71) и  в последующих я буду опускать числовые коэффициенты (m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')−1(n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Приведенная выше формула доказана только для случая, когда все значения  импульсов pi, qj различны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Точнее, мы должны предположить, что все векторы  v(pi) = ∇ε(pi), v(qj) = ∇ε(qj) различны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае когда функция ε(p) является  строго выпуклой, достаточно предположить, что pi, qj различны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это несуществен-  ное ограничение, но оно есть.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Формулу (71) следует понимать в смысле обобщенных  функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что следует в качестве пробных функций брать множество  функций fi(pi), gj(qj) с неперекрывающимися подмножествами U(fi), U(gj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Распишем теперь формулу (71) подробнее:  Smn(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fm|g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , gn) =  =  �  dmpdnq  �  fi(pi)  �  gj(qj)Smn(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , pm|q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , qn) =  = ⟨a+  in(g1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' a+  in(gn)θ, a+  out( ¯f1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' a+  out( ¯fm)θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  82  Вспомнив, что S-матрица рассеяния определена с помощью пределов при t → ±∞  и воспользовавшись формулами (70), мы приходим к следующему представлению:  Smn(f1, , fm|g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , gn) =  lim  t→∞,τ→−∞⟨θ| ˆB( ¯fmφ−1, t)∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ˆB( ¯f1φ−1, t)∗ ˆB(g1φ−1, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ˆB(gnφ−1, τ))|θ⟩ =  lim  t→∞,τ→−∞ ω(B( ¯fm ¯φ−1, t)∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' B( ¯f1 ¯φ−1, t)∗B(g1φ−1, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' B(gnφ−1, τ)),  где B(f, t)∗ =  �  dxB∗(x, t) ˜f(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно записать и более общую формулу  Smn(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fm|g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', gn) =  lim  ti→∞,  τj→−∞  ω(Bm( ¯fmφ−1  m , tm)∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' B1( ¯f1φ−1  1 , t1)∗Bm+1(g1φ−1  m+1, τ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Bm+n(gnφ−1  m+n, τn)),  справедливую, когда все Bi — разные хорошие операторы и Biθ = Φ(φi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Очень важное замечание состоит в том, что из условия неперекрывания, в пределе  ti → ∞, τj → −∞ порядок времен несущественен как в группе со звездочкой, так и  в группе без звездочки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это значит, что на больших временах я могу эти операторы  переставлять.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В частности, я могу их считать упорядоченными по времени и получу  то же, что появлялось у меня при введении функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, я могу  считать, что здесь стоит функция Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это конец доказательства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Чтобы быть совершенно аккуратным, нужно все-таки сказать, какая получится  функция Грина?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для того чтобы это установить, выразим согласно (67) операторы  ˆBk(f, t) через ˆBk(p, t) и получим в результате:  Smn(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , fm|g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , gn) =  �  dm+np  lim  ti→∞,  τj →−∞  ⟨θ|fm ¯φ−1  m eiεm(pm)tm ˆBm(pm, tm)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='f1 ¯φ−1  1 eiε1(p1)t1 ˆB1(p1, t1)∗×  g1φ−1  m+1e−iεm+1(pm+1)τ1 ˆBm+1(pm+1, τn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='gnφ−1  m+ne−iεm+n(pm+n)τn ˆBm+n(pm+n, τn)|θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для матричных элементов матрицы рассеяния получается следующая формула:  Smn(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , pm|pm+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' , pm+n) =  lim  t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=',tm→∞,  tm+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=',tm+n→−∞  ⟨θ|¯φ−1  m eiεm(pm)tm ˆBm(pm, tm)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='¯φ−1  1 eiε1(p1)t1 ˆB1(p1, t1)∗×  φ−1  m+1e−iεm+1(pm+1)tm+1 ˆBm+1(pm+1, tm+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='×  φ−1  m+ne−iεm+n(pm+n),tm+n ˆBm+n(pm+n, tm+n)|θ⟩,  где у меня стоят операторы ˆBm(pm, tm) и производится усреднение по состоянию θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это среднее совпадает с функцией Грина в (p, t)-представлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта формула мне  говорит, что если я исхожу из хороших операторов, матрица рассеяния выражается  через функции Грина в (p, t)-представлении, точнее через их асимптотику приti → ∞  и τj → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' У меня появились множители φ−1  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это ровно те самые Λ, которые вво-  дились для того, чтобы получить нормированные функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что я рассмат-  риваю асимптотику, означает, что в энергетическом представлении я беру функции  Грина on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, что у меня появились множители φ−1  i  означает, что я получаю  нормированные функции Грина, и это конец истории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  83  Я дал доказательство формулы для хороших операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из него, как я уже гово-  рил, можно сделать вывод для много более широкого класса операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я объясню  это, вернувшись снова к ситуации, когда есть только один тип частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Дело в том, что в подходе Лемана, Симанчика и Циммермана операторы Ai почти  произвольны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Нужно только чтобы проекция вектора ˆAiθ на одночастичные состоя-  ния была ненулевой, а проекция этого вектора на вектор θ обращалась в ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мне важно то, что в определении on-shell функции Грина эти операторы можно  заменить на операторы A′  i =  �  α(x, t)Ai(x, t)dxdt, которые осуществляют сглажива-  ние.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Легко проверить, что при этом не меняются нормированные on-shell функции  Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это первое замечание.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' А второе замечание состоит в том, что A′  i можно счи-  тать хорошими операторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Дело в том, что я могу взять α(x, t) таким образом,  чтобы носитель ее преобразования Фурье ˆα(p, ω) не пересекался с многочастичным  спектром и не содержал нуля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Я предполагаю, что одночастичный спектр не пере-  секается с многочастичным спектром.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') При этом я автоматически получу хороший  оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Действительно, оператор ˆα осуществляет умножение на α в (p, ω)-представлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Я уже говорил, что переход к сглаженному оператору описывается сверткой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' А в  (p, ǫ)-представлении свертка превращается в умножение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Если рассмотреть спектр  операторов энергии и импульса, то умножение на функцию в (p, ǫ)-представлении  убивает все точки спектра, где эта функция обращается в ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Функция, которую я  рассматриваю, убивает многочастичный спектр, и у меня получается, хороший опе-  ратор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  84  9  Лекция 9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Обобщенные функции Грина  В этой лекции я буду рассматривать обобщенные функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти функции есте-  ственно появляются в формализме L-функционалов, в формализме Келдыша, а так-  же и в других формализмах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хочу показать как инклюзивная матрица рассеяния  выражается через обобщенные функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого я заново докажу формулу  Лемана, Симанчика и Циммермана (LSZ), но доказательство проведу по-другому.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я  покажу как через обычные функции Грина выражается обычная матрица рассеяния,  но доказательства будут построены так, чтобы было ясно, что их можно повторить  и для обобщенных функций Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мои рассуждения во многом повторяют уже приведенные на прошлых лекциях,  но я постараюсь сделать так, чтобы эта лекция не зависела от двух предыдущих  лекций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я немного изменю обозначения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Временную переменную я буду обозначать буквой  τ и писать B(τ, x) вместо B(x, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Состояние ω, как и раньше, будет предполагаться  трансляционно инвариантным и стационарным, соответствующий вектор в предгиль-  бертовом пространстве H будет обозначаться θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обычная функция Грина в состоянии ω — это среднее значение произведения  операторов Bi ∈ A :  N = T(B1(τ1, x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Bn(τn, xn)),  стоящих в порядке уменьшения времени (хронологического произведения).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В обоб-  щенной функции Грина  Gn′n = ω(MN)  есть как хронологическое произведение N, в котором времена уменьшаются, так и  антихронологическое произведение  M = T opp(B′∗  1 (τ ′  1, x′  1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' B′∗  n′(τ ′  n′, x′  n′)),  в котором времена увеличиваются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я докажу, что инклюзивная матрица рассеяния  выражается через асимптотическое поведение обобщенных функций Грина в пред-  ставлении, когда аргументами являются импульс и время.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я объяснял уже, что по  общим свойствам преобразования Фурье это означает, что инклюзивная матрица рас-  сеяния выражается через полюса обобщенных функций Грина, где аргументами явля-  ются энергия и импульс.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Точнее говоря, она совпадает с нормализованной обобщенной  функцией Грина on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Матрица Мёллера  Я ограничиваюсь случаем, когда есть только один тип частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, рас-  сматривается обобщенная векторная функция Φ(p), являющаяся собственной, как  для оператора энергии, так и для оператора импульса:  HΦ(p) = ǫ(p)Φ(p),  PΦ(p) = pΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (72)  (Здесь можно добавить отвечающий типу частиц индекс, и тогда это будет общий  случай.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Еще одно определение, которое я буду использовать — это определение гладкого  оператора.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я говорю, что оператор ˆB, который задается формулой  ˆB =  �  dτdxα(τ, x) ˆA(τ, x),  α ∈ S,  85  это гладкий оператор, если α принадлежит пространству быстро убывающих гладких  функций S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если произвести сдвиг в пространстве и времени,  ˆB(τ, x) =  �  dτ ′dx′α(τ − τ ′, x − x′) ˆA(τ ′, x′),  получится гладкая функция от τ и x, так как свертку можно дифференцировать под  знаком интеграла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это важно: мне придется дифференцировать, и поэтому я требую,  чтобы все рассматриваемые операторы были гладкими.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для оператора ˆB(τ, x) можно взять преобразование Фурье:  ˆB(τ, x) =  �  dpeipx ˆB(τ, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Возникает вопрос о существовании преобразования Фурье, но всегда можно рассмат-  ривать ˆB(τ, p) в смысле обобщенных функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Точнее, этот оператор нужно рас-  сматривать как обычную функцию по τ и обобщенную по p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Исходным пунктом теории было трансляционно инвариантное стационарное со-  стояние ω, к которому может быть применена конструкцию Гельфанда, Наймарка  и Сигала (GNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта конструкция дает предгильбертово пространство H, в котором  лежат одночастичные физические состояния  �  dpf(p)Φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я потребую, чтобы опера-  тор ˆB переводил состояние θ (которое отвечает ω в пространстве H) в одночастичное  пространство:  ˆBθ =  �  dpφ(p)Φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я потребую также, чтобы волновая функция частицы φ(p) не обращалась в ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Это не обязательно, но для упрощения я хочу это потребовать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Тогда я буду гово-  рить, что оператор ˆB хороший.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Мне важно, чтобы хороший оператор был гладким,  но я уже сказал, что все операторы предполагаю гладкими.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Как уже было сказано, любой оператор, в том числе и хороший, можно сдвигать  по x и τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При сдвиге состояния по времени оно умножается на экспоненту от −iHτ,  а по пространству — на экспоненту от iPx, поэтому, действуя оператором ˆB(τ, x) на  θ, мы снова получим одночастичное состояние, только с некоторыми множителями:  ˆB(τ, x)θ =  �  dpe−iǫ(p)τeixpφ(p)Φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если сделать преобразование Фурье только по p, это выражение преобразуется сле-  дующим образом:  ˆB(τ, p)θ = e−iǫ(p)τφ(p)Φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Удобно ввести обозначение:  B(p, τ) = eiǫ(p)τ ˆB(τ, p),  где ǫ(p) — это закон дисперсии для рассматриваемой частицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь поменялись  местами p и τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если рассматривать действие оператора B(p, τ) на θ, экспонента со-  кратится и получится выражение независящее от τ:  B(p, τ)θ = φ(p)Φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь все это можно усреднить с какой-то гладкой функцией f(p), имеющей  компактный носитель: B(f) =  �  dpf(p) ˆB(0, p) =  �  dx ˜f(x) ˆB(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' После сдвига по τ  это выражение преобразуется в следующее:  B(f, τ) = TτB(T−τf)T−τ =  �  dpf(p)B(p, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  86  Очевидно, что B(f, τ)θ не зависит от τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Иными словами, выполняется условие  ˙B(f, τ)θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это – то, с чем я буду работать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Следующие выражения для вектора n-частичного состояния  Ψ(τ) = Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='., fn|τ) = B1(f1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='Bn(fn, τ)θ,  (73)  Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='., fn| ± ∞) =  lim  τ→±∞ Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='., fn|τ)  (74)  повторяют то, что было представлено на прошлой лекции, но здесь опущен индекс,  описывающий тип частиц, и считается, что все операторы Bi хорошие.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Как мы уви-  дим, последнее требование можно ослабить.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Теперь определим понятие in-состояния, устремляя τ → −∞, и аналогично опре-  делим out-состояние, устремляя τ → +∞ (предел берется в гильбертовом простран-  стве ¯H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти пределы описывают процесс рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они существуют при некоторых  условиях.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Самое простое — потребовать, чтобы коммутаторы операторов Bk и ˙Bj (где  точка обозначает дифференцирование по τ) вне зависимости от выбора операторов  B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', Bn стремились к нулю достаточно быстро при τ → ±∞ :  ||[ ˙Bk(fk, τ), Bl(fl, τ)]|| ≤  Ca  1 + |τ|a ,  (75)  где a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Достаточно потребовать, чтобы интеграл по τ от левой части сходился  абсолютно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Если это условие выполнено, то предел (74) существует.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вкратце я воспроизведу доказательство, изложенное на прошлой лекции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Доста-  точно в выражении (73) продифференцировать Ψ(τ) по времени, при этом по правилу  Лейбница образуется n слагаемых, каждое из которых содержит производную ˙Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все  замечательно в том случае, когда эта производная стоит на самом последнем месте:  примененная к θ она дает ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если производная стоит не на последнем месте, то  благодаря условиям на коммутаторы (75) можно переставить ее на последнее место  и получить ноль, но при этом нужно заплатить коммутаторами, которые становятся  малыми при τ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае разность Ψ(τ1) − Ψ(τ2), представимая в виде  интеграла от производной от ˙Ψ(τ), становится мала при τ1, τ2 → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' в силу полноты  гильбертова пространства можно применить признак Коши для обоснования суще-  ствования предела (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Данное рассуждение во многом повторяет то, что уже было на прошлой лекции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Оно еще будет применяться неоднократно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Основное в нем — малость коммутатора.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того чтобы проверить утверждение (75), говорящее, что встречающиеся в выра-  жении (73) операторы почти коммутируют при больших τ, нужно, во-первых, пред-  положить асимптотическую коммутативность (о которой я буду говорить немножко  позже) и, во-вторых, предположить, что фигурирующие в формуле (75) функции  fk(p) не перекрываются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что когда мы рассматривали поведение волновой  функции fk в зависимости от τ, мы видели, что при эволюции в x-пространстве важ-  ную роль играет множество возможных скоростей Ufk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Определяя множество Uf мы  берем градиент от закона дисперсии v = ∇ǫ(p) в тех точках, где f(p) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') После  этого берем замыкания ¯Ufk и требуем, чтобы эти множества не перекрывались.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Грубо  говоря, это означает, что частицы движутся в разных направлениях и существенные  носители волновых функций в координатном пространстве далеко отстоят друг от  друга.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это условие я буду все время накладывать.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Оно выполнено не всегда, но я  требую, чтобы оно было выполнено для семейств функций f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', fn, принадлежащих  всюду плотному подмножеству пространства Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Если наложить условие асимпто-  тической коммутативности, из него вытекает малость коммутаторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  87  Условие асимптотической коммутативности, прежде всего, означает, что комму-  татор операторов Bk при одном и том же времени, но в далеко отстоящих простран-  ственных точках будет мал.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Условие малости можно варьировать, но как минимум  мне нужно, чтобы этот коммутатор был мал в следующем смысле:  ∥[ ˆBk(τ, x), ˆBl(τ, x′)]∥ <  Ca  1 + |x − x′|a ,  (76)  где a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это условие должно быть выполнено не только для самих операторов,  но и для их производных по времени и по пространству.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Данный вариант несколько  отличается от приведенного на прошлой лекции — мне так удобнее.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Вспомним теперь, что все операторы ˆBk гладкие, то есть их можно получать пу-  тем сглаживания некоторых операторов ˆBk =  �  gk(t, x) ˆAk(t, x)dxdt, где gk ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' На  операторы ˆAk можно наложить условие сильной асимптотической коммутативности:  ∥[ ˆAk(t, x), ˆAl(t′, x′)]∥ <  Ca(t − t′)  1 + |x − x′|a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (77)  Это значит, что мы требуем, чтобы коммутаторы убывали быстрее любой степени,  когда пространственное расстояние стремится к бесконечности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В числителе долж-  на стоять полиномиальная функция Ca(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае условия на производные  накладывать не надо — они выводятся из приведенного условия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того чтобы вывести условие (76) из сильной асимптотической коммутативно-  сти, достаточно заметить, что его можно свести к интегралу от выражения, включа-  ющего произведение функций fk с различными индексами, у которых существенные  носители далеки друг от друга.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти интегралы будут малы потому, что при далеких  операторах коммутаторы стремятся к нулю быстрее любой степени.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Сделать оценку  в этом случае очень просто.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Можно ожидать, что сильная асимптотическая коммутативность имеет место в  том случае, когда в спектре гамильтониана есть щель, то есть, спектр принадлежит  лучу (ǫ, +∞), начинающемуся при каком-то положительном значении ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В случае  релятивистской теории это отвечает случаю, когда все частицы имеют массу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ко-  гда же масса нулевая, сильной асимптотической коммутативности не будет, но более  слабые условия (a > 1) выполняются в конформных теориях, где все аномальные  размерности > 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я определю понятие матрицы Мёллера следующим образом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я введу понятие  асимптотического пространства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Асимптотическое пространство Has определяется  как гильбертово пространство, в котором действует фоковское представление кано-  нических коммутационных соотношений:  [a(p), a+(p′)] = δ(p, p′),  [a(p), a(p′)] = [a+(p), a+(p′)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Я работаю с импульсными переменными.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Вместо того, чтобы рассматривать опера-  торные обобщенные функции a(p), a+(p′), можно рассматривать операторы a(f) =  �  dpf(p)a(p), a+(f) =  �  dpf(p)a+(p), где f ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В фоковском пространстве есть опе-  раторы сдвига по пространству и сдвига по времени — это очевидно, если вспомнить,  что это пространство может быть представлено как пополнение прямой суммы сим-  метрических степеней элементарного пространства h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Матрицы Мёллера определим  как отображения S± : Has → ¯H из асимптотического пространства Has в ¯H (в по-  полнение пространства GNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь нельзя ограничиваться предгильбертовым про-  странством — нужно рассматривать все гильбертово пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это отображение  определяется следующим образом:  Ψ(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='., fn| ± ∞) = S±(a+(g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+(gn)|0⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (78)  88  В левой части этой формулы стоит in-состояние — это асимптотическое состояние, за-  висящее от функций fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я хочу интерпретировать их как волновые функции частиц,  но для этого должно выполняться условие, чтобы одночастичное состояние в асимп-  тотическом пространстве Has переходило в одночастичное состояние в пространстве  ¯H с той же волновой функцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Чтобы это было выполнено, нужно наложить усло-  вие, чтобы функции fi и gi асимптотически были связаны между собой соотношением  fi = giφ−1  i , где, напомню, φi ̸= 0 ни в одной точке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Из формулы (78) могут быть выведены различные свойства матрицы Мёллера.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Одно из них заключается в том, что матрица Мёллера коммутирует с операторами  пространственных и временных сдвигов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это почти сразу следует из определений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если рассматривать эволюцию во времени in-состояния, определяемого выражением  (74), то ее можно свести к эволюции входящих в это выражение функций fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это в  точности означает, что асимптотическая динамика in-состояния Ψ сводится к дина-  мике функций fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Покажем, что формула (78) определяет матрицу Мёллера как изометричное отоб-  ражения S± : Has → ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Прежде всего докажем, что матрицы Мёллера не зависят от выбора хороших опе-  раторов B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Напомню, что операторы B в формуле (73) можно взять различными.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Изменим один из этих операторов — тот оператор Bn, который стоит на последнем  месте (заменим его другим хорошим оператором), оставляя функцию fn неизменной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Легко видеть, что при этом ничего не изменяется, потому что изменение последнего  оператора сводится к изменению одночастичного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' То, как меняется одноча-  стичное состояние, должно быть согласованно в асимптотическом пространстве Has  и в пространстве ¯H, поэтому изменение последнего оператора ничего не меняет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь изменим первый оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Поскольку по причине асимптотической комму-  тативности операторы Bi имеют исчезающе малые коммутаторы, я могу оператор B1  переставить на последнее место.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' За это нужно заплатить коммутаторами, но в пре-  деле это ничего не меняет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' После перестановки этого оператора, так же как и любого  другого, на последнее место можно применить предыдущие рассуждения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это дока-  зывает, что матрица Мёллера не зависит от выбора операторов Bi, использовавшихся  в определении (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того, чтобы матрица Мёллера была хорошо определена, в формуле (78) долж-  на быть симметрия по отношению к переменным gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта симметрия присутствует,  потому что коммутаторы малы и есть возможность переставлять операторы a+(gi)  (если есть сильная асимптотическая коммутативность и мы имеем дело с неперекры-  вающимися функциями).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если кроме асимптотической коммутативности наложить требование распадения  корреляций (кластеризации), то можно доказать, что матрицы Мёллера изометрич-  ны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они, напомню, были определены только для набора неперекрывающихся функ-  ций, но, раз они изометричны, то это ограниченные операторы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Их можно распро-  странить на все гильбертово пространство и поэтому операторы S±, представляют  собой изометрические вложения асимптотического пространства Has в пополненное  пространство ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из изометричности следует, в частности, что матрицы Мёллера не  зависят от случайностей построения (изометрический оператор не может быть мно-  гозначным).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Все перечисленные доказательства можно провести, не применяя свойства асимп-  тотической коммутативности, а используя только свойство кластеризации, но с асимп-  тотической коммутативностью все выглядит намного более понятно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  89  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='3  Матрица рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Формула LSZ  Теперь я хочу определить понятие матрицы рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это разумно, если в теории  есть интерпретация в терминах частиц, что подразумевает, что матрицы Мёллера яв-  ляются не только изометричными, но и унитарными операторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае они  определяют изоморфизм асимптотического пространства Has и пространства ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это  означает, грубо говоря, что любое (или почти любое) состояние с течением времени  распадается на частицы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Что-то похожее возникало при обсуждении солитонов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Перейдем теперь к вычислению матричных элементов матрицы рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пусть  имеются какое-то начальное состояние и конечное состояние, которые обозначаются,  соответственно, буквами i и f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Матрица рассеяния определяется формулой S = S∗  +S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Обращаю внимание на то, что S− задает отображение Has → ¯H, а S∗  + действует в про-  тивоположном направлении, и поэтому матрица рассеяния — это оператор в асимпто-  тическом пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я должен брать ее матричные элементы между состояниями  в асимптотическом пространстве: ⟨f|S|i⟩ = ⟨f|S∗  +S−|i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Рассмотрим состояния в асимптотическом пространстве, которые получаются из  фоковского вакуума с помощью применения некоторого количества операторов a+:  |i⟩ = a+(g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+(gn)|0⟩,  |f⟩ = a+(g′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+(g′  m)|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Выражая матричный элемент через операторы B, мы приходим к следующей фор-  муле:  ⟨f|S|i⟩ =  lim  τ′  k→+∞,  τj →−∞  ⟨θ|B′∗  m(f ′  m, τ ′  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B′∗  1 (f ′  1, τ ′  1)B1(f1, τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='Bn(fn, τn)|θ⟩,  (79)  где fj = gjφ−1  j , f ′  k = g′  kφ′−1  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь использованы формулы (73), (74) с небольшой  поправкой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Раньше при определении in-вектора считалось, что все времена в форму-  ле (73) одинаковы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Этого достаточно для определения матрицы Мёллера, но здесь  удобно (хотя и не обязательно) считать, что времена разные, но все стремятся, со-  ответственно, к плюс или минус бесконечности — предел при этом не изменится.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Доказательство этого может быть легко получено, но я его проводить не буду.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Очень важное наблюдение заключается в том, что в формуле (79) можно все вре-  мена упорядочить в порядке убывания.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ясно, что τ ′ я могу считать большим, чем τ,  потому что τ ′ → +∞, а τ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если же брать два времени τi1, τi2, то коммута-  тор соответствующих операторов исчезающе мал, поскольку соответствующие функ-  ции fi1, fi2 не перекрываются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае я могу поставить операторы в порядке  убывания времен и считать, что здесь появляется хронологическое произведение или,  иными словами, появляется функция Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того же, чтобы все было более аккуратно, я перейду к стандартному обоб-  щенному базису |i⟩ = |p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', pn⟩ , |f⟩ = |p′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', p′  m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Собственно говоря, в физи-  ке матричные элементы матрицы рассеяния нужно брать именно в таком базисе,  в котором имеются частицы с заданными импульсами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Перепишем теперь форму-  лу (79) в этом базисе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Стоящие в этой формуле операторы могут быть представ-  лены в виде B(f, τ) =  �  dpf(p)B(p, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' С другой стороны, напомню, что у опера-  тора B(p, τ) можно поменять местами аргументы, и за это заплатить экспонентой  B(p, τ) = eiǫ(p)τ ˆB(τ, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В результате для матричного элемента в импульсном базисе получается следую-  щее выражение:  ⟨f|S|i⟩ =  lim  τ ′  k→+∞,τj→−∞  ω(  �  e−iǫ(p′  k)τ ′  k( ¯φ′  k)−1 ˆB′∗  k (τ ′  k, p′  k)  �  eiǫ(pj)τj(φj)−1 ˆBj(τj, pj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  90  Если в этой формуле вынести численные множители за знак (линейного) функци-  онала ω, останется только ω( � ˆB′∗  k (τ ′  k, p′  k) ˆBj(τj, pj)) и можно считать, что времена  упорядочены, то есть, появляется функция Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это — то, что мне было нужно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Полученное выражение будет иметь некоторый предел, и это означает, что мат-  рица рассеяния выражена через асимптотику функции Грина в (τ, p)-представлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Как уже объяснялось, асимптотика в (ǫ, p)-представлении определяется полюсами по  энергетической переменной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это (для хороших операторов) — в точности то, что есть  в формуле LSZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Матрицы Мёллера коммутируют со сдвигами по времени и пространству и, сле-  довательно, с операторами энергии и импульса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если теория имеет интерпретацию в  терминах частиц, то в пространстве ¯H совместный спектр ˆH и ˆP совпадает со спек-  тром свободного бозона, поскольку должны выполняться соотношения:  a+  out(f)S+ = S+a+(f),  a+  out(g) =  lim  τ→+∞ B(f, τ),  где g = fφ, Bθ = Φ(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В доказательстве существования предела, определяющего in- (out)-состояние, пред-  положение о том, что ˆBj являются хорошими операторами, использовалось только  для вывода утверждения, что ˙ˆBj(f, τ)θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если теория имеет интерпретацию в  терминах частиц, можно доказать существование предела для любых гладких опера-  торов ˆBj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Будем предполагать, что проекция ˆBjθ на одночастичное пространство имеет вид  Φ(φj), где функция φj не обращающаяся в ноль.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Асимптотическая коммутативность  позволяет переместить множитель с производной по времени вправо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Докажем, что  полученное выражение мало для большого τ (достаточно доказать, что это — сумми-  руемая функция от τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я не предполагаю, что ˆBjθ, лежит в одночастичном пространстве, но мне важно,  чтобы при проектировании на одночастичное пространство получалось Φ(φj), где φj  — не обращающиеся в 0 функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В то время как для хорошего оператора требова-  лось, чтобы это выражение было равно одночастичному состоянию, здесь требуется  только чтобы таковой была проекция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Все будет хорошо, если только выполняется  условие ˙ˆBj(f, τ)θ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Перейдем теперь к выводу этого условия.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если есть представление в терминах частиц, любой вектор можно разложить по  векторам a+  in(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+  in(pr)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Я мог бы это сделать в асимптотическом пространстве  ¯  H⊣∫, и тогда у меня не было бы индекса in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Cамо пространство ¯H изоморфно асимп-  тотическому, если есть интерпретация в терминах частиц.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В таком случае имеет место  следующее разложение (справедливое для любого вектора):  �  r≥0  �  dp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='dprcr(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', pr)a+  in(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+  in(pr)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если оператор ˆBj гладкий, то функции cr(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', pr) для ˆBjθ также будут гладкими  (cr ∈ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это легко понять, так как гладкость оператора ˆBj подразумевает усреднение  и, соответственно, функция cr представляется интегралом, позволяющим доказать  гладкость.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для простоты формул предположим, что для среднего значения от глад-  кого оператора ˆBj выполняется условие ⟨θ| ˆBj|θ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тогда мы можем представить  ˆBj(f, τ)θ в виде  ˆBj(f, τ)θ =  �  dpφj(p)f(p)a+  in(p)θ+  �  r≥2  �  dpdp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='dpre−iτ(ε(p1)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='ε(pr)−ε(p))cr(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', pr)f(p)a+  in(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+  in(pr)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  91  В этой формуле суммирование должно быть по всем r, но при r = 1 в экспоненте  происходит сокращение, и зависимость от времени исчезает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' По этой причине выделе-  но суммирование по r ≥ 2, а случай r = 1 соответствует одночастичному состоянию,  для которого ответ известен, и он здесь представлен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Самое важное состоит в том, что  при τ → ∞ у слагаемых, для которых r ≥ 2 в экспоненте стоит большой показатель,  в то время как cr(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', pr) — хорошая гладкая функция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  После дифференцирования по τ показатель в экспоненте останется и при этом  возникнет некий множитель, который будет подавлен экспонентой, а так как cr —  хорошая функция и показатель в экспоненте большой, то в пределе τ → ∞ вклад  таких слагаемых исчезает.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Таким образом, формула LSZ доказана не только для хороших операторов, но и  для всех гладких операторов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Практически все операторы, что встречается в физике,  являются гладкими.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') На этом и завершается доказательство формулы LSZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='4  Инклюзивная матрица рассеяния  Теперь я хочу применить полученные результаты для того, чтобы выразить инклю-  зивную матрицу рассеяния в терминах обобщенных функций Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для этого я буду  рассматривать in-состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ранее я их рассматривал как векторы в гильбертовом  пространстве, но теперь я их буду рассматривать как состояния в смысле алгебраи-  ческого подхода.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Такие состояния — это положительные функционалы на алгебре, и  у меня есть связь этих функционалов с векторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если я применяю некоторый опе-  ратор, например B(f, τ) в гильбертовом пространстве, то в пространстве состояний я  должен применять оператор вида L(g, τ) = ˜B(f, τ)B(f, τ), где f = gφ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Напоминаю,  что элемент алгебры определяет два оператора на функционалах: я могу либо умно-  жать аргумент справа, либо умножать аргумент слева.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В последнем случае берется  сопряженный оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  В терминах таких операторов in-состояние, рассматриваемое как положительный  функционал, может быть записано в следующем виде:  ν =  lim  τ→−∞ L(g1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(gn, τ)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь рассмотрим следующее выражение:  ⟨1|L(g′  1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(g′  n′, τ ′)L(g1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(gn, τ)|ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В правой части формулы я, подействовав операторами L, получил in-состояние  при τ → −∞, а в левой части вычисляю какие-то значения при τ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это как раз  то, что делается при рассмотрении инклюзивных матриц рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я рассматриваю  in-состояние и вычисляю что-то при больших временах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Можно доказать существо-  вание предела Q этого выражения при τ ′ → +∞, τ → −∞ в предположении, что все  функции g′  i и gj не перекрываются:  Q =  lim  τ ′→+∞⟨1|L(g′  1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(g′  n′, τ ′)ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (Напомню, что асимптотическая коммутативность у меня всегда предполагается.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Используя соотношение ⟨1|σ⟩ = σ(1) и формулу limτ ′→+∞ B(f, τ ′) = a+  out(g), полу-  ченную на одной из предыдущих лекций, мы приходим к следующему выражению:  Q = ν(aout(g′  n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='aout(g′  1)a+  out(g′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='a+  out(g′  n′))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Через определенное таким образом выражение Q = Q(g′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', g′  n′, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='., gn) можно  вычислить инклюзивные сечения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Я назову это выражение инклюзивной матрицей  92  рассеяния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Здесь есть проблема, связанная с тем, что это нелинейное выражение по g  и g′ — это квадратичное (точнее, эрмитово) выражение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Всякое квадратичное выра-  жение можно заменить билинейной формой, а эрмитово выражение можно заменить  полуторалинейной формой — по одной переменной она будет линейной, по другой —  антилинейной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того, чтобы получить выражение, которое будет линейным (или где-то анти-  линейным), я введу обозначение L(˜g, g, τ) = ˜B( ˜f, τ)B(f, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Тут разведены перемен-  ные f и ˜f, и теперь то, что раньше рассматривалось как эрмитово выражение, будет  рассматриваться как полуторалинейное выражение от удвоенного числа переменных:  ρ(˜g′  1, g′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ˜g′  n′, g′  n′, ˜g1, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ˜gn, gn) =  lim  τ′→+∞,  τ→−∞  ⟨1|L(˜g′  1, g′  1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(˜g′  n′, g′  n′, τ ′)L(˜g1, g1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='L(˜gn, gn, τ)|ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Это выражение тоже будет называться инклюзивной матрицей рассеяния и его я  буду выражать через обобщенные функции Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Расписав его, приходим к формуле  ρ(˜g′  1, g′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ˜g′  n′, g′  n′, ˜g1, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ˜gn, gn) =  lim  τ′→+∞,  τ→−∞  ⟨1|B(f ′  1, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(f ′  n′, τ ′) ˜B( ˜f ′  n′, τ ′) ˜B( ˜f ′  1, τ ′)×  B(f1, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(fn, τ) ˜B( ˜fn, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ˜B( ˜f1, τ)|ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мне удобнее пользоваться несколько более общей формулой  ρ(˜g′  1, g′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ˜g′  n′, g′  n′, ˜g1, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ˜gn, gn) =  lim  τ′  j ,˜τ′  j→+∞,  τi,˜τi→−∞  ⟨1|B(f ′  1, τ ′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(f ′  n′, τ ′  n′) ˜B( ˜f ′  n′, ˜τ ′  n′) ˜B( ˜f ′  1, ˜τ ′  1)×  B(f1, τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='B(fn, τn) ˜B( ˜fn, ˜τn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' ˜B( ˜f1, ˜τ1)|ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мы можем считать, что в этом выражении времена упорядочены.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Часть времен  стремится к +∞, другая часть к −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Внутри каждой группы в силу асимптотической  коммутативности мы можем переставлять множители в любом порядке, в частности,  в порядке убывания времен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Применив формулу (36) мы видим, что то, что стоит  под знаком предела, можно выразить через обобщенную функцию Грина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Инклю-  зивная матрица рассеяния выражается через временную асимптотику этой функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Такие же рассуждения были раньше для обычной матрицы рассеяния, только число  аргументов удвоилось.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  93  10  Лекция 10  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='1  Удаление лишних состояний  В этой лекции я собираюсь рассказать о том, как можно представить квантовую  механику как классическую механику, в которой мы имеем возможность измерять  только часть наблюдаемых.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' С точки зрения физики это вполне естественно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Наши  приборы позволяют измерять какие-то наблюдаемые, но не исключено, что могут  появиться какие-то более совершенные приборы, позволяющие измерять и другие  вещи.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Я буду исходить из геометрического подхода к квантовой теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В геометриче-  ском подходе, описывая физическую теорию, мы начинаем с множества состояний,  представляющего собой ограниченное выпуклое множество C0-подмножество тополо-  гического векторного пространства L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Операторы эволюции должны принадлежать  некоторой группе V, которая состоит из автоморфизмов пространства состояний C0  (либо все автоморфизмы, либо часть их).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Оператор эволюции σA(t) удовлетворяет  уравнению  dσA(t)  dt  = AσA(t),  (80)  где A ∈ Lie(V) — это элемент алгебры Ли группы V (“гамильтониан”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это уравнение  должно иметь решение (то есть “гамильтониан” должен порождать однопараметриче-  скую подгруппу σA(t) группы V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Хотелось бы сказать, что “гамильтонианы” — это и  есть наблюдаемые в геометрическом подходе, но наблюдаемые должны давать какие-  то числа, и поэтому я буду говорить, что наблюдаемая- это пара (A, a) , где A ∈ Lie(V)  — это “гамильтониан”, а a -это линейный функционал, инвариантный относительно  группы σA(t), порожденной оператором A (это означает, что удовлетворяется условие  a(Az) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В обычной квантовой механике наблюдаемая определяется самосопряженным опе-  ратором ˆA, “гамильтониан” действует на матрицы плотности как коммутатор (с точ-  ностью до множителя i), функционал a определяется формулой a(K) = Tr ˆAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Группа V естественным образом действует на наблюдаемые: A преобразуется по  присоединенному представлению, a — как функция на L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Теперь исследуем, нет ли в нашей теории лишних состояний?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Когда есть два состо-  яния x, y ∈ C0, такие, что a(x) = a(y) для всякой наблюдаемой (A, a), то мы говорим,  что из этих двух состояний можно оставить только одно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если отождествить те состо-  яния, которые дают один и тот же ответ для всех наблюдаемых, то в результате будет  получена новая теория без лишних состояний эквивалентная, по существу, исходной  теории.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='2  Квантовая механика из классической механики  Применим эти соображения к случаю, когда в качестве исходной берется класси-  ческая теория.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этой теории чистые состояния — это точки фазового пространства  (симплектического многообразия) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Смешанные состояния — это распределения ве-  роятностей на M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' каждое смешанное состояние может быть однозначно представлено  как смесь чистых состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Физические наблюдаемые — это вещественные функ-  ции на M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Наблюдаемая a задает векторное поле A на M как гамильтоново вектор-  ное поле с гамильтонианом a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Идентифицируя векторное поле с дифференциальным  оператором первого порядка, мы можем выразить A в терминах скобки Пуассона:  Af = {a, f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Мы предполагаем, что, интегрируя это векторное поле, получаем од-  нопараметрическую группу σA(t) канонических преобразований (симплектоморфиз-  94  мы) многообразия M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта группа действует также и на смешанные состояния, и на  наблюдаемые, описывающие временную эволюцию этих объектов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Уравнение дви-  жения для плотности распределения вероятностей (уравнение Лиувилля) имеет вид  dρ  dt = −{a, ρ}, а уравнение движения для наблюдаемых имеет вид df  dt = {a, f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предположим, что наши устройства способны видеть только часть наблюдаемых  объектов и что множество Λ “наблюдаемых наблюдаемых” представляет собой ли-  нейное пространство, замкнутое относительно скобки Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Будем индексиро-  вать это множество элементами алгебры Ли, обозначаемой g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Отображение γ → aγ,  переводящее γ ∈ g в aγ ∈ Λ является изоморфизмом алгебр Ли g и Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Гамильтоновы векторные поля Aγ с гамильтонианами aγ определяют действие  алгебры Ли g на M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Предположение о том, что векторные поля Aγ генерируют одно-  мерные подгруппы, означает, что это действие индуцируется действием односвязной  группы Ли G, имеющей g в качестве алгебры Ли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Приведенные выше соображения могут быть применены также и к бесконечномер-  ным симплектическим многообразиям и к бесконечномерным алгебрам и группам Ли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Однако в бесконечномерном случае эти соображения не являются строгими.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Определим отображение моментов µ из M в g∨ как отображение x → µx, где  µx(γ) = aγ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Здесь x ∈ M, γ ∈ g, а g∨ обозначает пространство линейных функцио-  налов на g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Это отображение является G-эквивариантным относительно коприсоеди-  ненного действия G на g∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Иными словами оно коммутирует с преобразованиями из  группы G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Для каждого состояния классической системы (для каждого распределе-  ния вероятностей ρ на M) определим точку ν(ρ) ∈ g∨ как интеграл от µx по x ∈ M с  мерой ρ:  ν(ρ) =  �  x∈M  µxdρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Точка ν(ρ) принадлежит выпуклой оболочке N из µ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Выпуклая оболочка под-  множества E топологического векторного пространства определяется как наимень-  шее выпуклое замкнутое множество, содержащее E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Группа G естественным образом действует на пространстве классических состоя-  ний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Из G-эквивариантности отображения моментов следует, что отображение ν явля-  ется G-эквивариантным отображением этого пространства в g∨ с коприсоединенным  действием G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Будем говорить, что два классических состояния (два распределения вероятностей  ρ и ρ′) эквивалентны, если  �  x∈M  aγ(x)dρ =  �  x∈M  aγ(x)dρ′  (81)  для каждого γ ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Другими словами, мы говорим, что два состояния эквивалентны,  если вычисления с этими состояниями дают одинаковые результаты для каждого  гамильтониана aγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Наши устройства не могут различать эти два состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Докажем теперь следующее утверждение: два состояния ρ и ρ′ эквивалентны  тогда и только тогда, когда ν(ρ) = ν(ρ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Сначала заметим, что для каждого γ ∈ g  ν(ρ)(γ) =  �  x∈M  µx(γ)dρ =  �  x∈M  aγ(x)dρ  и точно так же  ν(ρ′)(γ) =  �  x∈M  µx(γ)dρ′ =  �  x∈M  aγ(x)dρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  В классической теории где допустимы только гамильтонианы из множества Λ = {aγ},  где γ ∈ g, должны быть отождествлены эквивалентные состояния (мы исключаем  95  избыточные состояния).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Отображение ν индуцирует биективное отображение про-  странства классов эквивалентности на множество N, полученное в виде выпуклой  оболочки множества µ(M) (“квантовые состояния”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' G-эквивариантность отображе-  ния ν означает, что эволюция классических состояний согласуется с эволюцией кван-  товых состояний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Применим теперь наши конструкции к комплексному проективному пространству  CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы определяем это пространство как сферу ||x|| = 1 в комплексном гильбертовом  пространстве H с отождествлениями x ∼ λx, где λ ∈ C, |λ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Группа U унитарных  операторов действует транзитивно на CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Существует единственная (с точностью до  постоянного множителя) U-инвариантная симплектическая структура на этом про-  странстве;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' это позволяет рассматривать комплексное проективное пространство как  однородное симплектическое многообразие.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Предположим, что мы можем наблюдать только гамильтонианы вида aC(x) =  ⟨x, Cx⟩ где C — самосопряженный оператор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Множество Λ таких гамильтонианов  является алгеброй Ли относительно скобки Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эта алгебра Ли изоморфна ал-  гебре Ли g самосопряженных операторов, где операция определяется как коммутатор,  умноженный на i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Однопараметрическая группа из унитарных операторов, соответ-  ствующих гамильтониану aC, задается формулой σ(t) = e−iCt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Отображение моментов преобразует точку x в линейный функционал на про-  странстве самосопряженных операторов, отображающий оператор C в TrKxC, где  Kx является проекцией H на вектор x, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Kx(z) = ⟨z, x⟩x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Напомню, что в наших  обозначениях точки комплексного проективного пространства представлены норми-  рованными векторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Выпуклая оболочка образа отображения моментов состоит  из положительно определенных самосопряженных операторов с единичным следом  (т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' она состоит из матриц плотности).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Мы видим, что, применяя нашу общую конструкцию к комплексному проектив-  ному пространству, мы получаем обычную квантовую механику.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае наши  соображения близки к “нелинейной квантовой механике” Вайнберга, предложившего  рассматривать классическую теорию на CP как деформацию квантовой механики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Богатый источник примеров описанной ранее конструкции дают орбиты группы  G в коприсоединенном представлении (в пространстве g∨ двойственном к алгебре Ли  g группы G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Эти орбиты играют важную роль в теории представлений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Они явля-  ются однородными симплектическими многообразиями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Квантуя их, можно получать  унитарные представления группы G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Для того чтобы ввести на орбите структуру симплектического многообразия, нуж-  но заметить, что для функций на g∨ определена скобка Пуассона.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Элементы алгебры  Ли g можно рассматривать как линейные функции на g∨ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' для них скобка Пуассо-  на определяется как операция в алгебре Ли.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Пользуясь свойствами скобки Пуассо-  на, можно определить скобку для произвольных гладких функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Пространство  g∨ получает структуру Пуассонова многоообразия, но это многообразие не являет-  ся симплектическим, потому что скобка Пуассона вырождена.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ограничивая скобку  Пуассона на орбиту, мы получаем невырожденную скобку, определяющую симплек-  тическую структуру.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Элементы алгебры Ли g определяют семейство Λ функций на орбите;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' мы при-  меним к нему описанную выше конструкцию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Она сильно упрощается, потому что  в рассматриваемом случае отображение моментов - это просто вложение орбиты в  g∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Мы видим, что считая в классической теории допустимыми только гамильтониа-  ны из семейства Λ мы получаем теорию, в которой множеством состояний является  выпуклая оболочка орбиты.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Группу V можно считать изоморфной группе G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Проиллюстрируем приведенные выше конструкции в случае, когда G является  группой U унитарных преобразований гильбертова пространства H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' В этом случае  96  мы можем идентифицировать элементы алгебры Ли g с ограниченными самосопря-  женными операторами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' При подходящем выборе топологии в g можно отождествить  двойственное пространство g∨ с линейным пространством самосопряженных опера-  торов имеющих след.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Для упрощения обозначений предположим, что гильбертово  пространство конечномерно, однако наши соображения могут быть применены и в  бесконечномерном случае.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Рассмотрим орбиты U в пространстве g∨ (в коприсоединенном представлении).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если dim H = n, орбита индексируется различными вещественными числами λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', λr  (собственными значениями операторов, принадлежащих орбите) и неотрицательными  целыми числами k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', kr (кратностями собственных значений).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Кратности долж-  ны удовлетворять условию k1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='kr = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Стационарная группа U(n) точки ор-  биты изоморфна прямому произведению групп U(ki), поэтому орбита гомеоморфна  U(n)/U(k1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' × U(kr) (многообразию флагов).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Если r = 2, то орбита гомеоморфна грассманиану.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Если r = 2, k1 = n − 1, k2 = 1,  мы получаем комплексное проективное пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Грассманиан Gk(H) опреде-  ляется как пространство всех k-мерных подпространств пространсва H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Его можно  рассматривать как симплектическое U-многообразие.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Ортонормированный базис k-  мерного подпространства определен с точностью до преобразования из унитарной  группы U(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Это означает, что точки Gk(H) описываются ортонормированными си-  стемами векторов φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', φk с идентификацией φ′  i ∼ ul  iφl, где ul  i — унитарная матрица.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Some problems (Mostly mathematical problems related to the material of my lectures)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' A complex vector space E is equipped with non-negative scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that  we can obtain pre Hilbert space factorizing E with respect to vectors with⟨x, x⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Check that these vectors constitute a linear subspace of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Density matrices are defined as positive-definite self-adjoint operators having unit trace  and acting in complex Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that the set of density matrices is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Check that extreme points of this set  are one-dimensional projectors KΨ(x) = ⟨x, Ψ⟩Ψ where ||Ψ|| = 1 (they are in one-to-one  correspondence with non-zero vectors of Hilbert space with identification Ψ ∼ λΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that the set of density matrices in two-dimensional Hilbert space is a three-  dimensional ball and set of of its extreme points is a two-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  The linear envelope T of the set of density matrices is the space of all self-adjoint  operators belonging to trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (A self-adjoint operator belongs to trace class if it  has discrete spectrum and the the series of its eigenvalues is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') We  consider T as a normed space with the norm ||T|| = � |λk| where λk are eigenvalues of  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' By definition an automorphism of the set of density matrices is a bicontinuous linear  operator in T generating a bicontinuous map of the set density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (One says that a  map is bicontinuous if it is continuous and has a continuous inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') It is obvious that a  unitary operator specifies an automorphism of the set of density matrices by the formula  T → UTU −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='Prove that automorphisms of the set of density matrices are in one-to-one correspondence  with unitary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  I do not know how to solve this problem (this does not mean that it is difficult, I did  not try).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Maybe this fact is proved somewhere, but I do not know any reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  In the next problems the term "operator"means "linear operator".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' It is convenient to  define etA where A is an operator as a solution of the equation  dU(t)  dt  = A · U(t)  97  with initial condition U(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' If (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', An) is a family of commuting operators we  define e  � tiAi as a product et1A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='etnAn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that eiaP = Tℏa, where P denotes the momentum operator P = ℏ  i ∇ and Ta  stands for translation operator transforming the function f(x) into a function f(x + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us define an operator CA acting in the space of operators by the formula CA(X) =  [A, X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (For definiteness one can assume that A and X are bounded operators in Hilbert  space, but this assumption is not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Prove that  etCA(X) = etAXe−tA,  or equivalently  etAXe−tA = X + t[A, X] + t2  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' [A, [A, X]] + t3  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' [A, [A, [A, X]]] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Differentiate these equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us assume that the commutator of operators X and Y is a number C (or, more  generally, an operator C, commuting with X and Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that  eXeY = eX+Y e  1  2C,  eXeY = eCeY eX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us assume that for an operator A acting in Banach space the norms of operators  etA where t ∈ R are uniformly bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' sup−∞<t<+∞ ||etA|| is finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that all  eigenvalues ofA are purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let A denote an operator acting in finite-dimensional complex vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Assume  that the norms of operators etA where t ∈ R are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that the  operator A is diagonalizable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' there exists a basis consisting of eigenvectors of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Use Jordan normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that all Jordan cells are one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us consider Grassmann algebra with generators ǫ1, ǫ2, ǫ3, ǫ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Calculate  (ǫ1 + ǫ2)(ǫ2 + ǫ3)(ǫ3 + ǫ4),  �  (ǫ1 + ǫ2)(ǫ2 + ǫ3)(ǫ3 + ǫ4)(ǫ4 + ǫ1)d4ǫ,  ∂  ∂ǫ2  (ǫ1ǫ2ǫ3),  cos(ǫ1ǫ2 + ǫ3ǫ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='Let us consider a unital associative algebra with commuting generators x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', xn and  anticommuting generators ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', ξn (tensor product of polynomial algebra and Grassmann  algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Prove that d2 = 0 and ∆2 = 0 where  d =  �  ξi  ∂  ∂xi  ,  ∆ =  � ∂  ∂xi  ∂  ∂ξi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Weyl algebra is defined as unital associative algebra with generators a∗  k, ak obeying  CCR( [ak, a∗  l ] = δk,l, [ak, al] = [a∗  k, a∗  l ] = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' An involution ∗ in Weyl algebra transforms ak  into a∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Fock representation of Weyl algebra=representation with cyclic vector |0⟩ obeying  ˆak|0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Scalar product is defined by the condition that ˆa∗  k is adjoint to ˆak  98  Fock space= completion of the space of Fock representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Poisson vector is defined by the formulaΨλ = eλˆa∗|0⟩ where λˆa∗ = � λkˆa∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  a) Check that Poisson vector is an eigenvector of the operator ˆak  b) Find scalar product of two Poisson vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us consider the Hamiltonian ˆH(t) = ω(t)ˆa∗ˆa (harmonic oscillator with time-  dependent frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  a) Let us suppose that ω(t) = ω0 + ω exp(−αt) for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Here α is a small positive  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=') Calculate the evolution of matrix entries of the density matrix in ˆH-representation  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' express matrix entries of K(t) in terms of matrix entries of K(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Here ˆH = ˆH(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  b) Assuming that ω is a random variable with given mean value and dispersion uniformly  distributed on some interval calculate the average of matrix entries of the density matrix  K(t) for t = const  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' A molecule is placed near a microwave oven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Give a rough estimate of decoherence  time for this molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Decoherence appears if the phase factors entering the expressions for non-diagonal  matrix entries of density matrix are changed significantly by the electric field of the oven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  You can calculate this change in perturbation theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' the first order contribution comes  from dipole momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' You can find the information about dipole momenta of ground  state and excited states on the web;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' you need only the order of magnitude of these momenta  and of electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us define the L-functional corresponding to the density matrix K by the the  formula  LK(α) = tre−αˆa∗eα∗ˆaK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  (We consider the case when there is only one degree of freedom, [ˆa, ˆa∗] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  Calculate the L-functional corresponding to the coherent state (to the normalized  Poisson vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Reminder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Every normalized vector Φ defines a density matrix KΦ and trAKΦ =  ⟨AΦ, Φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Coherent state is a normalized eigenvector of ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us assume that supp(φ) is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Then for large |τ| we have  |(Tτφ)(x)| < Cn(1 + |x|2 + τ 2)−n  where x  τ /∈ Uφ, the initial data φ = φ(x) is the Fourier transform of φ(k), and n is an  arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  Here supp(φ) is the closure of the set of points were φ(k) ̸= 0, Uφ is a set of all points  of the form ∇ǫ(k) where k belongs to a neighborhood of supp(φ),  Tτφ)(x) =  �  dkeikx−iǫ(k)τφ(k),  the function ǫ(k) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' A generalized function ρ(ǫ) is a sum of of the function  A  ǫ−E−i0 and a square integrable  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Find asymptotic behavior of its Fourier transform ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Let us consider a system with the Hamiltonian  ˆH =  �  dpǫ(p)ˆa∗(p)ˆa(p) and  momentum operator ˆP =  �  dpp)ˆa∗(p)ˆa(p) where ˆa, ˆa∗ obey CCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  a) Calculate time and space translations in the formalism of L-functionals  b) Find the operators ˆa(τ, p), ˆa∗(τ, p) and corresponding operators acing on L-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  c) Prove that the L-functional ω = exp(−  �  dpα∗(p)n(p)α(p)) specifies a stationary  translation-invariant state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' (Here n(p) is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=')  d) Calculate two-point GGreen functions in (τ, p)- and (ε, p)-representations for the  state ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  99  Список литературы  [1] Schwarz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Geometric approach to quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' SIGMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Symmetry, Integrability  and Geometry: Methods and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' 2020 Apr 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='16:020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  [2] Schwarz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Geometric and algebraic approaches to quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' arXiv  preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='09176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  [3] Schwarz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Scattering theory in algebraic approach to quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Associative  algebras, arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='08553, 2021  [4] Schwarz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Scattering theory in geometric approach to quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' arXiv  preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='08557, 2021  [5] Schwarz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Scattering theory in algebraic approach to quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Jordan  algebras (in preparation)  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Shvarts, New formulation of quantum theory, Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Nauk SSSR, 173, 793  (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  [7] Schwarz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Inclusive scattering matrix and scattering of quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' Nuclear  Physics B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content=' 2020 Jan 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='950:114869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
+page_content='  100' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQfTge-/content/2301.03804v1.pdf'}
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+Positivity-preserving entropy filtering for the ideal
+magnetohydrodynamics equations
+T. Dzanica,∗, F. D. Witherdena
+aDepartment of Ocean Engineering, Texas A&M University, College Station, TX 77843
+A R T I C L E I N F O
+Keywords:
+Discontinuous spectral element
+Ideal magnetohydrodynamics
+Shock capturing
+Positivity-preserving
+Entropy filtering
+A B S T R A C T
+In this work, we present a positivity-preserving adaptive filtering approach for discontinuous
+spectral element approximations of the ideal magnetohydrodynamics equations. This approach
+combines the entropy filtering method (Dzanic and Witherden, J. Comput. Phys., 468, 2022) for
+shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-
+free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting
+approach is introduced to guarantee that the positivity and entropy constraints remain satisfied
+by the discrete solution. Furthermore, a computationally efficient algorithm for solving the op-
+timization process for this nonlinear filtering approach is presented. The resulting scheme can
+robustly resolve strong discontinuities on general unstructured grids without tunable parameters
+while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown
+in numerical experiments on various problems including extremely magnetized blast waves and
+three-dimensional magnetohydrodynamic instabilities.
+1. Introduction
+The transport and interaction of a non-resistive conducting fluid and its electromagnetic field remain extensively
+investigated phenomena as they are instrumental in various applications ranging from the study of astrophysical accre-
+tion disks [1] and supernova remnants [2] to magnetic confinement fusion [3] and plasma physics [4]. These strongly
+nonlinear effects are governed by the equations of ideal magnetohydrodynamics (MHD), which are composed of a
+combination of the Euler equations of gas dynamics and Maxwell’s equations of electromagnetism. From this formu-
+lation, a strong coupling between the magnetic field and the conducting fluid can be observed, where the magnetic
+field induces a current in the fluid which, in turn, gives rise to a second, induced magnetic field. This interaction
+can introduce multi-scale, multi-physics behavior in the system, such that magnetohydrodynamic flows can become
+exceedingly complex.
+As a result of this complexity, the robust and accurate numerical approximation of ideal MHD can present many
+challenges. Since hyperbolic systems are known to produce discontinuities even with smooth initial conditions [5],
+the numerical scheme must be able to robustly resolve these discontinuities which, in the case of MHD, come in the
+form of hydrodynamic and magnetic shocks and contact waves. Furthermore, the approximation of the ideal MHD
+equations also requires an intrinsic constraint on the solution in the form of a solenoidal magnetic field which may
+not be satisfied by the scheme even if the magnetic field is initially solenoidal. Without a mechanism to enforce
+this constraint, unphysical dynamics can arise in the solution which can lead to numerical instabilities. The standard
+numerical schemes for approximating MHD flows are finite difference and finite volume methods, whose properties
+and robustness are well-established in the literature [6–11]. However, they possess certain drawbacks in that they
+are either difficult to extend to complex domains with unstructured grids or cannot recover high-order accuracy in a
+computationally efficient manner.
+A particular class of schemes which have more recently grown in popularity are high-order discontinuous spec-
+tral element methods (DSEM) as they possess the geometric flexibility of finite volume methods while retaining the
+arbitrarily high-order accuracy and efficiency of spectral methods. As such, they provide a promising avenue for
+significantly decreasing the computational cost and expanding the viability of simulating complex MHD problems.
+However, due to the presence of discontinuities in MHD, DSEM approximations of these systems may introduce spu-
+rious oscillations in the solution in the form of Gibbs phenomena. Without proper treatment, these oscillations can
+∗Corresponding author
+tdzanic@tamu.edu (T. Dzanic)
+ORCID(s): 0000-0003-3791-1134 (T. Dzanic); 0000-0003-2343-412X (F.D. Witherden)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 1 of 22
+arXiv:2301.03129v1  [math.NA]  9 Jan 2023
+
+aPositivity-preserving entropy filtering for the ideal MHD equations
+result in unphysical predictions or the failure of the numerical scheme altogether. To extend to use of DSEM to MHD,
+various numerical stabilization techniques have been proposed, ranging from artificial viscosity methods [12, 13] to
+limiting-type approaches [14, 15]. While these various methods may be sufficient to stabilize the solution in many
+cases, they may not guarantee that the solution will abide by physical constraints, may require problem-dependent tun-
+able parameters, can be computationally inefficient for general unstructured grids, or may be excessively dissipative in
+smooth regions of the flow.
+There is significant interest in the design of numerical schemes that are “provably robust” in the sense that they
+can guarantee that the solution will abide by certain physical constraints even in the presence of features such as
+discontinuities, the quintessential examples being positivity-preserving schemes for gas dynamics which guarantee the
+positivity of the density and internal energy/pressure. For DSEM, this property is typically achieved through some
+form of nonlinear limiting or filtering [14, 16–18]. However, designing schemes that possess this property without
+sacrificing the computational efficiency of DSEM for general unstructured grids and their advantageous scale-resolving
+properties in smooth flow regions can be challenging. In the context of MHD, this becomes even more difficult due
+to the additional complexity of the governing equations as well as the incorporation of differential constraints, namely
+solenoidal magnetic fields. As such, there is a need for numerical stabilization techniques for DSEM approximations
+of the ideal MHD equations that retain as many of these desirable properties as possible, namely that they:
+1. Guarantee that physical constraints of the solution are satisfied.
+2. Are compatible with numerical techniques for enforcing intrinsic constraints such as a solenoidal magnetic field.
+3. Do not require problem-dependent tunable parameters.
+4. Do not appreciably degrade the ability of the underlying DSEM to resolve smooth portions of the flow.
+5. Can be easily and efficiently implemented on general unstructured grids.
+In this work, we propose a nonlinear adaptive filtering approach as a numerical stabilization technique for DSEM
+approximations of the ideal MHD equations to address these points. The proposed technique can be considered as an
+extension of the entropy filtering approach originally introduced by the authors for shock capturing in gas dynamics to
+the ideal MHD system [17]. This technique relies on using the solution’s ability to preserve convex invariants of the
+system, namely positivity of the density and pressure and a discrete local minimum entropy principle, to compute the
+necessary filter strength to ensure a well-behaved solution in the vicinity of discontinuities. Extending this approach to
+the ideal MHD system presents several challenges, primarily stemming from the treatment of the divergence-free con-
+straint on the magnetic field. We utilize the eight-wave method of Powell et al. [7] which introduces non-conservative
+source terms in the equation proportional to the divergence of the magnetic field. As these non-conservative terms
+can conflict with the necessary assumptions of the entropy filtering approach, we present a modified set of conditions
+and introduce an operator splitting approach to the system which allows the filtering method to retain its positivity-
+preserving properties. Furthermore, as the original approach for performing the optimization process necessary in the
+adaptive filtering framework as presented in Dzanic and Witherden [17] was found to be quite computationally expen-
+sive, we develop a highly-efficient numerical approach which drastically reduces the overall computational cost. The
+resulting approach can robustly resolve strong hydrodynamic and magnetic discontinuities in the flow without appre-
+ciably degrading the accuracy of the underlying DSEM for smooth flows, does not require problem-dependent tunable
+parameters, and can be easily extended to unstructured grids with relatively low computational cost. The efficacy of
+the proposed method is demonstrated in a variety of numerical experiments including smooth transport, extremely
+magnetized blast waves, and three-dimensional magnetohydrodynamic instabilities computing using high-order ap-
+proximations on both structured and unstructured grids.
+The organization of this work is as follows. We present some preliminaries regarding DSEM approximations and
+the ideal MHD equations in Section 2. The entropy filtering approach for ideal MHD is then introduced in Section 3,
+and its numerical implementation and computational optimizations are presented in Section 4. Results for various test
+cases are then shown in Section 5, and conclusions are drawn in Section 6.
+2. Preliminaries
+2.1. Ideal magnetohydrodynamics
+The governing equations for the evolution of an ideal magnetohydrodynamic fluid can be given in the form of a
+hyperbolic conservation law as
+휕푡퐮 + 훁⋅퐅 (퐮) = 퐒푩(퐮),
+(1)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 2 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+where 퐮 = 퐮(퐱, 푡) ∈ ℝ푚 is the solution of some number of field variables 푚 defined over a 푑-dimensional spatial
+domain 퐱 ∈ ℝ푑 and time 푡, 퐅(퐮) ∈ ℝ푚×푑, and 퐒퐁(퐮) is an additional source term to be defined in Section 2.3 whose
+purpose is to ensure a solenoidal magnetic field. The solution and flux are given as
+퐮 =
+⎡
+⎢
+⎢
+⎢⎣
+휌
+흆풗
+푩
+퐸
+⎤
+⎥
+⎥
+⎥⎦
+and
+퐅 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+흆풗
+흆풗 ⊗ 퐯 + 퐈
+(
+푃 + 1
+2퐁⋅퐁
+)
+− 퐁 ⊗ 퐁
+풗 ⊗ 퐁 − 퐁 ⊗ 풗
+(
+퐸 + 푃 + 1
+2퐁⋅퐁
+)
+퐯 − 퐁(퐯⋅퐁)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+(2)
+where 휌 is the density, 흆풗 is the momentum, 퐸 is the total energy, 푃 = (훾 − 1)
+(
+퐸 − 1
+2휌퐯⋅퐯 − 1
+2퐁⋅퐁
+)
+is the pressure,
+퐁 is the magnetic field, and 훾 is the specific heat ratio. Furthermore, the symbol 퐈 denotes the identity matrix in ℝ푑×푑
+and 퐯 = 흆풗∕휌 denotes the velocity. The solution can be more conveniently expressed in terms of a vector of primitive
+variables as 퐪 = [휌, 퐯, 퐁, 푃]푇 , and auxiliary quantities representing the magnetic pressure and plasma-beta can be
+defined as 푃푏 = 1
+2(훾 − 1)퐁⋅퐁 and 훽 = 2푃∕(퐁⋅퐁), respectively.
+Due to the lack of magnetic monopoles, the MHD equations have an intrinsic constraint on the solution in the form
+of a solenoidal magnetic field, i.e.,
+훁⋅퐁 = 0.
+(3)
+Although this constraint must be satisfied analytically by the MHD equations, numerical approximations do not nec-
+essarily satisfy it even if the magnetic field is initially solenoidal. If this constraint is not enforced by the scheme,
+numerical instabilities may arise in addition to the non-physical nature of the approximation. Many approaches exist
+to enforce this condition on the magnetic field, including the use of solenoidal basis functions [19], projection meth-
+ods [20], constrained-transport schemes [6], divergence cleaning methods [21], and the eight-wave method [7]. An
+overview of the salient techniques is presented in Wu and Shu [15].
+The entropy solution of Eq. (1) satisfies an entropy inequality of the form
+휕푡휎(퐮) + 훁⋅횺(퐮) ≥ 0,
+(4)
+where (휎, 횺) is any numerical entropy-flux pair [22] that satisfies the relation
+휕퐮횺 = 휕퐮휎휕퐮퐅.
+Note that this inequality may be negated depending on which notation is used for the numerical entropy. In Dao and
+Nazarov [23], it was shown that the entropy solution (in a vanishing viscosity sense) of the ideal MHD system satisfies
+a minimum entropy principle on the specific physical entropy 휎 = 푃휌−훾 in the form
+휎 (퐮(퐱, 푡 + Δ푡)) ≥ min
+퐱
+휎 (퐮(퐱, 푡)) ,
+(5)
+where Δ푡 > 0. This property is identical to the minimum entropy principle in gas dynamics [24], and it should be
+satisfied by the solution in both smooth regions and in the vicinity of discontinuities.
+2.2. Discontinuous spectral element methods
+For nodal discontinuous spectral element approximations of Eq. (1), including discontinuous Galerkin [25] and flux
+reconstruction [26] schemes, the domain Ω is partitioned into 푁푒 elements Ω푘 such that Ω = ⋃
+푁푒 Ω푘 and Ω푖 ∩Ω푗 = ∅
+for 푖 ≠ 푗. With a slight abuse of notation, the solution 퐮(퐱) within each element Ω푘 is approximated in a nodal manner
+as
+퐮(퐱) =
+∑
+푖∈푆
+퐮푖휙푖(퐱),
+(6)
+where 퐱푖 ∀ 푖 ∈ 푆 is a set of solution nodes, 휙푖(퐱) are their associated nodal basis functions that possess the property
+휙푖(퐱푗) = 훿푖푗, and 푆 is the set of nodal indices for the stencil. For brevity, we utilize the notation that 퐮푖 = 퐮(퐱푖). The
+order of the approximation of the solution is denoted as ℙ푝 for some order 푝, where 푝 is the maximal order of 퐮(퐱).
+This approximation formally yields a convergence rate of at least 푝 + 1 [25].
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 3 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+The flux is approximated via the contribution of an interior term, denoted by the subscript Ω푘, and an interface
+term, denoted by the subscript 휕Ω푘, as
+퐅(퐮) ≈ 퐅Ω푘(퐮) + 퐅휕Ω푘(퐮).
+(7)
+For the interior component, the flux is computed through a collocation approach as
+퐅Ω푘(퐮) =
+∑
+푖∈푆
+퐅(퐮푖)휙푖(퐱),
+(8)
+such that the interior contribution to the divergence of the flux can be computed as
+훁⋅퐅Ω푘(퐮푖) =
+∑
+푗∈푆
+퐜푖푗퐅(퐮푗),
+where
+퐜푖푗 = ∇휙푖(퐱푗).
+(9)
+The interface component of the flux is formed over a set of interface nodes 퐱푖 ∈ 휕Ω푘 ∀ 푖 ∈ 퐼, where 퐼 is a set of nodal
+indices for the interface stencil. We assume that these interface nodes are a subset of the solution nodes (i.e., 퐼 ⊂ 푆)
+to avoid issues regarding interpolation for discontinuous solutions. At each interface node, there exist two values of
+the solution, 퐮−
+푖 and 퐮+
+푖 , representing the solution evaluated from the element of interest and the interface-adjacent
+element, respectively. The interface flux term can then be computed as
+퐅휕Ω푘(퐮푖) =
+∑
+푗∈퐼
+퐅(퐮−
+푗 , 퐮+
+푗 , 퐧푗)휙푗(퐱),
+(10)
+where 퐅(퐮−
+푖 , 퐮+
+푖 , 퐧푖) are the common interface flux values dependent on the interior and exterior values of the solution
+and their associated normal vectors 퐧푖 and 휙푖(퐱) are the interface bases. The common interface flux is generally
+computed using an approximate Riemann solver such as that of Rusanov [27]. The interface bases are dependent on
+the choice of spatial discretization, e.g., for flux reconstruction schemes, these terms can be given as
+휙푖(퐱) = 퐧푖⋅퐡푖(퐱) − 휙푖(퐱).
+(11)
+Here, 퐡푖 are a set of correction functions [28, 29] that posses the properties that
+퐧푖⋅퐡푗(퐱푖) = 훿푖푗
+and
+∑
+푖∈퐼
+퐡푖(퐱) ∈ RT푝,
+(12)
+where RT푝 is the Raviart–Thomas space [30] of order 푝. In this work, the flux reconstruction scheme with the equivalent
+discontinuous Galerkin correction functions [26] is used which recovers the nodal discontinuous Galerkin method [25].
+The interface contribution to the divergence of the flux can then be given as
+훁⋅퐅휕Ω푘(퐮푖) =
+∑
+푗∈퐼
+퐜푖푗퐅(퐮−
+푗 , 퐮+
+푗 , 퐧푗),
+where
+퐜푖푗 = ∇휙푖(퐱푗).
+(13)
+The semi-discrete form of Eq. (1) can then be given as
+휕푡퐮푖 = −
+(
+퐅휕Ω푘(퐮푖) + 훁⋅퐅휕Ω푘(퐮푖)
+)
++ 퐒퐁(퐮푖).
+(14)
+We assume that the spatial scheme satisfies the relation
+휕푡퐮 = − ∫휕Ω푘
+퐅 (퐱) ⋅ 퐧(퐱) d퐱 ≈ −
+∑
+푗∈퐼
+푚푗퐅(퐮−
+푗 , 퐮+
+푗 , 퐧푗)
+(15)
+where 푚푗 is the associated quadrature weight for 퐱푗 and 퐮 is the element-wise mean defined as
+퐮 = 1
+푉푘 ∫Ω푘
+퐮(퐱) d퐱
+and
+푉푘 = ∫Ω푘
+d퐱.
+(16)
+This assumption is appropriate for nodal discontinuous Galerkin schemes given appropriate quadrature and flux recon-
+struction schemes utilizing the equivalent discontinuous Galerkin correction functions.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 4 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+2.3. Eight-wave method
+A common method for enforcing a divergence-free magnetic field is to utilize the eight-wave method of Powell
+et al. [7]. This approach relies on an additional wave structure of the Riemann problem in MHD that arises when the
+magnetic field is not exactly solenoidal, and it can be utilized to force the magnetic field to a solenoidal state via a
+source term, given as
+퐒푩(퐮) = −
+⎡
+⎢
+⎢
+⎢⎣
+0
+푩
+풖
+풖⋅푩
+⎤
+⎥
+⎥
+⎥⎦
+훁⋅푩.
+(17)
+With the inclusion of this source term, the divergence of the magnetic field is typically suppressed to the order of mag-
+nitude of the approximation error [15]. As such, due to the simplicity of implementation and applicability to general
+unstructured grids, it remains a routine approach for robustly enforcing the divergence-free constraint on the solenoidal
+field. In addition, only this modified form of the ideal MHD equations is symmetrizable and Galilean invariant when
+the magnetic field is not exactly solenoidal [15]. However, as this form is non-conservative, it occasionally can cause
+inaccurate predictions around discontinuities in the flow (see Tóth [31]).
+The use of Powell’s method requires some clarification about the choice of the formulation for computing the
+divergence of the magnetic field. In the context of DSEM, there exist two formulations, a local divergence, consisting
+of just the interior component as
+훁⋅푩퐿(퐮푖) =
+∑
+푗∈푆
+퐜푖푗푩푗,
+(18)
+and a global divergence, consisting of both the interior component and the interface contribution as
+훁⋅푩퐺(퐮푖) =
+∑
+푗∈푆
+퐜푖푗푩푗 +
+∑
+푗∈퐼
+퐜푖푗푩푗,
+(19)
+where 푩푗 is a common interface value for the magnetic field, typically taken as the centered average of the interior and
+exterior values. Whereas the divergence-free constraint can be imposed on the local divergence through straightforward
+approaches such as projection to solenoidal bases, enforcing this constraint on the global divergence is typically more
+difficult as its domain of influence is not strictly contained within the element. It can be argued that the global approach
+is the “correct” choice as it is the one for which the space of the divergence is consistent with the space of the solution,
+but in practice, the local approach is typically sufficient. In this work, the global approach is used as the complexity of
+the two implementations is similar with Powell’s method.
+3. Methodology
+Due to the presence of discontinuities in MHD flows in the form of hydrodynamic and magnetic shocks, it is
+necessary to apply some sort of a numerical stabilization procedure to ensure robustness of the DSEM approximation.
+In Dzanic and Witherden [17], an adaptive filtering approach was introduced with goal of stabilizing the scheme by
+discretely enforcing convex constraints on the solution, given in the form of
+Γ(퐮푖) > 0 ∀ 푖 ∈ 푆,
+(20)
+where Γ(퐮) is some constraint functional. For a positivity-preserving scheme, these constraints are set as
+Γ1(퐮) = 휌
+and
+Γ2(퐮) = 푃,
+(21)
+corresponding to constraints on the positivity of density and pressure.
+While these constraints can ensure the positivity of these quantities, they are generally not restrictive enough to
+ensure that the solution remains well-behaved in the vicinity of discontinuities. It is necessary to attempt to form
+additional constraints on the solution that are restrictive enough to stabilize the solution in the vicinity of discontinuities
+without degrading the accuracy of the scheme in regions where the solution is smooth. By utilizing the fact that the
+minimum entropy principle presented in Section 2 should be satisfied by both smooth and discontinuous solutions, a
+third constraint on the solution is enforced corresponding to a discrete form of a local minimum entropy principle as
+Γ3(퐮) = 휎(퐮) − 휎min,
+(22)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 5 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+where 휎(퐮) = 푃 휌−훾 is the specific physical entropy and 휎min is some local minimum entropy bound. This minimum
+bound 휎min is computed in an element-wise manner as the discrete minima of the entropy functional across the element
+and its face neighbors prior to each stage of a temporal integration scheme, resulting in the enforcement of a discrete
+minimum entropy principle over the local domain of influence of the element (see Dzanic and Witherden [17], Section
+2 and 3). It was found in the context of gas dynamics that enforcing this constraint ensured well-behaved solutions in
+the vicinity of discontinuities while recovering high-order accuracy in smooth regions of the flow [17].
+3.1. Adaptive filtering
+The constraints are enforced by an adaptive filtering procedure, where the filtered solution ̃퐮 is given in terms of a
+filter kernel 퐻 applied to the solution, i.e.,
+̃퐮 = 퐻(퐮).
+(23)
+This filtering is performed in modal space given a modal decomposition of the solution in the form of
+퐮(퐱) =
+∑
+푖∈푆
+̂퐮푖휓푖(퐱),
+(24)
+where 휓푖(퐱) ∀ 푖 ∈ 푆 are a set of modal basis functions and ̂퐮푖 are their corresponding modes. We assume that this modal
+decomposition is chosen with respect to the unit measure (e.g., Legendre polynomials, Koornwinder polynomials, etc.).
+A discrete form of this change-of-basis operation can be given in terms of a Vandermonde matrix 퐕 as
+̂퐮 = 퐕−1퐮.
+(25)
+The filter kernel ̂
+퐻 is taken as a second-order exponential kernel in modal space, such that the filtered modal modes
+can be computed as
+̂
+퐻푖(̂퐮푖) = ̂퐮 exp(−휁푝2
+푖 ),
+(26)
+where 휁 is the filter strength and 푝푖 is the total order of the corresponding mode ̂퐮푖. It must be noted that the adaptive
+filtering approach is not restricted to this choice of filter and can be applied to any conservative filtering operation of
+one free variable that can recover both the unfiltered solution and the mean mode [17]. The filtering operation 퐻(퐮)
+can be cast in terms of a matrix-vector operation as
+̃퐮 = 퐻(퐮) = 퐕횲퐕−1퐮,
+(27)
+where 횲 is a diagonal matrix of 푝 + 1 unique values with its entries equal to 횲푖,푖 = exp(−휁푝2
+푖 ).
+The filter strength is computed via an element-wise nonlinear optimization process, taken as the minimum filter
+strength necessary such that the constraints are satisfied, i.e.,
+휁 = arg min
+휁 ≥ 0
+s.t. [Γ1
+(̃퐮(퐱푖)) > 0, Γ2
+(̃퐮(퐱푖)) > 0, Γ3
+(̃퐮(퐱푖)) > 0 ∀ 푖 ∈ 푆] .
+(28)
+Existence of a solution of 휁 is guaranteed if the element-wise mean of the solution satisfies the constraints, an assump-
+tion that will be explored in Section 3.2. As this optimization process is a function of a scalar free variable, its solution
+can be obtained using any root-bracketing approach. Furthermore, as it is nonlinear and non-convex, convergence to
+a local minima is sufficient in the case of multiple values of 휁 existing such that the constraints are satisfied exactly.
+While this optimization problem seems computationally demanding due to the element-wise matrix-vector operations
+necessary to compute the filtered solution each iteration of the solve, we present a numerical approach to solving
+this problem in Section 4.1 that is much more computationally efficient than the original methodology in Dzanic and
+Witherden [17].
+3.2. Extensions to MHD
+Extending the entropy filtering approach to the MHD system requires some modifications, with special care neces-
+sary in regards to the treatment of the source terms. The adaptive filtering operation naturally relies on that assumption
+that there exists a filter strength such that the constraints are satisfied, and it is trivial to show that a solution exists if the
+element-wise mean satisfies the constraints [17]. The ability of discontinuous Galerkin-type approaches to preserve
+convex invariants of hyperbolic systems on the element-wise mean is a well established in the literature, and the reader
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 6 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+is referred to a variety of works which utilize this property [15–17, 32–35]. However, the inclusion of the source term
+and the presence of entropy constraints introduces some caveats on this property of the scheme.
+Let the set 퐺 represent the set of solutions which satisfy the constraints (i.e., Γ1(퐮) > 0, Γ2(퐮) > 0, Γ3(퐮) > 0), and
+let the shorthand notation 퐮 ∈ 퐺 represent 퐮푖 ∈ 퐺 ∀ 푖 ∈ 푆. To ensure that the filter can recover a constraint-satisfying
+solution, it is necessary for the temporal update of the element-wise mean to preserve these invariants, i.e., for some
+time step 푛, if 퐮푛 ∈ 퐺, then 퐮푛+1 ∈ 퐺. For brevity, we consider a temporal update in the form of a forward Euler
+approximation, given as
+퐮푛+1 = 퐮푛 + Δ푡 [퐿1(퐮푛) + 퐿2(퐮푛)] ,
+(29)
+where
+퐿1(퐮) = −훁⋅퐅(퐮)
+and
+퐿2(퐮) = 퐒퐁(퐮).
+(30)
+Without an exactly solenoidal magnetic field, the property 퐮푛+1 ∈ 퐺 is not necessarily satisfied in this form under the
+standard assumptions posed in works such as Zhang and Shu [32] and the original presentation of entropy filtering
+for gas dynamics in Dzanic and Witherden [17], e.g., appropriate Riemann solver, CFL condition, strong stability
+preserving temporal integration. If we consider the set of solutions 퐺푃 which satisfy just the positivity constraints
+(i.e., 퐮 ∈ 퐺푃 if Γ1(퐮) > 0, Γ2(퐮) > 0), then the work of Wu and Shu [15] showed that the property 퐮푛+1 ∈ 퐺푃 is
+satisfied under a potentially more restrictive condition on the time step dependent on the discrete divergence of the
+magnetic field (see Theorem 3.1 in Wu and Shu [15]). Furthermore, if we neglect the source term and consider an
+intermediate temporal update as
+퐮∗ = 퐮푛 + Δ푡퐿1(퐮푛),
+(31)
+then the work of Bouchut et al. [36] (paired with the equivalency of the element-wise mean and Godunov methods
+presented in Zhang and Shu [32] and subsequent works) shows that this intermediate state satisfies the property 퐮∗ ∈ 퐺
+under the standard assumptions.
+These two observations motivate an operator splitting approach for the filter. Two separate filtering operations are
+considered, a more restrictive filter which enforces both the positivity and entropy constraints, denoted by 퐻푒[퐮], and
+a more relaxed filter that enforces only positivity constraints, denoted by 퐻푝[퐮]. As the assumption on the positivity
+and entropy constraints on the element-wise mean are satisfied by the intermediate state, the more restrictive filter can
+be applied, i.e.,
+̃퐮∗ = 퐻푒
+[퐮푛 + Δ푡퐿1(퐮푛)] .
+(32)
+Since the entropy constraints are the most restrictive constraint and the contribution of the source term is typically
+minimal compared to the divergence of the flux (since it is proportional to 훁⋅퐁), this filtering operation can usually
+mitigate the majority of the spurious oscillations in the vicinity of discontinuities. The contribution of the source term
+is then added onto this filtered state, after which the positivity constraints are then enforced again on the temporal
+update as
+̃퐮푛+1 = 퐻푝
+[̃퐮∗ + Δ푡퐿2(퐮푛)] .
+(33)
+A solution to this filtering optimization problem is also guaranteed to exist as the positivity of the element-wise mean
+is guaranteed [15].
+Several properties of this splitting approach must be noted. First, it is very rarely the case that the secondary filtering
+operation is necessary – the entropy constraints on ̃퐮∗ are typically restrictive enough to where ̃퐮∗ + Δ푡퐿2(퐮푛) retains
+its positivity-preserving properties, such that in most cases, the positivity constraints are typically just checked and no
+filtering is needed. However, to ensure that the scheme remains provably positivity-preserving, this secondary filtering
+operation must be included. Second, the splitting for the source term is calculated explicitly as 퐿2(퐮푛), not through
+a Strang-type splitting approach [37] as 퐿2(퐮∗). While the latter may potentially better approximate the necessary
+corrections to the solution for preserving a solenoidal magnetic field, these forms of splitting can introduce a limit on
+the temporal accuracy of the scheme and therefore are avoided. Finally, unless the linear filtering kernel which recovers
+the squeeze limiter of Zhang and Shu [32] is chosen (see Dzanic and Witherden [17], Remark 1), the divergence of
+the filtered magnetic field is not guaranteed to be equal or lower than the unfiltered state. As this work pertains to a
+nonlinear filter, it may introduce minor divergence errors similarly to any nonlinear limiting operation, but these are
+mitigated via the source term at the next temporal update with the explicit splitting approach such that its effects were
+found to be negligible.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 7 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+Extensions to higher-order strong stability preserving (SSP) schemes follow readily from this formulation, e.g., the
+temporal update for a third-order, three-stage SSP Runge–Kutta scheme, neglecting the notatioñ⋅ for brevity, is given
+as
+퐮(1) = 퐻푝
+[퐻푒
+[퐮푛 + Δ푡퐿1(퐮푛)] + Δ푡퐿2(퐮푛)] ,
+(34)
+퐮(2) = 퐻푝
+[
+퐻푒
+[3
+4퐮푛 + 1
+4퐮(1) + 1
+4Δ푡퐿1(퐮(1))
+]
++ 1
+4Δ푡퐿2(퐮(1))
+]
+,
+퐮푛+1 = 퐻푝
+[
+퐻푒
+[1
+3퐮푛 + 2
+3퐮(2) + 2
+3Δ푡퐿1(퐮(2))
+]
++ 2
+3Δ푡퐿2(퐮(2))
+]
+,
+where the entropy constraints for 퐻푒 are computed from the previous temporal stage (see Dzanic and Witherden [17],
+Appendix A).
+4. Implementation
+Ω푘
+Figure 1:
+Schematic of a two-dimensional ℙ2 triangular element Ω푘 showing interior solution points (red circles), interior
+interface flux/solution points (red circles, blue outline), and exterior interface flux points (blue circles).
+The governing equations and the adaptive filtering approach were implemented in PyFR [38], a high-order GPU-
+accelerated unstructured flux reconstruction solver. The solution nodes were distributed along the Gauss–Legendre–
+Lobatto quadrature points and 훼-optimized points [25] for tensor-product and simplex elements, respectively. An
+example of the solution and flux point distributions for a two-dimensional ℙ2 triangular element is shown in Fig. 1.
+Temporal integration was performed using a three-stage, third-order SSP Runge–Kutta scheme as presented in Eq. (34).
+Unless otherwise stated, common interface fluxes were computed using the Harten-Lax-van Leer contact (HLLC)
+Riemann solver of Li [39] and Gurski [40] with the Davis wavespeed estimate [41], although for most test cases, we
+observed negligible differences in comparison to Rusanov-type [27] and Harten-Lax-van Leer (HLL) [42] Riemann
+solvers. To avoid a vacuum state for the Riemann solver and apply a numerical tolerance to the entropy condition, the
+constraints were instead implemented as
+Γ1(퐮) = 휌 − 휖,
+Γ2(퐮) = 푃 − 휖,
+and
+Γ3(퐮) = 휎 − 휎min − 휖,
+where 휖 = 10−8 is a small constant taken as some arbitrary factor of the machine precision.
+Boundary conditions were enforced in a weak sense through the imposition of an exterior ghost state to the inter-
+face Riemann solver [43]. Three types of boundary conditions were considered in this work: 1) Dirichlet boundary
+conditions, where the exterior state is explicitly defined; 2) Neumann boundary conditions, where the exterior state is
+identical to the interior state; and 3) reflecting boundary conditions, where the exterior state is identical to the interior
+state with the normal component of the velocity and magnetic field negated.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 8 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+4.1. Filter optimization
+Each time the filtering operation is called, the constraints are first checked on the solution. If the solution satisfies
+the constraints, no filtering is applied, otherwise the filter strength is computed using the Illinois root-bracketing ap-
+proach [44] with a stopping tolerance of 10−8 and a maximum of 20 iterations. While the filter strength can be simply
+iterated by repeatedly evaluating the element-wise filtered solution as per Eq. (27) and computing the minima of the
+constraints, several optimizations can be performed to drastically decrease the computational cost of performing this
+filtering operation.
+First, instead of solving for 휁, it beneficial to solve for 푓 = exp(−휁) and utilize the relation
+exp(−휁푝2
+푖 ) = 푓 푝2
+푖 .
+This bounds the search space of the root-bracketing approach to 푓 ∈ [0, 1], and the evaluation of the filter coefficients
+reduces to simple integer powers of the argument 푓. Then, to avoid the costly computation of the matrix-vector
+product in Eq. (27) each iteration of the root-bracketing process, certain properties of the matrix 횲 can be exploited.
+As previously mentioned, 횲 is a diagonal matrix of 푝 + 1 unique values with its entries equal to 횲푖,푖 = exp(−휁푝2
+푖 ). If
+we define a set of diagonal matrices 퐈(푘) for 0 ≤ 푘 ≤ 푝 as
+퐈(푘)
+푖,푖 =
+{
+1,
+if푝푖 = 푘,
+0,
+else,
+(35)
+then the filtering operation can be equivalently represented as
+̃퐮 =
+푝
+∑
+푖=0
+푓 푝2
+푖 퐮(푘),
+(36)
+where
+퐮(푘) = 퐕퐈(푘)퐕−1퐮.
+(37)
+Note that the values 퐮(푘) are independent of the value of 푓, such that these values can be pre-computed and the fil-
+tered solution can be efficiently evaluated each iteration of the root-bracketing approach without having to repeatedly
+compute the matrix-vector product 퐕횲퐕−1퐮.
+This approach can be even further optimized by utilizing the fact that the nodal values of the solution can now
+be decoupled, such that the root-bracketing process can be applied across each solution node sequentially which is
+particularly beneficial for computing architectures where memory bandwidth is the bottleneck. In this sequential
+approach, each solution node 퐱푗 for 푗 ∈ 푆 solves for a value of 푓푗 such that ̃퐮푗 satisfies the constraints. It is trivial to
+show that if
+푓 = min
+푗∈푆 푓푗,
+then ̃퐮 satisfies the constraints at all nodes. It is therefore advantageous to then use 푓푗 as the upper bound for the root-
+bracketing process for the node 퐱푗+1 as the constraints can be checked for ̃퐮푗+1 using this upper bound and the root-
+bracketing process for that node can be skipped if they are satisfied. As the proposed algorithm requires effectively only
+one full evaluation of Eq. (27) irrespective of the number of iterations of the root-bracketing approach, the memory
+bandwidth requirements are significantly decreased, such that the filtering process becomes only a relatively small
+portion of the total compute time that is typically less than the cost of the evaluation of the divergence of the flux. An
+example of the implementation of this approach is provided in the electronic supplementary material of this work, and
+an evaluation of the efficiency improvements of this proposed algorithm in comparison to the original methodology in
+Dzanic and Witherden [17] which utilizes repeated evaluations of Eq. (27) is presented in Section 5.
+5. Results
+5.1. Near-vacuum convecting vortex
+To verify that the proposed scheme retains the high-order accuracy of the underlying DSEM for smooth solutions,
+the rate of convergence was calculated for the smooth magnetized convecting vortex problem introduced by Christlieb
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 9 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+et al. [45]. For this problem, the domain is taken as Ω = [−10, 10]2 with periodic boundary conditions discretized on
+a structured quadrilateral mesh, and the initial conditions are given as
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+1 − 푦훿푢
+1 + 푥훿푢
+−푦훿퐵
+푥훿퐵
+1 + 훿푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+(38)
+where
+훿푢 =
+휇
+√
+2휋
+휙(푟),
+훿퐵 = 휇
+2휋 휙(푟),
+훿푃 = −휇2(1 + 푟2)
+8휋2
+휙(푟)2,
+(39)
+and
+휙(푟) = exp(1 − 푟2),
+푟 =
+√
+푥2 + 푦2.
+(40)
+The specific heat ratio was set as 훾 = 5∕3. These conditions give a non-isentropic nature to the flow field, which
+allows for a proper assessment of the proposed entropy-based constraints for smooth flows where the filter should be
+primarily inactive. To make this problem more numerically challenging, the parameter 휇 is chosen such as to give
+a near-vacuum state for the pressure field [45]. This value was set as 휇 = 5.38948938512, which gives a minimum
+pressure value in the domain of approximately 2휖 = 2⋅10−8.
+The problem was solved until a non-dimensional time of 푡 = 0.05 using a fixed time step of Δ푡 = 1⋅10−4, after
+which the 퐿1 norm of the magnetic field error was computed as
+푒퐵 = 1
+퐴 ∫Ω
+||퐵푥 − 퐵exact
+푥
+|| + |||퐵푦 − 퐵exact
+푦
+||| d퐱,
+(41)
+where 퐴 = 202. The exact solution was computed through a translation of the initial conditions with a translation
+velocity of [1, 1], and the integration was computed using a 9th order Gauss–Legendre quadrature rule. The error
+with respect to the mesh resolution 푁푒 is shown for various approximation orders in Table 1 in addition to the rate of
+convergence. High-order convergence, on the order of 푝 to 푝 + 1, was observed for all approximation orders between
+ℙ2 and ℙ5. Furthermore, the ℙ2 results can be compared to the positivity-preserving third-order DG scheme in Wu and
+Shu [15] (Table 2), which can be considered as a subset of the adaptive filtering approach without entropy constraints
+(see Dzanic and Witherden [17], Remark 1). The proposed scheme gives marginally lower error even after removing
+the 1∕퐴 normalization factor.
+푁푒
+ℙ2
+ℙ3
+ℙ4
+ℙ5
+202
+2.29 × 10−4
+3.07 × 10−5
+3.30 × 10−6
+4.04 × 10−7
+252
+1.18 × 10−4
+1.17 × 10−5
+1.37 × 10−6
+9.36 × 10−8
+332
+5.39 × 10−5
+4.07 × 10−6
+3.39 × 10−7
+2.44 × 10−8
+402
+3.04 × 10−5
+2.02 × 10−6
+1.44 × 10−7
+9.55 × 10−9
+502
+1.59 × 10−5
+8.72 × 10−7
+5.02 × 10−8
+2.92 × 10−9
+672
+6.92 × 10−6
+2.99 × 10−7
+1.31 × 10−8
+6.51 × 10−10
+RoC
+2.89
+3.80
+4.62
+5.23
+Table 1:
+Convergence of the 퐿1 norm of the magnetic field error at 푡 = 0.05 with respect to mesh resolution 푁푒 for the
+near-vacuum convecting vortex problem with varying approximation order. Rate of convergence shown on bottom.
+5.2. Brio–Wu shock tube
+Extensions to flows with discontinuities was then performed through the shock tube problem of Brio and Wu [8]
+which includes features of the Riemann problem such as shock waves, contact discontinuities, rarefaction waves, and
+compound waves. For this problem, the domain is set as Ω = [0, 1] and the initial conditions are given by
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 10 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+{
+퐪푙,
+if 푥 ≤ 0.5,
+퐪푟,
+else,
+where
+퐪푙 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+0
+0
+0.75
+1
+1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+and
+퐪푟 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+0.125
+0
+0
+0.75
+−1
+0.1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+.
+(42)
+The specific heat ratio is set as 훾 = 2. The hydrodynamic components of this problem are identical to the Sod shock
+tube [46]. Although this problem is one-dimensional, it was instead solved on a one element wide two-dimensional
+mesh to facilitate the use of the vertical magnetic field component within the solver. Dirichlet boundary conditions were
+applied on the left/right boundaries while periodic boundary conditions were applied along the top/bottom boundaries.
+The problem was computed with a ℙ3 scheme using a coarser mesh of 200 elements and a finer mesh of 400
+elements with time steps of Δ푡 = 2⋅10−4 and 1⋅10−4, respectively. A reference solution was also computed using a
+highly-resolved ℙ0 scheme with 5⋅104 elements. The predicted density, pressure, and vertical magnetic field profiles
+at 푡 = 0.1 are shown in Fig. 2 for both the coarse and fine mesh. For all fields, both rarefaction waves and the shock
+were well-resolved, showing sub-element resolution without any noticeable spurious oscillations. Furthermore, similar
+behavior was observed for the contact and compound wave in the pressure and magnetic fields. Some minor oscillations
+were observed in the density profile in the region between the compound wave and contact discontinuity, although this
+behavior is not uncommon for some numerical schemes. The predicted density profile in that region converged to the
+reference results with increasing resolution, but minor undershoots in front of the contact discontinuity were observed.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+푥
+휌
+Reference
+푁푒 = 200
+푁푒 = 400
+(a) Density
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+푥
+푃
+(b) Pressure
+0
+0.2
+0.4
+0.6
+0.8
+1
+−1
+−0.5
+0
+0.5
+1
+푥
+퐵푦
+(c) Vertical magnetic field
+Figure 2:
+Density, pressure, and vertical magnetic field profiles for the Brio–Wu shock tube problem at 푡 = 0.1 computed
+using a ℙ3 FR scheme with 200 and 400 elements.
+5.3. Orszag–Tang vortex
+Two-dimensional flows with more complex features were then considered through the canonical Orszag–Tang
+vortex problem [47]. This case is a well-known model problem for evaluating a scheme’s ability to handle MHD
+shocks and shock interactions as well as predicting transition to supersonic MHD turbulence. The domain is set as
+Ω = [0, 1]2 with periodic boundary conditions, and the initial conditions are given as
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+25∕(36휋)
+− sin(2휋푦)
+sin(2휋푥)
+sin(2휋푦)∕
+√
+4휋
+− sin(4휋푥)∕
+√
+4휋
+5∕(12휋)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+.
+(43)
+The specific heat ratio is set as 훾 = 5∕3. Uniform meshes of various resolution were generated, and the problem
+was solved using a ℙ3 scheme. The contours of density at 푡 = 0.5 computed on meshes with 푁푒 = 642, 1282, and
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 11 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+2562 elements are shown in Fig. 3, computed using time steps of Δ푡 = 4⋅10−4, 2⋅10−4, and 1⋅10−4, respectively. The
+results show good prediction of the canonical flow field of the Orszag–Tang vortex, with better approximation of shock
+structure and small-scale flow features with increasing resolution. Minor spurious oscillations were observed in the
+density field at low resolutions, but these oscillations diminished with increasing mesh resolution, such that the flow
+field at 푁푒 = 2562 was virtually oscillation-free.
+(a) 푁푒 = 642
+(b) 푁푒 = 1282
+(c) 푁푒 = 2562
+Figure 3:
+Contours of density for the Orszag-Tang vortex at 푡 = 0.5 computed using a ℙ3 FR scheme with 642 (left),
+1282 (middle), and 2562 (right) elements.
+For a more quantitative comparison, the predicted pressure profile on the cross-section 푦 = 0.3125 at 푡 = 0.48 is
+shown in Fig. 4 in comparison to the results of Jiang and Wu [48] obtained using a high-order weighted essentially
+non-oscillator (WENO) scheme. It can be seen that similar observations can be made for the pressure field as with the
+density field, with minor spurious oscillations at lower resolutions that diminish with increasing resolution. The overall
+prediction of the pressure profile was in good agreement with the reference results at moderate and high resolutions,
+and strong discontinuities in the pressure field were generally well resolved at the sub-element level even at lower
+resolutions. At the highest resolution, there was very good agreement with the reference data with minor differences
+in the prediction of the location of some small-scale flow features on the left-hand side of the cross-section. Overall,
+the results showed good predictions of the various flow features of the Orszag–Tang vortex.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.1
+0.2
+0.3
+푥∕퐿
+푝
+Reference
+푁푒 = 642
+푁푒 = 1282
+푁푒 = 2562
+Figure 4:
+Pressure profile of the Orszag-Tang vortex on the cross-section 푦∕퐿 = 0.3125 at 푡 = 0.48 computed using a ℙ3
+FR scheme with 642, 1282, and 2562 elements. Numerical results of Jiang and Wu [48] shown for reference.
+5.4. Shock cloud interaction
+The proposed scheme was then evaluated on the shock cloud interaction problem of Dai and Woodward [9], con-
+sisting of a high-density cloud interacting with an impinging shock wave which results in strong discontinuities and
+the development of small-scale flow instabilities. The problem setup as described by Balbás and Tadmor [11] is solved
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 12 of 22
+
+0.4
+0.35
+0.3
+0.25
+0.2
+0.15
+0.1Positivity-preserving entropy filtering for the ideal MHD equations
+on the domain Ω = [0, 1]2 with the initial conditions given as
+퐪(퐱, 0) = [휌, 푢, 푣, 푤, 퐵푥, 퐵푦, 퐵푧, 푃]푇 =
+⎧
+⎪
+⎨
+⎪⎩
+퐪푐,
+if 푟′ ≤ 0.15,
+퐪푙,
+else if 푥 ≤ 0.6,
+퐪푟,
+else,
+(44)
+where the cloud state, left state, and right state are given as
+퐪푐 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+10
+0
+0
+0
+2.1826182
+−2.1826182
+1
+167.345
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+퐪푙 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+3.86859
+0
+0
+0
+2.1826182
+−2.1826182
+1
+167.345
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+and
+퐪푟 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+−11.2536
+0
+0
+0
+0.56418958
+0.56418958
+1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+(45)
+respectively. The specific heat ratio is set as 훾 = 5∕3. The cloud is centered at [0.8, 0.5] with a radius of 0.15, such
+that
+푟′ =
+√
+(푥 − 0.8)2 + (푦 − 0.5)2.
+To facilitate the use of the three-dimensional magnetic field within the solver, the problem is solved on a one
+element deep three-dimensional mesh. Additionally, while the original problem setup uses a [0, 1]2 domain with
+Neumann boundary conditions on the top/bottom boundaries, we instead extend the domain to [0, 1] × [−0.5, 1.5] and
+apply periodic boundary conditions on the top/bottom boundaries. As these boundaries on the extended domain are
+outside of the domain of influence of the shock cloud interaction over the time range of the simulation, the effect of this
+modified setup on the flow field is negligible but it helps alleviate any issues arising from numerical errors compounding
+at free boundaries. The remaining left and right boundary conditions were set as Neumann and Dirichlet, respectively,
+while periodicity was enforced along the 푧 direction.
+(a) Density
+(b) Pressure
+(c) Magnetic pressure
+Figure 5:
+Contours of density (left), pressure (middle), and magnetic pressure (right) on the subregion [0, 1]2 for the
+shock cloud interaction problem at 푡 = 0.06 computed using a ℙ2 FR scheme with 4002 elements.
+To perform a comparison of the proposed approach to a third-order DG scheme augmented with a WENO limiter
+presented in Wu and Shu [15], an identical problem setup is used with a ℙ2 scheme on 4002 mesh (with respect to
+the original domain size Ω = [0, 1]2). The predicted contours of density, pressure, and magnetic pressure at 푡 = 0.06
+computed using a time step of Δ푡 = 4⋅10−6 are shown in Fig. 5. The results show good resolution of the strong
+discontinuities in the various fields without any observable spurious oscillations. Furthermore, small-scale features in
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 13 of 22
+
+p
+2
+5
+10
+20
+40P
+100
+200
+300
+4000.2
+0.5
+2
+5
+10
+20
+50
+100Positivity-preserving entropy filtering for the ideal MHD equations
+the cloud region of the density and magnetic fields were not excessively dissipated, and the symmetry of the problem
+was well-preserved. A comparison to the method of Wu and Shu [15] (Fig. 1) is shown in Fig. 6. Note that some
+discrepancy in the color schemes between the two images may be present. The proposed scheme was roughly equally
+performative in terms of the resolution of discontinuities and marginally better at resolving small-scale flow features on
+the trailing side of the cloud. Without the positivity-preserving filtering approach, the scheme diverged due to negative
+pressure in the solution.
+(a) Entropy filter
+(b) DG with WENO limiter
+Figure 6:
+Comparison of the contours of density computed by the proposed entropy filtering approach (left) and the
+positivity-preserving DG scheme augmented with a WENO limiter of Wu and Shu [15] (right).
+5.5. Magnetized blast
+As a stress test for the positivity-preserving property of the proposed scheme for extreme flow conditions, a modified
+form of the magnetized blast wave problem of Zachary et al. [49] and Balsara and Spicer [10] was considered. In this
+problem, a blast wave is driven by a spherical overpressure region in the center of the domain surrounded by a low
+plasma-beta ambient state, resulting in strong magnetosonic shocks. The problem is solved on the periodic domain
+Ω = [−0.5, 0.5]2, and the initial conditions are given as
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+{
+퐪푒,
+if
+√
+푥2 + 푦2 ≤ 0.1,
+퐪푎,
+else,
+where
+퐪푒 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+0
+0
+퐵0
+0
+푃푒
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+and
+퐪푎 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+0
+0
+퐵0
+0
+푃푎
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+.
+(46)
+The specific heat ratio is set as 5∕3. While the original problem setup uses 푃푎 = 0.1, 푃푒 = 103, and 퐵0 = 100∕
+√
+4휋, we
+consider the much more extreme case presented in Wu and Shu [15] with 푃푎 = 0.1, 푃푒 = 104, and 퐵0 = 1000∕
+√
+4휋,
+resulting in a very large pressure ratio of 105 and a very small plasma-beta of 훽 ≈ 2.5⋅10−4. As these conditions are
+quite extreme, the scheme would diverge almost instantly in the absence of any positivity-preserving modifications.
+To verify that the proposed scheme can be easily extended to unstructured grids, the problem was solved on trian-
+gular meshes using a ℙ4 scheme. A coarse mesh and a fine mesh were generated, consisting of approximately 1.2⋅105
+elements with an average edge length of ℎ = 1∕200 and approximately 5⋅105 elements with an average edge length of
+ℎ = 1∕400, respectively. For this case, the HLL Riemann solver was used as it was found to be much better behaved
+in these extreme conditions than the HLLC Riemann solver, although both approaches were properly stabilized with
+the proposed entropy filtering method. The contours of density, velocity magnitude, pressure, and magnetic pressure
+at 푡 = 0.001 are shown in Fig. 7 and Fig. 8 for the coarse and fine meshes, respectively, computed using time steps of
+Δ푡 = 2⋅10−7 and Δ푡 = 1⋅10−7. Even with these extreme conditions on unstructured grids, the predicted solutions were
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 14 of 22
+
+d
+0
+10
+20
+30
+40Positivity-preserving entropy filtering for the ideal MHD equations
+(a) Density
+(b) Velocity Magnitude
+(c) Pressure
+(d) Magnetic Pressure
+Figure 7:
+Contours of density (top left), velocity magnitude (top right), pressure (bottom left), and magnetic pressure
+(bottom right) for the magnetized blast problem at 푡 = 0.001 computed using a ℙ4 scheme on an unstructured mesh with
+an average edge length of ℎ = 1∕200.
+well-behaved, and both the coarse and fine meshes showed excellent resolution of the various discontinuities in the
+velocity, pressure, and magnetic fields with sub-element resolution and no observable spurious oscillations. Further-
+more, the numerical width of the discontinuities decreased appropriately with increasing resolution. For the density
+field, minor spurious oscillations were observed, primarily at lower resolution and somewhat indicative of mesh im-
+printing, but the strength and distribution of these oscillations decreased with the finer mesh. These observations are
+consistent with the case of the Brio–Wu shock tube where the density field was marginally less well-behaved than the
+other fields. However, the predicted fields were still very good given such extreme conditions and an unstructured
+mesh, indicating that the proposed approach remains robust and accurate for such flows.
+To demonstrate the sub-element shock-resolving ability of the entropy filtering approach on unstructured grids, an
+enlarged view of the contours of pressure with the mesh overlaid is shown for the two meshes in Fig. 9. It can be seen
+that the shock was resolved well within the element, with the majority of the feature resolved across 1-2 solution nodes.
+Furthermore, this behavior persisted when the mesh resolution was increased, such that the discrete shock thickness
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 15 of 22
+
+3
+4
+5/vl
+O
+10
+20
+30
+40
+50P
+1000
+2000
+3000
+4000
+5000P
+23000
+24000
+25000
+26000
+27000
+28000Positivity-preserving entropy filtering for the ideal MHD equations
+(a) Density
+(b) Velocity Magnitude
+(c) Pressure
+(d) Magnetic Pressure
+Figure 8:
+Contours of density (top left), velocity magnitude (top right), pressure (bottom left), and magnetic pressure
+(bottom right) for the magnetized blast problem at 푡 = 0.001 computed using a ℙ4 scheme on an unstructured mesh with
+an average edge length of ℎ = 1∕400.
+decreased proportionally. Given the unstructured nature of the mesh and the resulting solution point distribution, the
+circular shape of the shock front was still well-represented, with even better approximation at higher mesh resolutions.
+These results indicate that the proposed approach can be extended to unstructured grids in a straightforward manner
+without appreciably sacrificing its efficiency or performance at resolving discontinuities.
+5.6. Three-dimensional Rayleigh–Taylor instability
+A final evaluation of the proposed approach and the extension to three-dimensional flows was performed through
+the simulation of a magnetized three-dimensional Rayleigh–Taylor instability. The problem consists of a denser gas
+resting on top of a lighter gas under the effect of a gravitational field initially in equilibrium with the pressure gradient.
+Instabilities arise in the form of “bubbles” of the lighter gas rising and “fingers” of the heavier gas descending, after
+which nonlinear momentum transport drives the flow to a turbulent mixing state. The problem is solved on the domain
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 16 of 22
+
+3
+4
+5I
+/vl
+10
+20
+30
+40
+50P
+1000
+2000
+3000
+4000
+5000P
+23000
+24000
+25000
+26000
+27000
+28000Positivity-preserving entropy filtering for the ideal MHD equations
+(a) Coarse mesh
+(b) Fine mesh
+Figure 9:
+Enlarged view of contours of pressure with mesh overlay for the magnetized blast problem at 푡 = 0.001 computed
+using a ℙ4 scheme on the coarse mesh (left) and fine mesh (right). Contour scale identical to Fig. 7 and Fig. 8.
+[−퐿∕2, 퐿∕2]2 × [−퐿, 퐿], where 퐿 = 1, and the initial conditions are given as
+퐪(퐱, 0) = [휌, 푢, 푣, 푤, 퐵푥, 퐵푦, 퐵푧, 푃]푇 =
+{
+퐪푙,
+if 푧 ≤ 0,
+퐪ℎ,
+else,
+(47)
+where
+퐪푙 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌푙
+0
+0
+푊 (푥, 푦, 푧)
+퐵0
+0
+0
+푃(푧)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+and
+퐪ℎ =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌ℎ
+0
+0
+푊 (푥, 푦, 푧)
+퐵0
+0
+0
+푃(푧)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+(48)
+for some vertical velocity perturbation 푊 (푥, 푦, 푧) and initial pressure distribution 푃 (푧). Periodic boundary conditions
+are enforced along the transverse (푥, 푦) directions while reflecting boundary conditions are enforced along the top and
+bottom boundaries. A gravitational field is added to the problem, given in the form of a source term as
+퐒퐆 = −[0, 0, 0, 휌푔, 0, 0, 0, 휌푤푔],
+(49)
+where 푔 = 1. The treatment of this gravitational field in the context of the entropy filter is taken simply as an additional
+term in the source term of Powell’s method, and as it not stiff for this problem, it does not appreciably affect the time
+step restrictions of the scheme.
+The parameters for the problem are taken similarly to a scaled form of the setup in Stone and Gardiner [50]. The
+densities of the light and heavy gases are taken as 휌푙 = 1 and 휌ℎ = 3, respectively, yielding an Atwood number of 1∕2.
+To enforce equilibrium in the flow, the initial pressure field is taken as
+푃(푧) = 푃0 − 휌푔푧,
+(50)
+where 푃0 = 10∕훾 for a specific heat ratio 훾 = 5∕3. To seed instabilities in the flow, perturbations were added in the
+form of a vertical velocity field component as
+푊 (푥, 푦, 푧) = 퐴 cos
+( 휋푧
+2퐿
+)
+sin
+(4휋푥
+퐿
+)
+sin
+(4휋푦
+퐿
+)
+,
+(51)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 17 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+where 퐴 = 0.05. This differs from the work of Stone and Gardiner [50] in that the transverse distribution of the
+perturbations is taken as a single deterministic mode instead of randomly-generated noise. As such, the predicted flow
+fields are expected to differ during the linear growth regime of the instability.
+The addition of a magnetic field significantly impacts the behavior of the Rayleigh–Taylor instability as it induces
+a stabilizing effect on the flow. In fact, linear stability analysis presents a cutoff magnetic field value,
+퐵푐 =
+√
+(휌ℎ − 휌푙)푔퐿 =
+√
+2,
+(52)
+above which the instability is completely damped by the magnetic field [50]. We consider three variations of this flow,
+a hydrodynamic case where 퐵0 = 0, a weakly magnetized case where 퐵0 = 0.1퐵푐, and a strongly magnetized case
+where 퐵0 = 0.5퐵푐. The term strongly magnetized is relative in the sense that the plasma-beta is still quite high, but
+the magnetic field can suppress almost all potential instability modes along its orientation.
+These three flow conditions were computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh with a time step of
+Δ푡 = 2⋅10−5. The flow, visualized in the form of a volume rendering of the density field, at various times is shown in
+Fig. 10, Fig. 11, and Fig. 12 for the hydrodynamic, weakly magnetized, and strongly magnetized cases, respectively.
+For the hydrodynamic case, the canonical flow pattern of the Rayleigh–Taylor instability was observed, with rising
+bubbles and descending figures. At later times, these features transitioned to a turbulent mixing state. When a weak
+magnetic field was applied, a significant degree of anisotropy was imparted on the flow, seen in the form of distortions
+in the bubbles and fingers aligned with the orientation of the magnetic field. Furthermore, the slowed growth rate of
+the instabilities due to the stabilizing effect of this weak magnetic field could be observed. For the strongly magnetized
+case, the magnetic field effectively damped all instabilities along the orientation of the field, such that the resulting
+flow only varied perpendicular to the orientation of the field. Given a long enough simulation time, this flow would
+be expected to transition to a three-dimensional turbulent mixing state as nonlinear transport effects overcome the
+stabilizing nature of the magnetic field. For all cases, the flow was numerically well-behaved, indicating that the
+proposed approach can be effectively applied to three-dimensional flows in both the magnetized and hydrodynamic
+regimes.
+(a) 푡 = 2
+(b) 푡 = 3
+(c) 푡 = 4
+(d) 푡 = 5
+Figure 10:
+Volume rendering of the density field for the hydrodynamic (퐵0 = 0) Rayleigh–Taylor instability problem at
+varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.
+To quantify the efficiency improvements of the proposed algorithm in comparison to the original approach in
+Dzanic and Witherden [17] which utilizes repeated evaluations of the matrix-vector product in Eq. (27), a runtime
+comparison between the two methods was performed for the case of 퐵0 = 0.1퐵푐. The cost was evaluated on 16
+NVIDIA V100 GPUs with respect to the wall-clock time elapsed until the simulation reached 푡 = 1, and the results
+are shown in Fig. 13. While the original approach required 61.4 GPU hours, the proposed approach required only
+25.3 GPU hours, a speedup factor of approximately 2.4 across the entire simulation time. Furthermore, this speedup
+is expected to increase with higher approximation orders due to the increased number of solution points per element.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 18 of 22
+
+2
+3
+dPositivity-preserving entropy filtering for the ideal MHD equations
+(a) 푡 = 2
+(b) 푡 = 3
+(c) 푡 = 4
+(d) 푡 = 5
+Figure 11:
+Volume rendering of the density field for the weakly magnetized (퐵0 = 0.1퐵푐) Rayleigh–Taylor instability
+problem at varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.
+(a) 푡 = 2
+(b) 푡 = 3
+(c) 푡 = 4
+(d) 푡 = 5
+Figure 12:
+Volume rendering of the density field for the strongly magnetized (퐵0 = 0.5퐵푐) Rayleigh–Taylor instability
+problem at varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.
+To confirm this, an identical comparison was performed using a ℙ4 approximation on a 푁푒 = 51 × 51 × 102 mesh,
+which results in approximately the same number of degrees of freedom. At this approximation order, the original
+approach required 257 GPU hours whereas the proposed approach required only 39.8 GPU hours, a speedup factor
+of approximately 6.5. These results indicate that the proposed algorithmic improvements both substantially decrease
+the overall computational cost of the entropy filtering approach and show much better scaling with respect to the
+approximation order.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 19 of 22
+
+2
+3
+d2
+3
+dPositivity-preserving entropy filtering for the ideal MHD equations
+ℙ3
+ℙ4
+0
+100
+200
+300
+25.3
+61.4
+39.8
+257
+GPU hours per characteristic time
+Proposed algorithm
+Original algorithm
+Figure 13:
+Comparison of the wall-clock time to reach 푡 = 1 for the weakly magnetized (퐵0 = 0.1퐵푐) Rayleigh–Taylor
+instability problem with a ℙ3 (left) and ℙ4 (right) scheme with the same number of degrees of freedom using the original
+algorithm of Dzanic and Witherden [17] and the proposed algorithm.
+6. Conclusions
+In this work, a positivity-preserving adaptive filtering approach was proposed for shock capturing in discontinuous
+spectral element approximations of the ideal magnetohydrodynamics equations. The proposed scheme can be consid-
+ered as an extension of the entropy filtering approach [17] introduced by the authors for the gas dynamics equations
+to the ideal magnetohydrodynamics system. By formulating convex invariants such as positivity of density and pres-
+sure and a local discrete minimum entropy principle as discrete constraints on the solution, the amount of filtering
+necessary to satisfy the constraints was computed as an element-wise scalar optimization problem. This approach was
+combined with the eight-wave method of Powell et al. [7] for enforcing a solenoidal magnetic field. As this method
+introduced non-conservative source terms to the system, an operator splitting approach was proposed and its effects
+on the assumptions necessitated by the adaptive filtering approach to guarantee the satisfaction of the constraints were
+analyzed. An improved algorithm for solving the optimization problem for the filter strength was also introduced which
+significantly improved the computational efficiency of the proposed method.
+The proposed scheme could robustly resolve strong discontinuities while recovering high-order accuracy in smooth
+regions of the flow and could be easily and efficiently implemented on general unstructured grids. The efficacy of the
+approach was shown in a variety of numerical experiments, ranging from simple transport and shock tubes to extremely
+magnetized blast waves and three-dimensional magnetohydrodynamic instabilities. Furthermore, the proposed algo-
+rithmic enhancements yielded significant improvements in the computational cost and showed much better scaling with
+respect to approximation order, reducing the total runtime of the simulations by a factor of 2.4 for ℙ3 approximations
+and 6.5 for ℙ4 approximations. Future improvements to the proposed scheme could focus on applying different filter
+kernels to various components on the solution, alternate methods for enforcing a divergence-free magnetic field, and
+anisotropic filtering approaches.
+Acknowledgements
+This work was supported in part by the U.S. Air Force Office of Scientific Research via grant FA9550-21-1-0190
+("Enabling next-generation heterogeneous computing for massively parallel high-order compressible CFD") of the
+Defense University Research Instrumentation Program (DURIP) under the direction of Dr. Fariba Fahroo.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf,len=977
+page_content='Positivity-preserving entropy filtering for the ideal  magnetohydrodynamics equations  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanica,∗, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Witherdena  aDepartment of Ocean Engineering, Texas A&M University, College Station, TX 77843  A R T I C L E I N F O  Keywords:  Discontinuous spectral element  Ideal magnetohydrodynamics  Shock capturing  Positivity-preserving  Entropy filtering  A B S T R A C T  In this work, we present a positivity-preserving adaptive filtering approach for discontinuous  spectral element approximations of the ideal magnetohydrodynamics equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This approach  combines the entropy filtering method (Dzanic and Witherden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', 468, 2022) for  shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-  free magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Due to the inclusion of non-conservative source terms, an operator-splitting  approach is introduced to guarantee that the positivity and entropy constraints remain satisfied  by the discrete solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, a computationally efficient algorithm for solving the op-  timization process for this nonlinear filtering approach is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The resulting scheme can  robustly resolve strong discontinuities on general unstructured grids without tunable parameters  while recovering high-order accuracy for smooth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The efficacy of the scheme is shown  in numerical experiments on various problems including extremely magnetized blast waves and  three-dimensional magnetohydrodynamic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Introduction  The transport and interaction of a non-resistive conducting fluid and its electromagnetic field remain extensively  investigated phenomena as they are instrumental in various applications ranging from the study of astrophysical accre-  tion disks [1] and supernova remnants [2] to magnetic confinement fusion [3] and plasma physics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' These strongly  nonlinear effects are governed by the equations of ideal magnetohydrodynamics (MHD), which are composed of a  combination of the Euler equations of gas dynamics and Maxwell’s equations of electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' From this formu-  lation, a strong coupling between the magnetic field and the conducting fluid can be observed, where the magnetic  field induces a current in the fluid which, in turn, gives rise to a second, induced magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This interaction  can introduce multi-scale, multi-physics behavior in the system, such that magnetohydrodynamic flows can become  exceedingly complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  As a result of this complexity, the robust and accurate numerical approximation of ideal MHD can present many  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Since hyperbolic systems are known to produce discontinuities even with smooth initial conditions [5],  the numerical scheme must be able to robustly resolve these discontinuities which, in the case of MHD, come in the  form of hydrodynamic and magnetic shocks and contact waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, the approximation of the ideal MHD  equations also requires an intrinsic constraint on the solution in the form of a solenoidal magnetic field which may  not be satisfied by the scheme even if the magnetic field is initially solenoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Without a mechanism to enforce  this constraint, unphysical dynamics can arise in the solution which can lead to numerical instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The standard  numerical schemes for approximating MHD flows are finite difference and finite volume methods, whose properties  and robustness are well-established in the literature [6–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' However, they possess certain drawbacks in that they  are either difficult to extend to complex domains with unstructured grids or cannot recover high-order accuracy in a  computationally efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  A particular class of schemes which have more recently grown in popularity are high-order discontinuous spec-  tral element methods (DSEM) as they possess the geometric flexibility of finite volume methods while retaining the  arbitrarily high-order accuracy and efficiency of spectral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As such, they provide a promising avenue for  significantly decreasing the computational cost and expanding the viability of simulating complex MHD problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  However, due to the presence of discontinuities in MHD, DSEM approximations of these systems may introduce spu-  rious oscillations in the solution in the form of Gibbs phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Without proper treatment, these oscillations can  ∗Corresponding author  tdzanic@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='edu (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic)  ORCID(s): 0000-0003-3791-1134 (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 0000-0003-2343-412X (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Witherden)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 1 of 22  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='03129v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='NA]  9 Jan 2023  aPositivity-preserving entropy filtering for the ideal MHD equations  result in unphysical predictions or the failure of the numerical scheme altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' To extend to use of DSEM to MHD,  various numerical stabilization techniques have been proposed, ranging from artificial viscosity methods [12, 13] to  limiting-type approaches [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' While these various methods may be sufficient to stabilize the solution in many  cases, they may not guarantee that the solution will abide by physical constraints, may require problem-dependent tun-  able parameters, can be computationally inefficient for general unstructured grids, or may be excessively dissipative in  smooth regions of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  There is significant interest in the design of numerical schemes that are “provably robust” in the sense that they  can guarantee that the solution will abide by certain physical constraints even in the presence of features such as  discontinuities, the quintessential examples being positivity-preserving schemes for gas dynamics which guarantee the  positivity of the density and internal energy/pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For DSEM, this property is typically achieved through some  form of nonlinear limiting or filtering [14, 16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' However, designing schemes that possess this property without  sacrificing the computational efficiency of DSEM for general unstructured grids and their advantageous scale-resolving  properties in smooth flow regions can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In the context of MHD, this becomes even more difficult due  to the additional complexity of the governing equations as well as the incorporation of differential constraints, namely  solenoidal magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As such, there is a need for numerical stabilization techniques for DSEM approximations  of the ideal MHD equations that retain as many of these desirable properties as possible, namely that they:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Guarantee that physical constraints of the solution are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Are compatible with numerical techniques for enforcing intrinsic constraints such as a solenoidal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Do not require problem-dependent tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Do not appreciably degrade the ability of the underlying DSEM to resolve smooth portions of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Can be easily and efficiently implemented on general unstructured grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  In this work, we propose a nonlinear adaptive filtering approach as a numerical stabilization technique for DSEM  approximations of the ideal MHD equations to address these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The proposed technique can be considered as an  extension of the entropy filtering approach originally introduced by the authors for shock capturing in gas dynamics to  the ideal MHD system [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This technique relies on using the solution’s ability to preserve convex invariants of the  system, namely positivity of the density and pressure and a discrete local minimum entropy principle, to compute the  necessary filter strength to ensure a well-behaved solution in the vicinity of discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Extending this approach to  the ideal MHD system presents several challenges, primarily stemming from the treatment of the divergence-free con-  straint on the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' We utilize the eight-wave method of Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' [7] which introduces non-conservative  source terms in the equation proportional to the divergence of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As these non-conservative terms  can conflict with the necessary assumptions of the entropy filtering approach, we present a modified set of conditions  and introduce an operator splitting approach to the system which allows the filtering method to retain its positivity-  preserving properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, as the original approach for performing the optimization process necessary in the  adaptive filtering framework as presented in Dzanic and Witherden [17] was found to be quite computationally expen-  sive, we develop a highly-efficient numerical approach which drastically reduces the overall computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The  resulting approach can robustly resolve strong hydrodynamic and magnetic discontinuities in the flow without appre-  ciably degrading the accuracy of the underlying DSEM for smooth flows, does not require problem-dependent tunable  parameters, and can be easily extended to unstructured grids with relatively low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The efficacy of  the proposed method is demonstrated in a variety of numerical experiments including smooth transport, extremely  magnetized blast waves, and three-dimensional magnetohydrodynamic instabilities computing using high-order ap-  proximations on both structured and unstructured grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The organization of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' We present some preliminaries regarding DSEM approximations and  the ideal MHD equations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The entropy filtering approach for ideal MHD is then introduced in Section 3,  and its numerical implementation and computational optimizations are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Results for various test  cases are then shown in Section 5, and conclusions are drawn in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Ideal magnetohydrodynamics  The governing equations for the evolution of an ideal magnetohydrodynamic fluid can be given in the form of a  hyperbolic conservation law as  휕푡퐮 + 훁⋅퐅 (퐮) = 퐒푩(퐮),  (1)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 2 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  where 퐮 = 퐮(퐱, 푡) ∈ ℝ푚 is the solution of some number of field variables 푚 defined over a 푑-dimensional spatial  domain 퐱 ∈ ℝ푑 and time 푡, 퐅(퐮) ∈ ℝ푚×푑, and 퐒퐁(퐮) is an additional source term to be defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3 whose  purpose is to ensure a solenoidal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The solution and flux are given as  퐮 =  ⎡  ⎢  ⎢  ⎢⎣  휌  흆풗  푩  퐸  ⎤  ⎥  ⎥  ⎥⎦  and  퐅 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢⎣  흆풗  흆풗 ⊗ 퐯 + 퐈  (  푃 + 1  2퐁⋅퐁  )  − 퐁 ⊗ 퐁  풗 ⊗ 퐁 − 퐁 ⊗ 풗  (  퐸 + 푃 + 1  2퐁⋅퐁  )  퐯 − 퐁(퐯⋅퐁)  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥⎦  ,  (2)  where 휌 is the density, 흆풗 is the momentum, 퐸 is the total energy, 푃 = (훾 − 1)  (  퐸 − 1  2휌퐯⋅퐯 − 1  2퐁⋅퐁  )  is the pressure,  퐁 is the magnetic field, and 훾 is the specific heat ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, the symbol 퐈 denotes the identity matrix in ℝ푑×푑  and 퐯 = 흆풗∕휌 denotes the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The solution can be more conveniently expressed in terms of a vector of primitive  variables as 퐪 = [휌, 퐯, 퐁, 푃]푇 , and auxiliary quantities representing the magnetic pressure and plasma-beta can be  defined as 푃푏 = 1  2(훾 − 1)퐁⋅퐁 and 훽 = 2푃∕(퐁⋅퐁), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Due to the lack of magnetic monopoles, the MHD equations have an intrinsic constraint on the solution in the form  of a solenoidal magnetic field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=',  훁⋅퐁 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (3)  Although this constraint must be satisfied analytically by the MHD equations, numerical approximations do not nec-  essarily satisfy it even if the magnetic field is initially solenoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' If this constraint is not enforced by the scheme,  numerical instabilities may arise in addition to the non-physical nature of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Many approaches exist  to enforce this condition on the magnetic field, including the use of solenoidal basis functions [19], projection meth-  ods [20], constrained-transport schemes [6], divergence cleaning methods [21], and the eight-wave method [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' An  overview of the salient techniques is presented in Wu and Shu [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The entropy solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (1) satisfies an entropy inequality of the form  휕푡휎(퐮) + 훁⋅횺(퐮) ≥ 0,  (4)  where (휎, 횺) is any numerical entropy-flux pair [22] that satisfies the relation  휕퐮횺 = 휕퐮휎휕퐮퐅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Note that this inequality may be negated depending on which notation is used for the numerical entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In Dao and  Nazarov [23], it was shown that the entropy solution (in a vanishing viscosity sense) of the ideal MHD system satisfies  a minimum entropy principle on the specific physical entropy 휎 = 푃휌−훾 in the form  휎 (퐮(퐱, 푡 + Δ푡)) ≥ min  퐱  휎 (퐮(퐱, 푡)) ,  (5)  where Δ푡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This property is identical to the minimum entropy principle in gas dynamics [24], and it should be  satisfied by the solution in both smooth regions and in the vicinity of discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Discontinuous spectral element methods  For nodal discontinuous spectral element approximations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (1), including discontinuous Galerkin [25] and flux  reconstruction [26] schemes, the domain Ω is partitioned into 푁푒 elements Ω푘 such that Ω = ⋃  푁푒 Ω푘 and Ω푖 ∩Ω푗 = ∅  for 푖 ≠ 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' With a slight abuse of notation, the solution 퐮(퐱) within each element Ω푘 is approximated in a nodal manner  as  퐮(퐱) =  ∑  푖∈푆  퐮푖휙푖(퐱),  (6)  where 퐱푖 ∀ 푖 ∈ 푆 is a set of solution nodes, 휙푖(퐱) are their associated nodal basis functions that possess the property  휙푖(퐱푗) = 훿푖푗, and 푆 is the set of nodal indices for the stencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For brevity, we utilize the notation that 퐮푖 = 퐮(퐱푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The  order of the approximation of the solution is denoted as ℙ푝 for some order 푝, where 푝 is the maximal order of 퐮(퐱).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  This approximation formally yields a convergence rate of at least 푝 + 1 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 3 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  The flux is approximated via the contribution of an interior term, denoted by the subscript Ω푘, and an interface  term, denoted by the subscript 휕Ω푘, as  퐅(퐮) ≈ 퐅Ω푘(퐮) + 퐅휕Ω푘(퐮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (7)  For the interior component, the flux is computed through a collocation approach as  퐅Ω푘(퐮) =  ∑  푖∈푆  퐅(퐮푖)휙푖(퐱),  (8)  such that the interior contribution to the divergence of the flux can be computed as  훁⋅퐅Ω푘(퐮푖) =  ∑  푗∈푆  퐜푖푗퐅(퐮푗),  where  퐜푖푗 = ∇휙푖(퐱푗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (9)  The interface component of the flux is formed over a set of interface nodes 퐱푖 ∈ 휕Ω푘 ∀ 푖 ∈ 퐼, where 퐼 is a set of nodal  indices for the interface stencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' We assume that these interface nodes are a subset of the solution nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', 퐼 ⊂ 푆)  to avoid issues regarding interpolation for discontinuous solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' At each interface node, there exist two values of  the solution, 퐮−  푖 and 퐮+  푖 , representing the solution evaluated from the element of interest and the interface-adjacent  element, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The interface flux term can then be computed as  퐅휕Ω푘(퐮푖) =  ∑  푗∈퐼  퐅(퐮−  푗 , 퐮+  푗 , 퐧푗)휙푗(퐱),  (10)  where 퐅(퐮−  푖 , 퐮+  푖 , 퐧푖) are the common interface flux values dependent on the interior and exterior values of the solution  and their associated normal vectors 퐧푖 and 휙푖(퐱) are the interface bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The common interface flux is generally  computed using an approximate Riemann solver such as that of Rusanov [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The interface bases are dependent on  the choice of spatial discretization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', for flux reconstruction schemes, these terms can be given as  휙푖(퐱) = 퐧푖⋅퐡푖(퐱) − 휙푖(퐱).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (11)  Here, 퐡푖 are a set of correction functions [28, 29] that posses the properties that  퐧푖⋅퐡푗(퐱푖) = 훿푖푗  and  ∑  푖∈퐼  퐡푖(퐱) ∈ RT푝,  (12)  where RT푝 is the Raviart–Thomas space [30] of order 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In this work, the flux reconstruction scheme with the equivalent  discontinuous Galerkin correction functions [26] is used which recovers the nodal discontinuous Galerkin method [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The interface contribution to the divergence of the flux can then be given as  훁⋅퐅휕Ω푘(퐮푖) =  ∑  푗∈퐼  퐜푖푗퐅(퐮−  푗 , 퐮+  푗 , 퐧푗),  where  퐜푖푗 = ∇휙푖(퐱푗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (13)  The semi-discrete form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (1) can then be given as  휕푡퐮푖 = −  (  퐅휕Ω푘(퐮푖) + 훁⋅퐅휕Ω푘(퐮푖)  )  + 퐒퐁(퐮푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (14)  We assume that the spatial scheme satisfies the relation  휕푡퐮 = − ∫휕Ω푘  퐅 (퐱) ⋅ 퐧(퐱) d퐱 ≈ −  ∑  푗∈퐼  푚푗퐅(퐮−  푗 , 퐮+  푗 , 퐧푗)  (15)  where 푚푗 is the associated quadrature weight for 퐱푗 and 퐮 is the element-wise mean defined as  퐮 = 1  푉푘 ∫Ω푘  퐮(퐱) d퐱  and  푉푘 = ∫Ω푘  d퐱.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (16)  This assumption is appropriate for nodal discontinuous Galerkin schemes given appropriate quadrature and flux recon-  struction schemes utilizing the equivalent discontinuous Galerkin correction functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 4 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Eight-wave method  A common method for enforcing a divergence-free magnetic field is to utilize the eight-wave method of Powell  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This approach relies on an additional wave structure of the Riemann problem in MHD that arises when the  magnetic field is not exactly solenoidal, and it can be utilized to force the magnetic field to a solenoidal state via a  source term, given as  퐒푩(퐮) = −  ⎡  ⎢  ⎢  ⎢⎣  0  푩  풖  풖⋅푩  ⎤  ⎥  ⎥  ⎥⎦  훁⋅푩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (17)  With the inclusion of this source term, the divergence of the magnetic field is typically suppressed to the order of mag-  nitude of the approximation error [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As such, due to the simplicity of implementation and applicability to general  unstructured grids, it remains a routine approach for robustly enforcing the divergence-free constraint on the solenoidal  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In addition, only this modified form of the ideal MHD equations is symmetrizable and Galilean invariant when  the magnetic field is not exactly solenoidal [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' However, as this form is non-conservative, it occasionally can cause  inaccurate predictions around discontinuities in the flow (see Tóth [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The use of Powell’s method requires some clarification about the choice of the formulation for computing the  divergence of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In the context of DSEM, there exist two formulations, a local divergence, consisting  of just the interior component as  훁⋅푩퐿(퐮푖) =  ∑  푗∈푆  퐜푖푗푩푗,  (18)  and a global divergence, consisting of both the interior component and the interface contribution as  훁⋅푩퐺(퐮푖) =  ∑  푗∈푆  퐜푖푗푩푗 +  ∑  푗∈퐼  퐜푖푗푩푗,  (19)  where 푩푗 is a common interface value for the magnetic field, typically taken as the centered average of the interior and  exterior values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Whereas the divergence-free constraint can be imposed on the local divergence through straightforward  approaches such as projection to solenoidal bases, enforcing this constraint on the global divergence is typically more  difficult as its domain of influence is not strictly contained within the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It can be argued that the global approach  is the “correct” choice as it is the one for which the space of the divergence is consistent with the space of the solution,  but in practice, the local approach is typically sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In this work, the global approach is used as the complexity of  the two implementations is similar with Powell’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Methodology  Due to the presence of discontinuities in MHD flows in the form of hydrodynamic and magnetic shocks, it is  necessary to apply some sort of a numerical stabilization procedure to ensure robustness of the DSEM approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  In Dzanic and Witherden [17], an adaptive filtering approach was introduced with goal of stabilizing the scheme by  discretely enforcing convex constraints on the solution, given in the form of  Γ(퐮푖) > 0 ∀ 푖 ∈ 푆,  (20)  where Γ(퐮) is some constraint functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For a positivity-preserving scheme, these constraints are set as  Γ1(퐮) = 휌  and  Γ2(퐮) = 푃,  (21)  corresponding to constraints on the positivity of density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  While these constraints can ensure the positivity of these quantities, they are generally not restrictive enough to  ensure that the solution remains well-behaved in the vicinity of discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It is necessary to attempt to form  additional constraints on the solution that are restrictive enough to stabilize the solution in the vicinity of discontinuities  without degrading the accuracy of the scheme in regions where the solution is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' By utilizing the fact that the  minimum entropy principle presented in Section 2 should be satisfied by both smooth and discontinuous solutions, a  third constraint on the solution is enforced corresponding to a discrete form of a local minimum entropy principle as  Γ3(퐮) = 휎(퐮) − 휎min,  (22)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 5 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  where 휎(퐮) = 푃 휌−훾 is the specific physical entropy and 휎min is some local minimum entropy bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This minimum  bound 휎min is computed in an element-wise manner as the discrete minima of the entropy functional across the element  and its face neighbors prior to each stage of a temporal integration scheme, resulting in the enforcement of a discrete  minimum entropy principle over the local domain of influence of the element (see Dzanic and Witherden [17], Section  2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It was found in the context of gas dynamics that enforcing this constraint ensured well-behaved solutions in  the vicinity of discontinuities while recovering high-order accuracy in smooth regions of the flow [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Adaptive filtering  The constraints are enforced by an adaptive filtering procedure, where the filtered solution ̃퐮 is given in terms of a  filter kernel 퐻 applied to the solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=',  ̃퐮 = 퐻(퐮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (23)  This filtering is performed in modal space given a modal decomposition of the solution in the form of  퐮(퐱) =  ∑  푖∈푆  ̂퐮푖휓푖(퐱),  (24)  where 휓푖(퐱) ∀ 푖 ∈ 푆 are a set of modal basis functions and ̂퐮푖 are their corresponding modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' We assume that this modal  decomposition is chosen with respect to the unit measure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', Legendre polynomials, Koornwinder polynomials, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  A discrete form of this change-of-basis operation can be given in terms of a Vandermonde matrix 퐕 as  ̂퐮 = 퐕−1퐮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (25)  The filter kernel ̂  퐻 is taken as a second-order exponential kernel in modal space, such that the filtered modal modes  can be computed as  ̂  퐻푖(̂퐮푖) = ̂퐮 exp(−휁푝2  푖 ),  (26)  where 휁 is the filter strength and 푝푖 is the total order of the corresponding mode ̂퐮푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It must be noted that the adaptive  filtering approach is not restricted to this choice of filter and can be applied to any conservative filtering operation of  one free variable that can recover both the unfiltered solution and the mean mode [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The filtering operation 퐻(퐮)  can be cast in terms of a matrix-vector operation as  ̃퐮 = 퐻(퐮) = 퐕횲퐕−1퐮,  (27)  where 횲 is a diagonal matrix of 푝 + 1 unique values with its entries equal to 횲푖,푖 = exp(−휁푝2  푖 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The filter strength is computed via an element-wise nonlinear optimization process, taken as the minimum filter  strength necessary such that the constraints are satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=',  휁 = arg min  휁 ≥ 0  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' [Γ1  (̃퐮(퐱푖)) > 0, Γ2  (̃퐮(퐱푖)) > 0, Γ3  (̃퐮(퐱푖)) > 0 ∀ 푖 ∈ 푆] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (28)  Existence of a solution of 휁 is guaranteed if the element-wise mean of the solution satisfies the constraints, an assump-  tion that will be explored in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As this optimization process is a function of a scalar free variable, its solution  can be obtained using any root-bracketing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, as it is nonlinear and non-convex, convergence to  a local minima is sufficient in the case of multiple values of 휁 existing such that the constraints are satisfied exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  While this optimization problem seems computationally demanding due to the element-wise matrix-vector operations  necessary to compute the filtered solution each iteration of the solve, we present a numerical approach to solving  this problem in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1 that is much more computationally efficient than the original methodology in Dzanic and  Witherden [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Extensions to MHD  Extending the entropy filtering approach to the MHD system requires some modifications, with special care neces-  sary in regards to the treatment of the source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The adaptive filtering operation naturally relies on that assumption  that there exists a filter strength such that the constraints are satisfied, and it is trivial to show that a solution exists if the  element-wise mean satisfies the constraints [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The ability of discontinuous Galerkin-type approaches to preserve  convex invariants of hyperbolic systems on the element-wise mean is a well established in the literature, and the reader  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 6 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  is referred to a variety of works which utilize this property [15–17, 32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' However, the inclusion of the source term  and the presence of entropy constraints introduces some caveats on this property of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Let the set 퐺 represent the set of solutions which satisfy the constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', Γ1(퐮) > 0, Γ2(퐮) > 0, Γ3(퐮) > 0), and  let the shorthand notation 퐮 ∈ 퐺 represent 퐮푖 ∈ 퐺 ∀ 푖 ∈ 푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' To ensure that the filter can recover a constraint-satisfying  solution, it is necessary for the temporal update of the element-wise mean to preserve these invariants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', for some  time step 푛, if 퐮푛 ∈ 퐺, then 퐮푛+1 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For brevity, we consider a temporal update in the form of a forward Euler  approximation, given as  퐮푛+1 = 퐮푛 + Δ푡 [퐿1(퐮푛) + 퐿2(퐮푛)] ,  (29)  where  퐿1(퐮) = −훁⋅퐅(퐮)  and  퐿2(퐮) = 퐒퐁(퐮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (30)  Without an exactly solenoidal magnetic field, the property 퐮푛+1 ∈ 퐺 is not necessarily satisfied in this form under the  standard assumptions posed in works such as Zhang and Shu [32] and the original presentation of entropy filtering  for gas dynamics in Dzanic and Witherden [17], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', appropriate Riemann solver, CFL condition, strong stability  preserving temporal integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' If we consider the set of solutions 퐺푃 which satisfy just the positivity constraints  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', 퐮 ∈ 퐺푃 if Γ1(퐮) > 0, Γ2(퐮) > 0), then the work of Wu and Shu [15] showed that the property 퐮푛+1 ∈ 퐺푃 is  satisfied under a potentially more restrictive condition on the time step dependent on the discrete divergence of the  magnetic field (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1 in Wu and Shu [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, if we neglect the source term and consider an  intermediate temporal update as  퐮∗ = 퐮푛 + Δ푡퐿1(퐮푛),  (31)  then the work of Bouchut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' [36] (paired with the equivalency of the element-wise mean and Godunov methods  presented in Zhang and Shu [32] and subsequent works) shows that this intermediate state satisfies the property 퐮∗ ∈ 퐺  under the standard assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  These two observations motivate an operator splitting approach for the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Two separate filtering operations are  considered, a more restrictive filter which enforces both the positivity and entropy constraints, denoted by 퐻푒[퐮], and  a more relaxed filter that enforces only positivity constraints, denoted by 퐻푝[퐮].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As the assumption on the positivity  and entropy constraints on the element-wise mean are satisfied by the intermediate state, the more restrictive filter can  be applied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=',  ̃퐮∗ = 퐻푒  [퐮푛 + Δ푡퐿1(퐮푛)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (32)  Since the entropy constraints are the most restrictive constraint and the contribution of the source term is typically  minimal compared to the divergence of the flux (since it is proportional to 훁⋅퐁), this filtering operation can usually  mitigate the majority of the spurious oscillations in the vicinity of discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The contribution of the source term  is then added onto this filtered state, after which the positivity constraints are then enforced again on the temporal  update as  ̃퐮푛+1 = 퐻푝  [̃퐮∗ + Δ푡퐿2(퐮푛)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (33)  A solution to this filtering optimization problem is also guaranteed to exist as the positivity of the element-wise mean  is guaranteed [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Several properties of this splitting approach must be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' First, it is very rarely the case that the secondary filtering  operation is necessary – the entropy constraints on ̃퐮∗ are typically restrictive enough to where ̃퐮∗ + Δ푡퐿2(퐮푛) retains  its positivity-preserving properties, such that in most cases, the positivity constraints are typically just checked and no  filtering is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' However, to ensure that the scheme remains provably positivity-preserving, this secondary filtering  operation must be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Second, the splitting for the source term is calculated explicitly as 퐿2(퐮푛), not through  a Strang-type splitting approach [37] as 퐿2(퐮∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' While the latter may potentially better approximate the necessary  corrections to the solution for preserving a solenoidal magnetic field, these forms of splitting can introduce a limit on  the temporal accuracy of the scheme and therefore are avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Finally, unless the linear filtering kernel which recovers  the squeeze limiter of Zhang and Shu [32] is chosen (see Dzanic and Witherden [17], Remark 1), the divergence of  the filtered magnetic field is not guaranteed to be equal or lower than the unfiltered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As this work pertains to a  nonlinear filter, it may introduce minor divergence errors similarly to any nonlinear limiting operation, but these are  mitigated via the source term at the next temporal update with the explicit splitting approach such that its effects were  found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 7 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  Extensions to higher-order strong stability preserving (SSP) schemes follow readily from this formulation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=', the  temporal update for a third-order, three-stage SSP Runge–Kutta scheme, neglecting the notatioñ⋅ for brevity, is given  as  퐮(1) = 퐻푝  [퐻푒  [퐮푛 + Δ푡퐿1(퐮푛)] + Δ푡퐿2(퐮푛)] ,  (34)  퐮(2) = 퐻푝  [  퐻푒  [3  4퐮푛 + 1  4퐮(1) + 1  4Δ푡퐿1(퐮(1))  ]  + 1  4Δ푡퐿2(퐮(1))  ]  ,  퐮푛+1 = 퐻푝  [  퐻푒  [1  3퐮푛 + 2  3퐮(2) + 2  3Δ푡퐿1(퐮(2))  ]  + 2  3Δ푡퐿2(퐮(2))  ]  ,  where the entropy constraints for 퐻푒 are computed from the previous temporal stage (see Dzanic and Witherden [17],  Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Implementation  Ω푘  Figure 1:  Schematic of a two-dimensional ℙ2 triangular element Ω푘 showing interior solution points (red circles), interior  interface flux/solution points (red circles, blue outline), and exterior interface flux points (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The governing equations and the adaptive filtering approach were implemented in PyFR [38], a high-order GPU-  accelerated unstructured flux reconstruction solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The solution nodes were distributed along the Gauss–Legendre–  Lobatto quadrature points and 훼-optimized points [25] for tensor-product and simplex elements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' An  example of the solution and flux point distributions for a two-dimensional ℙ2 triangular element is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Temporal integration was performed using a three-stage, third-order SSP Runge–Kutta scheme as presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Unless otherwise stated, common interface fluxes were computed using the Harten-Lax-van Leer contact (HLLC)  Riemann solver of Li [39] and Gurski [40] with the Davis wavespeed estimate [41], although for most test cases, we  observed negligible differences in comparison to Rusanov-type [27] and Harten-Lax-van Leer (HLL) [42] Riemann  solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' To avoid a vacuum state for the Riemann solver and apply a numerical tolerance to the entropy condition, the  constraints were instead implemented as  Γ1(퐮) = 휌 − 휖,  Γ2(퐮) = 푃 − 휖,  and  Γ3(퐮) = 휎 − 휎min − 휖,  where 휖 = 10−8 is a small constant taken as some arbitrary factor of the machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Boundary conditions were enforced in a weak sense through the imposition of an exterior ghost state to the inter-  face Riemann solver [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Three types of boundary conditions were considered in this work: 1) Dirichlet boundary  conditions, where the exterior state is explicitly defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 2) Neumann boundary conditions, where the exterior state is  identical to the interior state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' and 3) reflecting boundary conditions, where the exterior state is identical to the interior  state with the normal component of the velocity and magnetic field negated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 8 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Filter optimization  Each time the filtering operation is called, the constraints are first checked on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' If the solution satisfies  the constraints, no filtering is applied, otherwise the filter strength is computed using the Illinois root-bracketing ap-  proach [44] with a stopping tolerance of 10−8 and a maximum of 20 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' While the filter strength can be simply  iterated by repeatedly evaluating the element-wise filtered solution as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (27) and computing the minima of the  constraints, several optimizations can be performed to drastically decrease the computational cost of performing this  filtering operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  First, instead of solving for 휁, it beneficial to solve for 푓 = exp(−휁) and utilize the relation  exp(−휁푝2  푖 ) = 푓 푝2  푖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  This bounds the search space of the root-bracketing approach to 푓 ∈ [0, 1], and the evaluation of the filter coefficients  reduces to simple integer powers of the argument 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Then, to avoid the costly computation of the matrix-vector  product in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (27) each iteration of the root-bracketing process, certain properties of the matrix 횲 can be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  As previously mentioned, 횲 is a diagonal matrix of 푝 + 1 unique values with its entries equal to 횲푖,푖 = exp(−휁푝2  푖 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' If  we define a set of diagonal matrices 퐈(푘) for 0 ≤ 푘 ≤ 푝 as  퐈(푘)  푖,푖 =  {  1,  if푝푖 = 푘,  0,  else,  (35)  then the filtering operation can be equivalently represented as  ̃퐮 =  푝  ∑  푖=0  푓 푝2  푖 퐮(푘),  (36)  where  퐮(푘) = 퐕퐈(푘)퐕−1퐮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (37)  Note that the values 퐮(푘) are independent of the value of 푓, such that these values can be pre-computed and the fil-  tered solution can be efficiently evaluated each iteration of the root-bracketing approach without having to repeatedly  compute the matrix-vector product 퐕횲퐕−1퐮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  This approach can be even further optimized by utilizing the fact that the nodal values of the solution can now  be decoupled, such that the root-bracketing process can be applied across each solution node sequentially which is  particularly beneficial for computing architectures where memory bandwidth is the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In this sequential  approach, each solution node 퐱푗 for 푗 ∈ 푆 solves for a value of 푓푗 such that ̃퐮푗 satisfies the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It is trivial to  show that if  푓 = min  푗∈푆 푓푗,  then ̃퐮 satisfies the constraints at all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It is therefore advantageous to then use 푓푗 as the upper bound for the root-  bracketing process for the node 퐱푗+1 as the constraints can be checked for ̃퐮푗+1 using this upper bound and the root-  bracketing process for that node can be skipped if they are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As the proposed algorithm requires effectively only  one full evaluation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (27) irrespective of the number of iterations of the root-bracketing approach, the memory  bandwidth requirements are significantly decreased, such that the filtering process becomes only a relatively small  portion of the total compute time that is typically less than the cost of the evaluation of the divergence of the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' An  example of the implementation of this approach is provided in the electronic supplementary material of this work, and  an evaluation of the efficiency improvements of this proposed algorithm in comparison to the original methodology in  Dzanic and Witherden [17] which utilizes repeated evaluations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (27) is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Results  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Near-vacuum convecting vortex  To verify that the proposed scheme retains the high-order accuracy of the underlying DSEM for smooth solutions,  the rate of convergence was calculated for the smooth magnetized convecting vortex problem introduced by Christlieb  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 9 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For this problem, the domain is taken as Ω = [−10, 10]2 with periodic boundary conditions discretized on  a structured quadrilateral mesh, and the initial conditions are given as  퐪(퐱, 0) =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  휌  푢  푣  퐵푥  퐵푦  푃  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  1  1 − 푦훿푢  1 + 푥훿푢  −푦훿퐵  푥훿퐵  1 + 훿푃  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  (38)  where  훿푢 =  휇  √  2휋  휙(푟),  훿퐵 = 휇  2휋 휙(푟),  훿푃 = −휇2(1 + 푟2)  8휋2  휙(푟)2,  (39)  and  휙(푟) = exp(1 − 푟2),  푟 =  √  푥2 + 푦2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (40)  The specific heat ratio was set as 훾 = 5∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' These conditions give a non-isentropic nature to the flow field, which  allows for a proper assessment of the proposed entropy-based constraints for smooth flows where the filter should be  primarily inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' To make this problem more numerically challenging, the parameter 휇 is chosen such as to give  a near-vacuum state for the pressure field [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This value was set as 휇 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='38948938512, which gives a minimum  pressure value in the domain of approximately 2휖 = 2⋅10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The problem was solved until a non-dimensional time of 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='05 using a fixed time step of Δ푡 = 1⋅10−4, after  which the 퐿1 norm of the magnetic field error was computed as  푒퐵 = 1  퐴 ∫Ω  ||퐵푥 − 퐵exact  푥  || + |||퐵푦 − 퐵exact  푦  ||| d퐱,  (41)  where 퐴 = 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The exact solution was computed through a translation of the initial conditions with a translation  velocity of [1, 1], and the integration was computed using a 9th order Gauss–Legendre quadrature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The error  with respect to the mesh resolution 푁푒 is shown for various approximation orders in Table 1 in addition to the rate of  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' High-order convergence, on the order of 푝 to 푝 + 1, was observed for all approximation orders between  ℙ2 and ℙ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, the ℙ2 results can be compared to the positivity-preserving third-order DG scheme in Wu and  Shu [15] (Table 2), which can be considered as a subset of the adaptive filtering approach without entropy constraints  (see Dzanic and Witherden [17], Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The proposed scheme gives marginally lower error even after removing  the 1∕퐴 normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  푁푒  ℙ2  ℙ3  ℙ4  ℙ5  202  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='29 × 10−4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='07 × 10−5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='30 × 10−6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='04 × 10−7  252  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='18 × 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='17 × 10−5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='37 × 10−6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='36 × 10−8  332  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='39 × 10−5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='07 × 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='39 × 10−7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='44 × 10−8  402  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='04 × 10−5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='02 × 10−6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='44 × 10−7  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='55 × 10−9  502  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='59 × 10−5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='72 × 10−7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='02 × 10−8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='92 × 10−9  672  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='92 × 10−6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='99 × 10−7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='31 × 10−8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='51 × 10−10  RoC  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='89  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='80  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='62  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='23  Table 1:  Convergence of the 퐿1 norm of the magnetic field error at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='05 with respect to mesh resolution 푁푒 for the  near-vacuum convecting vortex problem with varying approximation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Rate of convergence shown on bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Brio–Wu shock tube  Extensions to flows with discontinuities was then performed through the shock tube problem of Brio and Wu [8]  which includes features of the Riemann problem such as shock waves, contact discontinuities, rarefaction waves, and  compound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For this problem, the domain is set as Ω = [0, 1] and the initial conditions are given by  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 10 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  퐪(퐱, 0) =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  휌  푢  푣  퐵푥  퐵푦  푃  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  =  {  퐪푙,  if 푥 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5,  퐪푟,  else,  where  퐪푙 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='75  1  1  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  and  퐪푟 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='125  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='75  −1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (42)  The specific heat ratio is set as 훾 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The hydrodynamic components of this problem are identical to the Sod shock  tube [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Although this problem is one-dimensional, it was instead solved on a one element wide two-dimensional  mesh to facilitate the use of the vertical magnetic field component within the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dirichlet boundary conditions were  applied on the left/right boundaries while periodic boundary conditions were applied along the top/bottom boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The problem was computed with a ℙ3 scheme using a coarser mesh of 200 elements and a finer mesh of 400  elements with time steps of Δ푡 = 2⋅10−4 and 1⋅10−4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' A reference solution was also computed using a  highly-resolved ℙ0 scheme with 5⋅104 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The predicted density, pressure, and vertical magnetic field profiles  at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 2 for both the coarse and fine mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For all fields, both rarefaction waves and the shock  were well-resolved, showing sub-element resolution without any noticeable spurious oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, similar  behavior was observed for the contact and compound wave in the pressure and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Some minor oscillations  were observed in the density profile in the region between the compound wave and contact discontinuity, although this  behavior is not uncommon for some numerical schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The predicted density profile in that region converged to the  reference results with increasing resolution, but minor undershoots in front of the contact discontinuity were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
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+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
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+page_content='8  1  푥  휌  Reference  푁푒 = 200  푁푒 = 400  (a) Density  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='8  1  푥  푃  (b) Pressure  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='8  1  −1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5  1  푥  퐵푦  (c) Vertical magnetic field  Figure 2:  Density, pressure, and vertical magnetic field profiles for the Brio–Wu shock tube problem at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1 computed  using a ℙ3 FR scheme with 200 and 400 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Orszag–Tang vortex  Two-dimensional flows with more complex features were then considered through the canonical Orszag–Tang  vortex problem [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This case is a well-known model problem for evaluating a scheme’s ability to handle MHD  shocks and shock interactions as well as predicting transition to supersonic MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The domain is set as  Ω = [0, 1]2 with periodic boundary conditions, and the initial conditions are given as  퐪(퐱, 0) =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  휌  푢  푣  퐵푥  퐵푦  푃  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  25∕(36휋)  − sin(2휋푦)  sin(2휋푥)  sin(2휋푦)∕  √  4휋  − sin(4휋푥)∕  √  4휋  5∕(12휋)  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (43)  The specific heat ratio is set as 훾 = 5∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Uniform meshes of various resolution were generated, and the problem  was solved using a ℙ3 scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The contours of density at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5 computed on meshes with 푁푒 = 642, 1282, and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 11 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  2562 elements are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 3, computed using time steps of Δ푡 = 4⋅10−4, 2⋅10−4, and 1⋅10−4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The  results show good prediction of the canonical flow field of the Orszag–Tang vortex, with better approximation of shock  structure and small-scale flow features with increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Minor spurious oscillations were observed in the  density field at low resolutions, but these oscillations diminished with increasing mesh resolution, such that the flow  field at 푁푒 = 2562 was virtually oscillation-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (a) 푁푒 = 642  (b) 푁푒 = 1282  (c) 푁푒 = 2562  Figure 3:  Contours of density for the Orszag-Tang vortex at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5 computed using a ℙ3 FR scheme with 642 (left),  1282 (middle), and 2562 (right) elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  For a more quantitative comparison, the predicted pressure profile on the cross-section 푦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3125 at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='48 is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 4 in comparison to the results of Jiang and Wu [48] obtained using a high-order weighted essentially  non-oscillator (WENO) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It can be seen that similar observations can be made for the pressure field as with the  density field, with minor spurious oscillations at lower resolutions that diminish with increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The overall  prediction of the pressure profile was in good agreement with the reference results at moderate and high resolutions,  and strong discontinuities in the pressure field were generally well resolved at the sub-element level even at lower  resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' At the highest resolution, there was very good agreement with the reference data with minor differences  in the prediction of the location of some small-scale flow features on the left-hand side of the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Overall,  the results showed good predictions of the various flow features of the Orszag–Tang vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3  푥∕퐿  푝  Reference  푁푒 = 642  푁푒 = 1282  푁푒 = 2562  Figure 4:  Pressure profile of the Orszag-Tang vortex on the cross-section 푦∕퐿 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3125 at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='48 computed using a ℙ3  FR scheme with 642, 1282, and 2562 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Numerical results of Jiang and Wu [48] shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Shock cloud interaction  The proposed scheme was then evaluated on the shock cloud interaction problem of Dai and Woodward [9], con-  sisting of a high-density cloud interacting with an impinging shock wave which results in strong discontinuities and  the development of small-scale flow instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The problem setup as described by Balbás and Tadmor [11] is solved  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 12 of 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1Positivity-preserving entropy filtering for the ideal MHD equations  on the domain Ω = [0, 1]2 with the initial conditions given as  퐪(퐱, 0) = [휌, 푢, 푣, 푤, 퐵푥, 퐵푦, 퐵푧, 푃]푇 =  ⎧  ⎪  ⎨  ⎪⎩  퐪푐,  if 푟′ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='15,  퐪푙,  else if 푥 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='6,  퐪푟,  else,  (44)  where the cloud state, left state, and right state are given as  퐪푐 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  10  0  0  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1826182  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1826182  1  167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='345  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  ,  퐪푙 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='86859  0  0  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1826182  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1826182  1  167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='345  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  ,  and  퐪푟 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  1  −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2536  0  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='56418958  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='56418958  1  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  ,  (45)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The specific heat ratio is set as 훾 = 5∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The cloud is centered at [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5] with a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='15, such  that  푟′ =  √  (푥 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='8)2 + (푦 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  To facilitate the use of the three-dimensional magnetic field within the solver, the problem is solved on a one  element deep three-dimensional mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Additionally, while the original problem setup uses a [0, 1]2 domain with  Neumann boundary conditions on the top/bottom boundaries, we instead extend the domain to [0, 1] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5] and  apply periodic boundary conditions on the top/bottom boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As these boundaries on the extended domain are  outside of the domain of influence of the shock cloud interaction over the time range of the simulation, the effect of this  modified setup on the flow field is negligible but it helps alleviate any issues arising from numerical errors compounding  at free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The remaining left and right boundary conditions were set as Neumann and Dirichlet, respectively,  while periodicity was enforced along the 푧 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (a) Density  (b) Pressure  (c) Magnetic pressure  Figure 5:  Contours of density (left), pressure (middle), and magnetic pressure (right) on the subregion [0, 1]2 for the  shock cloud interaction problem at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='06 computed using a ℙ2 FR scheme with 4002 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  To perform a comparison of the proposed approach to a third-order DG scheme augmented with a WENO limiter  presented in Wu and Shu [15], an identical problem setup is used with a ℙ2 scheme on 4002 mesh (with respect to  the original domain size Ω = [0, 1]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The predicted contours of density, pressure, and magnetic pressure at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='06  computed using a time step of Δ푡 = 4⋅10−6 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The results show good resolution of the strong  discontinuities in the various fields without any observable spurious oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, small-scale features in  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 13 of 22  p  2  5  10  20  40P  100  200  300  4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5  2  5  10  20  50  100Positivity-preserving entropy filtering for the ideal MHD equations  the cloud region of the density and magnetic fields were not excessively dissipated, and the symmetry of the problem  was well-preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' A comparison to the method of Wu and Shu [15] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 1) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Note that some  discrepancy in the color schemes between the two images may be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The proposed scheme was roughly equally  performative in terms of the resolution of discontinuities and marginally better at resolving small-scale flow features on  the trailing side of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Without the positivity-preserving filtering approach, the scheme diverged due to negative  pressure in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (a) Entropy filter  (b) DG with WENO limiter  Figure 6:  Comparison of the contours of density computed by the proposed entropy filtering approach (left) and the  positivity-preserving DG scheme augmented with a WENO limiter of Wu and Shu [15] (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Magnetized blast  As a stress test for the positivity-preserving property of the proposed scheme for extreme flow conditions, a modified  form of the magnetized blast wave problem of Zachary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' [49] and Balsara and Spicer [10] was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In this  problem, a blast wave is driven by a spherical overpressure region in the center of the domain surrounded by a low  plasma-beta ambient state, resulting in strong magnetosonic shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The problem is solved on the periodic domain  Ω = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5]2, and the initial conditions are given as  퐪(퐱, 0) =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  휌  푢  푣  퐵푥  퐵푦  푃  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  =  {  퐪푒,  if  √  푥2 + 푦2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1,  퐪푎,  else,  where  퐪푒 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  1  0  0  퐵0  0  푃푒  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  and  퐪푎 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  1  0  0  퐵0  0  푃푎  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (46)  The specific heat ratio is set as 5∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' While the original problem setup uses 푃푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1, 푃푒 = 103, and 퐵0 = 100∕  √  4휋, we  consider the much more extreme case presented in Wu and Shu [15] with 푃푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1, 푃푒 = 104, and 퐵0 = 1000∕  √  4휋,  resulting in a very large pressure ratio of 105 and a very small plasma-beta of 훽 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5⋅10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As these conditions are  quite extreme, the scheme would diverge almost instantly in the absence of any positivity-preserving modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  To verify that the proposed scheme can be easily extended to unstructured grids, the problem was solved on trian-  gular meshes using a ℙ4 scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' A coarse mesh and a fine mesh were generated, consisting of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='2⋅105  elements with an average edge length of ℎ = 1∕200 and approximately 5⋅105 elements with an average edge length of  ℎ = 1∕400, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For this case, the HLL Riemann solver was used as it was found to be much better behaved  in these extreme conditions than the HLLC Riemann solver, although both approaches were properly stabilized with  the proposed entropy filtering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The contours of density, velocity magnitude, pressure, and magnetic pressure  at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='001 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 8 for the coarse and fine meshes, respectively, computed using time steps of  Δ푡 = 2⋅10−7 and Δ푡 = 1⋅10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Even with these extreme conditions on unstructured grids, the predicted solutions were  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 14 of 22  d  0  10  20  30  40Positivity-preserving entropy filtering for the ideal MHD equations  (a) Density  (b) Velocity Magnitude  (c) Pressure  (d) Magnetic Pressure  Figure 7:  Contours of density (top left), velocity magnitude (top right), pressure (bottom left), and magnetic pressure  (bottom right) for the magnetized blast problem at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='001 computed using a ℙ4 scheme on an unstructured mesh with  an average edge length of ℎ = 1∕200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  well-behaved, and both the coarse and fine meshes showed excellent resolution of the various discontinuities in the  velocity, pressure, and magnetic fields with sub-element resolution and no observable spurious oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Further-  more, the numerical width of the discontinuities decreased appropriately with increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For the density  field, minor spurious oscillations were observed, primarily at lower resolution and somewhat indicative of mesh im-  printing, but the strength and distribution of these oscillations decreased with the finer mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' These observations are  consistent with the case of the Brio–Wu shock tube where the density field was marginally less well-behaved than the  other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' However, the predicted fields were still very good given such extreme conditions and an unstructured  mesh, indicating that the proposed approach remains robust and accurate for such flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  To demonstrate the sub-element shock-resolving ability of the entropy filtering approach on unstructured grids, an  enlarged view of the contours of pressure with the mesh overlaid is shown for the two meshes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' It can be seen  that the shock was resolved well within the element, with the majority of the feature resolved across 1-2 solution nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Furthermore, this behavior persisted when the mesh resolution was increased, such that the discrete shock thickness  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 15 of 22  3  4  5/vl  O  10  20  30  40  50P  1000  2000  3000  4000  5000P  23000  24000  25000  26000  27000  28000Positivity-preserving entropy filtering for the ideal MHD equations  (a) Density  (b) Velocity Magnitude  (c) Pressure  (d) Magnetic Pressure  Figure 8:  Contours of density (top left), velocity magnitude (top right), pressure (bottom left), and magnetic pressure  (bottom right) for the magnetized blast problem at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='001 computed using a ℙ4 scheme on an unstructured mesh with  an average edge length of ℎ = 1∕400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  decreased proportionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Given the unstructured nature of the mesh and the resulting solution point distribution, the  circular shape of the shock front was still well-represented, with even better approximation at higher mesh resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  These results indicate that the proposed approach can be extended to unstructured grids in a straightforward manner  without appreciably sacrificing its efficiency or performance at resolving discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Three-dimensional Rayleigh–Taylor instability  A final evaluation of the proposed approach and the extension to three-dimensional flows was performed through  the simulation of a magnetized three-dimensional Rayleigh–Taylor instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The problem consists of a denser gas  resting on top of a lighter gas under the effect of a gravitational field initially in equilibrium with the pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Instabilities arise in the form of “bubbles” of the lighter gas rising and “fingers” of the heavier gas descending, after  which nonlinear momentum transport drives the flow to a turbulent mixing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The problem is solved on the domain  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 16 of 22  3  4  5I  /vl  10  20  30  40  50P  1000  2000  3000  4000  5000P  23000  24000  25000  26000  27000  28000Positivity-preserving entropy filtering for the ideal MHD equations  (a) Coarse mesh  (b) Fine mesh  Figure 9:  Enlarged view of contours of pressure with mesh overlay for the magnetized blast problem at 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='001 computed  using a ℙ4 scheme on the coarse mesh (left) and fine mesh (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Contour scale identical to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  [−퐿∕2, 퐿∕2]2 × [−퐿, 퐿], where 퐿 = 1, and the initial conditions are given as  퐪(퐱, 0) = [휌, 푢, 푣, 푤, 퐵푥, 퐵푦, 퐵푧, 푃]푇 =  {  퐪푙,  if 푧 ≤ 0,  퐪ℎ,  else,  (47)  where  퐪푙 =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  휌푙  0  0  푊 (푥, 푦, 푧)  퐵0  0  0  푃(푧)  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  and  퐪ℎ =  ⎡  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢  ⎢⎣  휌ℎ  0  0  푊 (푥, 푦, 푧)  퐵0  0  0  푃(푧)  ⎤  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥⎦  ,  (48)  for some vertical velocity perturbation 푊 (푥, 푦, 푧) and initial pressure distribution 푃 (푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Periodic boundary conditions  are enforced along the transverse (푥, 푦) directions while reflecting boundary conditions are enforced along the top and  bottom boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' A gravitational field is added to the problem, given in the form of a source term as  퐒퐆 = −[0, 0, 0, 휌푔, 0, 0, 0, 휌푤푔],  (49)  where 푔 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The treatment of this gravitational field in the context of the entropy filter is taken simply as an additional  term in the source term of Powell’s method, and as it not stiff for this problem, it does not appreciably affect the time  step restrictions of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The parameters for the problem are taken similarly to a scaled form of the setup in Stone and Gardiner [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The  densities of the light and heavy gases are taken as 휌푙 = 1 and 휌ℎ = 3, respectively, yielding an Atwood number of 1∕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  To enforce equilibrium in the flow, the initial pressure field is taken as  푃(푧) = 푃0 − 휌푔푧,  (50)  where 푃0 = 10∕훾 for a specific heat ratio 훾 = 5∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' To seed instabilities in the flow, perturbations were added in the  form of a vertical velocity field component as  푊 (푥, 푦, 푧) = 퐴 cos  ( 휋푧  2퐿  )  sin  (4휋푥  퐿  )  sin  (4휋푦  퐿  )  ,  (51)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 17 of 22  Positivity-preserving entropy filtering for the ideal MHD equations  where 퐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This differs from the work of Stone and Gardiner [50] in that the transverse distribution of the  perturbations is taken as a single deterministic mode instead of randomly-generated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As such, the predicted flow  fields are expected to differ during the linear growth regime of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The addition of a magnetic field significantly impacts the behavior of the Rayleigh–Taylor instability as it induces  a stabilizing effect on the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' In fact, linear stability analysis presents a cutoff magnetic field value,  퐵푐 =  √  (휌ℎ − 휌푙)푔퐿 =  √  2,  (52)  above which the instability is completely damped by the magnetic field [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' We consider three variations of this flow,  a hydrodynamic case where 퐵0 = 0, a weakly magnetized case where 퐵0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1퐵푐, and a strongly magnetized case  where 퐵0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5퐵푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The term strongly magnetized is relative in the sense that the plasma-beta is still quite high, but  the magnetic field can suppress almost all potential instability modes along its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  These three flow conditions were computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh with a time step of  Δ푡 = 2⋅10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The flow, visualized in the form of a volume rendering of the density field, at various times is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 10, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 11, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 12 for the hydrodynamic, weakly magnetized, and strongly magnetized cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  For the hydrodynamic case, the canonical flow pattern of the Rayleigh–Taylor instability was observed, with rising  bubbles and descending figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' At later times, these features transitioned to a turbulent mixing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' When a weak  magnetic field was applied, a significant degree of anisotropy was imparted on the flow, seen in the form of distortions  in the bubbles and fingers aligned with the orientation of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, the slowed growth rate of  the instabilities due to the stabilizing effect of this weak magnetic field could be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For the strongly magnetized  case, the magnetic field effectively damped all instabilities along the orientation of the field, such that the resulting  flow only varied perpendicular to the orientation of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Given a long enough simulation time, this flow would  be expected to transition to a three-dimensional turbulent mixing state as nonlinear transport effects overcome the  stabilizing nature of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' For all cases, the flow was numerically well-behaved, indicating that the  proposed approach can be effectively applied to three-dimensional flows in both the magnetized and hydrodynamic  regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (a) 푡 = 2  (b) 푡 = 3  (c) 푡 = 4  (d) 푡 = 5  Figure 10:  Volume rendering of the density field for the hydrodynamic (퐵0 = 0) Rayleigh–Taylor instability problem at  varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  To quantify the efficiency improvements of the proposed algorithm in comparison to the original approach in  Dzanic and Witherden [17] which utilizes repeated evaluations of the matrix-vector product in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' (27), a runtime  comparison between the two methods was performed for the case of 퐵0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1퐵푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The cost was evaluated on 16  NVIDIA V100 GPUs with respect to the wall-clock time elapsed until the simulation reached 푡 = 1, and the results  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' While the original approach required 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='4 GPU hours, the proposed approach required only  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3 GPU hours, a speedup factor of approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='4 across the entire simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, this speedup  is expected to increase with higher approximation orders due to the increased number of solution points per element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 18 of 22  2  3  dPositivity-preserving entropy filtering for the ideal MHD equations  (a) 푡 = 2  (b) 푡 = 3  (c) 푡 = 4  (d) 푡 = 5  Figure 11:  Volume rendering of the density field for the weakly magnetized (퐵0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1퐵푐) Rayleigh–Taylor instability  problem at varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  (a) 푡 = 2  (b) 푡 = 3  (c) 푡 = 4  (d) 푡 = 5  Figure 12:  Volume rendering of the density field for the strongly magnetized (퐵0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5퐵푐) Rayleigh–Taylor instability  problem at varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  To confirm this, an identical comparison was performed using a ℙ4 approximation on a 푁푒 = 51 × 51 × 102 mesh,  which results in approximately the same number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' At this approximation order, the original  approach required 257 GPU hours whereas the proposed approach required only 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='8 GPU hours, a speedup factor  of approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' These results indicate that the proposed algorithmic improvements both substantially decrease  the overall computational cost of the entropy filtering approach and show much better scaling with respect to the  approximation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Dzanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 19 of 22  2  3  d2  3  dPositivity-preserving entropy filtering for the ideal MHD equations  ℙ3  ℙ4  0  100  200  300  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='3  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='4  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='8  257  GPU hours per characteristic time  Proposed algorithm  Original algorithm  Figure 13:  Comparison of the wall-clock time to reach 푡 = 1 for the weakly magnetized (퐵0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='1퐵푐) Rayleigh–Taylor  instability problem with a ℙ3 (left) and ℙ4 (right) scheme with the same number of degrees of freedom using the original  algorithm of Dzanic and Witherden [17] and the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Conclusions  In this work, a positivity-preserving adaptive filtering approach was proposed for shock capturing in discontinuous  spectral element approximations of the ideal magnetohydrodynamics equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The proposed scheme can be consid-  ered as an extension of the entropy filtering approach [17] introduced by the authors for the gas dynamics equations  to the ideal magnetohydrodynamics system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' By formulating convex invariants such as positivity of density and pres-  sure and a local discrete minimum entropy principle as discrete constraints on the solution, the amount of filtering  necessary to satisfy the constraints was computed as an element-wise scalar optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' This approach was  combined with the eight-wave method of Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' [7] for enforcing a solenoidal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' As this method  introduced non-conservative source terms to the system, an operator splitting approach was proposed and its effects  on the assumptions necessitated by the adaptive filtering approach to guarantee the satisfaction of the constraints were  analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' An improved algorithm for solving the optimization problem for the filter strength was also introduced which  significantly improved the computational efficiency of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  The proposed scheme could robustly resolve strong discontinuities while recovering high-order accuracy in smooth  regions of the flow and could be easily and efficiently implemented on general unstructured grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' The efficacy of the  approach was shown in a variety of numerical experiments, ranging from simple transport and shock tubes to extremely  magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Furthermore, the proposed algo-  rithmic enhancements yielded significant improvements in the computational cost and showed much better scaling with  respect to approximation order, reducing the total runtime of the simulations by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='4 for ℙ3 approximations  and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='5 for ℙ4 approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Future improvements to the proposed scheme could focus on applying different filter  kernels to various components on the solution, alternate methods for enforcing a divergence-free magnetic field, and  anisotropic filtering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='  Acknowledgements  This work was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
+page_content=' Air Force Office of Scientific Research via grant FA9550-21-1-0190  ("Enabling next-generation heterogeneous computing for massively parallel high-order compressible CFD") of the  Defense University Research Instrumentation Program (DURIP) under the direction of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQfXwQA/content/2301.03129v1.pdf'}
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+EgoDistill: Egocentric Head Motion Distillation
+for Efficient Video Understanding
+Shuhan Tan1
+Tushar Nagarajan1
+Kristen Grauman1,2
+1The University of Texas at Austin
+2FAIR, Meta AI
+{shuhan,tushar.nagarajan,grauman}@cs.utexas.edu
+Abstract
+Recent advances in egocentric video understanding
+models are promising, but their heavy computational ex-
+pense is a barrier for many real-world applications. To ad-
+dress this challenge, we propose EgoDistill, a distillation-
+based approach that learns to reconstruct heavy egocen-
+tric video clip features by combining the semantics from
+a sparse set of video frames with the head motion from
+lightweight IMU readings. We further devise a novel self-
+supervised training strategy for IMU feature learning. Our
+method leads to significant improvements in efficiency, re-
+quiring 200× fewer GFLOPs than equivalent video models.
+We demonstrate its effectiveness on the Ego4D and EPIC-
+Kitchens datasets, where our method outperforms state-of-
+the-art efficient video understanding methods.
+1. Introduction
+Recent advances in augmented and virtual reality
+(AR/VR) technology have the potential to change the way
+people interact with the digital world, much like the smart-
+phone did in the previous decade.
+A fundamental re-
+quirement for AR/VR systems is the ability to recognize
+user behavior from egocentric video captured from a head-
+mounted camera.
+Towards this goal, several egocentric
+video datasets have been proposed in recent years, spurring
+increasing attention of the research community [11,26,56].
+Recent advances in egocentric action recognition, antic-
+ipation, and retrieval focus on building powerful clip-based
+video models that operate on video clips of a few seconds
+at a time [12, 16, 18, 25, 43, 44, 54, 55]. Despite encourag-
+ing performance, these models typically process densely-
+sampled frames with temporally-aware operations, making
+them computationally heavy. This makes them impractical
+for AR/VR devices with constrained resources, or for real-
+time video applications that require low latency. How to
+efficiently perform egocentric video understanding is there-
+fore an important, yet unsolved problem.
++
+Camera motion (IMU)
+Video Frame
+EgoDistill
+Figure 1. Illustration of EgoDistill. Given a single video frame
+and camera motion from IMU, EgoDistill learns to reconstruct the
+more expensive dense video clip feature. With its lightweight in-
+put, EgoDistill significantly improves efficiency.
+To address this issue, we take inspiration from how an-
+imals perceive the world with ego-motion. Neuroscience
+research has found that during active movement, the animal
+visual cortex receives and utilizes head motion signals from
+the motor cortex for visual processing [27,52,53]. This indi-
+cates that head motion signals support an embodied agent’s
+efficient understanding of the egocentric visual stream. In-
+spired by this phenomenon, we explore the relationship be-
+tween human head motion and egocentric video for efficient
+video understanding. In practice, we consider head motion
+signals captured by the inertial measurement unit (IMU) of
+a head-mounted camera. IMU measures motion from an ac-
+celerometer and gyroscope and is widely available on pop-
+ular wearable devices. Prior work leverages IMU as an ex-
+tra modality for human action recognition [13,68,69], (e.g.,
+jumping, walking, standing) or as geometric cues for visual-
+inertial odometry [7,20,71].
+In contrast, we propose to achieve efficient video under-
+standing by drawing on IMU as a substitute for dense video
+frame observations. The intuition is as follows. A video
+clip contains two things: semantic content (appearance of
+objects, places, people) and dynamics (how the scene and
+the camera move).
+While densely sampled frames are
+sure to capture both of the above—as done by current clip
+arXiv:2301.02217v1  [cs.CV]  5 Jan 2023
+
+models [16, 17, 54]—we hypothesize they are sometimes
+overkill. For a short video clip, much of the semantic con-
+tent is intelligible from even a single frame; meanwhile, the
+head motion provides a good portion of the dynamics, im-
+plicitly revealing how the visual appearance changes across
+neighboring frames.
+Building on this insight, we introduce EgoDistill, an ap-
+proach that learns to reconstruct dense egocentric video
+clip features using temporally sparse visual observations
+(as few as one RGB frame) together with the head motion
+from IMU. Specifically, EgoDistill employs a new form of
+knowledge distillation from video models. During training,
+we train a lightweight model that takes sparsely sampled
+image(s) and IMU to approximate the video representation
+extracted by a powerful but expensive video model. We fur-
+ther improve the model with a novel self-supervised train-
+ing stage for IMU feature learning. During inference, we
+directly utilize the lightweight model for egocentric video
+recognition, leading to much higher efficiency. Our model
+is flexible to the target heavy video feature, as we demon-
+strate with multiple current leading egocentric video mod-
+els [16–18,54]. See Figure 1.
+Importantly, EgoDistill offers a major efficiency gain:
+processing low-dimensional IMU and a few frames is much
+more efficient compared to processing a dense stack of
+frames. In practice, EgoDistill uses 200× fewer GFLOPs
+than the original video model.
+We experiment on the largest available egocentric ac-
+tion recognition datasets: Ego4D [26] and EPIC-Kitchens-
+100 [11]. We show that IMU coupled with an image offers
+better cross-modality knowledge distillation performance
+than images alone or images with audio. For a typical 50-
+minute egocentric video, EgoDistill reduces inference time
+of the source video model from 25 minutes to 36 seconds.
+Moreover, with only 1-4 frames, our lightweight distillation
+model achieves a better accuracy-efficiency trade-off than
+state-of-the-art models for adaptively sampling video con-
+tent [50,65]. Notably, we surpass the accuracy of these fast
+approaches by a large margin while requiring 4× to 8× less
+computation.
+2. Related Work
+IMU for activity recognition. Recent work explores us-
+ing the IMU sensor on mobile devices for human activ-
+ity recognition of actions like walking, jumping, or sit-
+ting [3,47,59–61]. Normally, these models take input from
+IMU sensors mounted on human body joints [9, 42, 60],
+waist-mounted [41] or in-pocket smartphones [33]. See [64]
+for a survey. Abundant work in video recognition explores
+ways to learn from RGB coupled with other modalities—
+audio [1, 22, 38], optical flow [19, 57, 58] or both [32, 36,
+51]—but comparatively fewer use IMU [48], and unlike our
+work, they focus on third-person video [13, 68, 69] and do
+not target at model efficiency. Our idea is for IMU to help
+reconstruct more expensive video features, rather than sim-
+ply fuse IMU with RGB for multi-modal recognition.
+IMU for odometry. Inertial odometry aims to estimate the
+position and orientation of the camera-wearer with readings
+from the IMU sensor. Traditionally, methods rely on IMU
+double integration [4] or enhancements thereof [5, 35, 40].
+Recent data-driven methods automatically learn to per-
+form inertial odometry with supervised [30, 70] or self-
+supervised learning [7], or combine IMU and visual in-
+put for more robust estimates with visual-inertial odome-
+try [20, 71]. While IMU can convey geometric ego-motion
+to our learned model, our goal is to produce efficient ego-
+centric video features rather than to output odometry.
+Visual feature learning with IMU. IMU is also used to
+learn better vision features [14,15,34,63], e.g., to encourage
+image features that are equivariant with ego-motion [34],
+to predict an IMU-captured body part (leg, hand) [14, 15],
+or to predict video-IMU correspondence [63], for applica-
+tions like action recognition [15,63] and scene understand-
+ing [14, 34]. While these results reinforce that IMU can
+inject embodied motion into visual features, our idea to use
+head motion to infer pretrained video features for speedy
+video understanding is distinct.
+Efficient video recognition. Being crucial for mobile ap-
+plications, efficient video recognition has received increas-
+ing attention in recent years. Several studies focus on de-
+signing lightweight architectures [17, 30, 37, 62, 72] by re-
+ducing 3D CNN operations across densely sampled frames.
+Our idea is orthogonal to them as we focus on inputs with
+sparsely-sampled frames. As we show in experiments, our
+method is compatible with different video architectures.
+Another line of research achieves efficiency by adap-
+tively selecting video content to process.
+Some reduce
+temporal redundancy by adaptively selecting which video
+clip [39], frames [24, 49], and/or feature channel [50] to
+process and which to skip, while others reduce spatial re-
+dundancy, efficient recognition by dynamically selecting
+selecting for each frame a smaller but important region to
+process [66,67]. Other work dynamically selects tokens in
+video transformers among both the spatial and temporal di-
+mensions [65]. Our idea is complementary: rather than dy-
+namically subsample the available video content, we show
+how to infer “full” video features for every clip using static
+image(s) and motion data. Our results outperform state-of-
+the-art sampling models (cf. Sec. 4). In addition, we fo-
+cus on egocentric video, where head motion is particularly
+meaningful for inferring unobserved visual content. To our
+knowledge, ours is the first technique specifically aimed at
+accelerating egocentric video processing.
+Multimodal distillation. Knowledge distillation aims to
+transfer knowledge learned by an expensive model to a
+lightweight model [31]. Recent work explores multimodal
+
+distillation, e.g., transferring from a RGB model to a flow or
+depth model [23,28], from a 3D model to a 2D model [45],
+or from a visual model to audio model [2,21]. The Listen-
+ToLook model [22] incorporates both clip subsampling and
+video-to-audio distillation for fast activity recognition in
+third-person video. In contrast, we explore the relationship
+between the camera-wearer’s head motion and RGB sig-
+nals for egocentric video. Our experiments show EgoDis-
+till’s advantage over ListenToLook in terms of the speed-
+accuracy tradeoff on egocentric video datasets.
+3. Approach
+We introduce EgoDistill, which uses sparsely-sampled
+frames and head motion from IMU to approximate the fea-
+tures of heavy video models for efficient egocentric video
+understanding.
+We first introduce the egocentric action
+recognition task (Sec. 3.1). Then, we introduce our pipeline
+(Sec. 3.2), our distillation model and training objective
+(Sec. 3.3), and our self-supervised IMU feature learning
+(Sec. 3.4). Figure 2 overviews our approach.
+3.1. Egocentric action recognition
+Given a fixed-length video clip V ∈ RT ×H×W ×3 con-
+sisting of T RGB frames of size H×W and a set of C action
+classes, the task of action recognition is to output a score for
+each action class, representing its likelihood. Typically, this
+is done with a powerful but expensive video model Ω, that
+directly operates on all the available frames to output the C
+class logits Ω(V) ∈ RC. Ω is trained with standard classifi-
+cation loss:
+LACT =
+�
+Vi
+LCE(ci, σ(Ω(Vi))),
+(1)
+where Vi is the i-th video clip in the dataset, ci is the
+corresponding ground-truth action label, σ is the softmax
+function, and LCE is cross-entropy loss.
+Popular video
+recognition models use clips that are typically �2 seconds
+long [16, 18, 54]. For longer videos, scores are averaged
+across all clips it contains to infer the video action label.
+3.2. Efficient video inference with head motion
+Processing the video clip V for action recognition is
+computationally intensive; however, the computation cost
+can be modulated depending on how frames from the clip
+are used. On the one hand, clip-based models [16–18, 54]
+process most (or all) frames in a video clip V to achieve
+strong recognition performance, but come at a high com-
+putational cost.
+On the other hand, frame-level mod-
+els [24, 49, 51, 67] only process one (or a small number)
+of frames from V and are more efficient, but suffer a drop
+in performance as a result. Our goal is to train a frame-
+based model that can approximate heavy clip-based model
+performance while maintaining high efficiency.
+For this, we turn to head motion captured by IMU. Along
+with RGB frames, each video clip is paired with IMU mea-
+surements M that record the camera (head) motion during
+the video. Specifically, the IMU readings are composed of
+6-dimensional accelerometer and gyroscope measurements
+in the xyz axes, which encode strong temporal motion in-
+formation about camera pose changes (both translation and
+rotation) across frames.
+For short video clips, a set of sparsely sampled frames I
+often already captures most appearance information. Com-
+plementary to this, the IMU readings capture camera mo-
+tion information (see below for discussion on scene mo-
+tion). Moreover, IMU is very efficient to process due to its
+low dimensionality. By processing inputs from these two
+sources with a lightweight frame-based model, we can infer
+the semantic and dynamic features of a heavier clip-based
+video model.
+Given I and M, we train an efficient lightweight model
+Φ to approximate the output of video model Ω. Specifically,
+we train our EgoDistill model Φ that achieves
+Φ(I, M) ≈ Ω(V).
+(2)
+Such a lightweight model will be able to approximate the
+result of the heavy video model, while being much more ef-
+ficient. Our approach is agnostic to the specific video model
+Ω; in experiments, we demonstrate its versatility for Mo-
+tionFormer [54], MViT [16], SlowFast [18] and X3D [17].
+In practice, we uniformly sample N frames1 from V to
+obtain I. We can achieve a trade-off between efficiency and
+performance by changing the number of frames N. In our
+experiments we use very low values of N (1 to 4 frames).
+In the next section, we discuss how we train Φ.
+3.3. Video feature distillation with IMU
+We achieve the objective in Equation 2 via knowledge
+distillation [31], where we transfer knowledge learned by
+the expensive teacher model Ω to a lightweight student
+model Φ. Next we present the design of Φ and the train-
+ing objectives, followed by our self-supervised IMU feature
+pretraining stage in Sec. 3.4.
+We design Φ to be a two-stream model. For a video clip
+and associated IMU signal (I, M), we extract image fea-
+tures zI = fI(I) and IMU features zM = fM(M) using
+lightweight feature encoders fI, fM respectively. Then,
+we fuse zI and zM with a fusion network Π to obtain the
+fused VisIMU feature zφ = Π(zI, zM). Finally, a fully-
+connected layer uses the fused feature to predict class logits
+Φ(I, M) ∈ RC.
+The fused feature zφ contains semantic information from
+the image frame coupled with complementary motion infor-
+1Other frame sampling heuristics (e.g., selecting from the start or center
+of the video) performed equivalently or worse than uniform sampling.
+
+Video clip
+   Video model
+Frame
+IMU
+Image Encoder
+IMU Encoder
+Fusion Layer
+FC-Softmax
+FC-Softmax
+Frame
+IMU
+Image Encoder
+IMU Encoder
+Frame
+Image Encoder
+IMU  Predictor
+Figure 2. EgoDistill architecture. Left: Self-supervised IMU feature learning. Given start and end frames of a clip, we train the IMU
+encoder to anticipate visual changes. Right: Video feature distillation with IMU. Given image frame(s) and IMU, along with our pre-
+trained IMU encoder, our method trains a lightweight model with knowledge distillation to reconstruct the features from a heavier video
+model. When the input includes more than one image frame, the image encoder aggregates frame features temporally with a GRU.
+mation from IMU, allowing us to accurately reconstruct the
+video clip feature. See Figure 2.
+We train Φ with a combination of three losses, as fol-
+lows. First, we train Φ to approximate the original video
+feature zV from the video model Ω:
+L1 =
+�
+(zVi,zφi)
+∥zVi − zφi∥1 .
+(3)
+This cross-modal loss encourages the fused feature zφ to
+match the video feature, i.e., the combined features from
+the different modalities should match in the feature space.
+Training with L1 alone does not fully capture the clas-
+sification output of Ω. Therefore, we also train Φ with a
+knowledge distillation loss:
+LKD =
+�
+(Vi,Ii,Mi)
+DKL(σ(Ω(Vi)/τ), σ(Φ(Ii, Mi)/τ)),
+(4)
+where (Vi, Ii, Mi) represents the i-th clip in the dataset,
+DKL measures KL-divergence between the class logits from
+the teacher model Ω and student model Φ, and τ is a tem-
+perature parameter. Intuitively, LKD casts the output of the
+video teacher model as a soft target for training the student
+model. In this way, the student model learns to better gen-
+eralize by mimicking the output distribution of the heavy
+video model.
+Finally, to further encourage the features to preserve el-
+ements useful for activity understanding, we also compute
+an action classification loss:
+LGT =
+�
+(Ii,Mi)
+LCE(ci, σ(Φ(Ii, Mi))),
+(5)
+where ci is the ground-truth action label, following Equa-
+tion 1. The final training loss is a combination of these three
+loss functions:
+L = αLKD + (1 − α)LGT + βL1,
+(6)
+where α controls the balance between knowledge distilla-
+tion and activity training [31], and β controls the weight for
+feature space matching.
+Critically, processing a few image frame(s) and the low-
+dimensional IMU readings is substantially faster than pro-
+cessing the entire video. Once trained, our model can ap-
+proximate the behavior of the source video model for recog-
+nition tasks, with the key benefit of efficient egocentric
+video recognition.
+What kind of motion does our model preserve? Video
+motion decomposes into scene motion (e.g., how the objects
+and the camera wearer’s hands are moving on their own),
+and camera motion (i.e., how the camera wearer is moving
+their head). By itself, IMU would directly account only for
+camera motion, not scene motion. However, by learning to
+map from the RGB frame and IMU to the full video fea-
+ture, we are able to encode predictable scene motions tied
+to scene content, e.g., how does hand and object movement
+in subsequent frames relate to the camera wearer’s head mo-
+tion (see Figure 7). Moreover, our model is applied to rel-
+atively short clips (1-2 seconds) in sequence, which means
+the appearance content is regularly refreshed as we slide
+down to process the longer video.
+3.4. Self-supervised IMU feature learning
+The success of EgoDistill depends on how well the IMU
+feature encoder fM extracts useful camera motion informa-
+tion and associates it with the visual appearance change in
+the video clip. In this way EgoDistill can learn to antic-
+ipate unseen visual changes in the video with I and M.
+We design a self-supervised pretraining task to initialize the
+weights of fM to achieve this.
+
+Specifically, for each clip V, we obtain its first and last
+frames (I0, IT ) as well as the IMU M. We first extract
+visual features z0
+I, zT
+I and IMU feature zM with feature
+extractors fI and fM mentioned above. Then, we train
+a feature predictor h to predict the IMU feature ˆzM =
+h(z0
+I, zT
+I ). By connecting ˆzM—which is a function of im-
+age features only—with zM, we encourage fM to extract
+useful camera motion features specifically associated with
+the visual appearance changes. Note that those appearance
+changes may include scene motion. Therefore, we include
+an L1 loss to train fM, which encourages fM to extract mo-
+tion features accounting for scene motion in the full video.
+In sum, we train fM, h, and the fusion network Π using
+L1 and NCE loss [29]: Lpretrain = LNCE + L1, where
+LNCE =
+�
+i
+− log
+sim(ˆzMi, zMi)
+�
+j sim(ˆzMi, zMj).
+(7)
+We sample negative examples zMj from other instances
+in the same mini-batch for j
+̸=
+i, and sim(q, k)
+=
+exp( q·k
+|q||k|
+1
+τ ′ ) with temperature τ ′ = 0.12.
+To summarize, prior to the main training stage of Equa-
+tion 6, we pretrain the IMU feature extractor fM and fu-
+sion network Π. As we will show below, both pretraining
+losses result in IMU features that are consistent with visual
+changes and lead to better finetuning performance.
+4. Experiments
+We evaluate our approach for resource-efficient action
+recognition.
+4.1. Experimental setup
+Datasets.
+We experiment on two large-scale egocen-
+tric action recognition datasets.
+Ego4D [26] contains
+3,670 hours of egocentric videos of people performing di-
+verse tasks (from cooking to farming) across the globe.
+As action recognition is not part of the original Ego4D
+benchmark, we construct this task with annotations from
+the Hands+Objects temporal localization benchmark [26]
+(see Supp. for details).
+We include clips with paired
+IMU and audio3, and consider classes with at least 2 la-
+beled instances.
+This results in a 94-class action recog-
+nition dataset with 8.5k training videos and 3.6k evalua-
+tion videos. EPIC-Kitchens [11] contains 100 hours of
+egocentric videos capturing daily activities in kitchen en-
+vironments. We use annotations from the action recogni-
+tion benchmark. Similar to Ego4D, we select videos that
+have paired IMU and audio data, and split the resulting data
+by camera-wearer. This results in a 62-class action dataset
+2We keep the ImageNet-pretrained fI model frozen, as finetuning it
+leads to mode collapse.
+3We require audio to compare with the audio-based baseline [22].
+with 29k training videos and 6.2k evaluation videos. For
+both datasets, we use “verb” labels as the target for action
+recognition as they are well aligned to activity motions.
+Evaluation metrics. To measure action recognition per-
+formance, we report the per-video top-1 accuracy on the
+validation set. We densely sample clips from each video
+and average their predictions to compute accuracy.
+To
+benchmark efficiency, we measure computational cost with
+FLOPs (floating-point operations) during inference.
+Implementation details. In our main experiments, we
+use MotionFormer [54] as the video teacher model Ω due
+to its strong performance for egocentric video. For EPIC-
+Kitchens, we use the authors’ provided checkpoint.
+For
+Ego4D, we finetune the above model for 50 epochs with
+1e−4 learning rate and 64 batch size on the training set.
+We use 16-frame input with sample rate 4. For the stu-
+dent model Φ, we use a ResNet-18 as the image backbone
+fI and a 1D Dilated CNN [6] for the IMU backbone fM.
+The feature fusion module Π uses a concatenation operation
+following a two-layer fully-connected layer with hidden di-
+mension 1024. For each video clip, the input image(s) is
+resized to 224 × 224, and the IMU is a 422 × 6 matrix
+(around 2 seconds with 198Hz frequency), representing the
+accelerometer and gyroscope readings along the xyz axes.
+For the image input, we uniformly sample N frames from
+the video clip. If N > 1, we use fI to sequentially gen-
+erate features for each frame and aggregate them with a
+GRU module [10]. For both datasets, we first pretrain the
+model with the self-supervised objective (Section 3.4) for
+50 epochs with AdamW [46] using batch size 64 and learn-
+ing rate 1e−4. Then, we finetune all the models with the
+same setting (Equation 6). We set α = 0.95 and β = 1.0
+based on validation data. For Ego4D, we set τ = 10.0 and
+train the model for 150 epochs. For EPIC-Kitchens, we set
+τ = 1.0 and train for 50 epochs.
+4.2. Baselines
+We compare to the following methods:
+• AdaFuse [50] trains a lightweight policy network to
+adaptively compute (or skip) feature map channels for
+each frame during inference. We use the AdaFuseTSN
+R50
+model with the provided hyper-parameters.
+• STTS [65] trains a module to rank spatio-temporal
+tokens derived from videos in a transformer-based
+model, and selects only the top-K tokens to speed up
+inference.
+• ListenToLook [22]: uses the audio-based feature dis-
+tillation module from [22] following the same audio
+processing and model architecture.
+These methods represent recent advances in efficient
+video recognition models. AdaFuse represents state-of-the-
+
+2
+4
+6
+8
+10
+12
+14
+16
+Inference cost per video clip (GFLOPs)
+33
+34
+35
+36
+37
+38
+Ego4D accuracy (%)
+Ego4D
+2
+4
+6
+8
+10
+12
+14
+16
+Inference cost per video clip (GFLOPs)
+30
+35
+40
+45
+50
+EPIC-Kitchens accuracy (%)
+EPIC-Kitchens
+EgoDistill (ours)
+AdaFuse [50]
+STTS [65]
+ListenToLook [22]
+VisOnly-Distill
+VisIMU
+VisOnly
+Figure 3. Accuracy vs. efficiency for action recognition on Ego4D (left) and EPIC-Kitchens (right). EgoDistill outperforms state-of-
+the-art efficient video recognition methods that adaptively sample video content, while using 4× to 8× fewer GFLOPs.
+LKD
+L1
+LGT
+L1-pretrain
+LNCE-pretrain
+Ego4D
+EPIC-Kitchens
+✓
+34.15
+35.04
+✓
+✓
+✓
+✓
+35.51
+39.33
+✓
+✓
+✓
+✓
+37.71
+42.20
+✓
+✓
+✓
+✓
+37.46
+43.17
+✓
+✓
+✓
+36.99
+41.21
+✓
+✓
+✓
+✓
+37.26
+42.30
+✓
+✓
+✓
+✓
+37.49
+43.51
+✓
+✓
+✓
+✓
+✓
+37.95
+44.95
+Table 1. Ablation study of model components. We compare the
+accuracy of EgoDistill with different components under N = 1.
+art approaches that achieve efficiency by reducing tempo-
+ral redundancy in CNN models. STTS is one of the most
+recent approaches that efficiently reduces both spatial and
+temporal redundancy in ViT models, which achieves the
+state-of-the-art on Kinectics-400 [8]. ListenToLook also re-
+lies on distillation, but using audio rather than head motion.
+For each model we generate multiple versions with differ-
+ent computation budgets to plot accuracy vs. GFLOPs. We
+train all AdaFuse and STTS models with 4 input frames to
+align with the maximum frames used by our model.
+For
+AdaFuse, we use the only provided hyper-parameter in the
+paper.4 For STTS, we use three provided variants: T0
+0.5-
+S4
+0.7, T0
+0.8-S4
+0.9 and the full model without token selection.
+For ListenToLook we adopt the same efficiency-accuracy
+trade-off as our method, i.e., varying the number of input
+frames.
+In addition, we test variants of our method:
+• VisOnly-Distill is our model without the IMU branch
+and fusion layer but trained with the same loss func-
+tion. Performance of this model reveals the role of
+IMU in the process of distillation.
+• VisIMU is our model trained with only LGT in Equa-
+4Modifying hyper-parameters to control the accuracy-efficiency trade-
+off results in unstable training and unreliable performance.
+Source Model
+Ego4D
+EPIC-Kitchens
+Video
+EgoDistill
+VisOnly-D
+Video
+EgoDistill
+VisOnly-D
+MFormer [54]
+46.38
+37.95
+34.32
+77.28
+44.95
+37.20
+MViT [16]
+40.32
+36.46
+33.40
+53.38
+36.90
+31.22
+SlowFast [18]
+40.52
+33.29
+33.04
+58.34
+39.42
+33.47
+X3D [17]
+37.56
+33.57
+32.90
+52.28
+36.34
+31.71
+Table 2. Versatility to model architectures. EgoDistill outper-
+forms the baseline for multiple common architectures, showing
+the generality of our idea. “Video” refers to the more expensive
+source model. We show the model accuracy under N = 1.
+tion 5. It shows the effectiveness of distillation from
+the video model compared with directly training the
+features with action labels.
+• VisOnly is an image-only model trained with LGT,
+which serves as the baseline.
+4.3. Main Results
+Importance of IMU-guided distillation.
+Figure 3
+shows the accuracy vs. efficiency curves. Methods towards
+the top-left of the plot represent those with both high ac-
+curacy and efficiency.
+Our method achieves good accu-
+racy with low computational cost. Specifically, on EPIC-
+Kitchens, when N = 1, EgoDistill improves over VisOnly-
+Distill by 8.4% with only a small increase in computa-
+tion. This result shows the effectiveness of IMU for recon-
+structing egocentric video features. Compared to VisIMU,
+EgoDistill improves by 9.9%, showing the effectiveness of
+knowledge distillation from the video model. Importantly,
+this reveals that EgoDistill does not simply benefit from the
+extra IMU context; our idea to approximate video features is
+necessary for best results. We see similar results on Ego4D.
+Comparison with the state of the art. Figure 3 also
+shows that EgoDistill achieves better accuracy with less
+computation than existing efficient video recognition mod-
+els AdaFuse [50], STTS [65], and ListenToLook [22].
+
+pack
+sew
+iron
+spray
+unscrew
+paint
+dip
+water
+play
+file
+hit
+throw
+inspect
+pull
+insert
+clean
+close
+detach
+smooth
+cut
+press
+hang
+shuffle
+tighten
+20
+0
+20
+40
+60
+Accuracy improvement(%)
+Ego4D
+break
+mix
+drink
+shake
+dry
+pour
+close
+wash
+put
+cut
+turn-off
+apply
+open
+peel
+turn-on
+throw
+flip
+scrub
+insert
+move
+take
+empty
+fill
+scoop
+10
+0
+10
+20
+30
+40
+Accuracy improvement(%)
+EPIC-Kitchens
+Figure 4. Per-class accuracy improvement over VisOnly-Distill.
+Best and worst performing classes are shown.
+GFLOPs
+Runtime (ms)
+Parameters (M)
+Video [54]
+369.51
+10.70
+108.91
+AdaFuse [50]
+15.20
+2.04
+38.85
+STTS [65]
+7.19
+1.63
+36.63
+ListenToLook [22]
+3.10
+0.43
+25.53
+EgoDistill
+1.91
+0.25
+20.56
+Table 3. Efficiency analysis. Our approach is the most efficient.
+“Video” refers to the original (full-clip) feature. Lower is better.
+With N = 4 frames, EgoDistill surpasses STTS by 7.4%
+and AdaFuse by 4.2% on EPIC-Kitchens, with 2× fewer
+GFLOPs, and surpasses both methods by 2.1% on Ego4D.
+In addition, EgoDistill surpasses ListenToLook by 7.4%
+and 2.9% on EPIC-Kitchens and Ego4D respectively, which
+suggests that head motion is more informative than audio
+for feature reconstruction in egocentric video.
+4.4. Analysis
+Model component ablations. Table 1 ablates different
+design choices in our model, setting N = 1 for all exper-
+iments. We observe that training EgoDistill without L1,
+LKD or LGT deteriorates performance. Specifically, training
+without LKD leads to the largest performance drop, which
+indicates that knowledge distillation is an essential compo-
+nent in our approach. Training without L1 also leads to a
+significant performance drop, which shows the importance
+of our idea to align features from the different modalities.
+Further, our self-supervised pretraining stage is very effec-
+tive at training the IMU extractor to encode useful motion
+information that is consistent with visual feature change.
+Finally, we compare with a model that simply does multi-
+modal recognition with IMU (top row). The strong contrast
+here indicates the importance of our idea to use IMU to pre-
+dict video model features, as opposed to simply adding IMU
+as an additional input modality.
+Impact of teacher video model architecture. In our
+main experiments we use MotionFormer [54] as the teacher
+video model due to its strong performance on egocentric
+video tasks.
+To emphasize the generality of our idea,
+we show the performance of EgoDistill with other video
+teacher architectures in Table 2.
+Similar to the Motion-
+Former model, we train these models on each of the labeled
+Figure 5. Best (top) and worst (bottom) reconstructed videos.
+datasets, and then train our model using the resulting video
+models as the teacher. As expected, better video teacher
+models lead to better student model performance. More im-
+portantly, we observe consistent improvement by EgoDis-
+till over the VisOnly-Distill baseline on both datasets and
+with different video teacher models, highlighting our idea’s
+generality and versatility.
+Where does our model work best/worst? In Figure 3
+we saw that using IMU leads to an overall performance
+improvement on action recognition, indicating better video
+feature prediction capability. Next, we explore what kinds
+of clips are better reconstructed using EgoDistill. Figure 4
+shows the improvement of EgoDistill over the VisOnly-
+Distill model on Ego4D and EPIC-Kitchens split by action
+class. We observe that IMU is more useful for actions with
+predictable head motion (e.g., break, cut, close), and is less
+helpful for actions where head motion may be small or un-
+related (e.g., empty, fill, press).
+Figure 5 shows clip examples whose video features are
+best and worst reconstructed. We observe that the best re-
+constructed clips (top) contain moderate head motion that
+is predictive of scene motion and action semantics. For ex-
+ample, the camera wearer’s head moves slightly backwards
+while opening the cabinet. On the other hand, more poorly
+reconstructed clips tend to contain little head motion (third
+row)—in which case IMU is redundant to the RGB frame—
+or drastic head motion that is weakly correlated with the
+camera wearer’s activity and introduces blur to the frame
+(last row).
+Efficiency analysis. To compare the efficiency of dif-
+ferent models, aside from GFLOPs, we also compare their
+inference run-time and number of parameters. For run-time,
+we record the time spent to infer a single video clip’s label
+with a single A40 GPU, and take the average time over the
+full validation datasets of Ego4D and EPIC-Kitchens with
+batch-size of 32. Table 3 shows the results. EgoDistill runs
+much faster than the other methods. Notably, it reduces the
+GFLOPs of MotionFormer by nearly 200×. Furthermore,
+
+EgoDistill
+EgoDistill
+VisOnly-D
+EgoDistill
+EgoDistill
+EgoDistill
+VisOnly-D
+EgoDistill
+Figure 6. Retrieving video clips with EgoDistill. Given a query frame (bottom left) and a paired IMU segment (red camera frustums) ,
+we retrieve the nearest clip in the video dataset according to EgoDistill and visualize its (unobserved) frames (strip to the right). Compared
+to VisOnly-Distill, which outputs a single feature for a given input frame (bottom row), EgoDistill outputs a distinct feature by condi-
+tioning on IMU, showing its ability to preserve both semantic and motion during reconstruction. For instance, in the top-right example,
+EgoDistill retains the cabinet interaction semantics in the frame as well as the upward camera-motion in the IMU. Zoom in to view best.
+close: 0.88
+open: 0.01
+close: 0.18
+open: 0.34
+GT: close
+EgoDistill
+VisOnly-D
+put: 0.40
+take: 0.10
+put: 0.14
+take: 0.44
+GT: put
+EgoDistill
+VisOnly-D
+Figure 7. Anticipating scene motion with EgoDistill. For each clip, we show the head motion and video frames. Note, only the center
+frame (red border) is observed by the model. Action classification scores are shown on the right. EgoDistill can successfully anticipate
+scene motion and disambiguate the action semantics in the input frame. For example, in the top center frame, the image alone cannot reveal
+if the door is being opened or closed, whereas our feature, learned with head motion, recovers correlations with the scene motion (i.e., hand
+motion and door motion) to disambiguate “close” from “open”. A similar effect for “put” vs. “take” is seen in the second example.
+it runs 6.5× faster than STTS [65] while achieving 4.4%
+higher accuracy on EPIC-Kitchens.
+4.5. Qualitative Results
+What do EgoDistill features capture? To explore this,
+we pair a single input frame with different IMU clips as in-
+puts to EgoDistill, then retrieve the nearest video clip for
+each resulting anticipated video feature. Figure 6 illustrates
+this. We see that EgoDistill outputs video features that all
+involve interaction with the cabinet (right panel), and is able
+to use different IMU inputs to retrieve different video clips
+that show consistent camera motion. In contrast, VisOnly-
+Distill only retains the semantic context to retrieve a single
+clip. These results indicate that EgoDistill is able to approx-
+imate video features that capture both semantic and motion
+information. See Supp. for more (and animated) results.
+Is there evidence EgoDistill captures scene motion?
+Figure 7 shows how our features learned with head motion
+can nonetheless expose certain scene motion cues. EgoDis-
+till improves the accuracy over VisOnly-Distill on ambigu-
+ous categories (like close and put) by a large margin (20.3%
+and 10.4% on EPIC-Kitchens, 8.5% and 3.9% on Ego4D).
+See caption for details.
+5. Conclusion
+We present EgoDistill, the first model to explore ego-
+centric video feature approximation for fast recognition.
+Experiments on action recognition on Ego4D and EPIC-
+Kitchens demonstrate that our model achieves a good bal-
+ance between accuracy and efficiency, outperforming state-
+of-the-art efficient video understanding methods. Our ap-
+proach has great potential to accelerate video understand-
+ing for egocentric videos using a data stream that is already
+ubiquitous in egocentric cameras. In the future, we plan to
+investigate how to use head motion for long-term human
+activity understanding with room context and visual corre-
+spondence learning for multi-view videos.
+
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+The supplementary materials of this work consist of:
+A. Supplementary video.
+B. Dataset details.
+C. Implementation details.
+D. Additional analysis of our model.
+A. Supplementary Video
+In our supplementary video, we have a brief introduction
+of our work. More importantly, we show animated videos
+of Best and Worse reconstructed clips (Figure 5), Retriev-
+ing video clips with EgoDistill (Figure 6), and Anticipating
+scene motion with EgoDistill (Figure 7).
+Animated version of these figures better show head mo-
+tion and video dynamics. We recommend viewing the sup-
+plementary video for better understanding of our method
+and results.
+B. Dataset Details.
+We use two datasets in our experiments: Ego4D [26] and
+EPIC-Kitchens-100 [11]. In this section we describe more
+details about how we create our training and evaluation data.
+1. Ego4D [26] contains 3,670 hours of egocentric videos
+of people performing diverse tasks (from cooking to
+farming) across the globe. As action recognition is not
+part of the original Ego4D benchmark, we construct
+this task with annotations from the Hands+Objects
+temporal localization benchmark [26].
+Specifically,
+for each hand-objects interaction temporal annotation,
+we take the video clip between the pre-frame and post-
+frame of the annotation as input, and use the annotated
+verb for this interaction as label.
+We include clips with paired IMU and audio, and con-
+sider classes with at least 2 labeled instances, resulting
+in 94 action categories with 12.1k videos in total. In
+average, each clip has 2.2 second duration. Then, we
+randomly split data from each category into training
+and evaluation sets with 70%:30% ratio. Finally, we
+obtain a 94-class action recognition dataset with 8.5k
+training videos and 3.6k evaluation videos.
+2. EPIC-Kitchens [11] contains 100 hours of egocen-
+tric videos capturing daily activities in kitchen environ-
+ments. We use annotations from the action recognition
+benchmark in our experiment.
+We select videos that have paired IMU and audio data,
+and split the resulting data by camera-wearer, ensuring
+non-overlapping splits following the original bench-
+mark setting. Specifically, we take videos captured by
+camera-wearer id starting with P30, P35, P37 as evalu-
+ation videos and use all the remaining videos as train-
+ing videos. This results in a 62-class action dataset
+with 29k training videos and 6.2k evaluation videos.
+C. Implementation Details.
+IMU input processing. For each input clip, IMU in-
+put is a 422 × 6 matrix (around 2 seconds with 198Hz
+frequency), representing the accelerometer and gyroscope
+readings along the xyz axes. We observe that the raw IMU
+input has significant drifting and bias issues. This induces
+inconsistent correspondence between camera motion and
+IMU reading across different clips and videos. Therefore,
+for IMU reading of each clip, on each dimension we sep-
+arately subtract raw readings by the mean values on the
+corresponding dimension. This operation normalizes IMU
+readings in each dimension to have zero average value. In
+this way, our model can only focus on the temporal motion
+patterns in each clip.
+Audio input processing. For ListenToLook [22], we
+process the audio input in the same way mentioned in the
+paper. Specifically, we subsample the audio at 16kHZ, and
+compute STFT using Hann window size of 400 and hop
+length of 160. Please refer to [22] for more details.
+Model architecture. For the image backbone, we use
+the ImageNet-pretrained ResNet-18 model. For the IMU
+backbone, we use a 5-layer 1D Dilated CNN, as found ef-
+fective for IMU data processing [6]. We use the same net-
+work setting (kernel dimension, dilation gap and channel
+dimension) as in prior work [6]. The feature fusion model
+consists of a concatenation operation following two fully-
+connected layers with hidden dimension of 1024.
+Each
+layer except for the output layer is followed by a ReLU ac-
+tivation. The output dimension is the same as the teacher
+video model’s feature dimension (768 in the case of Mo-
+tionFormer). When N > 1, we use a one-layer GRU mod-
+ule to aggregate extracted features for each frame. We use
+a single-directioal GRU with hidden dimension of 512.
+Model training.
+We train our models in two stages.
+In the self-supervised IMU feature learning stage, we train
+random initialized IMU encoder fM, IMU predictor h and
+the fusion network Π with LNCE. Here the image encoder
+fI is a fixed ImageNet pretrained model. On both datasets,
+we train the model for 50 epochs with AdamW and batch
+size 64. The initial training rate is 1e−4. We decay the
+training rate by 0.1 at epoch 30 and epoch 40. In the sec-
+ond video feature distillation stage, we initialize the model
+with parameters obtained in the last stage and finetune. On
+both datasets, we use AdamW with batch size 64 and ini-
+tial learning rate 1e−4. On Ego4D, we train for 150 epochs.
+We decay the training rate by 0.1 at epoch 90 and epoch
+120. On EPIC-Kitchens, we train for 50 epochs. We decay
+the training rate by 0.1 at epoch 30 and epoch 40.
+
+Ego4D
+EPIC-Kitchens
+uniform
+38.46
+52.43
+random
+36.85
+48.48
+first
+38.68
+46.40
+last
+35.46
+41.72
+center
+37.04
+44.85
+Table 4. Effect of frame selection. We compare the accuracy of
+using different frame selection heuristics for EgoDistill when N =
+4. We observe that Uniform on average achieves better results.
+D. Analysis.
+Effect of frame selection.
+In Section 3.2, we men-
+tioned that we use uniform sampling to obtain the N frames
+from each video clip. In this section, we compare the per-
+formance of our work under uniform sampling with other
+heuristics. Specifically, we compare with random sampling,
+the first N frames, the last N frames and the center N
+frames. We show the results in Table 4 under N = 4. These
+results indicate that uniform sampling leads to the best per-
+formance on average. Intuitively, uniform sampling on av-
+erage leads to a broader coverage of both semantic contexts
+as well as scene motion.
+Why we set N to be small. In our experiments, we set
+N to be 1 to 4. Using larger N (e.g., 8 or 16) with densely
+sampled frames could lead to better results of all the meth-
+ods with more computational cost. Efficient video under-
+standing methods could benefit more as they have better
+temporal aggregation mechanisms given densely-sampled
+frames.
+However, the core purpose of our model is to
+deal with cases where we only use a few number of sam-
+ples. Therefore, our model is not comparable to video clip
+models under dense-frame setting. Furthermore, setting N
+to be a small number is very important in many applica-
+tions. As loading more image frames takes additional time
+and memory, applications with streaming videos or low-
+resource AR/VR devices will benefit from loading only a
+few frames.
+
diff --git a/JtE0T4oBgHgl3EQfSADl/content/tmp_files/load_file.txt b/JtE0T4oBgHgl3EQfSADl/content/tmp_files/load_file.txt
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@@ -0,0 +1,921 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf,len=920
+page_content='EgoDistill: Egocentric Head Motion Distillation  for Efficient Video Understanding  Shuhan Tan1  Tushar Nagarajan1  Kristen Grauman1,2  1The University of Texas at Austin  2FAIR, Meta AI  {shuhan,tushar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='nagarajan,grauman}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='edu  Abstract  Recent advances in egocentric video understanding  models are promising, but their heavy computational ex-  pense is a barrier for many real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' To ad-  dress this challenge, we propose EgoDistill, a distillation-  based approach that learns to reconstruct heavy egocen-  tric video clip features by combining the semantics from  a sparse set of video frames with the head motion from  lightweight IMU readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We further devise a novel self-  supervised training strategy for IMU feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our  method leads to significant improvements in efficiency, re-  quiring 200× fewer GFLOPs than equivalent video models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We demonstrate its effectiveness on the Ego4D and EPIC-  Kitchens datasets, where our method outperforms state-of-  the-art efficient video understanding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Introduction  Recent advances in augmented and virtual reality  (AR/VR) technology have the potential to change the way  people interact with the digital world, much like the smart-  phone did in the previous decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  A fundamental re-  quirement for AR/VR systems is the ability to recognize  user behavior from egocentric video captured from a head-  mounted camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Towards this goal, several egocentric  video datasets have been proposed in recent years, spurring  increasing attention of the research community [11,26,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Recent advances in egocentric action recognition, antic-  ipation, and retrieval focus on building powerful clip-based  video models that operate on video clips of a few seconds  at a time [12, 16, 18, 25, 43, 44, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Despite encourag-  ing performance, these models typically process densely-  sampled frames with temporally-aware operations, making  them computationally heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' This makes them impractical  for AR/VR devices with constrained resources, or for real-  time video applications that require low latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' How to  efficiently perform egocentric video understanding is there-  fore an important, yet unsolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  +  Camera motion (IMU)  Video Frame  EgoDistill  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Illustration of EgoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Given a single video frame  and camera motion from IMU, EgoDistill learns to reconstruct the  more expensive dense video clip feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' With its lightweight in-  put, EgoDistill significantly improves efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  To address this issue, we take inspiration from how an-  imals perceive the world with ego-motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Neuroscience  research has found that during active movement, the animal  visual cortex receives and utilizes head motion signals from  the motor cortex for visual processing [27,52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' This indi-  cates that head motion signals support an embodied agent’s  efficient understanding of the egocentric visual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In-  spired by this phenomenon, we explore the relationship be-  tween human head motion and egocentric video for efficient  video understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In practice, we consider head motion  signals captured by the inertial measurement unit (IMU) of  a head-mounted camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' IMU measures motion from an ac-  celerometer and gyroscope and is widely available on pop-  ular wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Prior work leverages IMU as an ex-  tra modality for human action recognition [13,68,69], (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=',  jumping, walking, standing) or as geometric cues for visual-  inertial odometry [7,20,71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In contrast, we propose to achieve efficient video under-  standing by drawing on IMU as a substitute for dense video  frame observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' The intuition is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' A video  clip contains two things: semantic content (appearance of  objects, places, people) and dynamics (how the scene and  the camera move).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  While densely sampled frames are  sure to capture both of the above—as done by current clip  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='02217v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='CV]  5 Jan 2023  models [16, 17, 54]—we hypothesize they are sometimes  overkill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For a short video clip, much of the semantic con-  tent is intelligible from even a single frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' meanwhile, the  head motion provides a good portion of the dynamics, im-  plicitly revealing how the visual appearance changes across  neighboring frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Building on this insight, we introduce EgoDistill, an ap-  proach that learns to reconstruct dense egocentric video  clip features using temporally sparse visual observations  (as few as one RGB frame) together with the head motion  from IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically, EgoDistill employs a new form of  knowledge distillation from video models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' During training,  we train a lightweight model that takes sparsely sampled  image(s) and IMU to approximate the video representation  extracted by a powerful but expensive video model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We fur-  ther improve the model with a novel self-supervised train-  ing stage for IMU feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' During inference, we  directly utilize the lightweight model for egocentric video  recognition, leading to much higher efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our model  is flexible to the target heavy video feature, as we demon-  strate with multiple current leading egocentric video mod-  els [16–18,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Importantly, EgoDistill offers a major efficiency gain:  processing low-dimensional IMU and a few frames is much  more efficient compared to processing a dense stack of  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In practice, EgoDistill uses 200× fewer GFLOPs  than the original video model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We experiment on the largest available egocentric ac-  tion recognition datasets: Ego4D [26] and EPIC-Kitchens-  100 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We show that IMU coupled with an image offers  better cross-modality knowledge distillation performance  than images alone or images with audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For a typical 50-  minute egocentric video, EgoDistill reduces inference time  of the source video model from 25 minutes to 36 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Moreover, with only 1-4 frames, our lightweight distillation  model achieves a better accuracy-efficiency trade-off than  state-of-the-art models for adaptively sampling video con-  tent [50,65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Notably, we surpass the accuracy of these fast  approaches by a large margin while requiring 4× to 8× less  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Related Work  IMU for activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Recent work explores us-  ing the IMU sensor on mobile devices for human activ-  ity recognition of actions like walking, jumping, or sit-  ting [3,47,59–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Normally, these models take input from  IMU sensors mounted on human body joints [9, 42, 60],  waist-mounted [41] or in-pocket smartphones [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' See [64]  for a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Abundant work in video recognition explores  ways to learn from RGB coupled with other modalities—  audio [1, 22, 38], optical flow [19, 57, 58] or both [32, 36,  51]—but comparatively fewer use IMU [48], and unlike our  work, they focus on third-person video [13, 68, 69] and do  not target at model efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our idea is for IMU to help  reconstruct more expensive video features, rather than sim-  ply fuse IMU with RGB for multi-modal recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  IMU for odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Inertial odometry aims to estimate the  position and orientation of the camera-wearer with readings  from the IMU sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Traditionally, methods rely on IMU  double integration [4] or enhancements thereof [5, 35, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Recent data-driven methods automatically learn to per-  form inertial odometry with supervised [30, 70] or self-  supervised learning [7], or combine IMU and visual in-  put for more robust estimates with visual-inertial odome-  try [20, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' While IMU can convey geometric ego-motion  to our learned model, our goal is to produce efficient ego-  centric video features rather than to output odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Visual feature learning with IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' IMU is also used to  learn better vision features [14,15,34,63], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', to encourage  image features that are equivariant with ego-motion [34],  to predict an IMU-captured body part (leg, hand) [14, 15],  or to predict video-IMU correspondence [63], for applica-  tions like action recognition [15,63] and scene understand-  ing [14, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' While these results reinforce that IMU can  inject embodied motion into visual features, our idea to use  head motion to infer pretrained video features for speedy  video understanding is distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Efficient video recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Being crucial for mobile ap-  plications, efficient video recognition has received increas-  ing attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Several studies focus on de-  signing lightweight architectures [17, 30, 37, 62, 72] by re-  ducing 3D CNN operations across densely sampled frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Our idea is orthogonal to them as we focus on inputs with  sparsely-sampled frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' As we show in experiments, our  method is compatible with different video architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Another line of research achieves efficiency by adap-  tively selecting video content to process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Some reduce  temporal redundancy by adaptively selecting which video  clip [39], frames [24, 49], and/or feature channel [50] to  process and which to skip, while others reduce spatial re-  dundancy, efficient recognition by dynamically selecting  selecting for each frame a smaller but important region to  process [66,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Other work dynamically selects tokens in  video transformers among both the spatial and temporal di-  mensions [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our idea is complementary: rather than dy-  namically subsample the available video content, we show  how to infer “full” video features for every clip using static  image(s) and motion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our results outperform state-of-  the-art sampling models (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In addition, we fo-  cus on egocentric video, where head motion is particularly  meaningful for inferring unobserved visual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' To our  knowledge, ours is the first technique specifically aimed at  accelerating egocentric video processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Multimodal distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Knowledge distillation aims to  transfer knowledge learned by an expensive model to a  lightweight model [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Recent work explores multimodal  distillation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', transferring from a RGB model to a flow or  depth model [23,28], from a 3D model to a 2D model [45],  or from a visual model to audio model [2,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' The Listen-  ToLook model [22] incorporates both clip subsampling and  video-to-audio distillation for fast activity recognition in  third-person video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In contrast, we explore the relationship  between the camera-wearer’s head motion and RGB sig-  nals for egocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our experiments show EgoDis-  till’s advantage over ListenToLook in terms of the speed-  accuracy tradeoff on egocentric video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Approach  We introduce EgoDistill, which uses sparsely-sampled  frames and head motion from IMU to approximate the fea-  tures of heavy video models for efficient egocentric video  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We first introduce the egocentric action  recognition task (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Then, we introduce our pipeline  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2), our distillation model and training objective  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='3), and our self-supervised IMU feature learning  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Figure 2 overviews our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Egocentric action recognition  Given a fixed-length video clip V ∈ RT ×H×W ×3 con-  sisting of T RGB frames of size H×W and a set of C action  classes, the task of action recognition is to output a score for  each action class, representing its likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Typically, this  is done with a powerful but expensive video model Ω, that  directly operates on all the available frames to output the C  class logits Ω(V) ∈ RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Ω is trained with standard classifi-  cation loss:  LACT =  �  Vi  LCE(ci, σ(Ω(Vi))),  (1)  where Vi is the i-th video clip in the dataset, ci is the  corresponding ground-truth action label, σ is the softmax  function, and LCE is cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Popular video  recognition models use clips that are typically �2 seconds  long [16, 18, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For longer videos, scores are averaged  across all clips it contains to infer the video action label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Efficient video inference with head motion  Processing the video clip V for action recognition is  computationally intensive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' however, the computation cost  can be modulated depending on how frames from the clip  are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' On the one hand, clip-based models [16–18, 54]  process most (or all) frames in a video clip V to achieve  strong recognition performance, but come at a high com-  putational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  On the other hand, frame-level mod-  els [24, 49, 51, 67] only process one (or a small number)  of frames from V and are more efficient, but suffer a drop  in performance as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our goal is to train a frame-  based model that can approximate heavy clip-based model  performance while maintaining high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  For this, we turn to head motion captured by IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Along  with RGB frames, each video clip is paired with IMU mea-  surements M that record the camera (head) motion during  the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically, the IMU readings are composed of  6-dimensional accelerometer and gyroscope measurements  in the xyz axes, which encode strong temporal motion in-  formation about camera pose changes (both translation and  rotation) across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  For short video clips, a set of sparsely sampled frames I  often already captures most appearance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Com-  plementary to this, the IMU readings capture camera mo-  tion information (see below for discussion on scene mo-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Moreover, IMU is very efficient to process due to its  low dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' By processing inputs from these two  sources with a lightweight frame-based model, we can infer  the semantic and dynamic features of a heavier clip-based  video model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Given I and M, we train an efficient lightweight model  Φ to approximate the output of video model Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically,  we train our EgoDistill model Φ that achieves  Φ(I, M) ≈ Ω(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  (2)  Such a lightweight model will be able to approximate the  result of the heavy video model, while being much more ef-  ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our approach is agnostic to the specific video model  Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' in experiments, we demonstrate its versatility for Mo-  tionFormer [54], MViT [16], SlowFast [18] and X3D [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In practice, we uniformly sample N frames1 from V to  obtain I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We can achieve a trade-off between efficiency and  performance by changing the number of frames N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In our  experiments we use very low values of N (1 to 4 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In the next section, we discuss how we train Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Video feature distillation with IMU  We achieve the objective in Equation 2 via knowledge  distillation [31], where we transfer knowledge learned by  the expensive teacher model Ω to a lightweight student  model Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Next we present the design of Φ and the train-  ing objectives, followed by our self-supervised IMU feature  pretraining stage in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We design Φ to be a two-stream model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For a video clip  and associated IMU signal (I, M), we extract image fea-  tures zI = fI(I) and IMU features zM = fM(M) using  lightweight feature encoders fI, fM respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Then,  we fuse zI and zM with a fusion network Π to obtain the  fused VisIMU feature zφ = Π(zI, zM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Finally, a fully-  connected layer uses the fused feature to predict class logits  Φ(I, M) ∈ RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  The fused feature zφ contains semantic information from  the image frame coupled with complementary motion infor-  1Other frame sampling heuristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', selecting from the start or center  of the video) performed equivalently or worse than uniform sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Video clip  Video model  Frame  IMU  Image Encoder  IMU Encoder  Fusion Layer  FC-Softmax  FC-Softmax  Frame  IMU  Image Encoder  IMU Encoder  Frame  Image Encoder  IMU  Predictor  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EgoDistill architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Left: Self-supervised IMU feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Given start and end frames of a clip, we train the IMU  encoder to anticipate visual changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Right: Video feature distillation with IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Given image frame(s) and IMU, along with our pre-  trained IMU encoder, our method trains a lightweight model with knowledge distillation to reconstruct the features from a heavier video  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' When the input includes more than one image frame, the image encoder aggregates frame features temporally with a GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  mation from IMU, allowing us to accurately reconstruct the  video clip feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We train Φ with a combination of three losses, as fol-  lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' First, we train Φ to approximate the original video  feature zV from the video model Ω:  L1 =  �  (zVi,zφi)  ∥zVi − zφi∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  (3)  This cross-modal loss encourages the fused feature zφ to  match the video feature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', the combined features from  the different modalities should match in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Training with L1 alone does not fully capture the clas-  sification output of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Therefore, we also train Φ with a  knowledge distillation loss:  LKD =  �  (Vi,Ii,Mi)  DKL(σ(Ω(Vi)/τ), σ(Φ(Ii, Mi)/τ)),  (4)  where (Vi, Ii, Mi) represents the i-th clip in the dataset,  DKL measures KL-divergence between the class logits from  the teacher model Ω and student model Φ, and τ is a tem-  perature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Intuitively, LKD casts the output of the  video teacher model as a soft target for training the student  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In this way, the student model learns to better gen-  eralize by mimicking the output distribution of the heavy  video model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Finally, to further encourage the features to preserve el-  ements useful for activity understanding, we also compute  an action classification loss:  LGT =  �  (Ii,Mi)  LCE(ci, σ(Φ(Ii, Mi))),  (5)  where ci is the ground-truth action label, following Equa-  tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' The final training loss is a combination of these three  loss functions:  L = αLKD + (1 − α)LGT + βL1,  (6)  where α controls the balance between knowledge distilla-  tion and activity training [31], and β controls the weight for  feature space matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Critically, processing a few image frame(s) and the low-  dimensional IMU readings is substantially faster than pro-  cessing the entire video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Once trained, our model can ap-  proximate the behavior of the source video model for recog-  nition tasks, with the key benefit of efficient egocentric  video recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  What kind of motion does our model preserve?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Video  motion decomposes into scene motion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', how the objects  and the camera wearer’s hands are moving on their own),  and camera motion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', how the camera wearer is moving  their head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' By itself, IMU would directly account only for  camera motion, not scene motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' However, by learning to  map from the RGB frame and IMU to the full video fea-  ture, we are able to encode predictable scene motions tied  to scene content, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', how does hand and object movement  in subsequent frames relate to the camera wearer’s head mo-  tion (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Moreover, our model is applied to rel-  atively short clips (1-2 seconds) in sequence, which means  the appearance content is regularly refreshed as we slide  down to process the longer video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Self-supervised IMU feature learning  The success of EgoDistill depends on how well the IMU  feature encoder fM extracts useful camera motion informa-  tion and associates it with the visual appearance change in  the video clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In this way EgoDistill can learn to antic-  ipate unseen visual changes in the video with I and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We design a self-supervised pretraining task to initialize the  weights of fM to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Specifically, for each clip V, we obtain its first and last  frames (I0, IT ) as well as the IMU M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We first extract  visual features z0  I, zT  I and IMU feature zM with feature  extractors fI and fM mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Then, we train  a feature predictor h to predict the IMU feature ˆzM =  h(z0  I, zT  I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' By connecting ˆzM—which is a function of im-  age features only—with zM, we encourage fM to extract  useful camera motion features specifically associated with  the visual appearance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Note that those appearance  changes may include scene motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Therefore, we include  an L1 loss to train fM, which encourages fM to extract mo-  tion features accounting for scene motion in the full video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In sum, we train fM, h, and the fusion network Π using  L1 and NCE loss [29]: Lpretrain = LNCE + L1, where  LNCE =  �  i  − log  sim(ˆzMi, zMi)  �  j sim(ˆzMi, zMj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  (7)  We sample negative examples zMj from other instances  in the same mini-batch for j  ̸=  i, and sim(q, k)  =  exp( q·k  |q||k|  1  τ ′ ) with temperature τ ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  To summarize, prior to the main training stage of Equa-  tion 6, we pretrain the IMU feature extractor fM and fu-  sion network Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' As we will show below, both pretraining  losses result in IMU features that are consistent with visual  changes and lead to better finetuning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Experiments  We evaluate our approach for resource-efficient action  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Experimental setup  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We experiment on two large-scale egocen-  tric action recognition datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Ego4D [26] contains  3,670 hours of egocentric videos of people performing di-  verse tasks (from cooking to farming) across the globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  As action recognition is not part of the original Ego4D  benchmark, we construct this task with annotations from  the Hands+Objects temporal localization benchmark [26]  (see Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We include clips with paired  IMU and audio3, and consider classes with at least 2 la-  beled instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  This results in a 94-class action recog-  nition dataset with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='5k training videos and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='6k evalua-  tion videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EPIC-Kitchens [11] contains 100 hours of  egocentric videos capturing daily activities in kitchen en-  vironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We use annotations from the action recogni-  tion benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Similar to Ego4D, we select videos that  have paired IMU and audio data, and split the resulting data  by camera-wearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' This results in a 62-class action dataset  2We keep the ImageNet-pretrained fI model frozen, as finetuning it  leads to mode collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  3We require audio to compare with the audio-based baseline [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  with 29k training videos and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2k evaluation videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For  both datasets, we use “verb” labels as the target for action  recognition as they are well aligned to activity motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' To measure action recognition per-  formance, we report the per-video top-1 accuracy on the  validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We densely sample clips from each video  and average their predictions to compute accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  To  benchmark efficiency, we measure computational cost with  FLOPs (floating-point operations) during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In our main experiments, we  use MotionFormer [54] as the video teacher model Ω due  to its strong performance for egocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For EPIC-  Kitchens, we use the authors’ provided checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  For  Ego4D, we finetune the above model for 50 epochs with  1e−4 learning rate and 64 batch size on the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We use 16-frame input with sample rate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For the stu-  dent model Φ, we use a ResNet-18 as the image backbone  fI and a 1D Dilated CNN [6] for the IMU backbone fM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  The feature fusion module Π uses a concatenation operation  following a two-layer fully-connected layer with hidden di-  mension 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For each video clip, the input image(s) is  resized to 224 × 224, and the IMU is a 422 × 6 matrix  (around 2 seconds with 198Hz frequency), representing the  accelerometer and gyroscope readings along the xyz axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  For the image input, we uniformly sample N frames from  the video clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' If N > 1, we use fI to sequentially gen-  erate features for each frame and aggregate them with a  GRU module [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For both datasets, we first pretrain the  model with the self-supervised objective (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4) for  50 epochs with AdamW [46] using batch size 64 and learn-  ing rate 1e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Then, we finetune all the models with the  same setting (Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='95 and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='0  based on validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For Ego4D, we set τ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='0 and  train the model for 150 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For EPIC-Kitchens, we set  τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='0 and train for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Baselines  We compare to the following methods:  AdaFuse [50] trains a lightweight policy network to  adaptively compute (or skip) feature map channels for  each frame during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We use the AdaFuseTSN  R50  model with the provided hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  STTS [65] trains a module to rank spatio-temporal  tokens derived from videos in a transformer-based  model, and selects only the top-K tokens to speed up  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  ListenToLook [22]: uses the audio-based feature dis-  tillation module from [22] following the same audio  processing and model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  These methods represent recent advances in efficient  video recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' AdaFuse represents state-of-the-  2  4  6  8  10  12  14  16  Inference cost per video clip (GFLOPs)  33  34  35  36  37  38  Ego4D accuracy (%)  Ego4D  2  4  6  8  10  12  14  16  Inference cost per video clip (GFLOPs)  30  35  40  45  50  EPIC-Kitchens accuracy (%)  EPIC-Kitchens  EgoDistill (ours)  AdaFuse [50]  STTS [65]  ListenToLook [22]  VisOnly-Distill  VisIMU  VisOnly  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' efficiency for action recognition on Ego4D (left) and EPIC-Kitchens (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EgoDistill outperforms state-of-  the-art efficient video recognition methods that adaptively sample video content, while using 4× to 8× fewer GFLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  LKD  L1  LGT  L1-pretrain  LNCE-pretrain  Ego4D  EPIC-Kitchens  ✓  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='15  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='04  ✓  ✓  ✓  ✓  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='51  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='33  ✓  ✓  ✓  ✓  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='71  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='20  ✓  ✓  ✓  ✓  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='46  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='17  ✓  ✓  ✓  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='99  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='21  ✓  ✓  ✓  ✓  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='26  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='30  ✓  ✓  ✓  ✓  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='49  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='51  ✓  ✓  ✓  ✓  ✓  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='95  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='95  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Ablation study of model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We compare the  accuracy of EgoDistill with different components under N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  art approaches that achieve efficiency by reducing tempo-  ral redundancy in CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' STTS is one of the most  recent approaches that efficiently reduces both spatial and  temporal redundancy in ViT models, which achieves the  state-of-the-art on Kinectics-400 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' ListenToLook also re-  lies on distillation, but using audio rather than head motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  For each model we generate multiple versions with differ-  ent computation budgets to plot accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' GFLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We  train all AdaFuse and STTS models with 4 input frames to  align with the maximum frames used by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  For  AdaFuse, we use the only provided hyper-parameter in the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4 For STTS, we use three provided variants: T0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='5-  S4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='7, T0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='8-S4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='9 and the full model without token selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  For ListenToLook we adopt the same efficiency-accuracy  trade-off as our method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', varying the number of input  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In addition, we test variants of our method:  VisOnly-Distill is our model without the IMU branch  and fusion layer but trained with the same loss func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Performance of this model reveals the role of  IMU in the process of distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  VisIMU is our model trained with only LGT in Equa-  4Modifying hyper-parameters to control the accuracy-efficiency trade-  off results in unstable training and unreliable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Source Model  Ego4D  EPIC-Kitchens  Video  EgoDistill  VisOnly-D  Video  EgoDistill  VisOnly-D  MFormer [54]  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='38  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='95  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='32  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='28  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='95  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='20  MViT [16]  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='32  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='46  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='40  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='38  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='90  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='22  SlowFast [18]  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='52  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='29  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='04  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='34  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='42  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='47  X3D [17]  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='56  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='57  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='90  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='28  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='34  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='71  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Versatility to model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EgoDistill outper-  forms the baseline for multiple common architectures, showing  the generality of our idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' “Video” refers to the more expensive  source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We show the model accuracy under N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' It shows the effectiveness of distillation from  the video model compared with directly training the  features with action labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  VisOnly is an image-only model trained with LGT,  which serves as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Main Results  Importance of IMU-guided distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Figure 3  shows the accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' efficiency curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Methods towards  the top-left of the plot represent those with both high ac-  curacy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Our method achieves good accu-  racy with low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically, on EPIC-  Kitchens, when N = 1, EgoDistill improves over VisOnly-  Distill by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4% with only a small increase in computa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' This result shows the effectiveness of IMU for recon-  structing egocentric video features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Compared to VisIMU,  EgoDistill improves by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='9%, showing the effectiveness of  knowledge distillation from the video model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Importantly,  this reveals that EgoDistill does not simply benefit from the  extra IMU context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' our idea to approximate video features is  necessary for best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We see similar results on Ego4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Comparison with the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Figure 3 also  shows that EgoDistill achieves better accuracy with less  computation than existing efficient video recognition mod-  els AdaFuse [50], STTS [65], and ListenToLook [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  pack  sew  iron  spray  unscrew  paint  dip  water  play  file  hit  throw  inspect  pull  insert  clean  close  detach  smooth  cut  press  hang  shuffle  tighten  20  0  20  40  60  Accuracy improvement(%)  Ego4D  break  mix  drink  shake  dry  pour  close  wash  put  cut  turn-off  apply  open  peel  turn-on  throw  flip  scrub  insert  move  take  empty  fill  scoop  10  0  10  20  30  40  Accuracy improvement(%)  EPIC-Kitchens  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Per-class accuracy improvement over VisOnly-Distill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Best and worst performing classes are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  GFLOPs  Runtime (ms)  Parameters (M)  Video [54]  369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='51  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='70  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='91  AdaFuse [50]  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='04  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='85  STTS [65]  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='63  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='63  ListenToLook [22]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='43  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='53  EgoDistill  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='25  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='56  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Efficiency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our approach is the most efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  “Video” refers to the original (full-clip) feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  With N = 4 frames, EgoDistill surpasses STTS by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4%  and AdaFuse by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2% on EPIC-Kitchens, with 2× fewer  GFLOPs, and surpasses both methods by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1% on Ego4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In addition, EgoDistill surpasses ListenToLook by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4%  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='9% on EPIC-Kitchens and Ego4D respectively, which  suggests that head motion is more informative than audio  for feature reconstruction in egocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Analysis  Model component ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Table 1 ablates different  design choices in our model, setting N = 1 for all exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We observe that training EgoDistill without L1,  LKD or LGT deteriorates performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically, training  without LKD leads to the largest performance drop, which  indicates that knowledge distillation is an essential compo-  nent in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Training without L1 also leads to a  significant performance drop, which shows the importance  of our idea to align features from the different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Further, our self-supervised pretraining stage is very effec-  tive at training the IMU extractor to encode useful motion  information that is consistent with visual feature change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Finally, we compare with a model that simply does multi-  modal recognition with IMU (top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' The strong contrast  here indicates the importance of our idea to use IMU to pre-  dict video model features, as opposed to simply adding IMU  as an additional input modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Impact of teacher video model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In our  main experiments we use MotionFormer [54] as the teacher  video model due to its strong performance on egocentric  video tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  To emphasize the generality of our idea,  we show the performance of EgoDistill with other video  teacher architectures in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Similar to the Motion-  Former model, we train these models on each of the labeled  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Best (top) and worst (bottom) reconstructed videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  datasets, and then train our model using the resulting video  models as the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' As expected, better video teacher  models lead to better student model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' More im-  portantly, we observe consistent improvement by EgoDis-  till over the VisOnly-Distill baseline on both datasets and  with different video teacher models, highlighting our idea’s  generality and versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Where does our model work best/worst?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In Figure 3  we saw that using IMU leads to an overall performance  improvement on action recognition, indicating better video  feature prediction capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Next, we explore what kinds  of clips are better reconstructed using EgoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Figure 4  shows the improvement of EgoDistill over the VisOnly-  Distill model on Ego4D and EPIC-Kitchens split by action  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We observe that IMU is more useful for actions with  predictable head motion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', break, cut, close), and is less  helpful for actions where head motion may be small or un-  related (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', empty, fill, press).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Figure 5 shows clip examples whose video features are  best and worst reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We observe that the best re-  constructed clips (top) contain moderate head motion that  is predictive of scene motion and action semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For ex-  ample, the camera wearer’s head moves slightly backwards  while opening the cabinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' On the other hand, more poorly  reconstructed clips tend to contain little head motion (third  row)—in which case IMU is redundant to the RGB frame—  or drastic head motion that is weakly correlated with the  camera wearer’s activity and introduces blur to the frame  (last row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Efficiency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' To compare the efficiency of dif-  ferent models, aside from GFLOPs, we also compare their  inference run-time and number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For run-time,  we record the time spent to infer a single video clip’s label  with a single A40 GPU, and take the average time over the  full validation datasets of Ego4D and EPIC-Kitchens with  batch-size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Table 3 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EgoDistill runs  much faster than the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Notably, it reduces the  GFLOPs of MotionFormer by nearly 200×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Furthermore,  EgoDistill  EgoDistill  VisOnly-D  EgoDistill  EgoDistill  EgoDistill  VisOnly-D  EgoDistill  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Retrieving video clips with EgoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Given a query frame (bottom left) and a paired IMU segment (red camera frustums) ,  we retrieve the nearest clip in the video dataset according to EgoDistill and visualize its (unobserved) frames (strip to the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Compared  to VisOnly-Distill, which outputs a single feature for a given input frame (bottom row), EgoDistill outputs a distinct feature by condi-  tioning on IMU, showing its ability to preserve both semantic and motion during reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For instance, in the top-right example,  EgoDistill retains the cabinet interaction semantics in the frame as well as the upward camera-motion in the IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Zoom in to view best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  close: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='88  open: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='01  close: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='18  open: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='34  GT: close  EgoDistill  VisOnly-D  put: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='40  take: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='10  put: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='14  take: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='44  GT: put  EgoDistill  VisOnly-D  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Anticipating scene motion with EgoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For each clip, we show the head motion and video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Note, only the center  frame (red border) is observed by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Action classification scores are shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EgoDistill can successfully anticipate  scene motion and disambiguate the action semantics in the input frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For example, in the top center frame, the image alone cannot reveal  if the door is being opened or closed, whereas our feature, learned with head motion, recovers correlations with the scene motion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', hand  motion and door motion) to disambiguate “close” from “open”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' A similar effect for “put” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' “take” is seen in the second example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  it runs 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='5× faster than STTS [65] while achieving 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4%  higher accuracy on EPIC-Kitchens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Qualitative Results  What do EgoDistill features capture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' To explore this,  we pair a single input frame with different IMU clips as in-  puts to EgoDistill, then retrieve the nearest video clip for  each resulting anticipated video feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Figure 6 illustrates  this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We see that EgoDistill outputs video features that all  involve interaction with the cabinet (right panel), and is able  to use different IMU inputs to retrieve different video clips  that show consistent camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In contrast, VisOnly-  Distill only retains the semantic context to retrieve a single  clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' These results indicate that EgoDistill is able to approx-  imate video features that capture both semantic and motion  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' See Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' for more (and animated) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Is there evidence EgoDistill captures scene motion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Figure 7 shows how our features learned with head motion  can nonetheless expose certain scene motion cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EgoDis-  till improves the accuracy over VisOnly-Distill on ambigu-  ous categories (like close and put) by a large margin (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='3%  and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='4% on EPIC-Kitchens, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='5% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='9% on Ego4D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  See caption for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Conclusion  We present EgoDistill, the first model to explore ego-  centric video feature approximation for fast recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Experiments on action recognition on Ego4D and EPIC-  Kitchens demonstrate that our model achieves a good bal-  ance between accuracy and efficiency, outperforming state-  of-the-art efficient video understanding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Our ap-  proach has great potential to accelerate video understand-  ing for egocentric videos using a data stream that is already  ubiquitous in egocentric cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In the future, we plan to  investigate how to use head motion for long-term human  activity understanding with room context and visual corre-  spondence learning for multi-view videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  References  [1] Relja Arandjelovi´c and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Look, listen and  learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 2017 IEEE International Conference on Computer Vi-  sion (ICCV), pages 609–617, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 2  [2] Yusuf Aytar, Carl Vondrick, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Sound-  net: Learning sound representations from unlabeled video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In Proceedings of the 30th International Conference on  Neural Information Processing Systems, NIPS’16, page  892–900, Red Hook, NY, USA, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Curran Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  3  [3] Anna Borowska-Terka and Pawel Strumillo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
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+page_content=' ECO: efficient convolutional network for online video  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In ECCV, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' 2  The supplementary materials of this work consist of:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Supplementary video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Dataset details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Additional analysis of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Supplementary Video  In our supplementary video, we have a brief introduction  of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' More importantly, we show animated videos  of Best and Worse reconstructed clips (Figure 5), Retriev-  ing video clips with EgoDistill (Figure 6), and Anticipating  scene motion with EgoDistill (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Animated version of these figures better show head mo-  tion and video dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We recommend viewing the sup-  plementary video for better understanding of our method  and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Dataset Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We use two datasets in our experiments: Ego4D [26] and  EPIC-Kitchens-100 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In this section we describe more  details about how we create our training and evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Ego4D [26] contains 3,670 hours of egocentric videos  of people performing diverse tasks (from cooking to  farming) across the globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' As action recognition is not  part of the original Ego4D benchmark, we construct  this task with annotations from the Hands+Objects  temporal localization benchmark [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Specifically,  for each hand-objects interaction temporal annotation,  we take the video clip between the pre-frame and post-  frame of the annotation as input, and use the annotated  verb for this interaction as label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We include clips with paired IMU and audio, and con-  sider classes with at least 2 labeled instances, resulting  in 94 action categories with 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1k videos in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In  average, each clip has 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2 second duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Then, we  randomly split data from each category into training  and evaluation sets with 70%:30% ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Finally, we  obtain a 94-class action recognition dataset with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='5k  training videos and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='6k evaluation videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' EPIC-Kitchens [11] contains 100 hours of egocen-  tric videos capturing daily activities in kitchen environ-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We use annotations from the action recognition  benchmark in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We select videos that have paired IMU and audio data,  and split the resulting data by camera-wearer, ensuring  non-overlapping splits following the original bench-  mark setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically, we take videos captured by  camera-wearer id starting with P30, P35, P37 as evalu-  ation videos and use all the remaining videos as train-  ing videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' This results in a 62-class action dataset  with 29k training videos and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2k evaluation videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  IMU input processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For each input clip, IMU in-  put is a 422 × 6 matrix (around 2 seconds with 198Hz  frequency), representing the accelerometer and gyroscope  readings along the xyz axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We observe that the raw IMU  input has significant drifting and bias issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' This induces  inconsistent correspondence between camera motion and  IMU reading across different clips and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Therefore,  for IMU reading of each clip, on each dimension we sep-  arately subtract raw readings by the mean values on the  corresponding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' This operation normalizes IMU  readings in each dimension to have zero average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In  this way, our model can only focus on the temporal motion  patterns in each clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Audio input processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For ListenToLook [22], we  process the audio input in the same way mentioned in the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically, we subsample the audio at 16kHZ, and  compute STFT using Hann window size of 400 and hop  length of 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Please refer to [22] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For the image backbone, we use  the ImageNet-pretrained ResNet-18 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' For the IMU  backbone, we use a 5-layer 1D Dilated CNN, as found ef-  fective for IMU data processing [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We use the same net-  work setting (kernel dimension, dilation gap and channel  dimension) as in prior work [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' The feature fusion model  consists of a concatenation operation following two fully-  connected layers with hidden dimension of 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Each  layer except for the output layer is followed by a ReLU ac-  tivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' The output dimension is the same as the teacher  video model’s feature dimension (768 in the case of Mo-  tionFormer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' When N > 1, we use a one-layer GRU mod-  ule to aggregate extracted features for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We use  a single-directioal GRU with hidden dimension of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We train our models in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In the self-supervised IMU feature learning stage, we train  random initialized IMU encoder fM, IMU predictor h and  the fusion network Π with LNCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Here the image encoder  fI is a fixed ImageNet pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' On both datasets,  we train the model for 50 epochs with AdamW and batch  size 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' The initial training rate is 1e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We decay the  training rate by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1 at epoch 30 and epoch 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In the sec-  ond video feature distillation stage, we initialize the model  with parameters obtained in the last stage and finetune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' On  both datasets, we use AdamW with batch size 64 and ini-  tial learning rate 1e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' On Ego4D, we train for 150 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  We decay the training rate by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1 at epoch 90 and epoch  120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' On EPIC-Kitchens, we train for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We decay  the training rate by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='1 at epoch 30 and epoch 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Ego4D  EPIC-Kitchens  uniform  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='46  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='43  random  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='85  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='48  first  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='68  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='40  last  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='46  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='72  center  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='04  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='85  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Effect of frame selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We compare the accuracy of  using different frame selection heuristics for EgoDistill when N =  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We observe that Uniform on average achieves better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Effect of frame selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='2, we men-  tioned that we use uniform sampling to obtain the N frames  from each video clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In this section, we compare the per-  formance of our work under uniform sampling with other  heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Specifically, we compare with random sampling,  the first N frames, the last N frames and the center N  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' We show the results in Table 4 under N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' These  results indicate that uniform sampling leads to the best per-  formance on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Intuitively, uniform sampling on av-  erage leads to a broader coverage of both semantic contexts  as well as scene motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  Why we set N to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' In our experiments, we set  N to be 1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Using larger N (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=', 8 or 16) with densely  sampled frames could lead to better results of all the meth-  ods with more computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Efficient video under-  standing methods could benefit more as they have better  temporal aggregation mechanisms given densely-sampled  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content='  However, the core purpose of our model is to  deal with cases where we only use a few number of sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Therefore, our model is not comparable to video clip  models under dense-frame setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' Furthermore, setting N  to be a small number is very important in many applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
+page_content=' As loading more image frames takes additional time  and memory, applications with streaming videos or low-  resource AR/VR devices will benefit from loading only a  few frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfSADl/content/2301.02217v1.pdf'}
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+Strong Reverse Saturation and Fast-Light in Ruby
+Akbar Safari,1, 2, ∗ Cara Selvarajah,1 Jenine Evans,1 Jeremy Upham,1 and Robert W. Boyd1, 3
+1Department of Physics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
+2Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA.
+3Institute of Optics, University of Rochester, Rochester, New York, 14627, USA.
+(Dated: February 1, 2023)
+We observe a strong reverse saturation of absorption in ruby at a wavelength of 473 nm. With
+an intensity-modulated laser, we observe that the peaks of the pulses appear more than a hundred
+microseconds earlier than the reference signal. A theoretical model based on coherent population
+oscillation would suggest a fast-light effect with an extremely large and negative group index of
+−(1.7±0.1)×106. We propose that this pulse advancement can also be described by time-dependent
+absorption of ruby. Our study helps to understand the nature of the fast- and slow-light effects in
+transition-metal-doped crystals such as ruby and alexandrite.
+Saturation of absorption is a well-understood nonlin-
+ear optical process in which an optical driving field pass-
+ing through an absorptive material experiences a de-
+crease in absorptivity as the intensity of the field in-
+creases [1]. This process is typical of most materials and
+is frequently employed for passively mode-locking or Q-
+switching lasers [2].
+However, there are conditions under which materials
+exhibit an increase in absorption with higher intensity at
+specific wavelengths. This response, called reverse sat-
+uration of absorption (RSA), requires particular condi-
+tions, including a more than two-level system and for
+an excited state to have a larger absorption cross-section
+than the ground state [3]. In addition, neither the first
+nor the second excited states should decay to other levels
+thereby trapping the population. Moreover, the incident
+light should saturate, or partially saturate, the first tran-
+sition only. This condition can be achieved easily when
+the lifetime of the first excited state is much longer than
+that of the second excited state. Experimental investiga-
+tion of RSA, particularly their temporal dynamics, will
+provide insight into their feasibility for power limiting [4]
+and fast-light applications in gravitational wave detec-
+tion [5], optical gyroscopes [6, 7], and more [8–10].
+Here, we show that ruby at room temperature ex-
+hibits a strong reverse saturation of absorption at the
+wavelength of 473 nm. Consequently, we observe what
+could appear to be a fast-light effect, where the peak of
+an intensity-modulated signal passing through the crys-
+tal reaches the detector earlier than that without the
+ruby crystal. This phenomenon had been explained ear-
+lier based on coherent population oscillation and hole
+burning [11–13]. While this model continues to be de-
+bated [11, 12, 14–20], we consider a simple model based
+on rate-equations that explains the seemingly fast-light
+effect without the need of hole or anti-hole burning. In
+the following, first, we confirm that ruby exhibits RSA.
+Then, we use rate equations to find the time-dependent
+population of the ground and excited states, and conse-
+quently, the time-dependent absorption of ruby. Finally,
+we show that the advancement of the peak intensities can
+be explained by time-dependent absorption.
+The reverse saturation of absorption is observed by
+measuring the transmission of a continuous-wave (CW)
+laser at wavelength 473 nm with a maximum power of
+500 mW. To achieve the desired intensities, the laser is
+focused to a beam waist of 36 µm at the center of the ruby
+crystal of length 20 mm. The results, shown in Fig. 1(a),
+show a clear reduction of the transmission as the laser
+power increases. The crystal is placed at a small angle
+to avoid any issues with the back reflection of the laser
+from the crystal faces. The Fresnel reflections are con-
+sidered in the calculation of absorption. We observe that
+the transmission through the ruby crystal decreases from
+68% in the linear regime, to 52% in the nonlinear (high
+power) regime.
+The relevant energy levels of ruby are drawn in the
+inset of Fig. 1(b). The laser excites the electrons from
+the ground state g to the excited state e. The excited
+electrons relax very rapidly to the meta-stable state g′
+by emitting phonons.
+These three levels are typically
+enough to describe the interaction of the Cr3+ ions with a
+green laser, for example. However, a blue laser at 473 nm,
+can excite the electrons from the meta-stable state to the
+second excited state, e′. Because the absorption cross-
+section of this transition, σ2, is larger than that of the
+first transition, σ1, the overall absorption rate increases
+with the intensity of the laser (as shown in Fig. 1(b)).
+Since states e and e′ have extremely short lifetimes,
+their populations are negligible and can be omitted in
+the rate equation. Therefore, the population density of
+the ground state Ng follows
+dNg
+dt
+= − I
+ℏω σ1Ng + N − Ng
+τ
+(1)
+where, I is the intensity of the laser with photon en-
+ergy ℏω, and N − Ng is the population density of the
+meta-stable state g′, with lifetime τ =5 ms [2]. σ1 is the
+absorption cross-section of the first transition from g to
+e. The density of the Cr3+ ions in the ruby crystal is
+N ≈ 4.75 × 1018 cm−3. For a CW excitation, Eq. (1) is
+solved in steady-state condition (dNg/dt = 0). Then, the
+arXiv:2301.13300v1  [physics.optics]  30 Jan 2023
+
+2
+(a)
+(b)
+Input power (W)
+Transmission (%)
+0.0
+0.1
+0.3
+0.2
+0.4
+0.5
+50
+55
+60
+65
+Intensity (W/cm )
+2
+0.20
+0.25
+0.35
+0.30
+0.40
+0
+×103
+6.0
+2.0×103
+4.0×103
+8.0×103
+1.0×104
+Absorption coefficient (cm  )
+-1
+20 mm
+72 μm
+rapid
+rapid
+τ =5ms
+(
+)
+(
+)
+(
+)
+(
+)
+g A
+4
+2
+E
+2
+T
+2
+2
+T
+2
+1
+e
+g'
+e'
+FIG. 1. Reverse saturation of absorption. (a) Experi-
+mental (open circles) and theoretical (solid line) transmission
+as a function of the input laser power for a CW (unmodulated)
+beam. The inset shows the beam profile and the position of
+ruby. (b) Theoretical absorption coefficient from Eq. (3). In-
+set: The relevant energy levels of ruby. The absorption cross-
+section of the second transition, g′ → e′, is larger than that of
+the first transition, g → e, which leads to reverse saturation
+of absorption.
+absorption coefficient is found from
+α =
+σ1Ng + σ3(N − Ng)
+(2)
+=
+N
+1+I/Is (σ1 − σ3) + σ3N,
+(3)
+where σ3 is the absorption cross-section of the second
+transition, and Is = ℏω/σ1τ is the saturation intensity.
+We note that since the populations of levels e and e′ are
+negligible, implementing the degeneracy factors of the
+states will not affect Eq. (3). Figure 1(b) plots the ab-
+sorption coefficient from Eq. (3) as a function of intensity.
+In order to test the validity of Eq. (3) with experi-
+mental data, we simulate the propagation of the laser
+through the ruby crystal and find the theoretical trans-
+mission as a function of the input power. We use the
+same beam profile as measured in the experiment (inset
+of Fig. 1(a)). Since the local intensity and the absorp-
+tion coefficient are interdependent, we adopt an iterative
+approach to calculate the intensity of the laser as it prop-
+agates through the crystal. The results, shown as a solid,
+red line in Fig. 1(a), exhibit an excellent agreement with
+the experimental data. Therefore, we extract the values
+of the absorption cross sections to be σ1 = 4.0×10−20 cm2
+and σ3 = 9.6 × 10−20 cm2. It is because the absorption
+cross-section of the second transition is nearly 2.5 times
+larger than that of the ground state at this wavelength
+CW laser
+(473 nm)
+EOM
+signal 
+generator
+Oscilloscope
+f=100mm
+ruby
+reference
+beamsplitter
+filter
+0
+5
+10
+15
+time (ms)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+117 μs
+(a)
+(b)
+input pulse
+output pulse
+normalized intensity
+FIG. 2.
+Fast-light experiment.
+(a) The experimental
+setup. An electro-optic modulator (EOM) is used to imprint a
+weak sinusoidal intensity modulation on the laser beam. The
+laser is focused in the ruby crystal. A spectral filter is used
+to block the fluorescence at 694 nm. (b) Upon propagation
+in ruby, the peak of the weakly modulated signal advances
+in time for approximately 117 µs compared to the reference
+signal.
+that enables the clear observation of reverse saturation
+of absorption in this experiment.
+Next, we investigate the fast-light effect and exam-
+ine how the temporal variations of a modulated beam
+are altered upon traveling through the reverse saturated
+medium. Following refs. [11, 12], we send the laser beam
+through an electro-optic device which imposes a 10% in-
+tensity modulation to the laser beam, Fig. 2(a).
+The
+electro-optic modulator is fed by a sinusoidal signal at
+frequency Ω/2π = 70 Hz. The beam is focused to a waist
+of 36 µm at the center of the ruby crystal. A spectral filter
+is used to filter out the fluorescence at 694 nm. The trans-
+mitted and reference signals are detected and compared
+on an oscilloscope. When operating at intensities where
+RSA is clearly visible, the peaks of the modulated signal
+appear to advance relative to the reference. Figure 2(b)
+shows 117 ± 6 µs pulse advancement for an average input
+power of 450 mW.
+These results appear to be consistent with similar ex-
+periments in ruby and alexandrite [11, 12], where slow-
+and fast-light effects were reported and attributed to
+coherent population oscillation. Were the same reason-
+ing used here, the RSA would lead to a spectral hill
+or anti-hole and the corresponding advancement of the
+pulse peak would indicate fast-light with a group index
+of −(1.7±0.1)×106. Here we examine another hypothe-
+sis to describe these results: a time-dependence to ruby’s
+response to the modulated signal.
+For an intensity-modulated laser beam, the input in-
+
+3
+time (ms)
+Absorption coefficient (m )
+-1
+Input intensity (10 W/cm )
+2
+3
+0
+10
+30
+20
+40
+50 1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+30.5
+31.0
+31.5
+32.0
+32.5
+FIG. 3. Ruby time-dependent response. Modulated in-
+put intensity (right vertical axis) and the corresponding time-
+dependent absorption coefficient (left vertical axis).
+When
+the frequency of modulation is small compared to 1/τ, the
+absorption oscillates with the driving intensity, however, with
+a time difference due to the finite lifetime of the meta-stable
+state. As the frequency of modulation increases, the oscilla-
+tion amplitude of the absorption coefficient, α1, decreases.
+tensity to the ruby crystal can be written as
+I(t) = I0 + I1e−iΩt + c.c.,
+(4)
+where, Ω is the angular frequency of the modulation, I0
+is the average intensity, and I1 is the amplitude of the
+modulation. In Refs. [11, 13, 14], in order to explain an
+anti-hole in the spectrum, the population of the ground
+state and the absorption were written as N(t) = N0 +
+N1e−iΩt+c.c., and α(t) = α0+α1e−iΩt+c.c., respectively.
+In other words, it was assumed that the population and
+the absorption follow the same time dependence as the
+driving intensity.
+Therefore, the plot of α0 + α1 as a
+function of the modulation frequency exhibited a narrow
+peak, which was interpreted as an anti-hole.
+However,
+the
+total
+absorption
+of
+ruby
+is
+time-
+dependent, with α0 + α1 merely showing the maximum
+absorption (see Fig. 3). When the period of the mod-
+ulation is long enough compared to the lifetime of the
+meta-stable state τ = 5 ms, the population can vary,
+following the driving intensity. Because the absorption
+cross-section of the meta-stable state is larger than that
+of the ground state, as the population of the meta-stable
+state increases, the total absorption, α(t), increases as
+well.
+Therefore, α1 is maximum at slow modulations.
+As the frequency of the modulation increases, the ampli-
+tude of α1 decreases because the population cannot keep
+up with the rapidly changing intensity.
+Therefore, α1
+decreases at fast modulations.
+Given
+that
+the
+driving
+intensity
+has
+the
+time-
+dependent form given by Eq. (4), the population and
+absorption may lag behind the driving intensity as the
+lifetime of the meta-stable state is relatively long. There-
+fore, we numerically solve the rate equation in Eq. (1)
+with the modulated intensity to find the population as
+a function of time.
+The absorption is calculated by
+α(t) = σ1Ng(t) + σ3(N − Ng(t)).
+Figure 3 plots the
+0
+5
+10
+15
+time (ms)
+input intensity (10 W/cm )
+2
+3
+3
+output intensity (10 W/cm )
+2
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.5
+1.0
+2.0
+3.0
+2.5
+120 μs
+2.0
+1.9
+2.1
+2.3
+2.2
+3
+output intensity (10 W/cm )
+2
+1.00
+1.05
+1.10
+1.15
+1.20
+input intensity (10 W/cm )
+2
+3
+0
+5
+10
+15
+time (ms)
+a) 10% modulation
+b) 50% modulation
+FIG. 4. Theoretical pulse advancement. The input and
+output (after propagation through ruby) intensity as a func-
+tion of time for (a) weak modulation and (b) strong mod-
+ulation calculated from time-dependent absorption of ruby.
+When the modulation is weak, the signal is advanced uni-
+formly.
+With strong modulations, the output signal is de-
+formed notably.
+driving intensity and the calculated absorption as a func-
+tion of time. There is clearly a time difference between
+the maximum of absorption and the maximum of the in-
+put intensity due to the finite lifetime of the meta-stable
+state. The time-dependent absorption lagging behind the
+modulated intensity reshapes the temporal form of the
+intensity as measured at the output of the crystal.
+Accounting for this time-dependent intensity and ab-
+sorption, we simulate the form of the output intensity as
+a function of time (Fig. 4). We observe that the peak
+of the output intensity appears earlier than the peak of
+the input intensity, resembling the experimental results
+of Fig. 2(b). We calculate a time-advancement of 120 µs,
+equivalent to a group index of −1.8 × 106, which are in
+excellent agreement with the experimental observations.
+For simplicity, we have ignored any change of absorp-
+tion along the ruby crystal due to the change of intensity
+through either diffraction or absorption. A more accu-
+rate quantitative result can be obtained by employing an
+iterative approach.
+Having such a theoretical model at hand, one can eas-
+ily calculate other responses of interest.
+For example,
+Fig. 4(b) shows that the output intensity is deformed
+when the amplitude of the modulation is large. There-
+fore, the pulse advancement will not be uniform across
+the entire period. This deformation can also be seen in
+our experimental results in Fig. 1(b).
+We also calcu-
+late the pulse advancement as a function of the input in-
+tensity, I0 (Fig. 5). Interestingly, the time-advancement
+
+4
+Input intensity (I /I )
+0
+s
+Pulse advancement (μs)
+0
+2
+4
+6
+8
+10
+12
+20
+40
+60
+0
+80
+100
+120
+FIG. 5. Theoretical pulse advancement as a function
+of intensity. The pulse advancement reaches the maximum
+at intensities about the saturation intensity. As the laser in-
+tensity increases further, the effect decreases gradually.
+reaches a maximum for intensities around the saturation
+intensity Is = ℏω/σ1τ, and decreases gradually for higher
+intensities.
+In summary, we observed a strong reverse saturation
+of absorption in ruby at a wavelength of 473 nm, which is
+a consequence of the excited state having an absorption
+cross-section larger than the ground state. Although the
+requirements for RSA are stringent, we showed that ruby
+can exhibit a strong RSA on the blue side of the visible
+spectrum. This effect has been used to demonstrate a
+pulse advancement which could indicate a fast-light ef-
+fect with a very large and negative group index.
+We
+showed that this observation can also be explained very
+well based on the sluggish response of ruby, consistent
+with the previous theories [17–20]. Investigating the rel-
+ative validity of the sluggish time-dependent absorption
+theory versus the coherent population oscillation theory
+will require further experimental investigations.
+How-
+ever, this experimental investigation does serve to bet-
+ter understand the nature of the pulse-delay and pulse-
+advancement in transition-metal-doped crystals, which is
+crucial for applications in optical delay lines and optical
+memories [8, 9, 21], optical gyroscopes [6, 7], and photon-
+drag [22, 23].
+Acknowledgment
+This work was supported by the Canada Excellence
+Research Chairs program and the National Science and
+Engineering Research Council of Canada (NSERC).
+∗ Akbar.Safari@gmail.com
+[1] R. W. Boyd, Nonlinear Optics (Academic Press, 2008).
+[2] Anthony E Siegman, Lasers (University Science Books,
+1986).
+[3] Methods of Laser Spectroscopy (Springer, Boston, MA,
+1986).
+[4] Lee W. Tutt and Thomas F. Boggess, “A review of op-
+tical limiting mechanisms and devices using organics,
+fullerenes, semiconductors and other materials,” Progress
+in Quantum Electronics 17, 299–338 (1993).
+[5] M.S. Shahriar and M. Salit, “Application of fast-light
+in gravitational wave detection with interferometers and
+resonators,” Journal of Modern Optics 55, 3133–3147
+(2008).
+[6] M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal,
+M. Messall, and K. Salit, “Ultrahigh enhancement in ab-
+solute and relative rotation sensing using fast and slow
+light,” Phys. Rev. A 75, 053807 (2007).
+[7] M. Salit, G. S. Pati, K. Salit,
+and M. S. Shahriar,
+“Fast-light for astrophysics: super-sensitive gyroscopes
+and gravitational wave detectors,” Journal of Modern
+Optics 54, 2425–2440 (2007).
+[8] Robert W. Boyd, “Slow and fast light: fundamentals and
+applications,” Journal of Modern Optics 56, 1908–1915
+(2009).
+[9] P. W. Milonni, Fast Light, Slow Light and Left-Handed
+Light (CRC Press, 2004).
+[10] Alexander M Akulshin and Russell J McLean, “Fast light
+in atomic media,” Journal of Optics 12, 104001 (2010).
+[11] Matthew S. Bigelow, Nick N. Lepeshkin, and Robert W.
+Boyd, “Observation of ultraslow light propagation in a
+ruby crystal at room temperature,” Phys. Rev. Lett. 90,
+113903 (2003).
+[12] Matthew S. Bigelow, Nick N. Lepeshkin, and Robert W.
+Boyd, “Superluminal and slow light propagation in a
+room-temperature solid,” Science 301, 200–202 (2003).
+[13] Michelle S. Malcuit, Robert W. Boyd, Lloyd W. Hill-
+man, Jerzy Krasinski,
+and C. R. Stroud, “Saturation
+and inverse-saturation absorption line shapes in alexan-
+drite,” J. Opt. Soc. Am. B 1, 73–75 (1984).
+[14] G. Piredda and R. Boyd, “Slow light by means of co-
+herent population oscillations: laser linewidth effects,”
+Journal of the European Optical Society - Rapid publi-
+cations 2 (2007).
+[15] Emma Wisniewski-Barker, Graham M Gibson, Sonja
+Franke-Arnold, Zhimin Shi, Paul Narum, Robert W
+Boyd, and Miles J Padgett, “Experimental investigation
+of the transient dynamics of slow light in ruby,” New
+Journal of Physics 16, 123054 (2014).
+[16] G G Kozlov, S V Poltavtsev, I I Ryzhov,
+and V S
+Zapasskii, “Comment on ‘evidence of slow-light effects
+from rotary drag of structured beams’,” New Journal of
+Physics 16, 038001 (2014).
+[17] V. S. Zapasski˘l and G. G. Kozlov, “A saturable absorber,
+coherent population oscillations, and slow light,” Optics
+and Spectroscopy 100, 419 (2006).
+[18] Bruno Macke and Bernard S´egard, “Slow light in sat-
+urable absorbers,” Phys. Rev. A 78, 013817 (2008).
+[19] Bruno Macke and Bernard S´egard, “From fast to slow
+light in a resonantly driven absorbing medium,” Phys.
+Rev. A 82, 023816 (2010).
+[20] Bruno Macke, Igor Razdobreev,
+and Bernard S´egard,
+“Slow light in saturable absorbers: Progress in the reso-
+lution of a controversy,” Phys. Rev. A 95, 063830 (2017).
+[21] P. Neveu, M.-A. Maynard, R. Bouchez, J. Lugani,
+R. Ghosh, F. Bretenaker, F. Goldfarb,
+and E. Brion,
+“Coherent population oscillation-based light storage,”
+
+5
+Phys. Rev. Lett. 118, 073605 (2017).
+[22] Sonja Franke-Arnold, Graham Gibson, Robert W. Boyd,
+and Miles J. Padgett, “Rotary photon drag enhanced by
+a slow-light medium,” Science 333, 65–67 (2011).
+[23] Akbar Safari, Israel De Leon, Mohammad Mirhosseini,
+Omar S. Maga˜na Loaiza, and Robert W. Boyd, “Light-
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+por,” Phys. Rev. Lett. 116, 013601 (2016).
+
diff --git a/MdFQT4oBgHgl3EQfVTaW/content/tmp_files/load_file.txt b/MdFQT4oBgHgl3EQfVTaW/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..790ac4fa28ecd9e45d98b62b784c2065f439e492
--- /dev/null
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@@ -0,0 +1,321 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf,len=320
+page_content='Strong Reverse Saturation and Fast-Light in Ruby  Akbar Safari,1, 2, ∗ Cara Selvarajah,1 Jenine Evans,1 Jeremy Upham,1 and Robert W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Boyd1, 3  1Department of Physics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  2Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  3Institute of Optics, University of Rochester, Rochester, New York, 14627, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  (Dated: February 1, 2023)  We observe a strong reverse saturation of absorption in ruby at a wavelength of 473 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' With  an intensity-modulated laser, we observe that the peaks of the pulses appear more than a hundred  microseconds earlier than the reference signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' A theoretical model based on coherent population  oscillation would suggest a fast-light effect with an extremely large and negative group index of  −(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='1)×106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' We propose that this pulse advancement can also be described by time-dependent  absorption of ruby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Our study helps to understand the nature of the fast- and slow-light effects in  transition-metal-doped crystals such as ruby and alexandrite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Saturation of absorption is a well-understood nonlin-  ear optical process in which an optical driving field pass-  ing through an absorptive material experiences a de-  crease in absorptivity as the intensity of the field in-  creases [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' This process is typical of most materials and  is frequently employed for passively mode-locking or Q-  switching lasers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  However, there are conditions under which materials  exhibit an increase in absorption with higher intensity at  specific wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' This response, called reverse sat-  uration of absorption (RSA), requires particular condi-  tions, including a more than two-level system and for  an excited state to have a larger absorption cross-section  than the ground state [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' In addition, neither the first  nor the second excited states should decay to other levels  thereby trapping the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Moreover, the incident  light should saturate, or partially saturate, the first tran-  sition only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' This condition can be achieved easily when  the lifetime of the first excited state is much longer than  that of the second excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Experimental investiga-  tion of RSA, particularly their temporal dynamics, will  provide insight into their feasibility for power limiting [4]  and fast-light applications in gravitational wave detec-  tion [5], optical gyroscopes [6, 7], and more [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Here, we show that ruby at room temperature ex-  hibits a strong reverse saturation of absorption at the  wavelength of 473 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Consequently, we observe what  could appear to be a fast-light effect, where the peak of  an intensity-modulated signal passing through the crys-  tal reaches the detector earlier than that without the  ruby crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' This phenomenon had been explained ear-  lier based on coherent population oscillation and hole  burning [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' While this model continues to be de-  bated [11, 12, 14–20], we consider a simple model based  on rate-equations that explains the seemingly fast-light  effect without the need of hole or anti-hole burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' In  the following, first, we confirm that ruby exhibits RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Then, we use rate equations to find the time-dependent  population of the ground and excited states, and conse-  quently, the time-dependent absorption of ruby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Finally,  we show that the advancement of the peak intensities can  be explained by time-dependent absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  The reverse saturation of absorption is observed by  measuring the transmission of a continuous-wave (CW)  laser at wavelength 473 nm with a maximum power of  500 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' To achieve the desired intensities, the laser is  focused to a beam waist of 36 µm at the center of the ruby  crystal of length 20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The results, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 1(a),  show a clear reduction of the transmission as the laser  power increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The crystal is placed at a small angle  to avoid any issues with the back reflection of the laser  from the crystal faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The Fresnel reflections are con-  sidered in the calculation of absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' We observe that  the transmission through the ruby crystal decreases from  68% in the linear regime, to 52% in the nonlinear (high  power) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  The relevant energy levels of ruby are drawn in the  inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The laser excites the electrons from  the ground state g to the excited state e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The excited  electrons relax very rapidly to the meta-stable state g′  by emitting phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  These three levels are typically  enough to describe the interaction of the Cr3+ ions with a  green laser, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' However, a blue laser at 473 nm,  can excite the electrons from the meta-stable state to the  second excited state, e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Because the absorption cross-  section of this transition, σ2, is larger than that of the  first transition, σ1, the overall absorption rate increases  with the intensity of the laser (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Since states e and e′ have extremely short lifetimes,  their populations are negligible and can be omitted in  the rate equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Therefore, the population density of  the ground state Ng follows  dNg  dt  = − I  ℏω σ1Ng + N − Ng  τ  (1)  where, I is the intensity of the laser with photon en-  ergy ℏω, and N − Ng is the population density of the  meta-stable state g′, with lifetime τ =5 ms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' σ1 is the  absorption cross-section of the first transition from g to  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The density of the Cr3+ ions in the ruby crystal is  N ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='75 × 1018 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' For a CW excitation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (1) is  solved in steady-state condition (dNg/dt = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Then, the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='13300v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='optics]  30 Jan 2023  2  (a)  (b)  Input power (W)  Transmission (%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='5  50  55  60  65  Intensity (W/cm )  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='40  0  ×103  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0×103  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0×103  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0×103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content="0×104  Absorption coefficient (cm  )  1  20 mm  72 μm  rapid  rapid  τ =5ms  (  )  (  )  (  )  (  )  g A  4  2  E  2  T  2  2  T  2  1  e  g'  e'  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Reverse saturation of absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (a) Experi-  mental (open circles) and theoretical (solid line) transmission  as a function of the input laser power for a CW (unmodulated)  beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The inset shows the beam profile and the position of  ruby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (b) Theoretical absorption coefficient from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' In-  set: The relevant energy levels of ruby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The absorption cross-  section of the second transition, g′ → e′, is larger than that of  the first transition, g → e, which leads to reverse saturation  of absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  absorption coefficient is found from  α =  σ1Ng + σ3(N − Ng)  (2)  =  N  1+I/Is (σ1 − σ3) + σ3N,  (3)  where σ3 is the absorption cross-section of the second  transition, and Is = ℏω/σ1τ is the saturation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  We note that since the populations of levels e and e′ are  negligible, implementing the degeneracy factors of the  states will not affect Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Figure 1(b) plots the ab-  sorption coefficient from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (3) as a function of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  In order to test the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (3) with experi-  mental data, we simulate the propagation of the laser  through the ruby crystal and find the theoretical trans-  mission as a function of the input power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' We use the  same beam profile as measured in the experiment (inset  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Since the local intensity and the absorp-  tion coefficient are interdependent, we adopt an iterative  approach to calculate the intensity of the laser as it prop-  agates through the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The results, shown as a solid,  red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 1(a), exhibit an excellent agreement with  the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Therefore, we extract the values  of the absorption cross sections to be σ1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0×10−20 cm2  and σ3 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='6 × 10−20 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' It is because the absorption  cross-section of the second transition is nearly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='5 times  larger than that of the ground state at this wavelength  CW laser  (473 nm)  EOM  signal  generator  Oscilloscope  f=100mm  ruby  reference  beamsplitter  filter  0  5  10  15  time (ms)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  117 μs  (a)  (b)  input pulse  output pulse  normalized intensity  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Fast-light experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  (a) The experimental  setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' An electro-optic modulator (EOM) is used to imprint a  weak sinusoidal intensity modulation on the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The  laser is focused in the ruby crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' A spectral filter is used  to block the fluorescence at 694 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (b) Upon propagation  in ruby, the peak of the weakly modulated signal advances  in time for approximately 117 µs compared to the reference  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  that enables the clear observation of reverse saturation  of absorption in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Next, we investigate the fast-light effect and exam-  ine how the temporal variations of a modulated beam  are altered upon traveling through the reverse saturated  medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Following refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' [11, 12], we send the laser beam  through an electro-optic device which imposes a 10% in-  tensity modulation to the laser beam, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  The  electro-optic modulator is fed by a sinusoidal signal at  frequency Ω/2π = 70 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The beam is focused to a waist  of 36 µm at the center of the ruby crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' A spectral filter  is used to filter out the fluorescence at 694 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The trans-  mitted and reference signals are detected and compared  on an oscilloscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' When operating at intensities where  RSA is clearly visible, the peaks of the modulated signal  appear to advance relative to the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Figure 2(b)  shows 117 ± 6 µs pulse advancement for an average input  power of 450 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  These results appear to be consistent with similar ex-  periments in ruby and alexandrite [11, 12], where slow-  and fast-light effects were reported and attributed to  coherent population oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Were the same reason-  ing used here, the RSA would lead to a spectral hill  or anti-hole and the corresponding advancement of the  pulse peak would indicate fast-light with a group index  of −(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='1)×106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Here we examine another hypothe-  sis to describe these results: a time-dependence to ruby’s  response to the modulated signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  For an intensity-modulated laser beam, the input in-  3  time (ms)  Absorption coefficient (m )  1  Input intensity (10 W/cm )  2  3  0  10  30  20  40  50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='25  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='5  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Ruby time-dependent response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Modulated in-  put intensity (right vertical axis) and the corresponding time-  dependent absorption coefficient (left vertical axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  When  the frequency of modulation is small compared to 1/τ, the  absorption oscillates with the driving intensity, however, with  a time difference due to the finite lifetime of the meta-stable  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' As the frequency of modulation increases, the oscilla-  tion amplitude of the absorption coefficient, α1, decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  tensity to the ruby crystal can be written as  I(t) = I0 + I1e−iΩt + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=',  (4)  where, Ω is the angular frequency of the modulation, I0  is the average intensity, and I1 is the amplitude of the  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' [11, 13, 14], in order to explain an  anti-hole in the spectrum, the population of the ground  state and the absorption were written as N(t) = N0 +  N1e−iΩt+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=', and α(t) = α0+α1e−iΩt+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  In other words, it was assumed that the population and  the absorption follow the same time dependence as the  driving intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Therefore, the plot of α0 + α1 as a  function of the modulation frequency exhibited a narrow  peak, which was interpreted as an anti-hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  However,  the  total  absorption  of  ruby  is  time-  dependent, with α0 + α1 merely showing the maximum  absorption (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' When the period of the mod-  ulation is long enough compared to the lifetime of the  meta-stable state τ = 5 ms, the population can vary,  following the driving intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Because the absorption  cross-section of the meta-stable state is larger than that  of the ground state, as the population of the meta-stable  state increases, the total absorption, α(t), increases as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Therefore, α1 is maximum at slow modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  As the frequency of the modulation increases, the ampli-  tude of α1 decreases because the population cannot keep  up with the rapidly changing intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Therefore, α1  decreases at fast modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Given  that  the  driving  intensity  has  the  time-  dependent form given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (4), the population and  absorption may lag behind the driving intensity as the  lifetime of the meta-stable state is relatively long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' There-  fore, we numerically solve the rate equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' (1)  with the modulated intensity to find the population as  a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  The absorption is calculated by  α(t) = σ1Ng(t) + σ3(N − Ng(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Figure 3 plots the  0  5  10  15  time (ms)  input intensity (10 W/cm )  2  3  3  output intensity (10 W/cm )  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='5  120 μs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='2  3  output intensity (10 W/cm )  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='20  input intensity (10 W/cm )  2  3  0  5  10  15  time (ms)  a) 10% modulation  b) 50% modulation  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Theoretical pulse advancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The input and  output (after propagation through ruby) intensity as a func-  tion of time for (a) weak modulation and (b) strong mod-  ulation calculated from time-dependent absorption of ruby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  When the modulation is weak, the signal is advanced uni-  formly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  With strong modulations, the output signal is de-  formed notably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  driving intensity and the calculated absorption as a func-  tion of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' There is clearly a time difference between  the maximum of absorption and the maximum of the in-  put intensity due to the finite lifetime of the meta-stable  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The time-dependent absorption lagging behind the  modulated intensity reshapes the temporal form of the  intensity as measured at the output of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Accounting for this time-dependent intensity and ab-  sorption, we simulate the form of the output intensity as  a function of time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' We observe that the peak  of the output intensity appears earlier than the peak of  the input intensity, resembling the experimental results  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' We calculate a time-advancement of 120 µs,  equivalent to a group index of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='8 × 106, which are in  excellent agreement with the experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  For simplicity, we have ignored any change of absorp-  tion along the ruby crystal due to the change of intensity  through either diffraction or absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' A more accu-  rate quantitative result can be obtained by employing an  iterative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Having such a theoretical model at hand, one can eas-  ily calculate other responses of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  For example,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 4(b) shows that the output intensity is deformed  when the amplitude of the modulation is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' There-  fore, the pulse advancement will not be uniform across  the entire period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' This deformation can also be seen in  our experimental results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  We also calcu-  late the pulse advancement as a function of the input in-  tensity, I0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Interestingly, the time-advancement  4  Input intensity (I /I )  0  s  Pulse advancement (μs)  0  2  4  6  8  10  12  20  40  60  0  80  100  120  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Theoretical pulse advancement as a function  of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' The pulse advancement reaches the maximum  at intensities about the saturation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' As the laser in-  tensity increases further, the effect decreases gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  reaches a maximum for intensities around the saturation  intensity Is = ℏω/σ1τ, and decreases gradually for higher  intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  In summary, we observed a strong reverse saturation  of absorption in ruby at a wavelength of 473 nm, which is  a consequence of the excited state having an absorption  cross-section larger than the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Although the  requirements for RSA are stringent, we showed that ruby  can exhibit a strong RSA on the blue side of the visible  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' This effect has been used to demonstrate a  pulse advancement which could indicate a fast-light ef-  fect with a very large and negative group index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  We  showed that this observation can also be explained very  well based on the sluggish response of ruby, consistent  with the previous theories [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content=' Investigating the rel-  ative validity of the sluggish time-dependent absorption  theory versus the coherent population oscillation theory  will require further experimental investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  How-  ever, this experimental investigation does serve to bet-  ter understand the nature of the pulse-delay and pulse-  advancement in transition-metal-doped crystals, which is  crucial for applications in optical delay lines and optical  memories [8, 9, 21], optical gyroscopes [6, 7], and photon-  drag [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
+page_content='  Acknowledgment  This work was supported by the Canada Excellence  Research Chairs program and the National Science and  Engineering Research Council of Canada (NSERC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFQT4oBgHgl3EQfVTaW/content/2301.13300v1.pdf'}
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+A new insight on positivity and contractivity of the
+Crank-Nicolson scheme for the heat equation∗
+I. Higueras1, T. Roldán1
+1Institute for Advanced Materials and Mathematics
+Public University of Navarre
+E-mails: higueras@unavarra.es, teo@unavarra.es.
+Abstract
+In this paper we study numerical positivity and contractivity in the infinite norm of
+Crank-Nicolson method when it is applied to the diffusion equation with homogeneous
+Dirichlet boundary conditions. For this purpose, the amplification matrices are written
+in terms of three kinds of Chebyshev-like polynomials, and necessary and sufficient
+bounds to preserve the desired qualitative properties are obtained. For each spatial
+mesh, we provide the equations that must be solved as well as the intervals that
+contain these bounds; consequently, they can be easily obtained by a bisection process.
+Besides, differences between numerical positivity and contractivity are highlighted.
+This problem has also been addressed by some other authors in the literature and
+some known results are recovered in our study. Our approach gives a new insight on
+the problem that completes the panorama and that can be used to study qualitative
+properties for other problems.
+Keywords: Positivity, Maximum norm contractivity, Monotonicity, Crank-Nicolson,
+Heat equation
+1
+Introduction
+We consider the numerical solution of the one-dimensional heat equation
+∂u(t, x)
+∂t
+= d ∂2u(t, x)
+∂x2
+,
+x ∈ [0, 1], t ≥ 0 ,
+(1.1)
+u(0, t) = u(1, t) = 0 ,
+t ≥ 0,
+(1.2)
+u(x, 0) = u0(x) ,
+x ∈ [0, 1] .
+(1.3)
+where u0 in (1.3) is a given function on [0, 1]. For the sake of simplicity, we consider the
+diffusion term d = 1 in the rest of the paper. Solutions u(x, t) for the linear parabolic
+∗Research project PID2019-109045GB-C31 funded by Agencia Estatal de Investigación, Ministerio de
+Ciencia e Innovación, Spain.
+1
+arXiv:2301.01066v1  [math.NA]  3 Jan 2023
+
+problem (1.1)-(1.3) have several qualitative properties that are relevant in the context of
+the physical model. In particular, the problem is positivity preserving, that is, for t ≥ 0,
+u0(x) ≥ 0
+⇒
+u(x, t) ≥ 0 ,
+(2)
+and the solutions are monotonically decreasing in time, i.e., for t2 ≥ t1 ≥ 0,
+max
+0≤x≤1 |u(t2, x)| ≤ max
+0≤x≤1 |u(t1, x)| .
+(3)
+In order to obtain numerical approximations with physical sense, properties (2)-(3) should
+be preserved in the discretization process. In this paper, we consider the Crank-Nicolson
+(CN) method, a method of lines approach where second order central finite differences
+in space are followed by a second order time-stepping method. Spatial discretization of
+(1.1)-(1.3) with second order central finite differences and mesh width h = 1/(m + 1),
+gives a semi-discrete linear differential system of the form
+w′(t) = Bhw(t) ,
+w(0) = w0 , t ≥ 0 ,
+(4)
+where Bh is a matrix of dimension m, that is positivity preserving with monotonically
+decreasing (in the maximum norm) solutions (see section 2 for details).
+Next, a time
+stepping method is used to obtain numerical approximations wn ≈ w(tn), where tn = nτ,
+and τ is the constant time stepsize used. In this paper, we consider approximations of the
+form
+wn = Am wn−1 , n ≥ 0 ,
+(5)
+where w0 is a known value, and Am is a matrix of dimension m that depends on the time
+stepping method. In particular, for Runge-Kutta methods, Am = φ(τBh), where φ is the
+stability function of the scheme.
+There is a vast list of references in the analysis of positivity and monotonicity decreasing
+time stepping schemes (see, e.g., [21, 26, 14, 16, 11, 4, 8, 12, 2, 6, 18]; see too [26, 6, 13]
+and the references therein). Depending on the context, monotonicity decreasing methods
+are also known as contractive, SSP (Strong Stability Preserving) or TVD (Total Variation
+Diminishing) schemes (see, e.g., [23, 21, 14, 8, 4, 6]). In this setting, stepsize restrictions
+of the form
+τ ≤ C τF E ,
+(6)
+are obtained, where C denotes the monotonicity threshold factor (also known as SSP-
+coefficient, radius of absolute monotonicity, contractivity radius, . . . ) of the time stepping
+method (see e.g.,[15, 27, 5, 23, 4]), and τF E is the stepsize restriction for the given qualita-
+tive property when forward Euler scheme is used to solve the specific ODE problem (see,
+e.g. [23, p. 379], [8, p. 201], [6, pp. 52-53]). In particular, for problem (4), τF E = h2/2
+for both positivity and contractivity (see, e.g., [26, p. 22]), and C = 1 for forward Euler
+method (see section 2 for details). Observe that with this approach, the stepsize restric-
+tion (6) is the same for all linear systems with the same τF E and for some problems this
+is not a sharp bound.
+A well known time stepping method is the θ-method, defined as
+wn = (I − θτBh)−1(I + (1 − θ)τBh)wn−1 ,
+θ ∈ [0, 1] .
+(7)
+2
+
+Observe that this method can be understood as the composition of a (1 − θ)τ-step with
+forward Euler method and a θτ-step with backward Euler scheme.
+In particular, for
+θ = 1/2, CN scheme is obtained. The radius of absolute monotonicity for the θ-method
+applied to linear problems is Cθ = 1/(1 − θ) [9]. Consequently, numerical positivity and
+contractivity can be ensured under the restriction (see [26, p. 135])
+τ
+h2 ≤
+1
+2(1 − θ) .
+However, If we look closer at the iteration matrix Am in (7), a sharper bound is possible.
+Different authors have studied numerical preservation of positivity and monotonicity for
+problem (1.1)-(1.3) with the θ-method (see, e.g., [16, 11, 2]). In [2, p. 72], the authors
+obtain that the numerical solution is positive if and only if
+τ
+h2 ≤ 1 −
+√
+1 − θ
+θ(1 − θ)
+,
+(8)
+whereas in [16, Remark 7.1] and [11, p. 456] it is shown that the numerical solution is
+contractive if and only if
+τ
+h2 ≤
+2 − θ
+4(1 − θ)2 .
+(9)
+On the following we will denote s = τ/h2 to the CFL coefficient. The contractivity result
+in [16] is obtained for the pure initial value problem
+∂u(t, x)
+∂t
+= d ∂2u(t, x)
+∂x2
+,
+x ∈ R, t ≥ 0 ,
+u(x, 0) = u0(x) ,
+x ∈ R .
+In particular, bound (9) for contractivity is valid for all m ≥ 1 [16, Theorem 4.1] [11].
+The bound τ/h2 ≤ 2/3 has also been obtained in [3, Theorem 1] in the analysis of the
+stability of CN method.
+In [2] and [11] results are based on the shape of the inverse
+matrix (I−θτBh)−1 of dimension m, whose entries can be expressed in terms of hyperbolic
+functions [20]; stepsize restrictions are obtained for each value of m, and bound (9) is valid
+for all m.
+Contributions of the paper
+The approach followed in this paper consists on the representation of the iteration matrix
+Am in (5) for the CN method in terms of three classes of polynomials, Pn(x), Cn(x) and
+Un(x), defined by iterations (26), (28) and (33), respectively; in particular, Un(x) are the
+Chebyshev polynomials of the second kind. In the three cases the iteration process is the
+same, but different initial values are considered. This formalism gives a new insight on
+the problem that allows us to improve some results in the literature.
+The contributions of this paper are the following ones. With regard to positivity, for
+any number of grid points m, we have obtained that:
+1. Crank-Nicolson method is positive if and only if s = τ/h2 ∈ (0, s(p)
+m ], with s(p)
+m =
+1/(cosh ω(p)
+m − 1), where ω(p)
+m is the unique positive root of equation (18). This root
+3
+
+lies in the narrow interval (log(2 +
+√
+2), log(2 +
+√
+3)] ≈ (1.22795, 1.31696]. Thus ω(p)
+m
+can be easily computed by solving (18) with bisection method. Proposition 2 shows
+the connection between the bounds s(p)
+m and polynomials Pn(x). Some values of s(p)
+m
+are shown in Table 1.
+2. The sequence of bounds (s(p)
+m ) is strictly monotonically increasing with all the terms
+in the interval [1, 2(2 −
+√
+2)). As a consequence, the CN method does not preserve
+positivity when the spatial mesh is refined (keeping s constant).
+3. In the limit case, when m tends to infinite, we recover the known bound, s ≲ 1.17
+for positivity ([13, p. 126],[2, Table 1]).
+With regard to contractivity, we have computed the value ∥Am(s)∥∞ for any number
+of grid points m (see Figure 3), and we have obtained that:
+1. Crank-Nicolson method is contractive if and only if s = τ/h2 ∈ (0, s(c)
+m ], with
+s(c)
+m = ∞ for m = 1, 2, 3. For m ≥ 4, s(c)
+m = 1/(cosh ω(c)
+m − 1), where ω(c)
+m is the
+unique positive root of equation (21) or (22), depending on the parity of m. This
+root lies in the interval
+� log
+�(3+
+√
+5+
+√
+−2 + 6
+√
+5)/4
+�, log 3
+� ≈ (0.767197, 1.09861].
+Thus ω(c)
+m can be easily computed by solving (21) or (22) with the bisection method.
+Some values of s(p)
+m
+are shown in Tables 2 and 3 for odd and even values of m,
+respectively. Some of these values can also be seen in Figure 3.
+2. ∥Am(s)∥∞ < 1 for s ∈ (0, s(c)
+m ) and ∥Am(s(c)
+m )∥∞ = 1 (see Figure 3), that resembles
+property (14) of the linear system (10).
+3. For m ≥ 4, the sequence (s(c)
+m ) is strictly monotonically decreasing with all the terms
+in the interval
+�3/2, 1 +
+√
+5
+�. As a consequence, CN method preserves contractivity
+when the spatial mesh is refined (keeping s constant).
+4. In the limit case, when m tends to infinite, we recover the known bound, τ/h2 ≤ 3/2
+for contractivity [16, Th. 4.1(Q3); Section 7.1], [11, Eq. (14)].
+The results in this paper complete and improve some results in the literature [2, 11].
+From equations (18) and (21)-(22), and the associated intervals, the computation of s(p)
+m
+and s(c)
+m for any value of m is straightforward with the bisection method. Besides, we
+obtain equations to compute bounds s(c)
+m both for odd and even values of m, whereas
+in [11], bounds s(c)
+m are only given for even m. From our approach we also get the correct
+value for s(c)
+3 . Figure 1 illustrates the differences between positivity and contractivity of
+CN method for the heat problem (1): if the scheme is positive for a given grid mesh m,
+then it is also contractive for any grid mesh.
+Scope of the paper
+The rest of the paper is organized as follows. In Section 2, we explain the CN discretization
+process; notation and definitions are also given in this section. In Section 3 we show the
+main results of the paper, namely: Theorems 1 and 2; Table 1, containing upper bounds
+4
+
+s(p)
+m for positivity; Tables 2 and 3 containing upper bounds s(c)
+m for contractivity (odd and
+even case); and Figure 1 showing sequences (s(p)
+m ) and (s(c)
+m ). An illustrative example is
+also given in Section 3. Section 4 contains some conclusions and ideas for future work.
+The proof of main results are given in Section 6. Previously, some technical material,
+needed for the proofs in Section 6, is included in Section 5.
+2
+Crank-Nicolson method for the heat equation
+In this paper we consider the Crank-Nicolson method, a method of lines approach where
+second order central finite differences in space are followed by a second order time-stepping
+method. Spatial discretization of heat equation (1) with second order central finite differ-
+ences and mesh width h = 1/(m + 1), gives the semi-discrete linear differential system
+w′(t) = Bhw(t) ,
+w(0) = w0 , t ≥ 0 ,
+(10)
+where Bh = (d/h2) tridiag(1, −2, 1) is a matrix of dimension m, w(t) ≈ (u(xi, t))m
+i=1,
+w0 = (u0(xi))m
+i=1, and xi = ih, i = 1, . . . , m, are the grid points.
+As the diffusion problem (1.1)-(1.3) is positivity preserving (2) and monotonically de-
+creasing (3), in order to obtain numerical approximations with these qualitative properties,
+problem (10) should also be positivity preserving and contractive in the maximum norm.
+An initial value problem
+w′(t) = f(t, w(t)),
+w(t0) = w0 , t ≥ 0 ,
+(11)
+is called positivity preserving (positive for short) if w0 ≥ 0 implies that w(t) ≥ 0 for t ≥ 0,
+where the inequalities should be understood component-wise. Problem (11) is said to be
+contractive in the maximum norm if its solution w(t) satisfy
+∥w(t2)∥∞ ≤ ∥w(t1)∥∞
+for t2 ≥ t1 ≥ 0 .
+It is well known that a linear problem,
+w′(t) = Aw(t) ,
+w(t0) = w0 , t ≥ 0 ,
+(12)
+where A = (aij) is an m × m matrix, is positive if and only if aij ≥ 0 for all i ̸= j [13,
+Theorem 7.2]. Matrix Bh in (10) satisfies this condition and thus problem (10) is positive.
+Observe that other spatial discretizations do not preserve positivity; indeed, there is an
+order barrier (q ≤ 2) from the requirement of positivity [13, p. 119].
+Contractivity of solutions of the linear problem (12) can be proven by using the concept
+of logarithmic norm of matrix A. This concept is an extremely useful tool to analyze the
+growth of solutions to ordinary differential equations because it can take negative values.
+The solutions of problem (12) are of the form w(t) = eAtw(0). If we consider a vector
+norm and its subordinate matrix norm, both denoted by ∥ · ∥, then
+∥w(t)∥ = ∥etAw(0)∥ ≤ ∥etA∥ ∥w(0)∥ ,
+(13)
+and contractivity is obtained if and only if ∥etA∥ ≤ 1. Given the set
+M =
+�
+δ ∈ R | ∥etA∥ ≤ etδ , t ≥ 0
+�
+,
+5
+
+it can be proven that µ∥·∥[A] = min(M), where µ∥·∥[A] stands for the logarithmic norm
+of matrix A in the norm ∥ · ∥ [22, Proposition 2.1] (see, e.g., [22, 24, 1] and the references
+therein for the definition and properties of logarithmic norms).
+From (13) and the definition of M, we get the inequalities
+∥w(t)∥ ≤ etµ∥·∥[A] ∥w(0)∥ ,
+t ≥ 0 ,
+∥etA∥ ≤ et µ∥·∥[A]
+t ≥ 0 .
+Thus, if µ∥·∥A] ≤ 0, the zero solution is stable and ∥etA∥ ≤ 1 for t ≥ 0; if µ∥·∥[A] < 0, then
+the zero solution is exponentially stable and ∥etA∥ < 1 for t > 0 [22, p. 634], [16, p. 2].
+For the maximum norm, the logarithmic norm of a matrix A = (aij) is given by
+µ∞[A] = max
+1≤i≤n
+�
+�aii +
+n
+�
+j=1
+|aij|
+�
+� .
+In particular, for matrix Bh in (10), as
+aii +
+n
+�
+j=1
+|aij| =
+�
+−1 ,
+i = 1, m ,
+0 ,
+i = 2, . . . , m − 1 ,
+we get µ∞[Bh] = 0, and thus
+∥etBh∥∞ ≤ 1,
+t ≥ 0 ,
+(14)
+that ensures that problem (10) is contractive in the maximum norm, that is,
+∥w(t)∥∞ ≤ ∥w(0)∥∞ ,
+t ≥ 0 .
+The time stepping process in Crank-Nicolson method with constant time step τ, gives
+the iteration
+wn = φ(τBh)wn−1 ,
+n ≥ 1 ,
+where
+φ(z) = 1 + 1
+2z
+1 − 1
+2z
+(15)
+is the stability function of the time integrator. On the following, we denote Am = φ(τBh)
+to the Crank-Nicolson iteration matrix of dimension m, that is,
+Am =
+�Im − τ
+2Bh
+�−1�Im + τ
+2Bh
+� =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1 + s − s
+2
+− s
+2
+1 + s ...
+...
+...
+− s
+2
+− s
+2 1 + s
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−1�
+�
+�
+�
+�
+�
+�
+�
+�
+1 − s
+s
+2
+s
+2
+1 − s ...
+...
+...
+s
+2
+s
+2 1 − s
+�
+�
+�
+�
+�
+�
+�
+�
+�
+. (16)
+Observe that the two matrices in (16), corresponding to half step with forward Euler
+and half step with backward Euler, commute because of the the linearity of the system
+(10). Besides, positivity and contractivity can be studied by analyzing these properties
+for forward and backward Euler separately.
+6
+
+Although there are no restrictions for positivity and contractivity with backward Euler
+applied to system (10), the restriction for positivity and contractivity with forward Euler
+is s ≤ 1 . This stepsize restriction for positivity is not sharp for problem (10); numerical
+experiments in [13, p.126] show that numerical positivity can be obtained for s ≲ 1.17.
+As it has been pointed out above, a closer look at the iteration matrix Am in (7) or (16),
+gives sharper bounds.
+3
+Main results
+In this section we show the main results of the paper concerning stepsize restrictions for
+positivity and contractivity in the maximum norm for the m-dimensional system (10).
+The proofs require some preliminary material about the structure of matrix Am and are
+given in section 6.
+On the following theorems, the positivity of the matrix Am means that all the entries
+of the matrix are non-negative; similarly, the contractivity in the maximum norm of the
+matrix Am means ∥Am∥∞ ≤ 1.
+Theorem 1. (Positivity of Crank Nicolson method)
+1. For m ∈ N, the matrix Am(s) in (16) is positive if and only if
+s ≤ s(p)
+m :=
+1
+cosh ω(p)
+m − 1
+,
+(17)
+where ω(p)
+m ∈
+� log(2 +
+√
+2), log(2 +
+√
+3)
+� is the unique positive root of equation
+coth(mω) sinh ω = 3 cosh ω − 4 .
+(18)
+2. The sequence (s(p)
+m ) is strictly monotonically increasing with all the terms in the
+narrow interval
+�1, 2(2 −
+√
+2)
+�. As a consequence, Crank Nicolson method preserves
+positivity when the spatial mesh is refined (keeping s constant).
+Remark 1.
+1. The sequence (s(p)
+m ) increasingly converges to the limit value s(p)
+∞ := 2(2 −
+√
+2) (see
+Table 1 and Figure 1). This value was also obtained in [2] with other techniques.
+Consequently, if
+s < s(p)
+∞ = 2(2 −
+√
+2) ≈ 1.17 ,
+(19)
+then there exists a natural number m0 such that the matrix Am is positive for any
+value of m ≥ m0.
+2. As ω(p)
+m ∈
+� log(2 +
+√
+2), log(2 +
+√
+3)
+� ≈ (1.22795, 1.31696], an approximated value
+can be easily computed by the bisection method.
+In Table 1 below we show the roots ω(p)
+m of equation (18) and the CFL restrictions s(p)
+m for
+positivity in (17) for different values of m.
+7
+
+m
+ω(p)
+m
+x(p)
+m = cosh ω(p)
+m
+s(p)
+m = 1/(x(p)
+m − 1)
+1
+log(2 +
+√
+3)
+2
+1
+2
+1.23590
+1 +
+√
+3/2 ≈ 1.86603
+2/
+√
+3 ≈ 1.15470
+3
+1.22864
+1.85464
+1.17009
+4
+1.22801
+1.85365
+1.17144
+...
+...
+...
+...
+7
+1.22795
+1.85355
+1.17157
+...
+...
+...
+...
+∞
+log(2 +
+√
+2)
+(6 +
+√
+2)/4
+2(2 −
+√
+2) ≈ 1.171572875
+Table 1: Roots ω(p)
+m of (18) and CFL restrictions s(p)
+m in (17) for positivity.
+Next, we give the results for contractivity in the infinite norm.
+Observe that the
+symmetry of matrix Am makes ∥Am∥∞ = ∥Am∥1, and the result is also valid for the
+1-norm.
+Theorem 2. (Contractivity of Crank Nicolson method)
+1. For m ∈ {1, 2, 3} the matrix Am(s) in (16) is contractive in the maximum norm for
+any value of s > 0.
+2. For m ∈ N, m ≥ 4 , the matrix Am(s) in (16) is contractive in the maximum norm
+if and only if
+s ≤ s(c)
+m :=
+1
+cosh ω(c)
+m − 1
+,
+(20)
+where ω(c)
+m
+∈
+� log
+�(3 +
+√
+5 +
+√
+−2 + 6
+√
+5)/4
+�, log 3
+� ≈ [0.767197, 1.09861) is the
+unique positive root of equation
+2 sinh (m − 1)ω
+4
+sinh (m + 1)ω
+4
+= sinh ω
+2 sinh (m + 1)ω
+2
+,
+(21)
+if m is odd, or equation
+sinh2 �ω
+2
+�
+sinh mω
+2
+�
+sinh (m + 2)ω
+2
+− sinh mω
+2
+�
+= sinh ω sinh (m + 1)ω
+2
+sinh mω
+4
+sinh (m − 2)ω
+4
+,
+(22)
+if m is even.
+3. The sequence (s(c)
+m ) is strictly monotonically decreasing with all the terms in the
+interval
+�3/2, 1 +
+√
+5
+�. As a consequence, Crank Nicolson method does not preserve
+contractivity when the spatial mesh is refined (keeping s constant).
+Remark 2. The sequence (s(c)
+m ) decreasingly converges to the limit value s(c)
+∞ := 3/2 (see
+Figures 1 and 3). Consequently, the matrix Am(s) is contractive for all m if and only if
+s ∈ (0, 3/2]. For infinite matrices the bound s(c)
+∞ := 3/2 has been obtained with different
+techniques in [3, 16].
+8
+
+In Tables 2 and 3 we give the CFL restrictions s(c)
+m in (20) for contractivity in the
+infinite norm. The values in Table 2 (odd case) and Table 3 (even case) have been ob-
+tained from equations (21) and (22), respectively.
+Observe that both, the roots ω(c)
+m
+of equation (21) (odd case) and the roots ω(c)
+m
+of equation (22) (even case), increas-
+ingly converge to the limit value log 3. Consequently these roots are in the narrow in-
+terval
+� log
+�(3 +
+√
+5 +
+√
+−2 + 6
+√
+5)/4
+�, log 3
+� ≈ [0.767197, 1.09861) and can be easily ob-
+tained with bisection method. The numeric values shown in tables 2 and 3 were obtained
+after 10 iterations with bisection method.
+s(c)
+m
+s(p)
+m
+Figure 1: Sequences s(p)
+m and s(c)
+m with the restrictions over CFL coefficient s for positivity
+and contractivity.
+Corollary 1. Consider the numerical integration of the m-dimensional problem (10) with
+the Crank-Nicolson method. Then if the method is positive for a given CFL coefficient,
+then it is also contractive in the infinity and 1 norms.
+Proof. It is straightforward from Theorems 1 and 2.
+The converse to the previous Corollary is not true as we can see in the following trivial
+example, where contractivity is preserved while positivity is violated.
+Example 1. Consider the diffusion equation in (1) with initial function
+u(x, 0) =
+�
+0
+for 0 < x < 7
+8 ,
+1
+for 7
+8 ≤ x < 1 ,
+giving discontinuities at x = 7/8 and x = 1 for t=0. From second order central differences
+with h = 1/8 we get approximations ω(t) = (ω1(t), . . . , ω7(t)) ≈ (u(x1, t), . . . , u(x7, t)).
+Application with τ = 0.025 of one Crank-Nicolson step, w1 = A7 w0, gives the vector
+w1 ≈ ω(τ)
+w1 = (0.0013, 0.0041, 0.0120, 0.0356, 0.1019, 0.2961, −0.1397) ,
+(23)
+where A7 is the Crank-Nicolson iteration matrix in (16) for the case m = 7 and w0 is the
+initial profile w0 = (0, 0, 0, 0, 0, 0, 1). Observe that ∥w1∥∞ = 0.2961 < ∥w0∥∞ = 1.
+9
+
+3
+3.0
+2
+s = 2(2- ~2)
+2.5
+2.0
+1.5
+1.0
+0
+2
+4
+6
+8
+10In this example the CFL coefficient s = τ/h2 = 1.6 is greater than the positivity bound
+s(p)
+7
+= 1.17157 (see Table 1), but it is lower than the contrativity one s(c)
+7
+= 1.61803 (see
+Table 2). Consequently, contractivity is preserved while we cannot ensure positivity. Ac-
+tually, as we can see in vector w1 in (23), negativity is not preserved.
+m
+ω(c)
+m
+x(c)
+m = cosh ω(c)
+m
+s(c)
+m = 1/(x(c)
+m − 1)
+3
+∞
+5
+2 arccsch 2 ≈ 0.962424
+3/2
+2
+7
+log
+�
+1+
+√
+5+√
+2(1+
+√
+5)
+�
+2
+≈ 1.06131
+(1 +
+√
+5)/2
+(1 +
+√
+5)/2 ≈ 1.61803
+9
+1.08707
+1.65139
+1.53518
+...
+...
+...
+...
+∞
+log 3 ≈ 1.09861
+5/3
+3/2
+Table 2: Positive root of (21) and bounds for contractivity (odd case).
+m
+ω(c)
+m
+x(c)
+m = cosh ω(c)
+m
+s(c)
+m = 1/(x(c)
+m − 1)
+4
+log
+� 1
+4(3 +
+√
+5 +
+√
+−2 + 6
+√
+5)
+�
+1
+4(3 +
+√
+5)
+1 +
+√
+5
+≈ 0.767197
+≈ 1.30902
+≈ 3.23607
+...
+...
+...
+...
+10
+1.09110
+1.65669
+1.52278
+...
+...
+...
+...
+20
+1.09855
+1.66658
+1.5002
+...
+...
+...
+...
+∞
+log 3 ≈ 1.09861
+5/3
+3/2
+Table 3: Positive root of (22) and bounds for contractivity (even case).
+4
+Conclusions and future work
+In this paper we have studied CFL restrictions when the Crank-Nicolson method is used
+to solve the heat equation (1) with Dirichlet boundary conditions.
+We have obtained
+bounds s(p)
+m for positivity and bounds s(c)
+m for contractivity for any value of the spatial
+discretization parameter m. To get these bounds we have represented the Crank-Nicolson
+iteration matrix Am in terms of some Chebyshev-like polynomials (26,28,33). We have
+obtained bounds for the θ-method (7) for the particular case θ = 1/2, but similar bounds
+can be obtained for other values of the parameter following the same ideas.
+We have seen that the positivity of matrix Am is determined by the largest root of
+polynomial Pm(x), and we have provided a narrow interval where this root can be found.
+Similarly, we have considered these polynomials to analyze the contractivity and we have
+10
+
+provided a narrow interval to get the corresponding bounds, both in the odd and even
+case.
+As far as we know, polynomials Pn(x) and Cn(x) have not been used previously in the
+literature. The strength of this idea can be used to prove qualitative properties for other
+problems. Furthermore, this approach can also be used for other discretizations of the
+heat equation (1.1) [10].
+5
+Preliminary material for the proofs of the main results
+In this section we introduce the notation, definitions and some results needed to prove
+the main results of the paper. In subsection 5.1 we express the Crank-Nicolson iteration
+matrix (16) in terms of rational functions. These functions can be written easily with the
+help of some Chebyshev-like polynomials Um, Pm and Cm. In subsection 5.2 we give the
+definition of these polynomials and we also add some results that will be used in the proofs
+of Section 6.
+5.1
+The Crank-Nicolson matrix Am in terms of rational functions
+A direct computation of the product (Im − τ
+2Bh)−1(Im + τ
+2Bh) in (16) gives us the entries
+of matrix Am expressed as rational functions, where the polynomials involved can be
+obtained recursively. These simplified closed expressions will make it easier to get bounds
+for positivity and contractivity.
+Example 2. For m = 3, a direct computation of the symmetric matrix A3 in (16) gives
+A3(s) =
+�
+�
+�
+�
+�
+2+2s−2s2−s3
+2+6s+5s2+s3
+2s
+2+4s+s2
+s2
+2+6s+5s2+s3
+2s
+2+4s+s2
+2−s2
+2+4s+s2
+2s
+2+4s+s2
+s2
+2+6s+5s2+s3
+2s
+2+4s+s2
+2+2s−2s2−s3
+2+6s+5s2+s3
+�
+�
+�
+�
+� .
+Remember s = τ/h2 denotes CFL coefficient. This matrix can be written even simpler if
+we consider the new variable x = 1 + 1/s. Observe that x > 1 when s > 0. With the help
+of a new kind of polynomials Un(x), Pn(x) and Cn(x), we can write A3(x) as
+A3(x) =
+�
+�
+�
+�
+�
+2x3−4x2+1
+2x3−x
+2(x−1)
+2x2−1
+x−1
+2x3−x
+2(x−1)
+2x2−1
+2x2−4x+1
+2x2−1
+2(x−1)
+2x2−1
+x−1
+2x3−x
+2(x−1)
+2x2−1
+2x3−4x2+1
+2x3−x
+�
+�
+�
+�
+� =
+1
+U3(x)
+�
+�
+�
+�
+P3(x)
+C2(x)
+C1(x)
+C2(x)
+C1(x) + P3(x)
+C2(x)
+C1(x)
+C2(x)
+P3(x)
+�
+�
+�
+� .
+Observe that A3(x) has been written just in terms of U3(x), P3(x), C1(x) and C2(x). We
+give the definition and all the details about these polynomials Un(x), Pn(x) and Cn(x)
+in the next subsection. Before, we extend the ideas in this simple example to the more
+general case of the matrix Am(x) for any value of m, although we have to distinguish
+between the odd case and the even case.
+11
+
+Proposition 1. Matrix Am(x) can be written in terms of polynomials Um(x), Pm(x) and
+Cn(x), n = 1, . . . , m − 1. If m is an odd number, Crank Nicolson matrix can be reduced to
+Am(x) =
+1
+Um
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Pm
+Cm−1
+...
+C m+1
+2
+...
+C2
+C1
+Cm−1
+Pm+Cm−2
+...
+C m−1
+2
++C m+3
+2
+...
+C1+C3
+C2
+...
+...
+...
+...
+...
+...
+...
+C m+1
+2
+C m−1
+2
++C m+3
+2
+...
+Pm+
+m−1
+2�
+n=1
+C2n−1
+...
+C m−1
+2
++C m+3
+2
+C m+1
+2
+...
+...
+...
+...
+...
+...
+...
+C2
+C1+C3
+...
+C m−1
+2
++C m+3
+2
+...
+Pm+Cm−2
+Cm−1
+C1
+C2
+...
+C m+1
+2
+...
+Cm−1
+Pm
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(24)
+where all the polynomials are evaluated at x = 1 + 1/s. If m is an even number, we write
+Am(x) =
+1
+Um
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Pm
+Cm−1
+...
+C m+2
+2
+C m
+2
+...
+C2
+C1
+Cm−1
+Pm+Cm−2
+...
+C m
+2 +C m+4
+2
+C m−2
+2
++C m+2
+2
+...
+C1+C3
+C2
+...
+...
+...
+...
+...
+...
+...
+...
+C m+2
+2
+C m
+2 +C m+4
+2
+...
+Pm+
+m−2
+2�
+n=1
+C2n
+m
+2�
+n=1
+C2n−1
+...
+C m−2
+2
++C m+2
+2
+C m
+2
+C m
+2
+C m−2
+2
++C m+2
+2
+...
+m
+2�
+n=1
+C2n−1
+Pm+
+m−2
+2�
+n=1
+C2n
+...
+C m
+2 +C m+4
+2
+C m+2
+2
+...
+...
+...
+...
+...
+...
+C2
+C1+C3
+... C m−2
+2
++C m+2
+2
+C m
+2 +C m+4
+2
+...
+Pm+Cm−2
+Cm−1
+C1
+C2
+...
+C m
+2
+C m+2
+2
+...
+Cm−1
+Pm
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(25)
+Proof. It is straightforward from the computation of the product (Im− τ
+2Bh)−1(Im+ τ
+2Bh)
+in (16) and the use of polynomials Un(x), Pn(x) and Cn(x), n = 1, . . . , m − 1, defined in
+the next subsection.
+Observe that Am(x) is bisymmetric, that is, it is symmetric on both diagonals. This im-
+plies that Am(x) is also centrosymmetric. Then the entries aij satisfy aij = an−i+1,n−j+1 ,
+for 1 ≤ i, j ≤ n. Consequently, if m is odd, the number of different entries in matrix Am
+is 1 + 3 + 5 + · · · + m = (m + 1)2/4, and, if m is even, this number is 2 + 4 + 6 + · · · + m =
+(m/2 + 1)m/2. For example, the number of different elements in matrix A3 in Example 2
+is 4, while this number is 6 for matrix A4 in Example 3 below.
+Observe also that the numerator of each entry aij in matrix Am(x) is a sum of some
+polynomials Pn(x), Cn(x), n = 1, . . . , m − 1, and the number of polynomials in this sum
+is equal to min{i, j, m − i + 1, m − j + 1}. Properties of polynomials Un(x), Pn(x) and
+Cn(x) will allow us to analyze positivity and contractivity of Crank Nicolson method in a
+quite simple way. In the next section, we study these properties.
+12
+
+Example 3. In Example 2 we have considered the odd case m = 3. Here, for completeness,
+we consider the even case m = 4. A direct computation of the symmetric matrix A4 in (16)
+gives
+A4(s) =
+1
+u4(s)
+�
+�
+�
+�
+p4(s)
+4s(4+8s+3s2)
+8s2(1+s)
+4s3
+4s(4+8s+3s2)
+16+32s+4s2−16s3−5s4
+16s(1+s)2
+8s2(1+s)
+8s2(1+s)
+16s(1+s)2
+16+32s+4s2−16s3−5s4
+4s(4+8s+3s2)
+4s3
+8s2(1+s)
+4s(4+8s+3s2)
+p4(s)
+�
+�
+�
+� ,
+where p4(s) = −5s4 − 24s3 − 4s2 + 32s + 16 and u4(s) = 5s4 + 40s3 + 84s2 + 64s + 16 .
+Now, with the help of variable x = 1 + 1/s, we can write
+A4(x) =
+1
+U4(x)
+�
+�
+�
+�
+�
+�
+�
+P4(x)
+C3(x)
+C2(x)
+C1(x)
+C3(x)
+P4(x) + C2(x)
+C1(x) + C3(x)
+C2(x)
+C2(x)
+C1(x) + C3(x)
+P4(x) + C2(x)
+C3(x)
+C1(x)
+C2(x)
+C3(x)
+P4(x)
+�
+�
+�
+�
+�
+�
+�
+.
+Observe that there are two central rows in the even case, but just one in the odd case.
+5.2
+Polynomials Un, Pn and Cn
+In this section we define the new polynomials Pn(x) and Cn(x). Together with the help
+of Chebyshev polynomials of the second kind Un(x) [7, 17, 19], we have got a simple way
+of writing Crank-Nicolson matrix Am(x). Besides, here we give some results concerning
+these polynomials, with particular interest in the distribution of their roots. Chebyshev
+polynomials of the second kind Un(x) belong to a general class of orthogonal polynomials
+and there are many works about the behaviour of their zeros [7, 17]. However, polynomials
+Pn(x) and Cn(x) do not belong to this class of orthogonal polynomials and, as far as we
+know, nothing is known about their roots.
+Chebyshev polynomials of second kind
+Chebyshev polynomial of second kind of degree n ≥ 0 is defined as
+Un(x) = sin((n + 1) arccos x)
+sin(arccos x)
+,
+x ∈ [−1, 1] ,
+or, in the angle variable ω, Un(cos ω) = sin((n + 1)ω)/sin ω, ω ∈ [0, π] . These polynomials
+can also be defined for any value of x ∈ R by the recurrence relation
+U0(x) = 1 ,
+U1(x) = 2x ,
+Un(x) = 2x Un−1(x) − Un−2(x) .
+(26)
+13
+
+It is possible to write the recurrence relation (26) in terms of the determinant of the
+tridiagonal matrix tridiag(1, 2x, 1) of dimension n.
+Un(x) =
+�����������
+2x
+1
+1
+2x
+...
+...
+...
+1
+1
+2x
+�����������
+(27)
+Recall that each polynomial Un(x) has n roots xn
+i = cos (iπ/(n + 1)) , i = 1, . . . , n, in the
+interval [−1, 1]. These roots are uniformly distributed in the angle variable ω = arccos x
+in the interval [0, π]. Notice that polynomials Un(x) defined in (26) are positive for x > 1.
+Polynomials Pn
+If we change the first two elements in the recursive relation (26), then a new family of
+polynomials can be defined
+P0(x) = −1 ,
+P1(x) = 2x − 4 ,
+Pn(x) = 2x Pn−1(x) − Pn−2(x) ,
+(28)
+where Pn(x) denotes the polynomial of degree n. In this case the recurrence relation (28)
+can also be written in terms of the determinant of a matrix of dimension n.
+Pn(x) =
+������������
+2x − 4
+−1
+1
+2x
+1
+...
+...
+...
+1
+2x
+1
+1
+2x
+������������
+.
+(29)
+In the next proposition we analyze the roots of each polynomial Pn(x). In the proof,
+shown in Section 6, it is relevant the fact that each polynomial Pn(x) can be written in
+terms of Chebyshev polynomials of second kind
+Pn(x) = 2Un−2(x) − 4Un−1(x) + Un(x) .
+(30)
+This equality is obtained by writing the determinant (29) in terms of the determinant (27),
+and the use of the recurrence relation (26).
+Proposition 2.
+The polynomial Pn(x) defined in (28) has exactly n − 1 roots xn
+i , i =
+1, . . . , n−1, in the interval (−1, 1), and an additional isolated root xn := xn
+n in the interval
+(6+
+√
+2
+4
+, 2]. Furthermore, the isolated root is xn = cosh ωn, where ωn is the unique root of
+the equation
+coth(nω) = 3 cosh ω − 4
+sinh ω
+,
+ω ∈ (0, ∞) .
+(31)
+Besides, x1 = 2 and the sequence of isolated roots (xn) decreasingly converges to the limit
+value x∞ = (6 +
+√
+2)/4.
+14
+
+Figure 2: Polynomials Pn(x) , n = 1, . . . , 4. Each polynomial has an isolated root in the
+interval (6+
+√
+2
+4
+, 2].
+As it is shown in the proof (see Section 6.3), for any value of n, the unique root ωn
+of equation (31) lies in the interval (ω∞, ω1] = (log(2 +
+√
+2), log(2 +
+√
+3)]. Consequently,
+the isolated root xn = cosh ωn of polynomial Pn(x) lies in the narrow interval (6+
+√
+2
+4
+, 2].
+Having the root well located makes it easy to approach it by any numerical method. In
+Table 1 we show some of these roots after 10 steps with bisection method.
+Polynomials Cn
+The polynomial Cn of degree n is defined as
+Cn(x) = Pn(x) + Un(x) ,
+n ∈ N,
+(32)
+where Pn is the polynomial of degree n defined above and Un is the Chebyshev polynomial
+of second kind of degree n. Consequently, all the properties of Cn are consequence of this
+definition, including its recursive definition
+C0(x) = 0 ,
+C1(x) = 4(x − 1) ,
+Cn(x) = 2x Cn−1(x) − Cn−2(x) .
+(33)
+Observe that the recursive formula (33) is the same as (28) for Pn and (26) for Un, with
+just different starting values C0 and C1. As in previous cases, it is worth writing Cn(x) in
+terms of a determinant
+Cn(x) =
+������������
+4(x − 1)
+0
+1
+2x
+1
+...
+...
+...
+1
+2x
+1
+1
+2x
+������������
+.
+(34)
+Proposition 3. Each polynomial Cn(x) has exactly n − 1 roots xn
+i , i = 1, . . . , n − 1, in
+the interval (−1, 1), and the additional isolated root xn = 1.
+15
+
+10
+5
+P1 (x)
+P2(x)
+-1.0
+0.5
+0.5
+1.0
+1.5
+2.0
+P3(x)
+.5
+P4(x)
+-10Proof. It is straightforward if we use the determinant (34), where we get
+Cn(x) = 4(x − 1)Un−1(x)
+(35)
+Consequently the roots of polynomial Cn are the isolated root xn = 1, and the n − 1 roots
+of the Chebyshev polynomial of second kind of degree n − 1.
+Corollary 2. If x > 1, then Cn(x) > 0 ∀n ∈ N .
+Proof. It is straightforward from the previous proposition.
+In the following Lemma we give some technical properties of polynomials Cn(x) that
+we will need in Section 6.
+Lemma 3. For the polynomial Cn of degree n defined in (32) or (33) the following prop-
+erties hold:
+1. Cn(1) = 0 , ∀n ∈ N .
+2. Cn(x) = 2Un−2(x) − 4Un−1(x) + 2Un(x) , ∀n ∈ N .
+3. If x > 1 , then 0 < Cn(x) ≤ Cn+1(x) ,
+∀n ∈ N .
+4. If x > 1 , then 2Cn(x) ≤ Cn−1(x) + Cn+1(x) ,
+∀n ∈ N .
+Proof. Part 1 is straightforward. Part 2 is also straightforward if we use the relation-
+ship (30) and definition (32).
+To prove part 3, as x > 1, we set x = cosh ω in (35), to obtain
+Cn(cosh ω) = 4(cosh ω − 1) sinh(nω)
+sinh ω
+> 0 .
+(36)
+As sinh ω is an increasing function, it holds sinh(nω) < sinh((n + 1)ω), and consequently
+Cn(x) ≤ Cn+1(x) .
+Finally, to prove 4, as
+sinh((n − 1)ω) + sinh((n + 1)ω) = 2 cosh ω sinh(nω) ≥ 2 sinh(nω) ,
+we can use again (36), for x > 1, to obtain that 2Cn(x) ≤ Cn−1(x) + Cn+1(x) ∀n ∈ N.
+6
+Proofs of theorems in Sections 3 and 5
+In this section we give the proofs of the main results in the paper.
+6.1
+Positivity of the Crank-Nicolson method
+Recall that the Crank Nicolson method is positive if and only if all the elements in matrix
+Am (24)-(25) are positive. The following lemma simplifies the proof of Theorem 1.
+Lemma 4. If x > 1, then all the elements in matrix Am(x) (24)-(25) are non-negative if
+and only if polynomial Pm(x) is non-negative.
+16
+
+Proof. Recall that polynomials Un(x) are positive for x > 1. Polynomials Cn(x) are also
+positive for x > 1 (see Corollary 2). Consequently, all extra-diagonal elements are positive.
+For the diagonal elements A(i,i)
+m (x), the positivity of polynomials Un(x) and Cn(x) for
+all n ∈ N implies that
+min
+i
+A(i,i)
+m (x) = A(1,1)
+m
+(x) = A(m,m)
+m
+(x) = Pm(x)
+Um(x).
+Then the analysis of the positivity of the elements in Am(x) is reduced to the positivity
+of polynomial Pm(x).
+Proof of Theorem 1
+1. From Lemma 4, we just have to study the positivity of polynomials Pm(x) for x > 1.
+From Proposition 2, Pm(x) has exactly m − 1 roots in the interval (−1, 1) and an
+additional isolated real root x(p)
+m = cosh(ω(p)
+m ) in the interval (6+
+√
+2
+4
+, 2], where ω(p)
+m is
+the unique root of equation (31), that is the same as (18). As x = 1 + 1/s, then the
+CFL coefficient is s = 1/(x − 1), and inequality (17) is obtained.
+2. In the limit case, Proposition 2 gives us the limit value x(p)
+∞ = (6 +
+√
+2)/4. Then
+for the CFL coefficient we get the bound s(p)
+∞ = 1/(x(p)
+∞ − 1) = 2(2 −
+√
+2), and
+inequality (19) is obtained.
+6.2
+Contractivity of the Crank-Nicolson method.
+Before writing the proof of Theorem 2, we need two technical lemmas.
+In Lemma 5
+below, we compute the maximum norm of matrix Am(x) in terms of polynomials Un(x),
+Pn(x) and Cn(x). Then, in Lemma 6 we get the inequalities needed for the contractivity
+condition ∥Am∥∞ ≤ 1. In contrast to positivity, in the analysis of the contractivity it is
+necessary to distinguish between the even and odd cases.
+Lemma 5.
+1. If m ≥ 3 is a natural odd number, then the maximum norm of Crank-Nicolson
+matrix is
+∥Am∥∞ =
+1
+Um
+�
+�
+�|Pm +
+m−1
+2
+�
+n=1
+C2n−1| + 2
+m−3
+2
+�
+i=0
+m−(2i+1)
+2
+�
+n=1
+C2n+i
+�
+�
+� .
+(37)
+2. If m ≥ 2 is a natural even number, then the norm of Crank-Nicolson matrix is
+∥Am∥∞ =
+1
+Um
+�
+�|Pm +
+m
+2 −1
+�
+n=1
+C2n| +
+m
+2 −1
+�
+i=1
+m−i
+�
+n=1+i
+Cn +
+m/2
+�
+n=1
+C2n−1
+�
+� .
+(38)
+Proof. Note that the norm in (37) is obtained from the sum of the elements in the central
+row of Am (odd case), while the norm in (38) is obtained from the sum of the elements
+17
+
+in any of the two symmetric central rows (even case). To get this result we will proof the
+following inequalities
+m
+�
+j=1
+|A(i,j)
+m | ≤
+m
+�
+j=1
+|A(i+1,j)
+m
+| ,
+i = 1, . . . , m−1
+2 (odd case) m−2
+2 (even case).
+For the sake of simplicity we will denote Ai
+m to the sum �m
+j=1|A(i,j)
+m |Um. Observe that the
+symmetry of matrices (24-25) makes Ai
+m = Am+1−i
+m
+, i = 1 , . . . , (m − 1)/2 (odd case) or
+i = 1 , . . . , m/2 (even case). Then, we will proof that
+A1
+m ≤ A2
+m ≤ · · · ≤ A
+m−1
+2
+m
+≤ A
+m+1
+2
+m
+(odd case)
+A1
+m ≤ A2
+m ≤ · · · ≤ A
+m−2
+2
+m
+≤ A
+m
+2m = A
+m+2
+2
+m
+(even case)
+Consequently, ∥Am∥∞ is obtained from the sum of the elements in the central row (odd
+case) or from the sum of the elements in any of the two central rows (even case).
+Recall that for a given matrix A with positive extra-diagonal elements the following
+equality trivially holds
+∥A∥∞ = max
+i
+�
+�
+m
+�
+j=1
+|aij|
+�
+� = max
+i
+�
+�|aii| +
+�
+j̸=i
+aij
+�
+� .
+(39)
+In our case Cn(x) ≥ 0 for x ≥ 1 (see Proposition 3). Consequently all extra-diagonal ele-
+ments of matrices (24) and (25) are non-negative and we can use (39) to compute ∥Am∥∞.
+For the first and the second row of matrix (24) or (25) we get the sums
+A1
+m = |Pm| +
+m−1
+�
+i=1
+Ci ,
+A2
+m = |Pm + Cm−2| +
+m−1
+�
+1
+Ci +
+m−1
+�
+2
+Ci − Cm−2 .
+Then, after cancelling terms, the difference A1
+m − A2
+m is
+A1
+m − A2
+m = |Pm| − |Pm + Cm−2| −
+m−1
+�
+2
+Ci + Cm−2 .
+Adding and subtracting Cm−2 in the term |Pm|, we can write
+A1
+m − A2
+m ≤ |Pm + Cm−2| + Cm−2 − |Pm + Cm−2| −
+m−1
+�
+2
+Ci + Cm−2
+= 2Cm−2 −
+m−1
+�
+2
+Ci ≤ Cm−3 + Cm−1 −
+m−1
+�
+2
+Ci = −
+m−4
+�
+2
+Ci − Cm−2 ≤ 0 ,
+where we have used 2Cm−2 ≤ Cm−3 + Cm−1 from item (4) in Lemma 3. Observe that
+property (3) in Lemma 3 allows us to finally write
+A1
+m − A2
+m ≤ −
+m−3
+�
+2
+Ci ≤ 0 .
+(40)
+18
+
+If we proceed in the same way for the difference A2
+m−A3
+m, after cancelling terms, and after
+adding and subtracting Cm−4 in the term |Pm + Cm−2|, we get the following inequality
+A2
+m − A3
+m ≤ 2Cm−4 −
+m−2
+�
+3
+Ci .
+Again, the use of properties (4) and (3), in this order, from Lemma 3 makes it possible to
+write an inequality analogous to (40)
+A2
+m − A3
+m ≤ Cm−5 + Cm−3 −
+m−2
+�
+3
+Ci ≤ −
+m−4
+�
+3
+Ci ≤ 0 .
+The proof follows in the same way for the odd and even case up to the last step when
+we achieve the central row (odd case) or the two central rows (even case). Then we have
+to consider two different cases:
+1. If m is an odd number, the last step consists in studying the difference A
+m−1
+2
+m
+−A
+m+1
+2
+n
+,
+where A
+m+1
+2
+n
+represents the sum of the elements in the central row. After cancelling
+terms, we can write
+A
+m−1
+2
+m
+− A
+m+1
+2
+m
+= |Pm +
+m−1
+2
+�
+n=2
+C2n−1| + C1 − |Pm +
+m−1
+2
+�
+n=1
+C2n−1| − C m+1
+2
+Adding and subtracting C1 in the term |Pm + � m−1
+2
+n=2 C2n−1|, we get
+A
+m−1
+2
+m
+− A
+m+1
+2
+m
+≤ |Pm +
+m−1
+2
+�
+n=1
+C2n−1| + 2C1 − |Pm +
+m−1
+2
+�
+n=1
+C2n−1| − C m+1
+2
+= 2C1 − C m+1
+2
+≤ C0 + C2 − C m+1
+2
+= C2 − C m+1
+2
+≤ 0 ,
+where we have used 2C1 ≤ C0 + C2 from property (4) in Lemma 3.
+Consequently, if m ≥ 3 is an odd number, the maximum value of Ai
+m is obtained in
+the central row A
+m+1
+2
+m
+and we can conclude
+∥Am∥∞ = A
+m+1
+2
+m
+Um
+=
+1
+Um
+�
+�
+�|Pm +
+m−1
+2
+�
+n=1
+C2n−1| + 2
+m−3
+2
+�
+i=0
+m−(2i+1)
+2
+�
+n=1
+C2n+i
+�
+�
+� .
+2. If m is an even number, then there is not a central row but two central symmet-
+ric rows A
+m
+2m and A
+m+2
+2
+m
+, and the maximum value is obtained at any of these two
+files.
+Now, in the last step of the proof, if m ≥ 4, we have to write the differ-
+ence A
+m−2
+2
+m
+− A
+m
+2m . Note that for the simple case m = 2, it holds A1
+m = A2
+m. After
+cancelling terms, we can write
+A
+m−2
+2
+m
+− A
+m
+2m = |Pm +
+m
+2 −1
+�
+n=2
+C2n| + C2 − |Pm +
+m
+2 −1
+�
+n=1
+C2n| − C m
+2 − C m+2
+2
+19
+
+Adding and subtracting C2 in the term |Pm + � m
+2 −1
+n=2 C2n|, we get
+A
+m−2
+2
+m
+− A
+m
+2m ≤ |Pm +
+m
+2 −1
+�
+n=1
+C2n| + 2C2 − |Pm +
+m
+2 −1
+�
+n=1
+C2n| − C m
+2 − C m+2
+2
+= 2C2 − C m
+2 − C m+2
+2
+≤ C1 + C3 − C m
+2 − C m+2
+2
+≤ 0 ,
+where we have used 2C2 ≤ C1 + C3 from property (4) in Lemma 3.
+Consequently, if m is an even number, the maximum value of Ai
+m is obtained in the
+row A
+m
+2m and we can conclude
+∥Am∥∞ = max
+i
+�
+�|aii| +
+�
+j̸=i
+aij
+�
+� =
+1
+Um
+�
+�|Pm +
+m
+2 −1
+�
+n=1
+C2n| +
+m
+2 −1
+�
+i=1
+m−i
+�
+n=1+i
+Cn +
+m/2
+�
+n=1
+C2n−1
+�
+�
+Once we have got the maximum norm of matrix Am in terms of polynomials Um, Pm
+and Cm, we can get bounds s(c)
+m for contractivity for any value of m if we are able to
+solve the corresponding inequality ∥Am∥∞ ≤ 1. This is done in the following lemma. In
+Figure 3 we have plot ∥Am(s)∥∞ for some values of m, and we have also added some
+contractivity bounds s(c)
+m .
+s(c)
+4
+s(c)
+5
+s(c)
+7
+∥A1(s)∥
+∥A2(s)∥
+∥A3(s)∥
+∥A4(s)∥
+∥A5(s)∥
+∥A7(s)∥
+∥A9(s)∥
+∥A21(s)∥
+Figure 3: ∥Am(s)∥ for different values of m ∈ {1, 2, 3, 4, 5, 7, 9, 21}. When m ∈ {1, 2, 3}
+it holds ∥Am(s)∥ < 1, for all s > 0. For m ≥ 4, ∥Am(s)∥ cuts the line s = 1 at s(c)
+m .
+The sequence (s(c)
+m ) is strictly monotonically decreasing with all the terms in the inter-
+val
+�3/2, 1 +
+√
+5
+�.
+Lemma 6.
+1. (Ood case) If m is a natural odd number, then for the maximum norm of Crank-
+Nicolson matrix we have
+∥Am∥∞ ≤ 1
+⇐⇒
+2 sinh (m−1)ω
+4
+sinh (m+1)ω
+4
+sinh (m+1)ω
+2
+≤ sinh ω
+2
+20
+
+1.1
+0.9
+0.75
+0.5
+0.25
+3/2
+2
+1+ V5where ω = arccosh(1 + 1/s). In the limit, when m → ∞, we get contractivity if and
+only if
+e−ω/2 ≤ sinh ω
+2
+(41)
+2. (Even case) If m is a natural even number, then we have
+∥Am∥∞ ≤ 1
+⇐⇒
+sinh mω
+2
+�
+sinh (m+2)ω
+2
+− sinh mω
+2
+�
+sinh (m+1)ω
+2
+sinh mω
+4 sinh (m−2)ω
+4
+≥ sinh ω
+sinh2 ω
+2
+In the limit, when m → ∞, we get contractivity if and only if
+2 (−1 + eω) ≥ sinh ω
+sinh2 ω
+2
+(42)
+If ω > 0, inequalities (41) and (42) are equivalent, and they are true iff ω ≥ log 3. In the
+variable s this is equivalent to the known restriction s ≤ 3/2.
+Proof.
+1. Remember that, in the odd case, the norm in (37) is obtained from the sum of the
+elements in the central row of Am. In that case, the diagonal element in the central
+row Pm + � m−1
+2
+n=1 C2n−1 can be written in closed form as
+Pm +
+m−1
+2
+�
+n=1
+C2n−1 = −Um +
+m+1
+2
+�
+n=1
+C2n−1 = −sinh((m + 1)ω)
+sinh ω
++ 4(x − 1) sinh2 (m+1)ω
+2
+sinh2 ω
+,
+where we have changed Pm = Cm − Um and we have considered the angle variable
+ω = arccosh x. For any value of m, the unique positive root ωm of this diagonal
+element lies in the interval (log 3, log(2 +
+√
+3)]. This root can easily be obtained
+from the simplified equation in the variable s = (cosh ω − 1)−1
+√
+1 + 2s = 2 tanh (1 + m) arccosh(1 + 1/s)
+2
+.
+In this variable, the unique positive root sm lies in the interval [1, 3/2). Observe that
+the sequence of roots (sm) increasingly converges to the limit value s∞ = 3/2. When
+s ∈ (0, sm) the diagonal element Pm + � m−1
+2
+n=1 C2n−1 is positive, and, from (37), we
+easily obtain ∥Am∥∞ < 1.
+On the other hand, when s ∈ [sm, ∞), we have Pm + � m−1
+2
+n=1C2n−1 ≤ 0 , and, from (37),
+the inequality for contractivity is
+∥Am∥∞ =
+1
+Um
+�
+�
+�−Pm −
+m−1
+2
+�
+n=1
+C2n−1 + 2
+m−3
+2
+�
+i=0
+m−(2i+1)
+2
+�
+n=1
+C2n+i
+�
+�
+� ≤ 1 .
+This is equivalent to
+−
+m−1
+2
+�
+n=1
+C2n−1 + 2
+m−3
+2
+�
+i=0
+m−(2i+1)
+2
+�
+n=1
+C2n+i ≤ Cm ,
+21
+
+or
+m+1
+2
+�
+n=1
+C2n−1 − 2
+m−3
+2
+�
+i=0
+m−(2i+1)
+2
+�
+n=1
+C2n+i ≥ 0 .
+As x > 1, in the angle variable ω = arccosh x > 0, we can write Cm(cosh ω) =
+4(x − 1) sinh(mω)/ sinh ω. Consequently, the previous inequality is reduced to
+m+1
+2
+�
+n=1
+sinh((2n − 1)ω) − 2
+m−3
+2
+�
+i=0
+m−(2i+1)
+2
+�
+n=1
+sinh((2n + i)ω) ≥ 0 .
+If we use the closed formulas for the expansions �
+k sinh(kω), �
+k sinh(2kω), and
+�
+k sinh((2k − 1)ω), then we can write the previous inequality as
+4(x − 1) sinh (m+1)ω
+2
+sinh2 ω
+�
+sinh (m + 1)ω
+2
+− 2sinh (m−1)ω
+4
+sinh (m+1)ω
+4
+sinh ω
+2
+�
+≥ 0 .
+And, for x > 1, this is true if and only if
+2 sinh (m−1)ω
+4
+sinh (m+1)ω
+4
+sinh (m+1)ω
+2
+≤ sinh ω
+2 .
+(43)
+In this way, for any value of m, we get contractivity if and only if ω ≥ ω(c)
+m , where
+ω(c)
+m is the unique positive root of the corresponding equality equation. Finally, going
+back to variable s, we get contractivity if and only if s ≤ s(c)
+m := 1/(cosh ω(c)
+m − 1).
+Computing the limit in (43), when m → ∞, we get contractivity if and only if
+e−ω/2 ≤ sinh ω
+2 .
+And this is true if and only if ω ≥ log 3. This is x = cosh ω ≥ 5/3 or s ≤ 3/2.
+2. In the even case, the norm in (38) is obtained from the sum of the elements in any
+of the two central symmetric rows. The diagonal element Pm + � m
+2 −1
+n=1 C2n in any of
+this central rows can be written in closed form as
+Pm+
+m
+2 −1
+�
+n=1
+C2n = −Um +
+m
+2
+�
+n=1
+C2n = −sinh((m + 1)ω)
+sinh ω
++ 4(x − 1) sinh mω
+2 sinh (m+2)ω
+2
+sinh2 ω
+,
+where ω = arccosh x and x = 1 + 1/s. Again, the unique positive root sm of this
+central diagonal element lies in the interval [1, 3/2). For any value of m, this root
+can easily be obtained from the simplified equation
+√
+1 + 2s = 4 sinh mω
+2 sinh (m+2)ω
+2
+sinh((m + 1)ω)
+.
+Observe that this sequence of roots (sm)m increasingly converges to the limit value
+s∞ = 3/2. When s ∈ (0, sm) the diagonal element Pm + � m
+2 −1
+n=1 C2n is positive, and,
+22
+
+from (38), we easily obtain ∥Am∥∞ < 1. On the other hand, when s ∈ [sm, ∞), the
+diagonal element is negative, and, from (38), the inequality for contractivity is
+∥Am∥∞ =
+1
+Um
+�
+�−Pm −
+m
+2 −1
+�
+n=1
+C2n +
+m
+2 −1
+�
+i=1
+m−i
+�
+n=1+i
+Cn +
+m/2
+�
+n=1
+C2n−1
+�
+� ≤ 1 .
+This is equivalent to
+−
+m
+2 −1
+�
+n=1
+C2n +
+m
+2 −1
+�
+i=1
+m−i
+�
+n=1+i
+Cn +
+m/2
+�
+n=1
+C2n−1 ≤ Cm ,
+or
+m
+2
+�
+n=1
+C2n −
+m
+2 −1
+�
+i=1
+m−i
+�
+n=1+i
+Cn −
+m/2
+�
+n=1
+C2n−1 ≥ 0 .
+As in the odd case, now we can use the angle variable ω to reduce the previous
+inequality.
+m
+2
+�
+n=1
+sinh(2nω) −
+m
+2 −1
+�
+i=1
+m−i
+�
+n=1+i
+sinh(nω) −
+m/2
+�
+n=1
+sinh((2n − 1)ω) ≥ 0 .
+Finally, the closed formulas for the expansions �
+k sinh(kω), �
+k sinh(2kω), and
+�
+k sinh((2k − 1)ω), allow us to reduce the inequality to
+4(x−1)
+sinh ω
+�
+sinh mω
+2
+sinh ω
+�
+sinh (m+2)ω
+2
+− sinh mω
+2
+�
+− sinh (m+1)ω
+2
+sinh mω
+4 sinh (m−2)ω
+4
+sinh2 ω
+2
+�
+≥ 0 .
+And, for x > 1, this is true if and only if
+sinh mω
+2
+�
+sinh (m+2)ω
+2
+− sinh mω
+2
+�
+sinh (m+1)ω
+2
+sinh mω
+4 sinh (m−2)ω
+4
+≥ sinh ω
+sinh2 ω
+2
+.
+(44)
+In this way, for any value of m, we get contractivity if and only if ω ≥ ω(c)
+m , where ω(c)
+m
+is the unique positive root of the corresponding equality equation in (44). Finally,
+going back to variable s, we get contractivity if and only if s ≤ 1/(cosh ω(c)
+m − 1).
+Computing the limit in (44), when m → ∞, we get contractivity if and only if
+2 (−1 + eω) ≥ sinh ω
+sinh2 ω
+2
+.
+And this is true if and only if ω ≥ log 3. This is x = cosh ω ≥ 5/3 or s ≤ 3/2.
+Proof of Theorem 2
+1. For m ∈ {1, 2, 3} the proof is straightforward.
+23
+
+2. For m ∈ N, m > 3 , the proof is straightforward from the previous lemma. The
+inequality for contractivity in the variable s = 1/(cosh ω − 1) is
+s ≤ s(c)
+m :=
+1
+cosh ω(c)
+m − 1
+where ω(c)
+m is the unique positive root of the equation from (43), if m is odd, or from
+(44) if m is even .
+3. It is also straightforward from the previous lemma. In the limit, when m → ∞, we
+get inequality (41) from (43), and inequality (42) from (44). These two inequalities
+(41) and (42) are equivalent if ω > 0, and they are true if and only if ω ≥ log 3. In
+the variable s this is equivalent to the known restriction s ≤ 3/2.
+6.3
+Proof of the remaining results
+Proof of Proposition 2
+We divide the proof into two parts, the case x ∈ (−1, 1) and the case x > 1.
+1. If x ∈ (−1, 1), then the angular variable ω = arccos x, ω ∈ (0, π), and equality (30),
+allow us to write the polynomial Pm in closed form as
+Pm(cos ω) = 2 sin((m − 1)ω) − 4 sin(mω) + sin((m + 1)ω)
+sin ω
+(45)
+If we convert all the angles in the numerator to the angle mω, we can write
+= (3 cos ω − 4) sin(mω)
+sin ω
+− cos(mω) ,
+(46)
+Then, from (46), we get that the roots of Pm(cos ω) in (0, π) are the roots of the
+equation
+tan(mω) =
+sin ω
+3 cos ω − 4 ,
+ω ∈ (0, π) .
+(47)
+The function on the right hand side, f(ω) := sin ω/(3 cos ω − 4), is continuous and
+bounded in the interval [0, π]. It is decreasing in the interval (0, 2 arctan(1/
+√
+7)) and
+increasing in the interval (2 arctan(1/
+√
+7), π). Its maximum value f(0) = f(π) = 0
+is obtained in the boundary, while the minimum value is f(2 arctan(1/
+√
+7)) =
+−1/
+√
+7 (see Figure 4). Thus, we can affirm that tan(mω) meets m−1 times the func-
+tion f(ω), and consequently there are m−1 roots 0 < ωm
+m−1 < ωm
+m−2 < · · · < ωm
+1 < π ,
+in the interval (0, π). Now, going back to the variable x = cos ω, we can affirm that
+the polynomial Pm(x) has m − 1 roots xm
+i = cos ωm
+i
+in the interval (−1, 1). As the
+cosine function is decreasing in the interval (0, π), we can write the m − 1 roots of
+Pm(x) as
+−1 < xm
+1 < xm
+2 < · · · < xm
+m−1 < 1 .
+24
+
+Figure 4: Roots 0 < ωm
+m−1 < ωm
+m−2 < · · · < ωm
+1 < π , of equation (47)
+2. If x > 1, we consider the variable ω = arccosh x, ω ∈ (0, ∞). Again, with the help
+of equality (30), we can write the polynomial Pm in closed form as
+Pm(cosh(ω)) = 2 sinh((m − 1)ω) − 4 sinh(mω) + sinh((m + 1)ω)
+sinh ω
+(48)
+= (3 cosh ω − 4) sinh(mω)
+sinh ω
+− cosh(mω) ,
+(49)
+where, as in the previous case, we have rewritten the numerator in the terms of the
+angle mω. Then, from (49), the roots of Pm(cosh(ω)) in (0, ∞) are the roots of the
+equation
+coth(mω) = 3 cosh ω − 4
+sinh ω
+,
+ω ∈ (0, ∞) .
+(50)
+The function on the right, g(ω) := (3 cosh ω−4)/ sinh ω, is continuous and increasing
+in the interval (0, ∞) to the limit value of 3. On the left, for any value of m, the
+function coth(mω) is continuous and decreasing to the limit value of 1 (see Figure 5).
+Thus, we can affirm that, for any value of m, equation (50) has a unique root ωm in
+the interval (0, ∞) . Now, going back to the variable x = cosh ω, we can affirm that
+the polynomial Pm(x) has a unique root xm = cosh ωm in the interval (1, ∞).
+Figure 5: Left hand side and right hand side of equation (50)
+In the limit, when m tends to infinity, for any value of ω > 0, we have coth(mω)↘1.
+Consequently, from (50), the sequence of roots (ωm) decreasingly converges to the
+limit value ω∞ = log(2 +
+√
+2), this is the positive solution of the limit equation
+25
+
+2
+1
+-
+一
+-
+-
+1
+-
+一
+1
+1
+1
+1
+/
+/
+/
+/
+/
+/
+/
+/
+/
+1
+/
+-
+-
+-
+/
+/
+1
+/
+1
+/
+-
+tan(m w)
+/
+/
+/
+1.0
+1.5
+3.0 /
+4.5
+(m)}
+-/ -
+/
+/
+1
+/
+-
+1
+/
+-
+-
+/
+1
+1
+/
+/
+/
+-1
+/
+/
+/
+1
+1
+-
+-
+1
+1
+-
+1
+1
+1
+-2
+1
+1
+11.6
+1.4
+tan(5 w)
+1.2
+tan(3 w)
+tan(2 w)
+1.0
+(m)6
+0.8
+0.0
+0.5
+1.0
+1.53 cosh ω − 4 = sinh ω. Now, in the variable x = cosh ω, we can affirm that each
+polynomial Pm(x) has a positive root xm = cosh ωm in the interval (1, ∞).
+As
+the cosh function is increasing in the interval (0, ∞), the sequence of roots (xm)
+decreasingly converges to the limit value x∞ = (6 +
+√
+2)/4.
+x∞ = cosh ω∞ = cosh log(2 +
+√
+2) = (6 +
+√
+2)/4 ≈ 1.85355 .
+References
+[1] Desoer, C., and Haneda, H.
+The measure of a matrix as a tool to analyze
+computer algorithms for circuit analysis. IEEE Transactions on Circuit Theory 19,
+5 (1972), 480–486.
+[2] Faragó, I., Korotov, S., and Szabó, T.
+Non-negativity preservation of the
+discrete nonstationary heat equation in 1d and 2d. Aplimat-Journal of applied math-
+ematics 3, 2 (2010), 61.
+[3] Faragó, I., and Palencia, C. Sharpening the estimate of the stability constant
+in the maximum-norm of the crank–nicolson scheme for the one-dimensional heat
+equation. Applied numerical mathematics 42, 1-3 (2002), 133–140.
+[4] Ferracina, L., and Spijker, M. N. Stepsize restrictions for the total-variation-
+diminishing property in general runge–kutta methods. SIAM Journal on Numerical
+Analysis 42, 3 (2004), 1073–1093.
+[5] Gottlieb, S., and Gottlieb, L.-A. J.
+Strong stability preserving properties
+of runge–kutta time discretization methods for linear constant coefficient operators.
+Journal of Scientific Computing 18, 1 (2003), 83–109.
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+Runge-Kutta and multistep time discretizations. World Scientific, 2011.
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+try of polynomial sequences ii: Type (1, 0). Journal of Mathematical Analysis and
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+[8] Higueras, I. On strong stability preserving time discretization methods. Journal of
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+[10] Higueras, I., and Roldán, T. On the positivity and contractivity of crank-nicolson
+method applied to the periodic diffusion problem. ArXiv (2022).
+[11] Horváth, R. Maximum norm contractivity in the numerical solution of the one-
+dimensional heat equation. Applied numerical mathematics 31, 4 (1999), 451–462.
+[12] Horváth, Z. On the positivity step size threshold of runge–kutta methods. Applied
+Numerical Mathematics 53, 2-4 (2005), 341–356.
+26
+
+[13] Hundsdorfer, W., and Verwer, J.
+Numerical Solution of Time-Dependent
+Advection-Diffusion-Reaction Equations. Springer, 2003.
+[14] Kraaijevanger, J. Contractivity of Runge-Kutta methods. BIT Numerical Math-
+ematics 31, 3 (1991), 482–528.
+[15] Kraaijevanger, J. F. B. M. Contractivity of runge-kutta methods. BIT Numerical
+Mathematics 31, 3 (1991), 482–528.
+[16] Kraaijevanger, J. F. B. M.
+Maximum norm contractivity of discretization
+schemes for the heat equation. Applied numerical mathematics 9, 6 (1992), 475–492.
+[17] Liu, L. L., and Wang, Y. A unified approach to polynomial sequences with only
+real zeros. Advances in Applied Mathematics 38, 4 (2007), 542–560.
+[18] Nüßlein, S., Ranocha, H., and Ketcheson, D. I. Positivity-preserving adaptive
+runge–kutta methods. Communications in Applied Mathematics and Computational
+Science 16, 2 (2021), 155–179.
+[19] Qi, F., Cernanová, V., and Semenov, Y. S.
+Some tridiagonal determinants
+related to central delannoy numbers, the chebyshev polynomials, and the fibonacci
+polynomials. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys 81, 1
+(2019), 123–136.
+[20] Rózsa, P. Linear Algebra and its Applications. Muszaki Konyvkiado, (in Hungarian),
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+[21] Shu, C.-W. Total-variation-diminishing time discretizations. SIAM Journal on Sci-
+entific and Statistical Computing 9, 6 (1988), 1073–1084.
+[22] Söderlind, G. The logarithmic norm. history and modern theory. BIT Numerical
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+377–392.
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+On logarithmic norms.
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+(1975), 741–753.
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+and reliable fuel cell modelling. PhD thesis, Eötvös Loránd University, Budapest,
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+27
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+28
+
diff --git a/PtAzT4oBgHgl3EQfIvs5/content/tmp_files/load_file.txt b/PtAzT4oBgHgl3EQfIvs5/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a991289776dd537769cd56d0a8d0e092a184d08d
--- /dev/null
+++ b/PtAzT4oBgHgl3EQfIvs5/content/tmp_files/load_file.txt
@@ -0,0 +1,917 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf,len=916
+page_content='A new insight on positivity and contractivity of the  Crank-Nicolson scheme for the heat equation∗  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Higueras1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Roldán1  1Institute for Advanced Materials and Mathematics  Public University of Navarre  E-mails: higueras@unavarra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='es, teo@unavarra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Abstract  In this paper we study numerical positivity and contractivity in the infinite norm of  Crank-Nicolson method when it is applied to the diffusion equation with homogeneous  Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For this purpose, the amplification matrices are written  in terms of three kinds of Chebyshev-like polynomials, and necessary and sufficient  bounds to preserve the desired qualitative properties are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For each spatial  mesh, we provide the equations that must be solved as well as the intervals that  contain these bounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' consequently, they can be easily obtained by a bisection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Besides, differences between numerical positivity and contractivity are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  This problem has also been addressed by some other authors in the literature and  some known results are recovered in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Our approach gives a new insight on  the problem that completes the panorama and that can be used to study qualitative  properties for other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Keywords: Positivity, Maximum norm contractivity, Monotonicity, Crank-Nicolson,  Heat equation  1  Introduction  We consider the numerical solution of the one-dimensional heat equation  ∂u(t, x)  ∂t  = d ∂2u(t, x)  ∂x2  ,  x ∈ [0, 1], t ≥ 0 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1)  u(0, t) = u(1, t) = 0 ,  t ≥ 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2)  u(x, 0) = u0(x) ,  x ∈ [0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3)  where u0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3) is a given function on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For the sake of simplicity, we consider the  diffusion term d = 1 in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Solutions u(x, t) for the linear parabolic  ∗Research project PID2019-109045GB-C31 funded by Agencia Estatal de Investigación, Ministerio de  Ciencia e Innovación, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='01066v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='NA]  3 Jan 2023  problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3) have several qualitative properties that are relevant in the context of  the physical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In particular, the problem is positivity preserving, that is, for t ≥ 0,  u0(x) ≥ 0  ⇒  u(x, t) ≥ 0 ,  (2)  and the solutions are monotonically decreasing in time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', for t2 ≥ t1 ≥ 0,  max  0≤x≤1 |u(t2, x)| ≤ max  0≤x≤1 |u(t1, x)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (3)  In order to obtain numerical approximations with physical sense, properties (2)-(3) should  be preserved in the discretization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In this paper, we consider the Crank-Nicolson  (CN) method, a method of lines approach where second order central finite differences  in space are followed by a second order time-stepping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Spatial discretization of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3) with second order central finite differences and mesh width h = 1/(m + 1),  gives a semi-discrete linear differential system of the form  w′(t) = Bhw(t) ,  w(0) = w0 , t ≥ 0 ,  (4)  where Bh is a matrix of dimension m, that is positivity preserving with monotonically  decreasing (in the maximum norm) solutions (see section 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Next, a time  stepping method is used to obtain numerical approximations wn ≈ w(tn), where tn = nτ,  and τ is the constant time stepsize used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In this paper, we consider approximations of the  form  wn = Am wn−1 , n ≥ 0 ,  (5)  where w0 is a known value, and Am is a matrix of dimension m that depends on the time  stepping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In particular, for Runge-Kutta methods, Am = φ(τBh), where φ is the  stability function of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  There is a vast list of references in the analysis of positivity and monotonicity decreasing  time stepping schemes (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', [21, 26, 14, 16, 11, 4, 8, 12, 2, 6, 18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' see too [26, 6, 13]  and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Depending on the context, monotonicity decreasing methods  are also known as contractive, SSP (Strong Stability Preserving) or TVD (Total Variation  Diminishing) schemes (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', [23, 21, 14, 8, 4, 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In this setting, stepsize restrictions  of the form  τ ≤ C τF E ,  (6)  are obtained, where C denotes the monotonicity threshold factor (also known as SSP-  coefficient, radius of absolute monotonicity, contractivity radius, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' ) of the time stepping  method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=',[15, 27, 5, 23, 4]), and τF E is the stepsize restriction for the given qualita-  tive property when forward Euler scheme is used to solve the specific ODE problem (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 379], [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 201], [6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 52-53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In particular, for problem (4), τF E = h2/2  for both positivity and contractivity (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', [26, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 22]), and C = 1 for forward Euler  method (see section 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Observe that with this approach, the stepsize restric-  tion (6) is the same for all linear systems with the same τF E and for some problems this  is not a sharp bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  A well known time stepping method is the θ-method, defined as  wn = (I − θτBh)−1(I + (1 − θ)τBh)wn−1 ,  θ ∈ [0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (7)  2  Observe that this method can be understood as the composition of a (1 − θ)τ-step with  forward Euler method and a θτ-step with backward Euler scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In particular, for  θ = 1/2, CN scheme is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The radius of absolute monotonicity for the θ-method  applied to linear problems is Cθ = 1/(1 − θ) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently, numerical positivity and  contractivity can be ensured under the restriction (see [26, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 135])  τ  h2 ≤  1  2(1 − θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  However, If we look closer at the iteration matrix Am in (7), a sharper bound is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Different authors have studied numerical preservation of positivity and monotonicity for  problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3) with the θ-method (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', [16, 11, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 72], the authors  obtain that the numerical solution is positive if and only if  τ  h2 ≤ 1 −  √  1 − θ  θ(1 − θ)  ,  (8)  whereas in [16, Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1] and [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 456] it is shown that the numerical solution is  contractive if and only if  τ  h2 ≤  2 − θ  4(1 − θ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (9)  On the following we will denote s = τ/h2 to the CFL coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The contractivity result  in [16] is obtained for the pure initial value problem  ∂u(t, x)  ∂t  = d ∂2u(t, x)  ∂x2  ,  x ∈ R, t ≥ 0 ,  u(x, 0) = u0(x) ,  x ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In particular, bound (9) for contractivity is valid for all m ≥ 1 [16, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1] [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The bound τ/h2 ≤ 2/3 has also been obtained in [3, Theorem 1] in the analysis of the  stability of CN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In [2] and [11] results are based on the shape of the inverse  matrix (I−θτBh)−1 of dimension m, whose entries can be expressed in terms of hyperbolic  functions [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' stepsize restrictions are obtained for each value of m, and bound (9) is valid  for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Contributions of the paper  The approach followed in this paper consists on the representation of the iteration matrix  Am in (5) for the CN method in terms of three classes of polynomials, Pn(x), Cn(x) and  Un(x), defined by iterations (26), (28) and (33), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' in particular, Un(x) are the  Chebyshev polynomials of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the three cases the iteration process is the  same, but different initial values are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This formalism gives a new insight on  the problem that allows us to improve some results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The contributions of this paper are the following ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' With regard to positivity, for  any number of grid points m, we have obtained that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Crank-Nicolson method is positive if and only if s = τ/h2 ∈ (0, s(p)  m ], with s(p)  m =  1/(cosh ω(p)  m − 1), where ω(p)  m is the unique positive root of equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This root  3  lies in the narrow interval (log(2 +  √  2), log(2 +  √  3)] ≈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='22795, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='31696].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Thus ω(p)  m  can be easily computed by solving (18) with bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Proposition 2 shows  the connection between the bounds s(p)  m and polynomials Pn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Some values of s(p)  m  are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The sequence of bounds (s(p)  m ) is strictly monotonically increasing with all the terms  in the interval [1, 2(2 −  √  2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As a consequence, the CN method does not preserve  positivity when the spatial mesh is refined (keeping s constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the limit case, when m tends to infinite, we recover the known bound, s ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='17  for positivity ([13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 126],[2, Table 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  With regard to contractivity, we have computed the value ∥Am(s)∥∞ for any number  of grid points m (see Figure 3), and we have obtained that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Crank-Nicolson method is contractive if and only if s = τ/h2 ∈ (0, s(c)  m ], with  s(c)  m = ∞ for m = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ≥ 4, s(c)  m = 1/(cosh ω(c)  m − 1), where ω(c)  m is the  unique positive root of equation (21) or (22), depending on the parity of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This  root lies in the interval  � log  �(3+  √  5+  √  −2 + 6  √  5)/4  �, log 3  � ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='767197, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='09861].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Thus ω(c)  m can be easily computed by solving (21) or (22) with the bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Some values of s(p)  m  are shown in Tables 2 and 3 for odd and even values of m,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Some of these values can also be seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' ∥Am(s)∥∞ < 1 for s ∈ (0, s(c)  m ) and ∥Am(s(c)  m )∥∞ = 1 (see Figure 3), that resembles  property (14) of the linear system (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ≥ 4, the sequence (s(c)  m ) is strictly monotonically decreasing with all the terms  in the interval  �3/2, 1 +  √  5  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As a consequence, CN method preserves contractivity  when the spatial mesh is refined (keeping s constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the limit case, when m tends to infinite, we recover the known bound, τ/h2 ≤ 3/2  for contractivity [16, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1(Q3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1], [11, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' (14)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The results in this paper complete and improve some results in the literature [2, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  From equations (18) and (21)-(22), and the associated intervals, the computation of s(p)  m  and s(c)  m for any value of m is straightforward with the bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Besides, we  obtain equations to compute bounds s(c)  m both for odd and even values of m, whereas  in [11], bounds s(c)  m are only given for even m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' From our approach we also get the correct  value for s(c)  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Figure 1 illustrates the differences between positivity and contractivity of  CN method for the heat problem (1): if the scheme is positive for a given grid mesh m,  then it is also contractive for any grid mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Scope of the paper  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In Section 2, we explain the CN discretization  process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' notation and definitions are also given in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In Section 3 we show the  main results of the paper, namely: Theorems 1 and 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Table 1, containing upper bounds  4  s(p)  m for positivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Tables 2 and 3 containing upper bounds s(c)  m for contractivity (odd and  even case);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' and Figure 1 showing sequences (s(p)  m ) and (s(c)  m ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' An illustrative example is  also given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Section 4 contains some conclusions and ideas for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The proof of main results are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Previously, some technical material,  needed for the proofs in Section 6, is included in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2  Crank-Nicolson method for the heat equation  In this paper we consider the Crank-Nicolson method, a method of lines approach where  second order central finite differences in space are followed by a second order time-stepping  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Spatial discretization of heat equation (1) with second order central finite differ-  ences and mesh width h = 1/(m + 1), gives the semi-discrete linear differential system  w′(t) = Bhw(t) ,  w(0) = w0 , t ≥ 0 ,  (10)  where Bh = (d/h2) tridiag(1, −2, 1) is a matrix of dimension m, w(t) ≈ (u(xi, t))m  i=1,  w0 = (u0(xi))m  i=1, and xi = ih, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , m, are the grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  As the diffusion problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3) is positivity preserving (2) and monotonically de-  creasing (3), in order to obtain numerical approximations with these qualitative properties,  problem (10) should also be positivity preserving and contractive in the maximum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  An initial value problem  w′(t) = f(t, w(t)),  w(t0) = w0 , t ≥ 0 ,  (11)  is called positivity preserving (positive for short) if w0 ≥ 0 implies that w(t) ≥ 0 for t ≥ 0,  where the inequalities should be understood component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Problem (11) is said to be  contractive in the maximum norm if its solution w(t) satisfy  ∥w(t2)∥∞ ≤ ∥w(t1)∥∞  for t2 ≥ t1 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  It is well known that a linear problem,  w′(t) = Aw(t) ,  w(t0) = w0 , t ≥ 0 ,  (12)  where A = (aij) is an m × m matrix, is positive if and only if aij ≥ 0 for all i ̸= j [13,  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Matrix Bh in (10) satisfies this condition and thus problem (10) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe that other spatial discretizations do not preserve positivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' indeed, there is an  order barrier (q ≤ 2) from the requirement of positivity [13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Contractivity of solutions of the linear problem (12) can be proven by using the concept  of logarithmic norm of matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This concept is an extremely useful tool to analyze the  growth of solutions to ordinary differential equations because it can take negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The solutions of problem (12) are of the form w(t) = eAtw(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If we consider a vector  norm and its subordinate matrix norm, both denoted by ∥ · ∥, then  ∥w(t)∥ = ∥etAw(0)∥ ≤ ∥etA∥ ∥w(0)∥ ,  (13)  and contractivity is obtained if and only if ∥etA∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Given the set  M =  �  δ ∈ R | ∥etA∥ ≤ etδ , t ≥ 0  �  ,  5  it can be proven that µ∥·∥[A] = min(M), where µ∥·∥[A] stands for the logarithmic norm  of matrix A in the norm ∥ · ∥ [22, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1] (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', [22, 24, 1] and the references  therein for the definition and properties of logarithmic norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  From (13) and the definition of M, we get the inequalities  ∥w(t)∥ ≤ etµ∥·∥[A] ∥w(0)∥ ,  t ≥ 0 ,  ∥etA∥ ≤ et µ∥·∥[A]  t ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Thus, if µ∥·∥A] ≤ 0, the zero solution is stable and ∥etA∥ ≤ 1 for t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' if µ∥·∥[A] < 0, then  the zero solution is exponentially stable and ∥etA∥ < 1 for t > 0 [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 634], [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  For the maximum norm, the logarithmic norm of a matrix A = (aij) is given by  µ∞[A] = max  1≤i≤n  �  �aii +  n  �  j=1  |aij|  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In particular, for matrix Bh in (10), as  aii +  n  �  j=1  |aij| =  �  −1 ,  i = 1, m ,  0 ,  i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , m − 1 ,  we get µ∞[Bh] = 0, and thus  ∥etBh∥∞ ≤ 1,  t ≥ 0 ,  (14)  that ensures that problem (10) is contractive in the maximum norm, that is,  ∥w(t)∥∞ ≤ ∥w(0)∥∞ ,  t ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The time stepping process in Crank-Nicolson method with constant time step τ, gives  the iteration  wn = φ(τBh)wn−1 ,  n ≥ 1 ,  where  φ(z) = 1 + 1  2z  1 − 1  2z  (15)  is the stability function of the time integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' On the following, we denote Am = φ(τBh)  to the Crank-Nicolson iteration matrix of dimension m, that is,  Am =  �Im − τ  2Bh  �−1�Im + τ  2Bh  � =  �  �  �  �  �  �  �  �  �  1 + s − s  2  − s  2  1 + s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  − s  2  − s  2 1 + s  �  �  �  �  �  �  �  �  �  −1�  �  �  �  �  �  �  �  �  1 − s  s  2  s  2  1 − s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  s  2  s  2 1 − s  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' (16)  Observe that the two matrices in (16), corresponding to half step with forward Euler  and half step with backward Euler, commute because of the the linearity of the system  (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Besides, positivity and contractivity can be studied by analyzing these properties  for forward and backward Euler separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  6  Although there are no restrictions for positivity and contractivity with backward Euler  applied to system (10), the restriction for positivity and contractivity with forward Euler  is s ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This stepsize restriction for positivity is not sharp for problem (10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' numerical  experiments in [13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='126] show that numerical positivity can be obtained for s ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  As it has been pointed out above, a closer look at the iteration matrix Am in (7) or (16),  gives sharper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  3  Main results  In this section we show the main results of the paper concerning stepsize restrictions for  positivity and contractivity in the maximum norm for the m-dimensional system (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The proofs require some preliminary material about the structure of matrix Am and are  given in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  On the following theorems, the positivity of the matrix Am means that all the entries  of the matrix are non-negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' similarly, the contractivity in the maximum norm of the  matrix Am means ∥Am∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' (Positivity of Crank Nicolson method)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ∈ N, the matrix Am(s) in (16) is positive if and only if  s ≤ s(p)  m :=  1  cosh ω(p)  m − 1  ,  (17)  where ω(p)  m ∈  � log(2 +  √  2), log(2 +  √  3)  � is the unique positive root of equation  coth(mω) sinh ω = 3 cosh ω − 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (18)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The sequence (s(p)  m ) is strictly monotonically increasing with all the terms in the  narrow interval  �1, 2(2 −  √  2)  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As a consequence, Crank Nicolson method preserves  positivity when the spatial mesh is refined (keeping s constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The sequence (s(p)  m ) increasingly converges to the limit value s(p)  ∞ := 2(2 −  √  2) (see  Table 1 and Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This value was also obtained in [2] with other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Consequently, if  s < s(p)  ∞ = 2(2 −  √  2) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='17 ,  (19)  then there exists a natural number m0 such that the matrix Am is positive for any  value of m ≥ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As ω(p)  m ∈  � log(2 +  √  2), log(2 +  √  3)  � ≈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='22795, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='31696], an approximated value  can be easily computed by the bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In Table 1 below we show the roots ω(p)  m of equation (18) and the CFL restrictions s(p)  m for  positivity in (17) for different values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  7  m  ω(p)  m  x(p)  m = cosh ω(p)  m  s(p)  m = 1/(x(p)  m − 1)  1  log(2 +  √  3)  2  1  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='23590  1 +  √  3/2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='86603  2/  √  3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='15470  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='22864  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='85464  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='17009  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='22801  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='85365  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='17144  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='22795  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='85355  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='17157  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  ∞  log(2 +  √  2)  (6 +  √  2)/4  2(2 −  √  2) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='171572875  Table 1: Roots ω(p)  m of (18) and CFL restrictions s(p)  m in (17) for positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Next, we give the results for contractivity in the infinite norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe that the  symmetry of matrix Am makes ∥Am∥∞ = ∥Am∥1, and the result is also valid for the  1-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' (Contractivity of Crank Nicolson method)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ∈ {1, 2, 3} the matrix Am(s) in (16) is contractive in the maximum norm for  any value of s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ∈ N, m ≥ 4 , the matrix Am(s) in (16) is contractive in the maximum norm  if and only if  s ≤ s(c)  m :=  1  cosh ω(c)  m − 1  ,  (20)  where ω(c)  m  ∈  � log  �(3 +  √  5 +  √  −2 + 6  √  5)/4  �, log 3  � ≈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='767197, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='09861) is the  unique positive root of equation  2 sinh (m − 1)ω  4  sinh (m + 1)ω  4  = sinh ω  2 sinh (m + 1)ω  2  ,  (21)  if m is odd, or equation  sinh2 �ω  2  �  sinh mω  2  �  sinh (m + 2)ω  2  − sinh mω  2  �  = sinh ω sinh (m + 1)ω  2  sinh mω  4  sinh (m − 2)ω  4  ,  (22)  if m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The sequence (s(c)  m ) is strictly monotonically decreasing with all the terms in the  interval  �3/2, 1 +  √  5  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As a consequence, Crank Nicolson method does not preserve  contractivity when the spatial mesh is refined (keeping s constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The sequence (s(c)  m ) decreasingly converges to the limit value s(c)  ∞ := 3/2 (see  Figures 1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently, the matrix Am(s) is contractive for all m if and only if  s ∈ (0, 3/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For infinite matrices the bound s(c)  ∞ := 3/2 has been obtained with different  techniques in [3, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  8  In Tables 2 and 3 we give the CFL restrictions s(c)  m in (20) for contractivity in the  infinite norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The values in Table 2 (odd case) and Table 3 (even case) have been ob-  tained from equations (21) and (22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe that both, the roots ω(c)  m  of equation (21) (odd case) and the roots ω(c)  m  of equation (22) (even case), increas-  ingly converge to the limit value log 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently these roots are in the narrow in-  terval  � log  �(3 +  √  5 +  √  −2 + 6  √  5)/4  �, log 3  � ≈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='767197, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='09861) and can be easily ob-  tained with bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The numeric values shown in tables 2 and 3 were obtained  after 10 iterations with bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  s(c)  m  s(p)  m  Figure 1: Sequences s(p)  m and s(c)  m with the restrictions over CFL coefficient s for positivity  and contractivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consider the numerical integration of the m-dimensional problem (10) with  the Crank-Nicolson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Then if the method is positive for a given CFL coefficient,  then it is also contractive in the infinity and 1 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' It is straightforward from Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The converse to the previous Corollary is not true as we can see in the following trivial  example, where contractivity is preserved while positivity is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consider the diffusion equation in (1) with initial function  u(x, 0) =  �  0  for 0 < x < 7  8 ,  1  for 7  8 ≤ x < 1 ,  giving discontinuities at x = 7/8 and x = 1 for t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' From second order central differences  with h = 1/8 we get approximations ω(t) = (ω1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , ω7(t)) ≈ (u(x1, t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , u(x7, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Application with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='025 of one Crank-Nicolson step, w1 = A7 w0, gives the vector  w1 ≈ ω(τ)  w1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0013, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0041, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0120, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0356, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1019, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2961, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1397) ,  (23)  where A7 is the Crank-Nicolson iteration matrix in (16) for the case m = 7 and w0 is the  initial profile w0 = (0, 0, 0, 0, 0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Observe that ∥w1∥∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2961 < ∥w0∥∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  9  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  2  s = 2(2- ~2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  0  2  4  6  8  10In this example the CFL coefficient s = τ/h2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='6 is greater than the positivity bound  s(p)  7  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='17157 (see Table 1), but it is lower than the contrativity one s(c)  7  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='61803 (see  Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently, contractivity is preserved while we cannot ensure positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Ac-  tually, as we can see in vector w1 in (23), negativity is not preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  m  ω(c)  m  x(c)  m = cosh ω(c)  m  s(c)  m = 1/(x(c)  m − 1)  3  ∞  5  2 arccsch 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='962424  3/2  2  7  log  �  1+  √  5+√  2(1+  √  5)  �  2  ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='06131  (1 +  √  5)/2  (1 +  √  5)/2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='61803  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='08707  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='65139  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='53518  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  ∞  log 3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='09861  5/3  3/2  Table 2: Positive root of (21) and bounds for contractivity (odd case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  m  ω(c)  m  x(c)  m = cosh ω(c)  m  s(c)  m = 1/(x(c)  m − 1)  4  log  � 1  4(3 +  √  5 +  √  −2 + 6  √  5)  �  1  4(3 +  √  5)  1 +  √  5  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='767197  ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='30902  ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='23607  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='09110  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='65669  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='52278  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='09855  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='66658  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5002  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  ∞  log 3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='09861  5/3  3/2  Table 3: Positive root of (22) and bounds for contractivity (even case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  4  Conclusions and future work  In this paper we have studied CFL restrictions when the Crank-Nicolson method is used  to solve the heat equation (1) with Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  We have obtained  bounds s(p)  m for positivity and bounds s(c)  m for contractivity for any value of the spatial  discretization parameter m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' To get these bounds we have represented the Crank-Nicolson  iteration matrix Am in terms of some Chebyshev-like polynomials (26,28,33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' We have  obtained bounds for the θ-method (7) for the particular case θ = 1/2, but similar bounds  can be obtained for other values of the parameter following the same ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  We have seen that the positivity of matrix Am is determined by the largest root of  polynomial Pm(x), and we have provided a narrow interval where this root can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Similarly, we have considered these polynomials to analyze the contractivity and we have  10  provided a narrow interval to get the corresponding bounds, both in the odd and even  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  As far as we know, polynomials Pn(x) and Cn(x) have not been used previously in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The strength of this idea can be used to prove qualitative properties for other  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Furthermore, this approach can also be used for other discretizations of the  heat equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  5  Preliminary material for the proofs of the main results  In this section we introduce the notation, definitions and some results needed to prove  the main results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1 we express the Crank-Nicolson iteration  matrix (16) in terms of rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' These functions can be written easily with the  help of some Chebyshev-like polynomials Um, Pm and Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2 we give the  definition of these polynomials and we also add some results that will be used in the proofs  of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1  The Crank-Nicolson matrix Am in terms of rational functions  A direct computation of the product (Im − τ  2Bh)−1(Im + τ  2Bh) in (16) gives us the entries  of matrix Am expressed as rational functions, where the polynomials involved can be  obtained recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' These simplified closed expressions will make it easier to get bounds  for positivity and contractivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m = 3, a direct computation of the symmetric matrix A3 in (16) gives  A3(s) =  �  �  �  �  �  2+2s−2s2−s3  2+6s+5s2+s3  2s  2+4s+s2  s2  2+6s+5s2+s3  2s  2+4s+s2  2−s2  2+4s+s2  2s  2+4s+s2  s2  2+6s+5s2+s3  2s  2+4s+s2  2+2s−2s2−s3  2+6s+5s2+s3  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Remember s = τ/h2 denotes CFL coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This matrix can be written even simpler if  we consider the new variable x = 1 + 1/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Observe that x > 1 when s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' With the help  of a new kind of polynomials Un(x), Pn(x) and Cn(x), we can write A3(x) as  A3(x) =  �  �  �  �  �  2x3−4x2+1  2x3−x  2(x−1)  2x2−1  x−1  2x3−x  2(x−1)  2x2−1  2x2−4x+1  2x2−1  2(x−1)  2x2−1  x−1  2x3−x  2(x−1)  2x2−1  2x3−4x2+1  2x3−x  �  �  �  �  � =  1  U3(x)  �  �  �  �  P3(x)  C2(x)  C1(x)  C2(x)  C1(x) + P3(x)  C2(x)  C1(x)  C2(x)  P3(x)  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe that A3(x) has been written just in terms of U3(x), P3(x), C1(x) and C2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' We  give the definition and all the details about these polynomials Un(x), Pn(x) and Cn(x)  in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Before, we extend the ideas in this simple example to the more  general case of the matrix Am(x) for any value of m, although we have to distinguish  between the odd case and the even case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  11  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Matrix Am(x) can be written in terms of polynomials Um(x), Pm(x) and  Cn(x), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If m is an odd number, Crank Nicolson matrix can be reduced to  Am(x) =  1  Um  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  Pm  Cm−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m+1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C2  C1  Cm−1  Pm+Cm−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m−1  2  +C m+3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C1+C3  C2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m+1  2  C m−1  2  +C m+3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Pm+  m−1  2�  n=1  C2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m−1  2  +C m+3  2  C m+1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C2  C1+C3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m−1  2  +C m+3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Pm+Cm−2  Cm−1  C1  C2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m+1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Cm−1  Pm  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ,  (24)  where all the polynomials are evaluated at x = 1 + 1/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If m is an even number, we write  Am(x) =  1  Um  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  Pm  Cm−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m+2  2  C m  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C2  C1  Cm−1  Pm+Cm−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m  2 +C m+4  2  C m−2  2  +C m+2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C1+C3  C2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m+2  2  C m  2 +C m+4  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Pm+  m−2  2�  n=1  C2n  m  2�  n=1  C2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m−2  2  +C m+2  2  C m  2  C m  2  C m−2  2  +C m+2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  m  2�  n=1  C2n−1  Pm+  m−2  2�  n=1  C2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m  2 +C m+4  2  C m+2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C2  C1+C3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' C m−2  2  +C m+2  2  C m  2 +C m+4  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Pm+Cm−2  Cm−1  C1  C2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  C m  2  C m+2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Cm−1  Pm  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (25)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' It is straightforward from the computation of the product (Im− τ  2Bh)−1(Im+ τ  2Bh)  in (16) and the use of polynomials Un(x), Pn(x) and Cn(x), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , m − 1, defined in  the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe that Am(x) is bisymmetric, that is, it is symmetric on both diagonals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This im-  plies that Am(x) is also centrosymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Then the entries aij satisfy aij = an−i+1,n−j+1 ,  for 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently, if m is odd, the number of different entries in matrix Am  is 1 + 3 + 5 + · · · + m = (m + 1)2/4, and, if m is even, this number is 2 + 4 + 6 + · · · + m =  (m/2 + 1)m/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For example, the number of different elements in matrix A3 in Example 2  is 4, while this number is 6 for matrix A4 in Example 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe also that the numerator of each entry aij in matrix Am(x) is a sum of some  polynomials Pn(x), Cn(x), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , m − 1, and the number of polynomials in this sum  is equal to min{i, j, m − i + 1, m − j + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Properties of polynomials Un(x), Pn(x) and  Cn(x) will allow us to analyze positivity and contractivity of Crank Nicolson method in a  quite simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the next section, we study these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  12  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In Example 2 we have considered the odd case m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Here, for completeness,  we consider the even case m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' A direct computation of the symmetric matrix A4 in (16)  gives  A4(s) =  1  u4(s)  �  �  �  �  p4(s)  4s(4+8s+3s2)  8s2(1+s)  4s3  4s(4+8s+3s2)  16+32s+4s2−16s3−5s4  16s(1+s)2  8s2(1+s)  8s2(1+s)  16s(1+s)2  16+32s+4s2−16s3−5s4  4s(4+8s+3s2)  4s3  8s2(1+s)  4s(4+8s+3s2)  p4(s)  �  �  �  � ,  where p4(s) = −5s4 − 24s3 − 4s2 + 32s + 16 and u4(s) = 5s4 + 40s3 + 84s2 + 64s + 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Now, with the help of variable x = 1 + 1/s, we can write  A4(x) =  1  U4(x)  �  �  �  �  �  �  �  P4(x)  C3(x)  C2(x)  C1(x)  C3(x)  P4(x) + C2(x)  C1(x) + C3(x)  C2(x)  C2(x)  C1(x) + C3(x)  P4(x) + C2(x)  C3(x)  C1(x)  C2(x)  C3(x)  P4(x)  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe that there are two central rows in the even case, but just one in the odd case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2  Polynomials Un, Pn and Cn  In this section we define the new polynomials Pn(x) and Cn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Together with the help  of Chebyshev polynomials of the second kind Un(x) [7, 17, 19], we have got a simple way  of writing Crank-Nicolson matrix Am(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Besides, here we give some results concerning  these polynomials, with particular interest in the distribution of their roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Chebyshev  polynomials of the second kind Un(x) belong to a general class of orthogonal polynomials  and there are many works about the behaviour of their zeros [7, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' However, polynomials  Pn(x) and Cn(x) do not belong to this class of orthogonal polynomials and, as far as we  know, nothing is known about their roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Chebyshev polynomials of second kind  Chebyshev polynomial of second kind of degree n ≥ 0 is defined as  Un(x) = sin((n + 1) arccos x)  sin(arccos x)  ,  x ∈ [−1, 1] ,  or, in the angle variable ω, Un(cos ω) = sin((n + 1)ω)/sin ω, ω ∈ [0, π] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' These polynomials  can also be defined for any value of x ∈ R by the recurrence relation  U0(x) = 1 ,  U1(x) = 2x ,  Un(x) = 2x Un−1(x) − Un−2(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (26)  13  It is possible to write the recurrence relation (26) in terms of the determinant of the  tridiagonal matrix tridiag(1, 2x, 1) of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Un(x) =  �����������  2x  1  1  2x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1  1  2x  �����������  (27)  Recall that each polynomial Un(x) has n roots xn  i = cos (iπ/(n + 1)) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , n, in the  interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' These roots are uniformly distributed in the angle variable ω = arccos x  in the interval [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Notice that polynomials Un(x) defined in (26) are positive for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Polynomials Pn  If we change the first two elements in the recursive relation (26), then a new family of  polynomials can be defined  P0(x) = −1 ,  P1(x) = 2x − 4 ,  Pn(x) = 2x Pn−1(x) − Pn−2(x) ,  (28)  where Pn(x) denotes the polynomial of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In this case the recurrence relation (28)  can also be written in terms of the determinant of a matrix of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Pn(x) =  ������������  2x − 4  −1  1  2x  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1  2x  1  1  2x  ������������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (29)  In the next proposition we analyze the roots of each polynomial Pn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the proof,  shown in Section 6, it is relevant the fact that each polynomial Pn(x) can be written in  terms of Chebyshev polynomials of second kind  Pn(x) = 2Un−2(x) − 4Un−1(x) + Un(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (30)  This equality is obtained by writing the determinant (29) in terms of the determinant (27),  and the use of the recurrence relation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The polynomial Pn(x) defined in (28) has exactly n − 1 roots xn  i , i =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , n−1, in the interval (−1, 1), and an additional isolated root xn := xn  n in the interval  (6+  √  2  4  , 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Furthermore, the isolated root is xn = cosh ωn, where ωn is the unique root of  the equation  coth(nω) = 3 cosh ω − 4  sinh ω  ,  ω ∈ (0, ∞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (31)  Besides, x1 = 2 and the sequence of isolated roots (xn) decreasingly converges to the limit  value x∞ = (6 +  √  2)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  14  Figure 2: Polynomials Pn(x) , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Each polynomial has an isolated root in the  interval (6+  √  2  4  , 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  As it is shown in the proof (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3), for any value of n, the unique root ωn  of equation (31) lies in the interval (ω∞, ω1] = (log(2 +  √  2), log(2 +  √  3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently,  the isolated root xn = cosh ωn of polynomial Pn(x) lies in the narrow interval (6+  √  2  4  , 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Having the root well located makes it easy to approach it by any numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In  Table 1 we show some of these roots after 10 steps with bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Polynomials Cn  The polynomial Cn of degree n is defined as  Cn(x) = Pn(x) + Un(x) ,  n ∈ N,  (32)  where Pn is the polynomial of degree n defined above and Un is the Chebyshev polynomial  of second kind of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently, all the properties of Cn are consequence of this  definition, including its recursive definition  C0(x) = 0 ,  C1(x) = 4(x − 1) ,  Cn(x) = 2x Cn−1(x) − Cn−2(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (33)  Observe that the recursive formula (33) is the same as (28) for Pn and (26) for Un, with  just different starting values C0 and C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As in previous cases, it is worth writing Cn(x) in  terms of a determinant  Cn(x) =  ������������  4(x − 1)  0  1  2x  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1  2x  1  1  2x  ������������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (34)  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Each polynomial Cn(x) has exactly n − 1 roots xn  i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , n − 1, in  the interval (−1, 1), and the additional isolated root xn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  15  10  5  P1 (x)  P2(x)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  P3(x)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  P4(x)  10Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' It is straightforward if we use the determinant (34), where we get  Cn(x) = 4(x − 1)Un−1(x)  (35)  Consequently the roots of polynomial Cn are the isolated root xn = 1, and the n − 1 roots  of the Chebyshev polynomial of second kind of degree n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If x > 1, then Cn(x) > 0 ∀n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' It is straightforward from the previous proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In the following Lemma we give some technical properties of polynomials Cn(x) that  we will need in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For the polynomial Cn of degree n defined in (32) or (33) the following prop-  erties hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Cn(1) = 0 , ∀n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Cn(x) = 2Un−2(x) − 4Un−1(x) + 2Un(x) , ∀n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If x > 1 , then 0 < Cn(x) ≤ Cn+1(x) ,  ∀n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If x > 1 , then 2Cn(x) ≤ Cn−1(x) + Cn+1(x) ,  ∀n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Part 1 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Part 2 is also straightforward if we use the relation-  ship (30) and definition (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  To prove part 3, as x > 1, we set x = cosh ω in (35), to obtain  Cn(cosh ω) = 4(cosh ω − 1) sinh(nω)  sinh ω  > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (36)  As sinh ω is an increasing function, it holds sinh(nω) < sinh((n + 1)ω), and consequently  Cn(x) ≤ Cn+1(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Finally, to prove 4, as  sinh((n − 1)ω) + sinh((n + 1)ω) = 2 cosh ω sinh(nω) ≥ 2 sinh(nω) ,  we can use again (36), for x > 1, to obtain that 2Cn(x) ≤ Cn−1(x) + Cn+1(x) ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  6  Proofs of theorems in Sections 3 and 5  In this section we give the proofs of the main results in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1  Positivity of the Crank-Nicolson method  Recall that the Crank Nicolson method is positive if and only if all the elements in matrix  Am (24)-(25) are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The following lemma simplifies the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If x > 1, then all the elements in matrix Am(x) (24)-(25) are non-negative if  and only if polynomial Pm(x) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  16  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Recall that polynomials Un(x) are positive for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Polynomials Cn(x) are also  positive for x > 1 (see Corollary 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently, all extra-diagonal elements are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  For the diagonal elements A(i,i)  m (x), the positivity of polynomials Un(x) and Cn(x) for  all n ∈ N implies that  min  i  A(i,i)  m (x) = A(1,1)  m  (x) = A(m,m)  m  (x) = Pm(x)  Um(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Then the analysis of the positivity of the elements in Am(x) is reduced to the positivity  of polynomial Pm(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Proof of Theorem 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' From Lemma 4, we just have to study the positivity of polynomials Pm(x) for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  From Proposition 2, Pm(x) has exactly m − 1 roots in the interval (−1, 1) and an  additional isolated real root x(p)  m = cosh(ω(p)  m ) in the interval (6+  √  2  4  , 2], where ω(p)  m is  the unique root of equation (31), that is the same as (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As x = 1 + 1/s, then the  CFL coefficient is s = 1/(x − 1), and inequality (17) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the limit case, Proposition 2 gives us the limit value x(p)  ∞ = (6 +  √  2)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Then  for the CFL coefficient we get the bound s(p)  ∞ = 1/(x(p)  ∞ − 1) = 2(2 −  √  2), and  inequality (19) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2  Contractivity of the Crank-Nicolson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Before writing the proof of Theorem 2, we need two technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In Lemma 5  below, we compute the maximum norm of matrix Am(x) in terms of polynomials Un(x),  Pn(x) and Cn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Then, in Lemma 6 we get the inequalities needed for the contractivity  condition ∥Am∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In contrast to positivity, in the analysis of the contractivity it is  necessary to distinguish between the even and odd cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If m ≥ 3 is a natural odd number, then the maximum norm of Crank-Nicolson  matrix is  ∥Am∥∞ =  1  Um  �  �  �|Pm +  m−1  2  �  n=1  C2n−1| + 2  m−3  2  �  i=0  m−(2i+1)  2  �  n=1  C2n+i  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (37)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If m ≥ 2 is a natural even number, then the norm of Crank-Nicolson matrix is  ∥Am∥∞ =  1  Um  �  �|Pm +  m  2 −1  �  n=1  C2n| +  m  2 −1  �  i=1  m−i  �  n=1+i  Cn +  m/2  �  n=1  C2n−1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (38)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Note that the norm in (37) is obtained from the sum of the elements in the central  row of Am (odd case), while the norm in (38) is obtained from the sum of the elements  17  in any of the two symmetric central rows (even case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' To get this result we will proof the  following inequalities  m  �  j=1  |A(i,j)  m | ≤  m  �  j=1  |A(i+1,j)  m  | ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , m−1  2 (odd case) m−2  2 (even case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  For the sake of simplicity we will denote Ai  m to the sum �m  j=1|A(i,j)  m |Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Observe that the  symmetry of matrices (24-25) makes Ai  m = Am+1−i  m  , i = 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , (m − 1)/2 (odd case) or  i = 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' , m/2 (even case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Then, we will proof that  A1  m ≤ A2  m ≤ · · · ≤ A  m−1  2  m  ≤ A  m+1  2  m  (odd case)  A1  m ≤ A2  m ≤ · · · ≤ A  m−2  2  m  ≤ A  m  2m = A  m+2  2  m  (even case)  Consequently, ∥Am∥∞ is obtained from the sum of the elements in the central row (odd  case) or from the sum of the elements in any of the two central rows (even case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Recall that for a given matrix A with positive extra-diagonal elements the following  equality trivially holds  ∥A∥∞ = max  i  �  �  m  �  j=1  |aij|  �  � = max  i  �  �|aii| +  �  j̸=i  aij  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (39)  In our case Cn(x) ≥ 0 for x ≥ 1 (see Proposition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently all extra-diagonal ele-  ments of matrices (24) and (25) are non-negative and we can use (39) to compute ∥Am∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  For the first and the second row of matrix (24) or (25) we get the sums  A1  m = |Pm| +  m−1  �  i=1  Ci ,  A2  m = |Pm + Cm−2| +  m−1  �  1  Ci +  m−1  �  2  Ci − Cm−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Then, after cancelling terms, the difference A1  m − A2  m is  A1  m − A2  m = |Pm| − |Pm + Cm−2| −  m−1  �  2  Ci + Cm−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Adding and subtracting Cm−2 in the term |Pm|, we can write  A1  m − A2  m ≤ |Pm + Cm−2| + Cm−2 − |Pm + Cm−2| −  m−1  �  2  Ci + Cm−2  = 2Cm−2 −  m−1  �  2  Ci ≤ Cm−3 + Cm−1 −  m−1  �  2  Ci = −  m−4  �  2  Ci − Cm−2 ≤ 0 ,  where we have used 2Cm−2 ≤ Cm−3 + Cm−1 from item (4) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Observe that  property (3) in Lemma 3 allows us to finally write  A1  m − A2  m ≤ −  m−3  �  2  Ci ≤ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (40)  18  If we proceed in the same way for the difference A2  m−A3  m, after cancelling terms, and after  adding and subtracting Cm−4 in the term |Pm + Cm−2|, we get the following inequality  A2  m − A3  m ≤ 2Cm−4 −  m−2  �  3  Ci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Again, the use of properties (4) and (3), in this order, from Lemma 3 makes it possible to  write an inequality analogous to (40)  A2  m − A3  m ≤ Cm−5 + Cm−3 −  m−2  �  3  Ci ≤ −  m−4  �  3  Ci ≤ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The proof follows in the same way for the odd and even case up to the last step when  we achieve the central row (odd case) or the two central rows (even case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Then we have  to consider two different cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If m is an odd number, the last step consists in studying the difference A  m−1  2  m  −A  m+1  2  n  ,  where A  m+1  2  n  represents the sum of the elements in the central row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' After cancelling  terms, we can write  A  m−1  2  m  − A  m+1  2  m  = |Pm +  m−1  2  �  n=2  C2n−1| + C1 − |Pm +  m−1  2  �  n=1  C2n−1| − C m+1  2  Adding and subtracting C1 in the term |Pm + � m−1  2  n=2 C2n−1|, we get  A  m−1  2  m  − A  m+1  2  m  ≤ |Pm +  m−1  2  �  n=1  C2n−1| + 2C1 − |Pm +  m−1  2  �  n=1  C2n−1| − C m+1  2  = 2C1 − C m+1  2  ≤ C0 + C2 − C m+1  2  = C2 − C m+1  2  ≤ 0 ,  where we have used 2C1 ≤ C0 + C2 from property (4) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Consequently, if m ≥ 3 is an odd number, the maximum value of Ai  m is obtained in  the central row A  m+1  2  m  and we can conclude  ∥Am∥∞ = A  m+1  2  m  Um  =  1  Um  �  �  �|Pm +  m−1  2  �  n=1  C2n−1| + 2  m−3  2  �  i=0  m−(2i+1)  2  �  n=1  C2n+i  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If m is an even number, then there is not a central row but two central symmet-  ric rows A  m  2m and A  m+2  2  m  , and the maximum value is obtained at any of these two  files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Now, in the last step of the proof, if m ≥ 4, we have to write the differ-  ence A  m−2  2  m  − A  m  2m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Note that for the simple case m = 2, it holds A1  m = A2  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' After  cancelling terms, we can write  A  m−2  2  m  − A  m  2m = |Pm +  m  2 −1  �  n=2  C2n| + C2 − |Pm +  m  2 −1  �  n=1  C2n| − C m  2 − C m+2  2  19  Adding and subtracting C2 in the term |Pm + � m  2 −1  n=2 C2n|, we get  A  m−2  2  m  − A  m  2m ≤ |Pm +  m  2 −1  �  n=1  C2n| + 2C2 − |Pm +  m  2 −1  �  n=1  C2n| − C m  2 − C m+2  2  = 2C2 − C m  2 − C m+2  2  ≤ C1 + C3 − C m  2 − C m+2  2  ≤ 0 ,  where we have used 2C2 ≤ C1 + C3 from property (4) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Consequently, if m is an even number, the maximum value of Ai  m is obtained in the  row A  m  2m and we can conclude  ∥Am∥∞ = max  i  �  �|aii| +  �  j̸=i  aij  �  � =  1  Um  �  �|Pm +  m  2 −1  �  n=1  C2n| +  m  2 −1  �  i=1  m−i  �  n=1+i  Cn +  m/2  �  n=1  C2n−1  �  �  Once we have got the maximum norm of matrix Am in terms of polynomials Um, Pm  and Cm, we can get bounds s(c)  m for contractivity for any value of m if we are able to  solve the corresponding inequality ∥Am∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This is done in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In  Figure 3 we have plot ∥Am(s)∥∞ for some values of m, and we have also added some  contractivity bounds s(c)  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  s(c)  4  s(c)  5  s(c)  7  ∥A1(s)∥  ∥A2(s)∥  ∥A3(s)∥  ∥A4(s)∥  ∥A5(s)∥  ∥A7(s)∥  ∥A9(s)∥  ∥A21(s)∥  Figure 3: ∥Am(s)∥ for different values of m ∈ {1, 2, 3, 4, 5, 7, 9, 21}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' When m ∈ {1, 2, 3}  it holds ∥Am(s)∥ < 1, for all s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ≥ 4, ∥Am(s)∥ cuts the line s = 1 at s(c)  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  The sequence (s(c)  m ) is strictly monotonically decreasing with all the terms in the inter-  val  �3/2, 1 +  √  5  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' (Ood case) If m is a natural odd number, then for the maximum norm of Crank-  Nicolson matrix we have  ∥Am∥∞ ≤ 1  ⇐⇒  2 sinh (m−1)ω  4  sinh (m+1)ω  4  sinh (m+1)ω  2  ≤ sinh ω  2  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='25  3/2  2  1+ V5where ω = arccosh(1 + 1/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the limit, when m → ∞, we get contractivity if and  only if  e−ω/2 ≤ sinh ω  2  (41)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' (Even case) If m is a natural even number, then we have  ∥Am∥∞ ≤ 1  ⇐⇒  sinh mω  2  �  sinh (m+2)ω  2  − sinh mω  2  �  sinh (m+1)ω  2  sinh mω  4 sinh (m−2)ω  4  ≥ sinh ω  sinh2 ω  2  In the limit, when m → ∞, we get contractivity if and only if  2 (−1 + eω) ≥ sinh ω  sinh2 ω  2  (42)  If ω > 0, inequalities (41) and (42) are equivalent, and they are true iff ω ≥ log 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the  variable s this is equivalent to the known restriction s ≤ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Remember that, in the odd case, the norm in (37) is obtained from the sum of the  elements in the central row of Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In that case, the diagonal element in the central  row Pm + � m−1  2  n=1 C2n−1 can be written in closed form as  Pm +  m−1  2  �  n=1  C2n−1 = −Um +  m+1  2  �  n=1  C2n−1 = −sinh((m + 1)ω)  sinh ω  + 4(x − 1) sinh2 (m+1)ω  2  sinh2 ω  ,  where we have changed Pm = Cm − Um and we have considered the angle variable  ω = arccosh x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For any value of m, the unique positive root ωm of this diagonal  element lies in the interval (log 3, log(2 +  √  3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This root can easily be obtained  from the simplified equation in the variable s = (cosh ω − 1)−1  √  1 + 2s = 2 tanh (1 + m) arccosh(1 + 1/s)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  In this variable, the unique positive root sm lies in the interval [1, 3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Observe that  the sequence of roots (sm) increasingly converges to the limit value s∞ = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' When  s ∈ (0, sm) the diagonal element Pm + � m−1  2  n=1 C2n−1 is positive, and, from (37), we  easily obtain ∥Am∥∞ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  On the other hand, when s ∈ [sm, ∞), we have Pm + � m−1  2  n=1C2n−1 ≤ 0 , and, from (37),  the inequality for contractivity is  ∥Am∥∞ =  1  Um  �  �  �−Pm −  m−1  2  �  n=1  C2n−1 + 2  m−3  2  �  i=0  m−(2i+1)  2  �  n=1  C2n+i  �  �  � ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  This is equivalent to  −  m−1  2  �  n=1  C2n−1 + 2  m−3  2  �  i=0  m−(2i+1)  2  �  n=1  C2n+i ≤ Cm ,  21  or  m+1  2  �  n=1  C2n−1 − 2  m−3  2  �  i=0  m−(2i+1)  2  �  n=1  C2n+i ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  As x > 1, in the angle variable ω = arccosh x > 0, we can write Cm(cosh ω) =  4(x − 1) sinh(mω)/ sinh ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Consequently, the previous inequality is reduced to  m+1  2  �  n=1  sinh((2n − 1)ω) − 2  m−3  2  �  i=0  m−(2i+1)  2  �  n=1  sinh((2n + i)ω) ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  If we use the closed formulas for the expansions �  k sinh(kω), �  k sinh(2kω), and  �  k sinh((2k − 1)ω), then we can write the previous inequality as  4(x − 1) sinh (m+1)ω  2  sinh2 ω  �  sinh (m + 1)ω  2  − 2sinh (m−1)ω  4  sinh (m+1)ω  4  sinh ω  2  �  ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  And, for x > 1, this is true if and only if  2 sinh (m−1)ω  4  sinh (m+1)ω  4  sinh (m+1)ω  2  ≤ sinh ω  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (43)  In this way, for any value of m, we get contractivity if and only if ω ≥ ω(c)  m , where  ω(c)  m is the unique positive root of the corresponding equality equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Finally, going  back to variable s, we get contractivity if and only if s ≤ s(c)  m := 1/(cosh ω(c)  m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Computing the limit in (43), when m → ∞, we get contractivity if and only if  e−ω/2 ≤ sinh ω  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  And this is true if and only if ω ≥ log 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This is x = cosh ω ≥ 5/3 or s ≤ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the even case, the norm in (38) is obtained from the sum of the elements in any  of the two central symmetric rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The diagonal element Pm + � m  2 −1  n=1 C2n in any of  this central rows can be written in closed form as  Pm+  m  2 −1  �  n=1  C2n = −Um +  m  2  �  n=1  C2n = −sinh((m + 1)ω)  sinh ω  + 4(x − 1) sinh mω  2 sinh (m+2)ω  2  sinh2 ω  ,  where ω = arccosh x and x = 1 + 1/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Again, the unique positive root sm of this  central diagonal element lies in the interval [1, 3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For any value of m, this root  can easily be obtained from the simplified equation  √  1 + 2s = 4 sinh mω  2 sinh (m+2)ω  2  sinh((m + 1)ω)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Observe that this sequence of roots (sm)m increasingly converges to the limit value  s∞ = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' When s ∈ (0, sm) the diagonal element Pm + � m  2 −1  n=1 C2n is positive, and,  22  from (38), we easily obtain ∥Am∥∞ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' On the other hand, when s ∈ [sm, ∞), the  diagonal element is negative, and, from (38), the inequality for contractivity is  ∥Am∥∞ =  1  Um  �  �−Pm −  m  2 −1  �  n=1  C2n +  m  2 −1  �  i=1  m−i  �  n=1+i  Cn +  m/2  �  n=1  C2n−1  �  � ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  This is equivalent to  −  m  2 −1  �  n=1  C2n +  m  2 −1  �  i=1  m−i  �  n=1+i  Cn +  m/2  �  n=1  C2n−1 ≤ Cm ,  or  m  2  �  n=1  C2n −  m  2 −1  �  i=1  m−i  �  n=1+i  Cn −  m/2  �  n=1  C2n−1 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  As in the odd case, now we can use the angle variable ω to reduce the previous  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  m  2  �  n=1  sinh(2nω) −  m  2 −1  �  i=1  m−i  �  n=1+i  sinh(nω) −  m/2  �  n=1  sinh((2n − 1)ω) ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Finally, the closed formulas for the expansions �  k sinh(kω), �  k sinh(2kω), and  �  k sinh((2k − 1)ω), allow us to reduce the inequality to  4(x−1)  sinh ω  �  sinh mω  2  sinh ω  �  sinh (m+2)ω  2  − sinh mω  2  �  − sinh (m+1)ω  2  sinh mω  4 sinh (m−2)ω  4  sinh2 ω  2  �  ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  And, for x > 1, this is true if and only if  sinh mω  2  �  sinh (m+2)ω  2  − sinh mω  2  �  sinh (m+1)ω  2  sinh mω  4 sinh (m−2)ω  4  ≥ sinh ω  sinh2 ω  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (44)  In this way, for any value of m, we get contractivity if and only if ω ≥ ω(c)  m , where ω(c)  m  is the unique positive root of the corresponding equality equation in (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Finally,  going back to variable s, we get contractivity if and only if s ≤ 1/(cosh ω(c)  m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Computing the limit in (44), when m → ∞, we get contractivity if and only if  2 (−1 + eω) ≥ sinh ω  sinh2 ω  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  And this is true if and only if ω ≥ log 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' This is x = cosh ω ≥ 5/3 or s ≤ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Proof of Theorem 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ∈ {1, 2, 3} the proof is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' For m ∈ N, m > 3 , the proof is straightforward from the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' The  inequality for contractivity in the variable s = 1/(cosh ω − 1) is  s ≤ s(c)  m :=  1  cosh ω(c)  m − 1  where ω(c)  m is the unique positive root of the equation from (43), if m is odd, or from  (44) if m is even .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' It is also straightforward from the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In the limit, when m → ∞, we  get inequality (41) from (43), and inequality (42) from (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' These two inequalities  (41) and (42) are equivalent if ω > 0, and they are true if and only if ω ≥ log 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' In  the variable s this is equivalent to the known restriction s ≤ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='3  Proof of the remaining results  Proof of Proposition 2  We divide the proof into two parts, the case x ∈ (−1, 1) and the case x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If x ∈ (−1, 1), then the angular variable ω = arccos x, ω ∈ (0, π), and equality (30),  allow us to write the polynomial Pm in closed form as  Pm(cos ω) = 2 sin((m − 1)ω) − 4 sin(mω) + sin((m + 1)ω)  sin ω  (45)  If we convert all the angles in the numerator to the angle mω, we can write  = (3 cos ω − 4) sin(mω)  sin ω  − cos(mω) ,  (46)  Then, from (46), we get that the roots of Pm(cos ω) in (0, π) are the roots of the  equation  tan(mω) =  sin ω  3 cos ω − 4 ,  ω ∈ (0, π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (47)  The function on the right hand side, f(ω) := sin ω/(3 cos ω − 4), is continuous and  bounded in the interval [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' It is decreasing in the interval (0, 2 arctan(1/  √  7)) and  increasing in the interval (2 arctan(1/  √  7), π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Its maximum value f(0) = f(π) = 0  is obtained in the boundary, while the minimum value is f(2 arctan(1/  √  7)) =  −1/  √  7 (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Thus, we can affirm that tan(mω) meets m−1 times the func-  tion f(ω), and consequently there are m−1 roots 0 < ωm  m−1 < ωm  m−2 < · · · < ωm  1 < π ,  in the interval (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Now, going back to the variable x = cos ω, we can affirm that  the polynomial Pm(x) has m − 1 roots xm  i = cos ωm  i  in the interval (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' As the  cosine function is decreasing in the interval (0, π), we can write the m − 1 roots of  Pm(x) as  −1 < xm  1 < xm  2 < · · · < xm  m−1 < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  24  Figure 4: Roots 0 < ωm  m−1 < ωm  m−2 < · · · < ωm  1 < π , of equation (47)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' If x > 1, we consider the variable ω = arccosh x, ω ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Again, with the help  of equality (30), we can write the polynomial Pm in closed form as  Pm(cosh(ω)) = 2 sinh((m − 1)ω) − 4 sinh(mω) + sinh((m + 1)ω)  sinh ω  (48)  = (3 cosh ω − 4) sinh(mω)  sinh ω  − cosh(mω) ,  (49)  where, as in the previous case, we have rewritten the numerator in the terms of the  angle mω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Then, from (49), the roots of Pm(cosh(ω)) in (0, ∞) are the roots of the  equation  coth(mω) = 3 cosh ω − 4  sinh ω  ,  ω ∈ (0, ∞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  (50)  The function on the right, g(ω) := (3 cosh ω−4)/ sinh ω, is continuous and increasing  in the interval (0, ∞) to the limit value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' On the left, for any value of m, the  function coth(mω) is continuous and decreasing to the limit value of 1 (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Thus, we can affirm that, for any value of m, equation (50) has a unique root ωm in  the interval (0, ∞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Now, going back to the variable x = cosh ω, we can affirm that  the polynomial Pm(x) has a unique root xm = cosh ωm in the interval (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Figure 5: Left hand side and right hand side of equation (50)  In the limit, when m tends to infinity, for any value of ω > 0, we have coth(mω)↘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  Consequently, from (50), the sequence of roots (ωm) decreasingly converges to the  limit value ω∞ = log(2 +  √  2), this is the positive solution of the limit equation  25  2  1 一 1 一  1  1  1  1  /  /  /  /  /  /  /  /  /  1  / /  /  1  /  1  / tan(m w)  /  /  /  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0 /  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  (m)}  / -  /  /  1  / 1  / /  1  1  /  /  /  1  /  /  /  1  1 1  1 1  1  1  2  1  1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='4  tan(5 w)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='2  tan(3 w)  tan(2 w)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  (m)6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='53 cosh ω − 4 = sinh ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Now, in the variable x = cosh ω, we can affirm that each  polynomial Pm(x) has a positive root xm = cosh ωm in the interval (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  As  the cosh function is increasing in the interval (0, ∞), the sequence of roots (xm)  decreasingly converges to the limit value x∞ = (6 +  √  2)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  x∞ = cosh ω∞ = cosh log(2 +  √  2) = (6 +  √  2)/4 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='85355 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  References  [1] Desoer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', and Haneda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
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+page_content='  SIAM Journal on Numerical Analysis 12, 5  (1975), 741–753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  [25] Szabó, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Qualitative properties of some discretized partial differential equations  and reliable fuel cell modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' PhD thesis, Eötvös Loránd University, Budapest,  Hungary, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  [26] Thomée, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Finite difference methods for linear parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Handbook of  numerical analysis 1 (1990), 5–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' in: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Ciarlet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Lions (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' ), Handbook of  Numerical Analysis, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' 1, 1990, North-Holland, Amsterdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  [27] Van de Griend, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', and Kraaijevanger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Absolute monotonicity of rational  functions occurring in the numerical solution of initial value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Numerische  Mathematik 49, 4 (1986), 413–424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  27  [28] Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=', and Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' Root geometry of polynomial sequences iii: Type  (1, 1) with positive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content=' arXiv preprint arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='06105 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
+page_content='  28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfIvs5/content/2301.01066v1.pdf'}
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+Draft version January 6, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+A broad-line quasar with unexplained extreme velocity offsets: post-shock outflow?
+Vadim Rusakov
+,1, 2 Charles L. Steinhardt
+,2, 3 Malte Schramm
+,4 Andreas L. Faisst
+,5
+Daniel Masters
+,5 Bahram Mobasher
+,6 and Petchara Pattarakijwanich
+7
+1Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, København Ø 2100, Denmark rusakov124@gmail.com
+2Cosmic Dawn Center (DAWN)
+3Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, København Ø 2100, Denmark
+4Graduate school of Science and Engineering, Saitama Univ., 255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama 338-8570, JAPAN
+5California institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA
+6Department of Physics and Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA
+7Department of Physics, Faculty of Science, Mahidol University, 272 Rama IV Road, Ratchathewi, Bangkok, Thailand
+ABSTRACT
+The quasar SDSS 0956+5128 exhibits three distinct velocity components with large offsets in emis-
+sion: the systemic velocity of [O ii], [O iii] and [Ne iii] narrow lines have redshift z = 0.7142; broad
+Mg ii line is shifted by −1200 km s−1 with respect to the narrow lines; broad Hα, Hβ lines are at
+−4100 km s−1. We present new Hubble Space Telescope spectra of Lyα and C iv emission lines and
+high-resolution images of the quasar. The offsets of these lines are consistent with the velocity com-
+ponent of the Balmer emission, and the photometry in optical and near-infrared wavelengths does not
+show any signs of recent mergers in the host galaxy or irregularities in the location of the quasar. The
+data do not confirm predictions of the previous most-likely hypotheses involving a special orientation
+and morphology of the quasar disk, such as in the recoiling black hole scenario, neither it is consistent
+with accretion disk winds. Instead, based on the cumulative evidence, we propose a new scenario, in
+which the broad line region is in the state of outflow caused by a strong shock wave, with a supernova
+as a possible event for producing the shock ejecta.
+Keywords: Quasars(1319) — Shocks(2086) — Supernovae(1668)
+1. INTRODUCTION
+Although there initially appeared to be several sub-
+types of active galactic nuclei (AGN) and quasars (QSO)
+(Antonucci 1993), it has been believed for approximately
+three decades that nearly all observed AGN are con-
+sistent with being similar objects observed from differ-
+ent lines of sight (Urry & Padovani 1995; Miller et al.
+1991; Bailey et al. 1988; Lawrence & Elvis 1982; An-
+tonucci & Miller 1985; other references in Antonucci
+1993). One consequence is that all quasar spectra ex-
+hibit the same set of significant spectral lines: (1) broad
+emission (or absorption) lines (BEL) including C iv,
+Lyα, Hα, Hβ and Mg ii, among others, with Doppler
+widths of ∼ 104 km s−1 (Vanden Berk et al. 2001; Mur-
+ray et al. 1995; Wills et al. 1993), that are normally
+virialized (Shapovalova et al. 2001; Dietrich et al. 1998;
+Korista et al. 1995); (2) narrow emission lines, including
+[O iii]λλ4959,5007, [Ne ii]λ3869 and [O ii]λ3727 with
+widths of ∼ 500 − 1000 km s−1; and (3) a wide range
+of possible narrow emission or absorption lines from the
+host galaxy, with widths of < 500 km s−1. These ob-
+servations have led to a physical picture in which this
+emission originates, respectively, from a broad-line re-
+gion (BLR) ∼ 1 pc from the central supermassive black
+hole, a narrow-line region (NLR) ∼ 103 pc away, and
+the remainder of the host galaxy, which might extend
+to ∼ 104 − 105 pc out.
+Although inflows or outflows
+might skew the profiles of these lines (cf. Strateva et al.
+2003; Eracleous et al. 1995; Gaskell 1982), they all em-
+anate from objects in the same host galaxy, and thus
+are likely centered at the same Doppler velocity (equiv-
+alently, redshift) with respect to the observer.
+However, a single quasar among the more than 105
+quasars found in Sloan Digital Sky Survey (SDSS) ap-
+pears to be entirely incompatible with this model. SDSS
+J095632.49+512823.92 (hereafter, SDSS 0956+5128) ex-
+hibits three distinct and non-typical features that in
+combination make this object unique (Steinhardt et al.
+2012) (hereafter, S12).
+arXiv:2301.01782v1  [astro-ph.HE]  4 Jan 2023
+
+ID2
+Rusakov et al.
+At first, there are three significantly different veloc-
+ity components, corresponding to z = 0.714, z = 0.707,
+and z = 0.690. The narrow line emission of [O ii], [O
+iii] and [Ne iii] provides the systemic redshift of the
+galaxy z = 0.714. The blueshift of broad Balmer emis-
+sion lines and Mg ii from the host galaxy is ∼ 4100
+and ∼ 1200 km s−1, respectively. It is not surprising
+on its own to observe three components. For example,
+such objects as I Zw 1 are known to exhibit more than
+two velocity systems with blueshifts from the systemic
+redshift (Laor et al. 1997; Vestergaard & Wilkes 2001).
+However, unlike such quasars, the broad emission lines
+in SDSS 0956+5128 are not double-peaked or strongly
+skewed (as in the examples that were studied in detail in
+Eracleous et al. 1995; Tsalmantza et al. 2011; Eracleous
+et al. 2012). They are symmetric and completely offset
+and therefore appear to be consistent with some kind of
+physical offset of the BLR, such as an outflow.
+Only a handful of objects studied in outflows have
+been observed with nearly as high offsets in Hβ, Hα and
+Mg ii.
+In fact, offsets of the similar magnitude have
+been found more often in higher-ionization lines, such
+as Si iv, C iv or higher, and even so in the extreme tail
+of their offset distribution, as shown by Yu et al. (2021)
+in SDSS DR7 (Shen et al. 2011) and other quasar sam-
+ples.
+The objects with symmetricaly-offset lines have
+been studied in connection with the recoiling black holes
+(BH; Bonning et al. 2007). In fact, no recoil candidates
+have been identified with the BLR velocity as high as in
+the SDSS 0956+5128 which is almost twice the second
+highest offset (Chiaberge et al. 2017).
+In addition, the broad Mg ii line, which is typically
+found to be consistent with the systemic velocity (Shen
+et al. 2016; Hewett & Wild 2010), is offset by −1200
+km s−1 from the host lines, and thus by +2900 km s−1
+from Hα and Hβ. No other quasar spectra in the SDSS
+that have coverage of Balmer lines and Mg ii are known
+to exhibit a strong velocity offset between these lines in
+the BLR, including the candidates for BH recoils (eg.,
+Bonning et al. 2007).
+Indeed, such an offset between
+Mg ii and hydrogen lines, which have similar ionization
+potentials, should not be possible if both are emitted
+from nearby parts of the BLR.
+It is worth noting that originally S12 reported that
+Hα and Hβ broad lines are asymmetric, while Mg ii
+is symmetric, contrary to the claim that all lines are
+symmetric in this work. This is because there does not
+appear to be a strong asymmetry indicative of different
+velocity components in the Balmer lines. It may rather
+be attributed to the clumpy nature of the BLR leading
+to a weak skew of the shapes of broad lines (Risaliti et al.
+2005, 2002; references in Elitzur & Shlosman 2006).
+Several mechanisms or explanations were considered
+in S12 to describe SDSS 0956+5128, but none seems
+capable of producing all of the observed features. The
+ideas included multiple objects along the line of sight,
+accretion disk winds, special morphological configura-
+tions or a recoiling black hole. It is unclear which other
+known mechanism might be responsible for the behavior
+observed in SDSS 0956+5128. However, one way to dis-
+tinguish between possible mechanisms is that different
+explanations make distinct predictions for the velocities
+of higher-ionization lines.
+To examine the applicability of different scenarios, ad-
+ditional observations of high-ionization lines, like C iv,
+are needed. Combining it with the high-resolution pho-
+tometry of the central region allows a test of these hy-
+potheses.
+Here, new Hubble Space Telescope (HST) observations
+of C iv and Lyα lines and photometry of the central
+region of the host galaxy are presented and analyzed.
+Based on these data, we attempt to distinguish between
+the previously proposed mechanisms. As none of them
+appears to match these observations, a new mechanism
+is proposed to explain SDSS 0956+5128.
+New HST measurements of the previously unobserved
+C iv and Lyα BELs in the UV spectrum (§ 2) and pho-
+tometric observations of the host galaxy at high resolu-
+tion (§ 3) are presented in the following sections. The
+broad UV lines are symmetric and appear to have the
+offset consistent with the Balmer lines. No irregulari-
+ties are seen in the photometry of the central region of
+the galaxy. While the QSO is seen as spatially offset,
+it is within 3σ resolution. § 4 reviews hypotheses that
+were previously proposed to explain SDSS 0956+5128,
+such as a combination of the recoiling BH scenario
+and a double-peaked emitter, and describes why it is a
+challenging problem. Instead, the cumulative evidence
+shows a peculiar velocity pattern of the BLR consistent
+with a strong shock outflow from the central region. Dis-
+cussion of the observations and a new hypothesis along
+with final thoughts are presented in § 5.
+This work adopts a flat ΛCDM cosmology with Ωm =
+0.3, ΩΛ = 0.7, and H0 = 70 km s−1 Mpc−1 throughout.
+2. HST SPECTRUM
+The
+near
+ultraviolet
+(NUV)
+spectrum
+of
+SDSS
+0956+5128 was recorded with the Space Telescope Imag-
+ing Spectrograph (STIS) onboard the HST (Proposal
+15872; Steinhardt et al. 2019). The first order G230L
+long slit grating was used allowing for spatially re-
+solved spectroscopy with 0.025′′/pix (∼ 180 pc/pix at
+z = 0.7142) and low-to-medium spectral resolution with
+R ∼ 500 − 1010 (translates to 600 − 300 km s−1 reso-
+
+3
+lution at short to long wavelengths) in the NUV. The
+HST pipeline CALSTIS (Sohn 2019) used to reduce the
+spectrum allowed the detection of C iv and Lyα lines.
+The NUV spectrum allowed the detection of C iv and
+Lyα lines.
+The background in the spectrum was fitted with the
+exponential function in linear wavelength assuming that
+the continuum is dominated by the quasar emission.
+The wavelength windows used for the fit (1290 − 1460,
+1580−1810 ˚A) were expected to have little or no contam-
+ination from prominent line emission or Fe ii,iii pseudo-
+continuum described in Vestergaard & Wilkes (2001)
+and were similar to the clear quasar-continuum windows
+in Francis et al. (1991). The slope of the model contin-
+uum was used to validate the quasar nature of the spec-
+trum. Calculated as Fν1/Fν2 = (ν1/ν2)β, it was found
+to be β = −1.18±0.01, which loosely agrees with the ob-
+served distribution of slopes of typical quasars, although
+in the redder part (Davis et al. 2007). The model was
+used for the calculation as a cross-check, as the observed
+spectrum did not cover the wavelengths normally used
+above 1850˚A.
+The background-subtracted C iv and Lyα are shown
+in Figure 1. The systemic Lyα line is best fit with a
+Gaussian component at z = 0.7135 ± 0.0003, at 2.3σ
+deviation from the systemic redshift measured in S12.
+The line profile is strongly asymmetric, with the most
+likely contribution from a blueshifted broad Lyα compo-
+nent. This component is best fit at z = 0.6909 ± 0.0026
+with FWHM ∼ 11511 ± 1781 km s−1. This profile is
+consistent with the observations of Balmer lines in S12,
+where the narrow peaks are accompanied by the com-
+pletely blue-shifted broad emission. When considering
+other possible contributions to this blueshifted excess, Si
+iii λ1206.5 and O v λ1218 lines coincide with the pro-
+file at z = 0.707 and z = 0.690, respectively (associated
+with two offsets identified in S12). However, neither the
+expected narrow width of these lines, nor the expected
+low flux can explain the strong broad excess that is best
+described by the broad Lyα emission.
+Additionally, there appears to be a broad excess of
+flux in the red wing of the profile, which could not be
+caused by either Si iii, O v, Fe ii or Fe iii lines at any
+one of the involved redshifts. Including this component
+of the profile improves the fit. However, the widths and
+positions of the broad Lyα and the unknown component
+become less constrained, as reflected in their uncertain-
+ties (see Table 1). The small bump at rest wavelength
+1240 ˚A suggests that the unknown broad component
+may be the offset N v emission. With this, the fit of the
+whole profile was produced with χ2
+ν = 0.77, where the
+suspected broad N v line had FWHM = 2907 ± 2862
+km s−1 at z = 0.6914 ± 0.0051. There is however no
+distinct narrow N v λ1240 component. The offset of the
+broad Lyα and N v components is at 1.4 and 3.2σ levels,
+respectively.
+The redshift of C iv is measured at z = 0.6907±0.0008
+with χ2
+ν = 0.69 per degree of freedom. Although due to
+low spectral resolution and spectral purity the line ap-
+pears to be strongly affected by the noise, it is above 4.2σ
+noise level. The spectrum was resampled to improve its
+signal to noise ratio. No other distinct emission or ab-
+sorption features could be observed; however, the line
+can usually be contaminated by some narrow features.
+As the left wing of the profile at 1490 − 1520 ˚A (rest
+frame) could be affected by the Fe ii and Fe iii and Si
+ii at 1531 ˚A emission at the systemic redshift (Vester-
+gaard & Wilkes 2001), the width of the C iv fit was
+cross-checked by estimating the BH mass.
+The em-
+pirical mass estimator from Dalla Bont`a et al. (2020)
+for single epoch measurements yields log(MBH/M⊙) =
+8.80 ± 0.16, which is 1.0σ away from the Mg ii-based
+estimate of log(MBH/M⊙) = 8.65 from S12. The re-
+lations based on the C iv line width are less well con-
+strained than those made with the Hβ line mainly due
+to the lack of observations in the UV. The literature
+on the most commonly identified agreements and differ-
+ences in the estimators has been summarized in Dalla
+Bont`a et al. (2020). Their empirical estimator was cal-
+ibrated against identified correlations in the residuals
+with the reverberation mapping measurements. In this
+specific case if the Fe emission acted to decrease the
+width of the C iv profile, this would only make the dis-
+agreement with other estimators worse. Therefore, the
+cross-check above is used as the main argument for the
+broad C iv line producing dominant contribution to the
+profile. It is concluded that the C iv profile is observed
+as emission coming entirely from the BLR with the ve-
+locity offset. However, it should be stated that applying
+mass estimators to strong outflows makes the estimates
+not trustworthy and speculative.
+3. HST IMAGES
+As part of the same proposal (15872), ACS/WFC
+camera of the HST was used to take 6 dithered images
+in F606W and 4 in F850LP filters to produce the com-
+bined images totaling 2075 and 2300 second exposures,
+respectively.
+The individual exposures were aligned,
+background-subtracted and drizzled by grizli pipeline
+(Brammer & Matharu 2021). The final products repre-
+sent the combined mosaics of these dithered images for
+each filter.
+
+4
+Rusakov et al.
+Figure 1. Lyα λ1215 and C iv λ1549 spectrum profiles. Vertical dotted lines indicate the systemic redshift of line centroids.
+Lyα profile shows significant flux excess at ∼ 2055 ˚A in the observed frame, fitted here as the offset broad Lyα emission. The
+excess of flux in the long-wavelength tail is suggested to arise from the offset N v broad emission, as there is a hint at the narrow
+N v λ1240 line, which is unresolved in this spectrum. The spectrum is resampled to ∼ 3 ˚A per bin. C iv profile does not appear
+affected by strong emission or absorption features when resampled to ∼ 6 ˚A per bin. Verifying the BH mass using the C iv
+estimator suggests the width of the fitted profile is the real C iv width (see text). The line appears completely offset from the
+host quasar redshift.
+Table 1. Measurements of lines in NUV spectrum (this work), optical and NIR (S12).
+The offsets are stated with respect to the systemic redshift (z = 0.7142) reported in
+S12.
+Line
+Component
+Redshift
+FWHM (km s−1)
+Offset (km s−1)
+N v λ1240?
+BEL
+0.6914 ± 0.0051
+4917 ± 4840
+-3996 ± 893
+C iv λ1549
+BEL
+0.6907 ± 0.0008
+12491 ± 953
+-4107 ± 143
+Lyα λ1215
+BEL
+0.6909 ± 0.0026
+11511 ± 1781
+-4079 ± 457
+Hα λ6563
+BEL
+0.690
+∼7200
+-4100
+Hβ λ4861
+BEL
+0.690
+∼7200
+-4100
+Mg ii λ2798
+BEL
+0.7071 ± 0.0006
+12800 ± 490
+-1200
+Lyα λ1215
+NEL
+0.7135 ± 0.0003
+1978 ± 565
+-123 ± 48
+[O iii] λ4959
+NEL
+0.714
+[O iii] λ5007
+NEL
+0.714
+[O ii] λ3727
+NEL
+0.714
+[Ne iii] λ3881
+NEL
+0.714
+
+[A]
+Irest-frame
+1160
+1200
+1240
+1450
+1500
+1550
+1600
+1.5
+CIV
+Lya
+6
+cm
+1.0
+一
+[erg 
+0.5
+2
+NV
+X
+0
+0.0
+1950
+2000
+2050
+2100
+2150
+2500
+2600
+2700
+入[A]5
+The reconstruction of the quasar emission was per-
+formed by using an effective PSF (ePSF) constructed
+with photutils package (Bradley et al. 2020) and based
+on 6 foreground stars in the images utilised here. It is
+an empirical model based on the selection of stars in
+the images (Anderson & King 2000; Anderson 2016).
+It is produced by simply measuring the flux of stellar
+sources in the vicinity of the target at individual pixels
+and it represents a map of fractional flux produced by a
+point source given the optics and the detector sensitiv-
+ity, i.e. the instrumental PSF scaled by the pixel sensi-
+tivity map. This method was shown to be more precise
+and numerically efficient than deconvolving the photom-
+etry and instrumental effects, assuming and then fitting
+an analytical function. The advantages are particularly
+justified, when only a single point source is investigated.
+Figure 2 shows the residuals after fitting and subtract-
+ing the best-fit PSF from the images. The fit was per-
+formed using the 2D image profile fitting code IMFIT
+(Erwin 2015) with reduced χ2
+r = 2.83 for F606W and
+χ2
+r = 1.36 for F850LP images.
+With the pixel scale
+of 0.03′′ (0.22 kpc per pixel) the images allowed to re-
+solve the host structure around the AGN. The figure
+shows both the residuals in terms of the instrumental
+read-errors and the residuals in units of Poisson σ de-
+fined using the average of the counts in the observed
+and model images. These maps indicate that the resid-
+ual host emission around the AGN is most significant in
+the F606W band (up to 5σ, Poisson), while the emis-
+sion in F850LP (up to 2 − 3σ) is shallower and more
+extended outside of the central region than in F606W.
+This is roughly in agreement with the relative amount
+of flux of the model SED emission of the host galaxy at
+the respective wavelengths presented in S12.
+The residuals in either filter do not indicate any dis-
+ruption or deformation in the central structure of the
+apparently elliptical light profile, which would be ex-
+pected from a recent galaxy merger. However, it was not
+possible to perform a detailed study of the host galaxy
+profile due to the overall shallow photometry. The cyan
+‘x’-marks indicate the best-fit centers of the quasar emis-
+sion, which are misplaced from the center of the isopho-
+tal host emission by 1.1 (0.24 kpc) and 1.4 (0.31 kpc)
+pixels in F606W and F850LP, respectively. Given the
+resolution, the offset up to ∼ 1.5 (3σ) pixels is allowed,
+which makes the best-fit QSO location consistent with
+the center of the host emission.
+4. PREVIOUSLY PROPOSED EXPLANATIONS
+There exist several mechanisms that are responsible
+for creating different distinct velocity components in
+BLRs of quasars.
+However, none of them appears to
+provide a full explanation for the observed features of
+SDSS 0956+5128. Those include outflows, quasar mor-
+phology and orientation or quasar motion with respect
+to its host that could correspond to one or both of the
+velocity components in SDSS 0956+5128. Specifically,
+S12 considered multiple objects along the line of sight,
+double-peaked emitter profile and a recoiling black hole.
+Another considered mechanism is accretion disk winds
+that may be responsible for velocity profiles typically
+seen in most of the QSOs with outflows. This section
+describes how the previous and new evidence from SDSS
+0956+5128 fits within these scenarios and shows that
+none of them are capable of explaining the observation
+completely.
+4.1. Multiple objects along the line of sight
+Perhaps the simplest explanation for multiple velocity
+components would be multiple objects along the same
+line of sight. However, this explanation cannot produce
+the broad lines observed in SDSS 0956+5128, There is
+a unique set of narrow lines, consistent in their red-
+shift, and two velocity components, represented by ei-
+ther broad Mg ii or Balmer lines, but not both. With
+two objects along the line of sight, two sets of narrow
+lines would have to be present, even if there was a spe-
+cific combination with a strong Mg ii broad emission
+and very weak Balmer lines in one object and a very
+weak Mg ii and strong Balmer emission in the other.
+In addition, the narrow lines argue against this expla-
+nation. Narrow Balmer lines at the systemic velocity
+indicate that the central region of the presumed host
+galaxy is illuminated by the quasar. The broad Balmer
+lines are bluer than the host. Thus, if along the same
+line of sight, they would need to be in front of the host
+and unable to produce these features. Therefore, a sce-
+nario with three distinct components in the same system
+has to be considered: emission of the host (z = 0.714)
+in the NLR, Mg ii (z = 0.707) and Balmer (z = 0.690)
+emission in the BLR.
+4.2. Disk winds
+In the context of a single QSO system, one common
+mechanism thought to be responsible for outflows in the
+BLR is accretion disk winds (e.g., models by Murray
+et al. 1995; Mathews & Blumenthal 1977). In support
+of these models, Gaskell (1985) found evidence that BLR
+clouds could be radiatively accelerated. It was proposed
+and modeled that radiation pressure could be respon-
+sible for accelerating the gas clouds radially outwards
+(Emmering et al. 1992; Everett 2005).
+In such a scenario, the velocity of the outflow is high-
+est for the high-ionization material of the BLR and de-
+
+6
+Rusakov et al.
+Figure 2. HST/ACS images of SDSS 0956+5128: observed image (first column); model PSF (second column); positive
+residuals after subtracting the PSF in units of instrumental σ-noise (third column); residuals after subtracting the PSF in
+units of Poisson σ (fourth column). The top and bottom rows shows photometry in F606W and F850LP filters, respectively.
+Contours show the residual host emission without strong eccentricity or possible post-merger disruption (smoothed with a
+3.3-pixel Gaussian kernel). The outer-most isophotes extend to the 50th percentile in F606W and F850LP indicating that the
+emission is more centrally concentrated in F606W and more extended in F850LP. The best-fit centers of the quasar emission
+are marked with the cyan crosses, where the PSF centers are offset by 1.1 (F606W) and 1.4 (F850LP) pixels from the isophotal
+centers. This offset is within the 3σ offset of ∼ 1.5 pixels). The scale is 0.03 arcsec per pixel (0.22 kpc).
+creases for species with lower-ionization energies. Exam-
+ples of such profiles can be seen in Meyer et al. (2019);
+Shen et al. (2016); Marziani et al. (2010, 1996); Broth-
+erton et al. (1994). As the ionization potentials are cor-
+related with the radial distances of emission lines, the
+lines like N v, C iv and other high-ionization lines ex-
+perience some of the largest relative velocities reaching
+several thousands km per second (Yu et al. 2021). The
+intermediate- to low-ionization emission, including hy-
+drogen lines and Mg ii is typically seen accelerated to at
+most a few hundred km s−1. Often Hα, Hβ and Mg ii
+are consistent with the systemic velocity to within ∼ 200
+km s−1 (Shen et al. 2016; Hewett & Wild 2010).
+Although such a negative velocity gradient could pro-
+duce a difference in emission line velocities, the differ-
+ence seen between the hydrogen lines and Mg ii in SDSS
+0956+5128 is too steep and too high for the lines of
+such similar ionization potentials. For instance, Meyer
+et al. (2019) show that the offset velocities of Mg ii and
+other low-ionization lines are typically identical in spec-
+troscopically observed SDSS quasars, with velocity shifts
+well within a few hundred km s−1 at z < 7. Moreover,
+in the rare cases where Mg ii blueshifts of ∼ 103 km s−1
+are reported, they are generally connected with other
+mechanisms (as in the recoiling black hole candidate,
+3C 186, in Chiaberge et al. 2017).
+The similarity between ionization potentials of Hβ
+and Mg ii alone may not be indicative of their physi-
+cal proximity, especially if the lines originate from fully
+and partially-ionized regions and their emission is driven
+by different excitation mechanisms. However, the corre-
+lation between the radius of the species in the BLR and
+the accompanying continuum luminosity (at λ = 5100˚A
+and 3000 ˚A) indicates that they are likely coming from
+the adjacent gas shells in the BLR. Figure 3 shows es-
+timates with several of such empirical relations. They
+may not be applicable to all quasars, because they are
+based on different sample selections. However, any pair
+of radius estimates show that Hβ and Mg ii here are
+closely spaced and Hβ is closer to the center on average.
+
+Observed
+PSF
+Residuals  (instrumental) Residuals  (Poisson
+0
+20
+4060 80
+-12-
+-30
+12
+F606W
+51°28'24.5"
+24.0"
+Declination (J2000)
+23.5"
+23.0"
+F850LP
+51028'24.5″
+24.0"
+23.5"
+23.0″
+1 kpc
+gh56m32.55s
+32.45s 32.40s
+9h56m32.55s
+32.45s 32.40s
+gh56m32.55s
+32.45s 32.40s
+gh56m32.55s
+32.45s 32.40s
+Right ascention (J2000)7
+Thus, if a negative velocity gradient is responsible for
+the extreme offsets of the hydrogen and Mg ii lines in
+SDSS 0956+5128, there should be an even larger differ-
+ence when Mgii is compared with high-ionization lines.
+For example, such trend was found in the offsets of emis-
+sion lines of I Zw 1 (Laor et al. 1997; Vestergaard &
+Wilkes 2001) and other quasars (Corbin 1990; Espey
+et al. 1989; Wilkes 1986). However, as shown in § 2,
+this is inconsistent with the new HST observations pre-
+sented in this work, especially with the C iv, possibly N
+v and the hydrogen lines being symmetric and part of
+the same velocity system at −4100 km s−1.
+4.3. QSO jets
+Alternatively, QSO jets that provide a way of losing
+the angular momentum for the supermassive black hole
+(SMBH) can cause QSO outflows (Zheng et al. 1990).
+Even though the jets are highly collimated, they may
+cause outflows in the BLR assuming a high covering
+factor of the clouds.
+It was shown in Zheng et al.
+(1990) that double-peaked Balmer emission can be pro-
+duced in such AGN models.
+However, this does not
+explain the symmetric single-peaked lines observed in
+SDSS 0956+5128.
+4.4. Double-peaked emitter
+In another scenario, large line shifts of over 4000
+km s−1 observed in some galaxies can be described by
+models with non-axisymmetric accretion disks (Strateva
+et al. 2003): flattened, eccentric disks (or other forms of
+asymmetries) and a preferred inclination angle (Chen &
+Halpern 1989; Eracleous et al. 1995). The problem in
+this case is that BELs in such systems produce double or
+asymmetric profiles. S12 reported that the Balmer lines
+in SDSS 0956+5128 have asymmetric profiles, with a
+faint, broad component and that Mg ii, in contrast, is
+symmetric. However, the asymmetry appeared only in
+one of three independent observations (S12).
+If the model of a double-peaked emitter is allowed to
+have sufficiently many parameters, it is possible to pro-
+duce a reasonable fit to the spectrum in most cases.
+However, in this case one double-peaked emitter is not
+able to explain scenarios in which different lines show
+different offsets. At least two eccentric emitter compo-
+nents are required to provide an explanation for the two
+offsets, but still not sufficient to produce a symmetric
+(non-double) Mg ii, Lyα and C iv (and possibly N v),
+unless these lines have double profiles with the second
+component being very weak.
+4.5. Recoiling black hole
+Finally, S12 suggested that SDSS 0956+5128 may be a
+recoiling BH in a post-merger galaxy, in which the BLR
+is a combination of eccentric and circular components. A
+SMBH resulting from coalescence of two smaller SMBHs
+can receive a kick in some direction, depending on the ro-
+tation properties of the system (Campanelli et al. 2007b;
+Schnittman & Buonanno 2007; Loeb 2007).
+It was shown that such a BH can still exhibit symmet-
+ric BELs with the offset of over 1000 km s−1 (Merritt
+et al. 2006; Loeb 2007). However, an offset of ∼ 4000
+km s−1 would be difficult to produce. Some analytical
+and numerical considerations limit the maximum recoil
+velocity to ∼ 4000 km s−1 (Baker et al. 2008; Campan-
+elli et al. 2007a,b) or even lower (Healy et al. 2014),
+while others can reach up to ∼ 5000 km s−1 for special
+configurations (Lousto et al. 2012). In observations, one
+potential candidate is the object CID-42 (Civano et al.
+2010, 2012; Blecha et al. 2013), with the offset of up
+to 1300 km s−1 detected in the broad Balmer emission.
+Another candidate for a SMBH recoil with more BEL
+detections, 3C 186 Chiaberge et al. (2017), shows the
+highest known offset of ∼ 2100 km s−1, which is second
+only to SDSS 0956+5128 with the offset nearly twice as
+high (∼ 4100 km s−1). In addition, the spectroscopic
+velocity profile of 3C 186 appears to be constant with
+respect to the ionization potential of C iv, C iii] and
+Lyα, as well as Mg ii.
+In SDSS 0956+5128, C iv (and possibly N v) and the
+hydrogen lines are consistent with this scenario; how-
+ever, Mg ii is not. Given the similarity of the ionization
+potentials of hydrogen and Mg ii lines, it seems very
+unlikely that the Mg ii region could be moving ∼ 2900
+km s−1 slower than the rest of the recoiling BLR. One
+possibility is to assume that Mg ii is consistent with
+the recoil scenario and all of the BLR is offset by at
+least the offset of Mg ii (−1200 km s−1). Then even
+slightly higher-ionization lines, including the hydrogen
+lines, could arise due to an eccentric emitter disk that in-
+troduces additional offset on top of the recoil and causes
+asymmetry of the Balmer lines. Therefore, in this sce-
+nario, the Balmer and Mg ii emission must come from
+separate locations, with the latter being further out and
+the higher-ionization lines, like C iv or N v, must also
+have the same asymmetry, as reported for the Balmer
+lines in S12.
+In support of the idea of the recoiling BH, S12 showed
+using ground-based photometry that the peak of inten-
+sity in SDSS 0956+5128 was possibly offset from the
+center of light in the host galaxy, although the data had
+low resolution and systematic effects in PSF fitting were
+not ruled out. Additionally, photometric decomposition
+showed that the host is a dusty galaxy, which can be
+consistent with various states of galaxy evolution, in-
+cluding a possible post-merger. S12 showed that based
+
+8
+Rusakov et al.
+on the timescale for the BLR to sustain emission after
+the quasar accretion was disrupted, the recoil could have
+occurred in the past 140 Myr. This could allow for the
+host galaxy to preserve the evidence of a recent merger
+on a less than dynamical time, detectable in the follow-
+up resolved observations.
+However, the new evidence presented from HST does
+not hold any indications of a BH recoil that could re-
+sult from a previous galaxy merger. It is shown in § 3
+that the QSO, fitted with a PSF, is spatially consis-
+tent with location of the isophotal center of the resid-
+ual galaxy emission and no strong irregularities in the
+structure are seen that could result from a recent galaxy
+merger, although the photometry is not deep enough to
+accurately fit the host morphology.
+It remains possi-
+ble that the strong signatures of the merger could have
+been erased to the point of being undetected at the given
+resolution. Nevertheless, new observations of the high-
+ionization line C iv and low-ionization Lyα appear to be
+consistent with the offset of the Balmer lines, they are
+symmetric (see Section 2). The symmetry is contrary to
+the expectation from the BH recoil hypothesis. It was
+also shown by S12 that the system lies on the standard
+MBH −Mbulge galaxy mass relation, with the virial mass
+of the SMBH of log(MBH/M⊙) = 8.65 (S12) lying close
+to the empirical relation in the plane of the galaxy lumi-
+nosity and the black hole mass. Therefore, this does not
+provide any sharp indications that the SMBH does not
+belong to the host. In this context, the evidence sug-
+gests that the offsets in SDSS 0956+5128 do not appear
+to be caused by the motion of the quasar, but are rather
+intrinsic to the QSO.
+In § 5, we argue that the radial velocity profile in
+the BLR of SDSS 0956+5128 is reminiscent of the post-
+shock outflow and use the idea of star formation in the
+accretion disks (eg., from Goodman & Tan 2004) to sup-
+port this mechanism for causing strong outflows in the
+BLR of quasars.
+5. INTERPRETATION
+The observations of SDSS 0956+5128 appear to be in-
+consistent with known physical mechanisms and suggest
+that an outflow with very distinct velocity signature is
+responsible.
+The event that produced the outflow must have an
+origin at the BH or close to it.
+This appears to be
+the case, because the symmetry of the BELs suggests
+a spherically uniform outflow. Additionally, the whole
+BLR appears to be affected, where the radial velocity
+is constant across a large fraction of the BLR, between
+the high-ionization region and low-ionization hydrogen
+lines. Finally, there appears to be a drop-off of the out-
+Figure 3. Compilation of various calculations of the radii
+of Hβ and Mg ii obtained using the respective luminosities
+λL5100 and λL3000 of the continuum in SDSS 0956+5128. In-
+dividual estimates are shown with 1-σ uncertainties. The av-
+erage radii are shown as vertical dashed lines with their corre-
+sponding shaded uncertainty bars. The estimates are based
+on the empirical radius-luminosity relations for quasars.
+Some of the studies use very different sample selections, such
+as different accretion rates
+˙
+M in Du et al. (2018), to define
+the relations.
+Nevertheless, they produce similar radii for
+the two species here, where Hβ is on average is closer to the
+center than Mg ii.
+flow velocity starting from the location where Mg ii is
+produced. Therefore, it may be safe to assume any ori-
+gin of such event within the inner boundary of the BLR,
+with its effects limited to the BLR outskirts.
+Clearly, some extreme physical conditions must be at
+play. The event has to be consistent with the outflow
+energetically and with the observed radial velocity pro-
+file, as indicated by the offsets of the broad lines and
+their expected radial distances. The upper limit on the
+time of the event can be placed by using the velocity of
+the outflow.
+Below, a shock wave from an energetic explosion is
+postulated as a mechanism for causing an outflow. It
+is shown that this mechanism can be consistent with
+the observed features in SDSS 0956+5128.
+Then the
+plausibility of the extreme physical conditions due to a
+supernova explosion is considered, along with a discus-
+sion of whether stars can exist in close proximity of a
+SMBH and how they can appear there in the first place.
+5.1. Post-shock outflow
+The velocity profile of the outflow as a function of
+radius in the BLR of SDSS 0956+5128 is strongly rem-
+iniscent of that of a “shocked” material (Taylor 1950),
+as it shown in this section. It is expected that the radial
+profile in terms of increasing radial distances is as fol-
+lows: C iv, Lyα, Hα, Hβ, Mg ii, where Mg ii must be at
+least as far out as the Balmer line emission. Based on
+the spectroscopic observations made in Section 2 here
+
+Hβ
+Mg IH
+Fonseca Alvarez+2020
+U+2022
+Du+2018 (high M)
+Du+2018 (low M)
+Bentz+2013
+Prince+2022
+Trakhtenbrot & Netzer (2012)
+Czerny+2019
+Vestergaard & Osmer (2009)
+Zajacek+2020
+0.03
+0.06
+0.09
+0.12
+0.15
+R [pc]9
+and in S12, the outflow velocity profile is constant at
+∆V ≈ −4100 km s−1 starting at the location of the C
+iv line and extending out to the hydrogen lines. Then
+the radial velocity drops to −1200 km s−1 at the loca-
+tion of Mg ii. Assuming the velocities estimated from
+several spectra across an extended timeline are accurate,
+there would be only one explanation for different offset
+velocities of the Balmer lines and Mg ii: the latter line
+must originate at larger radii due to its lower ionization
+energy. In this case, the overall velocity profile is char-
+acteristic of a post-shock outflow, such as the outflows
+in the interstellar medium (ISM) produced by supernova
+(SN) explosions.
+Assuming a uniform density of the medium, the evolu-
+tion of shocks can be described as a three-phase process
+that starts with a free expansion at high pressure rela-
+tive to the surroundings, during which the mass of the
+medium swept up by the shock wave along the direction
+of travel is insignificant compared to the mass of the
+shock material itself. Thus, the energy and momentum
+of the wave are conserved and the kinetic energy Ek is:
+Ek = MejV 2
+ej/2,
+(1)
+where Mej and Vej are the mass and velocity of the
+ejecta.
+As the shock shell expands in radius, it is opposed
+by a larger mass of the ambient medium.
+When the
+two masses become equal, the shock enters the Sedov-
+Taylor phase (Taylor 1950). There, the ejecta start to
+lose its momentum, while the matter is too heated to
+irradiate, so the system remains adiabatic. At this stage,
+the expansion of the shock radius RS with time t is found
+to depend entirely on the initial kinetic energy Ek and
+density of the medium ρ0:
+Rs(t) ∝ E1/5
+k
+ρ−1/5
+0
+t2/5.
+(2)
+Finally, as the temperature of the shock drops, C, N,
+O ions start to recombine helping to cool the shock ma-
+terial efficiently, which starts to loose its energy radia-
+tively until the flow becomes subsonic and merges with
+the surrounding medium.
+This evolution can be matched with the velocity com-
+ponents observed in SDSS 0956+5128, as illustrated by
+the velocity and distance profiles of a shock wave in Fig-
+ure 4.
+In this interpretation, the velocities presented
+here sample the first two phases: C iv, Lyα, Hβ and Hα
+represent the region in the free expansion phase, while
+Mg ii is in the Sedov-Taylor region. Also, in the analysis
+above the high-ionization N v line1 was possibly identi-
+fied with the offset placing it consistently with the lines
+in the first phase. This phase is sampled well with 4 or
+5 lines probably spanning a significant fraction of the
+BLR radial profile. However, the second phase is only
+seen with one BEL (Mg ii). Fortunately, there have been
+two observations of this line 7 years apart to confirm the
+detection.
+The velocity profile of the shock stages shown in Fig-
+ure 4 is annotated with the offset velocities of the corre-
+sponding broad lines. While the source energy sets the
+initial velocity of the flow, the density of the BLR sets
+the distance and time scale of the shock profile. Here,
+it was assumed that the density is uniform throughout
+the region and is equal to the lower limit of ne = 109
+cm−3 in the BLR (Osterbrock 1989; Kwan & Krolik
+1981). Higher densities act to shrink the distance and
+time scales.
+The velocity that the ambient BLR material gains
+when it crosses over the shock boundary depends on the
+ratio between the speed of sound and the shock. The
+speed of sound in the BLR gas is cs ≈ 10 km s−1, assum-
+ing the BLR is in photoionization equilibrium and has
+the uniform temperature of T = 104 K, which is the min-
+imum required for photoionization (Osterbrock 1989).
+Therefore, the outflows observed in SDSS 0956+5128
+(∼ 4100 and 1200 km s−1) are strongly supersonic.
+Hence, the shock wave should be expected even more
+so, such that the limit as the Mach number M → ∞
+for M = v/cs can be safely used. In the reference frame
+with the shock at rest, the velocity of the post-shock
+material (v2, downstream) can be related to the veloc-
+ity of the pre-shock material (v1, upstream) using the
+Rankine-Hugoniot jump condition:
+v1
+v2
+=
+(γ + 1)M2
+(γ + 1) + (γ − 1)(M2 − 1),
+(3)
+where γ = 5/3 is the adiabatic index for an ideal
+monoatomic gas. The assumption here is that the flow is
+adiabatic and the entropy is constant across the shock
+boundary, which holds for the first two stages of the
+shock wave, before it becomes subsonic and its viscos-
+ity cannot be neglected leading to the start of radia-
+tive cooling. At M → ∞, v2 → 0.25v1. Therefore, we
+would expect the shock to have traversed the locations
+of the observed broad emission at v1 ≈ 16400 km s−1
+and v1 ≈ 4800 km s−1 to produce the high (v2 ∼ 4100
+km s−1) and low (v2 ∼ 1200 km s−1) observed velocity
+components in the BLR, respectively.
+1 N v is the line with the highest ionization potential in our sample,
+which places it closest to the SMBH.
+
+10
+Rusakov et al.
+Interestingly, under the assumption of the BLR gas
+density of ne = 109 cm−3 two key model predictions are
+in agreement with the independent expectations from
+the empirical radius-luminosity (RL) relations.
+First, the model correctly predicts the location of the
+Mg ii gas. This prediction is made by using the time
+when the calculated downstream flow (black solid line;
+left panel in Fig. 4) matches the observed velocity off-
+set of Mg ii (orange solid line): t ≈ 27 yr. Based on
+this time the predicted radius of the Mg ii line is 0.131
+pc (blue arrow; right panel). This closely agrees with
+the mean of the estimates from various RL relations:
+RMgII = 0.117 ± 0.005 pc (see Figure 3; Prince et al.
+2022; Zajaˇcek et al. 2020; Czerny et al. 2019; Trakhten-
+brot & Netzer 2012; Vestergaard & Osmer 2009).
+Second, the location of the boundary between the first
+two phases (≈ 0.057 pc) agrees with the location of Hβ
+from RL relations (blue solid line; right panel).
+The
+average of several RL estimators yields RHβ = 0.082 ±
+0.006 pc (see Figure 3; U et al. 2022; Fonseca Alvarez
+et al. 2020; Du et al. 2018; Bentz et al. 2013). This is
+consistent with the observation that the hydrogen lines
+are part of the high velocity component.
+These two
+predictions are possible only for a narrow range of BLR
+gas densities around ne = 109 cm−3 and for a constant
+density as a function of radius, where the time scale
+goes as t ∝ n−1/3
+e
+in the first phase and t ∝ n−1/5
+e
+in the
+second.
+Predicting the mass and type of the candidate super-
+nova star is beyond the scope of this work, but it is
+possible to estimate an order of magnitude mass of the
+ejecta and their kinetic energy. The ejecta mass can be
+obtained by locating the boundary between the first two
+shock regions. This radial distance encloses the mass of
+the swept up gas comparable to the mass of the shocked
+ejecta, as this is the approximate condition for termina-
+tion of the first phase.
+The boundary must lie between the hydrogen lines
+(eg., Hβ) and Mg ii, which are close spatially (see Fig-
+ure 3), but correspond to different velocity components
+in the BLR. Assuming Hβ is at the boundary (as shown
+in the right panel of Figure 4), we can use the estimated
+radius of the boundary of 0.057 pc and the BLR density
+of ne = 109 cm−3 to estimate the ejecta mass.
+This
+produces Mej ∼ 2 × 104 M⊙.
+With the velocities of the BEL offsets in the first
+phase, the kinetic energy of the event is EK = 5 × 1055
+erg2. Here, it is assumed that most of this mass is con-
+2 This energy is 104 times higher than the kinetic energy of the
+ejecta of a core-collapse SN explosion of an 8 M⊙ star.
+tained in a thin shell at the radius of Hβ. Therefore, the
+exploded star must have had an initial mass M∗ ≥ Mej.
+Such objects have not been observed before, although
+theoretical considerations of instability growth in the
+accretion disks of quasars suggest a possibility for forma-
+tion of stars with masses at least as high as 102−107 M⊙
+(Goodman & Tan 2004).
+5.2. Origin of stars in AGN
+If a stellar explosion produced the observed features
+in the spectrum of SDSS 0956+5128, how could a star
+appear in the vicinity of a SMBH in the first place?
+And how could it be nearly as massive as 2 × 104 M⊙?
+The star might be formed either in the proximity of the
+SMBH or in a host galaxy and then captured into the
+accretion disk. One way or another, it probably owes its
+large mass to the abundance of material in the accretion
+disk of a SMBH.
+The growth of supermassive stars may stem from the
+capture of stars from the host galaxy. The closest galac-
+tic source of stars is the bulge that encircles the quasar.
+Stochastic encounters can result in significant changes
+of stellar velocities on a relaxation timescale that could
+lead to stars being deflected off their orbits. A “lucky”
+star brought to the central object with the mass 108.6
+M⊙ would not risk being disrupted due to tidal forces,
+unless it sinks into the black hole. The reason is the tidal
+radius of BHs with masses > 108 M⊙ becomes smaller
+than the Schwarzschild radius. In fact, sinking into the
+BH would be a more likely outcome, as otherwise reach-
+ing an exact stable orbit around the BH would require
+acquiring a very specific velocity making it very unlikely.
+Following on the idea of disk-star interactions as a
+potential mechanism for removal of angular momentum
+from AGNs (Ostriker 1983), Syer et al. (1991) investi-
+gated a possibility for stars to be captured by the ac-
+cretion disk for favorable inclinations of stellar orbits.
+This mechanism was characterised by shorter than the
+relaxation timescale for delivering stars into the disk. It
+could be possible as the growth of the gaseous disk and
+clouds would be expected to extend as far as to the edge
+of influence of the BH for the stellar bulge. Stars would
+then be dragged down by the viscous forces to the stable
+circular orbit over time. If such stellar migration is real,
+then according to the predictions of Artymowicz et al.
+(1993), after the QSO acquires a disk and turns on, in
+the span of 108 yr there can be as many as 104 captured
+stars. They would be dragged down the disk, enmassed
+by the disk material to as high as ∼ 102 M⊙ and evolve
+on their respective main-sequence timescale leading to
+Type II SNe. For the case of SDSS 0956+5128 a single
+core collapse supernova with such initial mass would not
+
+11
+Figure 4. Velocity (left) and distance (right) profiles of a shock wave produced by an explosion of a star associated with the
+ejecta mass Mej = 2×104M⊙ and kinetic energy EK = 5×1055 erg. The BLR medium has a uniform density (ne = 109 cm−3);
+other main assumptions are stated in the text. Shadowed regions represent different stages of a shock wave traveling through
+the medium: free expansion (momentum and energy conserving); Sedov-Taylor phase (adiabatic; energy conserving); snowplow
+phase (radiative and momentum conserving). Left panel: two observed offset velocities of BELs are indicated by the horizontal
+solid lines and correspond to different stages by color, as deduced from their predicted downstream flow velocities (black solid
+line). The predicted downstream velocity profile is calculated only for the first two phases, before the shock becomes nearly
+sonic and valid approximations break. Right panel: predicted radial profile of the shock wave (black dashed line); the observed
+radii of Hβ and Mg ii are obtained using the mean of the estimates from the RL relations and shown by the horizontal lines
+with the same colors as before
+; the yellow arrow indicates the Mg ii radius predicted from the time when the shock reaches Mg ii location, i.e. the time when
+the downstream flow coincides with the observed velocity of Mg ii in the left panel; the blue arrow show the location of the
+boundary between the first two phases of the shock flow.
+be able to explain the observed BLR offsets. Although
+∼ 102 − 103 such SNe simultaneously would, this could
+hardly be a plausible scenario. In addition, it was ar-
+gued that the stars might more likely be destroyed by
+the release of energy for circularization of their orbit in
+the accretion disk, as a very limited set of conditions
+for entry must be met to stay bound (Goodman & Tan
+2004).
+On the other hand, Goodman & Tan (2004) consid-
+ered in situ star formation and growth of massive stars.
+The main argument that stars can and possibly should
+form in BH accretion disks was that starting from a
+distance of ∼ 103 RS, where RS is the Schwarzschild ra-
+dius, the disks are dominated by self-gravity and are
+expected to fragment via local dynamical instability.
+Goodman & Tan (2004) considered fragmentation at the
+exact boundary, where the radiation pressure equals the
+gas pressure, as at this location, any fragments would be
+well-separated and therefore likely to grow via accretion
+of the surrounding gas. The authors showed that given
+the strong accretion in the disk, the star is more likely
+to grow than fragment.
+The properties of the protostar would be defined by
+the instability.
+As mentioned previously, the initial
+overdensity masses could reach anywhere in the range
+102−107M⊙ and are likely to grow by further accretion.
+The mass growth continues until the gap roughly of the
+size of the protostar’s Roche lobe is cleared in the accre-
+tion disk. Finally, the material in the Roche lobe would
+contract until the star reaches the main sequence. Such
+stars are likely to be supported by radiation pressure
+and experience significant mass loss, although the levels
+are not known for the very high masses over 102 M⊙.
+In the theoretical framework of Goodman & Tan
+(2004), the mass of a protostar in a disk of 108.6 M⊙
+SMBH is expected to be ∼ 105M⊙, if it forms at the
+edge of the low self-gravity of the disk. The likely out-
+come of the SNe of stars with mass > 102 M⊙ is a com-
+plete collapse to a BH (Fryer et al. 2001), although an
+instability from e+e− pair production may occur that
+
+101
+104
+NVCIV
+Lyα Hα Hβ
+100
+Mg II
+103
+Mg II
+S
+10-1
+Hβ
+R
+U
+10-2
+101
+10-3
+Shock
+Downstream BLR
+100
+-2
+-1
+0
+1
+2
+3
+4
+5
+6
+7
+-2
+-1
+0
+1
+2
+3
+4
+5
+6
+7
+log1o(t [yr])12
+Rusakov et al.
+would convert all of the stellar mass to ejecta at explo-
+sion. Therefore the scenario in which a star with an ini-
+tial mass 105 M⊙ evolves into a less massive ∼ 104 M⊙
+star by mass loss and causes a pair-instability supernova
+that ejects all of the material in an explosion would be
+consistent with the observation in SDSS 0956+5128.
+Following Goodman & Tan (2004), if the star is stable
+on the main sequence, it can be expected to migrate
+inwards, preserving the disk gap, and reach the tidal
+radius on the time (∼ 105 yr) shorter than the main
+sequence time (∼ 106 yr).
+It would be of particular
+interest here to know whether the star is expected to
+disintegrate once on the main sequence or if it survives
+for some parameters. Future numerical simulations of
+such processes would be of particular interest to answer
+these questions.
+If star formation in accretion disks is common, how
+frequently would an SDSS 0956+5128-like event occur?
+Goodman & Tan (2004) estimate that if star formation
+in a QSO happens at a rate of one per viscous time,
+then given the approximate number density of QSOs at
+redshifts 0.5 < z < 1.5, 10−5 massive stars per QSO per
+year should form, for a total of a few stars per year in a
+survey the size of SDSS.
+However, most of those stars will not produce an ob-
+servable supernova. Goodman & Tan (2004) argued that
+sinking into the SMBH is the typical outcome. Fresh,
+hydrogen-rich material from the accretion disk can mix
+into the convective stellar core, so that the main se-
+quence lifetime becomes longer than the timescale for
+reaching the event horizon. Thus, their deaths would
+not be observable.
+Perhaps if the accretion disk becomes sparse towards
+either near the Eddington luminosity or towards the end
+of a quasar’s lifetime, there might be just a high enough
+density to form a massive star, yet not enough additional
+material to sustain that star until it reaches the event
+horizon. The resulting supernovae might even provide
+a turnoff mechanism, so that quasars die with a bang
+rather than with a whimper. As SDSS 0956+5128 lies
+at z = 0.7, below which relatively few quasars are ob-
+served, it is likely to turn off in the near future. It is
+therefore plausible to associate the extreme features of
+SDSS 0956+5128 with a short-lived turnoff mechanism.
+If such a process lasts for ∼ 103 − 104 yr (based on the
+timeline of the shock wave in Fig. 4), finding one exam-
+ple in all of SDSS is plausible.
+6. DISCUSSION
+It is argued here that none of the common and less
+so mechanisms, except for the shock wave, are likely to
+produce a spectroscopic signature observed in the BLR
+of SDSS 0956+5128.
+The scenarios that involve spe-
+cial morphology, alignment of multiple objects, radia-
+tion output of the QSO or SMBH mergers each have
+at least one sharp inconsistency with the evidence (see
+Section 4). Instead, this work proposes the idea of shock
+waves to explain the observed spectroscopic offsets. The
+toy model requires the tuning of only the BLR den-
+sity parameter to match the velocity offsets in SDSS
+0956+5128, as well as the location of Mg ii line.
+To support or reject the idea of the supernova shock
+causing the outflow in the BLR it is necessary to collect
+more observational evidence. Below is the list with a
+couple of possibilities.
+• Other offset lines? The shock mechanism predicts
+that there must be a continuum of velocity compo-
+nents in the second phase of the shocked BLR (see
+Section 5.1 and Figure 4), as opposed to only one
+that was observed. Although the most prominent
+BEL in the second phase (Mg ii) was already ob-
+served, lower-ionization lines in the NUV, like O i,
+Si ii or C ii are not detected in the HST spectrum
+here. Instead, they could be targeted with higher
+signal-to-noise spectra. Additionally, Fe Kα line
+in the X-rays would be expected to have the high
+velocity offset of 4100 km s−1, as the other lines of
+that velocity component. Probably, the direction
+of the offset and the line symmetry would depend
+on the position relative to the potential SN explo-
+sion site in the accretion disk. If any of these lines
+are inconsistent with the prediction, this will rule
+out the hypothesis.
+• Traces of neutrinos and electromagnetic emission.
+From another perspective, SN shocks act as neu-
+trino production sites, through pion decays as a
+result of proton-proton interactions. It was sug-
+gested that the diffuse neutrino background may
+be substantially contributed to by AGN stellar ex-
+plosions (Zhu et al. 2021). In order to detect these
+events, it was shown (though for much less mas-
+sive stars) that short bursts in the diffuse neutrino
+background and the ensuing lasting electromag-
+netic emission can be used. Therefore, such signa-
+ture may be used to search for potential candidates
+for follow-up observations or for testing models of
+neutrino and electromagnetic emission produced
+by energetic stellar explosions.
+After all, the shock hypothesis here is an analytical so-
+lution based on the simple assumptions of shock-medium
+interaction and lacks a more detailed insight. Therefore,
+numerical modeling of the involved processes would be
+
+13
+required to test the idea more robustly. Below are the
+suggestions for possible experiments and predictions.
+• Downstream velocity as function of time.
+The
+BLR gas with different offsets starts mixing in the
+snowplow phase altering the velocity of the BLR
+outflow. If one can calculate the outflow velocity
+of every one of the observed emission lines as a
+function of time, it will be possible to predict the
+blueshift that the lines should have at a time of a
+possible future observation. Besides, it will also be
+possible to predict how the strength of the broad
+line emission will change over time, as the emitting
+gas moves away from its ionization zone. Addi-
+tionally, the density of the BLR assumed here cor-
+responds to a rather conservative lower limit and
+it has a uniform radial profile, which can be tested
+by modeling the broad line emission observed in
+SDSS 0956+5128.
+• Numerical simulation of stars in a strong gravi-
+tational potential. One of the main problems in
+the shock hypothesis is what caused it. The only
+possibility to create a shock wave of the required
+energy known to us is a potential supernova. If
+the outflow profile in SDSS 0956+5128 is a signa-
+ture of an SN (see Section 5), then the mass of the
+ejecta is estimated to be ∼ 104 M⊙ and the kinetic
+energy is Ek ∼ 1055 erg.
+It is unclear whether
+stars can be “safely” delivered to accretion disks
+of SMBHs and then grow to such extent by accre-
+tion, but there exist theoretical models that pre-
+dict formation of such supermassive stars in these
+extreme environments in the first place. To shed
+more light on the possibility of the supermassive
+stars leading to SNe shocks, numerical simulations
+of their formation and evolution are required. In
+this regard, the most important questions are: (1)
+whether a stellar explosion is close to being spher-
+ical in a strong gravitational field (to offset a BLR
+entirely rather than in the plane of the disk); (2)
+whether a proto-star can remain stable to reach
+and stay on the stellar main sequence; (3) if the
+star’s main sequence lifetime is shorter than the
+timescale for spiralling down into the SMBH; (4)
+if the “spiralling down” timescale wins, whether
+it may be reversed in special cases when the QSO
+shuts down accretion and turns off.
+Solving the problem posed by SDSS 0956+5128 may
+shed new light on the evolution of quasars and forma-
+tion of massive stars in extreme environments. There
+is a potential for studying the physics of star formation
+and evolution in QSO disks, as potentially a few stars
+per year may form in an SDSS-sized survey of quasars.
+In addition, if stellar explosions are shown to be possible,
+this may improve the understanding of extreme stellar
+evolution and BLR outflows. However, it is less clear
+how frequent supernovae may be. If the offset emission
+of the BLR in SDSS 0956+5128 is the result of an SN, it
+is likely only one such event in the SDSS, as no similar
+candidates have been reported. Although, new multi-
+line observations of the evolved quasars with strong out-
+flows may reveal more of these objects in the future. On
+the other hand, it is believed that they are not observ-
+able and more likely to cross the event horizon before
+they explode, because their lifetimes are prolonged by
+the freshly accreted gas. Although, this may not be the
+case in sparse accretion disks of the old quasars, where
+little gas may not be enough to support the stars before
+sinking into the SMBH. Therefore, if the occurrence rate
+is low because the SN-induced velocity offsets of BLRs
+in old quasars are transient on some timescale, then it is
+possible that SNe are more frequent and could also play
+an important role in turning off QSOs.
+We would like to greatly thank Gabriel Brammer for
+help with the data processing, Martin Pessah, Darach
+Watson, Marianne Vestergaard, Johan Fynbo, Lise
+Christensen, Jeremy Goodman, Adam Jermyn, Ersilia
+Guarini and Kasper Elm Heintz for helpful and insight-
+ful discussions. The Cosmic Dawn Center (DAWN) is
+funded by the Danish National Research Foundation un-
+der grant No. 140.
+This work made use of data stored at Barbara A.
+Mikulski Archive for Space Telescopes (MAST), NASA
+Astrophysics Data System Bibliographic Services and
+the NASA/IPAC Extragalactic Database (NED). The
+HST photometry and spectroscopy used in this paper
+can be found in MAST: 10.17909/6g1c-9b66.
+Facilities: HST
+Software:
+astropy (Astropy Collaboration et al.
+2018,
+2013),
+astroquery
+(Ginsburg
+et
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+2019),
+photutils (Bradley et al. 2020), matplotlib (Hunter
+2007), numpy (Harris et al. 2020), scipy (Virtanen et al.
+2020), Source Extractor (Bertin & Arnouts 1996).
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+
diff --git a/RtAzT4oBgHgl3EQf0P7C/content/tmp_files/load_file.txt b/RtAzT4oBgHgl3EQf0P7C/content/tmp_files/load_file.txt
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+page_content='Draft version January 6, 2023  Typeset using LATEX twocolumn style in AASTeX631  A broad-line quasar with unexplained extreme velocity offsets: post-shock outflow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Vadim Rusakov  ,1, 2 Charles L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Steinhardt  ,2, 3 Malte Schramm  ,4 Andreas L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Faisst  ,5  Daniel Masters  ,5 Bahram Mobasher  ,6 and Petchara Pattarakijwanich  7  1Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, København Ø 2100, Denmark rusakov124@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='com  2Cosmic Dawn Center (DAWN)  3Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, København Ø 2100, Denmark  4Graduate school of Science and Engineering, Saitama Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
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+page_content=' Saitama City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Saitama 338-8570,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' JAPAN  5California institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1200 E California Blvd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' USA  6Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Riverside,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 900 University Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Riverside,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' CA 92521,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' USA  7Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Faculty of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Mahidol University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 272 Rama IV Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Ratchathewi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Bangkok,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Thailand  ABSTRACT  The quasar SDSS 0956+5128 exhibits three distinct velocity components with large offsets in emis-  sion: the systemic velocity of [O ii],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' [O iii] and [Ne iii] narrow lines have redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7142;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' broad  Mg ii line is shifted by −1200 km s−1 with respect to the narrow lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' broad Hα, Hβ lines are at  −4100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' We present new Hubble Space Telescope spectra of Lyα and C iv emission lines and  high-resolution images of the quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The offsets of these lines are consistent with the velocity com-  ponent of the Balmer emission, and the photometry in optical and near-infrared wavelengths does not  show any signs of recent mergers in the host galaxy or irregularities in the location of the quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  data do not confirm predictions of the previous most-likely hypotheses involving a special orientation  and morphology of the quasar disk, such as in the recoiling black hole scenario, neither it is consistent  with accretion disk winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Instead, based on the cumulative evidence, we propose a new scenario, in  which the broad line region is in the state of outflow caused by a strong shock wave, with a supernova  as a possible event for producing the shock ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Keywords: Quasars(1319) — Shocks(2086) — Supernovae(1668)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' INTRODUCTION  Although there initially appeared to be several sub-  types of active galactic nuclei (AGN) and quasars (QSO)  (Antonucci 1993), it has been believed for approximately  three decades that nearly all observed AGN are con-  sistent with being similar objects observed from differ-  ent lines of sight (Urry & Padovani 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Bailey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Lawrence & Elvis 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' An-  tonucci & Miller 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' other references in Antonucci  1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' One consequence is that all quasar spectra ex-  hibit the same set of significant spectral lines: (1) broad  emission (or absorption) lines (BEL) including C iv,  Lyα, Hα, Hβ and Mg ii, among others, with Doppler  widths of ∼ 104 km s−1 (Vanden Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Mur-  ray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Wills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1993), that are normally  virialized (Shapovalova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Dietrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Korista et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2) narrow emission lines, including  [O iii]λλ4959,5007, [Ne ii]λ3869 and [O ii]λ3727 with  widths of ∼ 500 − 1000 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' and (3) a wide range  of possible narrow emission or absorption lines from the  host galaxy, with widths of < 500 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' These ob-  servations have led to a physical picture in which this  emission originates, respectively, from a broad-line re-  gion (BLR) ∼ 1 pc from the central supermassive black  hole, a narrow-line region (NLR) ∼ 103 pc away, and  the remainder of the host galaxy, which might extend  to ∼ 104 − 105 pc out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Although inflows or outflows  might skew the profiles of these lines (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Strateva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Eracleous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Gaskell 1982), they all em-  anate from objects in the same host galaxy, and thus  are likely centered at the same Doppler velocity (equiv-  alently, redshift) with respect to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  However, a single quasar among the more than 105  quasars found in Sloan Digital Sky Survey (SDSS) ap-  pears to be entirely incompatible with this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' SDSS  J095632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='49+512823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='92 (hereafter, SDSS 0956+5128) ex-  hibits three distinct and non-typical features that in  combination make this object unique (Steinhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  2012) (hereafter, S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='01782v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='HE]  4 Jan 2023  ID2  Rusakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  At first, there are three significantly different veloc-  ity components, corresponding to z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='714, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='707,  and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The narrow line emission of [O ii], [O  iii] and [Ne iii] provides the systemic redshift of the  galaxy z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The blueshift of broad Balmer emis-  sion lines and Mg ii from the host galaxy is ∼ 4100  and ∼ 1200 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It is not surprising  on its own to observe three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' For example,  such objects as I Zw 1 are known to exhibit more than  two velocity systems with blueshifts from the systemic  redshift (Laor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Vestergaard & Wilkes 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  However, unlike such quasars, the broad emission lines  in SDSS 0956+5128 are not double-peaked or strongly  skewed (as in the examples that were studied in detail in  Eracleous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Tsalmantza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Eracleous  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' They are symmetric and completely offset  and therefore appear to be consistent with some kind of  physical offset of the BLR, such as an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Only a handful of objects studied in outflows have  been observed with nearly as high offsets in Hβ, Hα and  Mg ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In fact, offsets of the similar magnitude have  been found more often in higher-ionization lines, such  as Si iv, C iv or higher, and even so in the extreme tail  of their offset distribution, as shown by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2021)  in SDSS DR7 (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2011) and other quasar sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The objects with symmetricaly-offset lines have  been studied in connection with the recoiling black holes  (BH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Bonning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In fact, no recoil candidates  have been identified with the BLR velocity as high as in  the SDSS 0956+5128 which is almost twice the second  highest offset (Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In addition, the broad Mg ii line, which is typically  found to be consistent with the systemic velocity (Shen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Hewett & Wild 2010), is offset by −1200  km s−1 from the host lines, and thus by +2900 km s−1  from Hα and Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' No other quasar spectra in the SDSS  that have coverage of Balmer lines and Mg ii are known  to exhibit a strong velocity offset between these lines in  the BLR, including the candidates for BH recoils (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=',  Bonning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Indeed, such an offset between  Mg ii and hydrogen lines, which have similar ionization  potentials, should not be possible if both are emitted  from nearby parts of the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  It is worth noting that originally S12 reported that  Hα and Hβ broad lines are asymmetric, while Mg ii  is symmetric, contrary to the claim that all lines are  symmetric in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This is because there does not  appear to be a strong asymmetry indicative of different  velocity components in the Balmer lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It may rather  be attributed to the clumpy nature of the BLR leading  to a weak skew of the shapes of broad lines (Risaliti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  2005, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' references in Elitzur & Shlosman 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Several mechanisms or explanations were considered  in S12 to describe SDSS 0956+5128, but none seems  capable of producing all of the observed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  ideas included multiple objects along the line of sight,  accretion disk winds, special morphological configura-  tions or a recoiling black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It is unclear which other  known mechanism might be responsible for the behavior  observed in SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, one way to dis-  tinguish between possible mechanisms is that different  explanations make distinct predictions for the velocities  of higher-ionization lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  To examine the applicability of different scenarios, ad-  ditional observations of high-ionization lines, like C iv,  are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Combining it with the high-resolution pho-  tometry of the central region allows a test of these hy-  potheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Here, new Hubble Space Telescope (HST) observations  of C iv and Lyα lines and photometry of the central  region of the host galaxy are presented and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Based on these data, we attempt to distinguish between  the previously proposed mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' As none of them  appears to match these observations, a new mechanism  is proposed to explain SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  New HST measurements of the previously unobserved  C iv and Lyα BELs in the UV spectrum (§ 2) and pho-  tometric observations of the host galaxy at high resolu-  tion (§ 3) are presented in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  broad UV lines are symmetric and appear to have the  offset consistent with the Balmer lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' No irregulari-  ties are seen in the photometry of the central region of  the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' While the QSO is seen as spatially offset,  it is within 3σ resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' § 4 reviews hypotheses that  were previously proposed to explain SDSS 0956+5128,  such as a combination of the recoiling BH scenario  and a double-peaked emitter, and describes why it is a  challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Instead, the cumulative evidence  shows a peculiar velocity pattern of the BLR consistent  with a strong shock outflow from the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Dis-  cussion of the observations and a new hypothesis along  with final thoughts are presented in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  This work adopts a flat ΛCDM cosmology with Ωm =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='3, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7, and H0 = 70 km s−1 Mpc−1 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' HST SPECTRUM  The  near  ultraviolet  (NUV)  spectrum  of  SDSS  0956+5128 was recorded with the Space Telescope Imag-  ing Spectrograph (STIS) onboard the HST (Proposal  15872;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Steinhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The first order G230L  long slit grating was used allowing for spatially re-  solved spectroscopy with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='025′′/pix (∼ 180 pc/pix at  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7142) and low-to-medium spectral resolution with  R ∼ 500 − 1010 (translates to 600 − 300 km s−1 reso-  3  lution at short to long wavelengths) in the NUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  HST pipeline CALSTIS (Sohn 2019) used to reduce the  spectrum allowed the detection of C iv and Lyα lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The NUV spectrum allowed the detection of C iv and  Lyα lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The background in the spectrum was fitted with the  exponential function in linear wavelength assuming that  the continuum is dominated by the quasar emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The wavelength windows used for the fit (1290 − 1460,  1580−1810 ˚A) were expected to have little or no contam-  ination from prominent line emission or Fe ii,iii pseudo-  continuum described in Vestergaard & Wilkes (2001)  and were similar to the clear quasar-continuum windows  in Francis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The slope of the model contin-  uum was used to validate the quasar nature of the spec-  trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Calculated as Fν1/Fν2 = (ν1/ν2)β, it was found  to be β = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='01, which loosely agrees with the ob-  served distribution of slopes of typical quasars, although  in the redder part (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The model was  used for the calculation as a cross-check, as the observed  spectrum did not cover the wavelengths normally used  above 1850˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The background-subtracted C iv and Lyα are shown  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The systemic Lyα line is best fit with a  Gaussian component at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0003, at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='3σ  deviation from the systemic redshift measured in S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The line profile is strongly asymmetric, with the most  likely contribution from a blueshifted broad Lyα compo-  nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This component is best fit at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6909 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0026  with FWHM ∼ 11511 ± 1781 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This profile is  consistent with the observations of Balmer lines in S12,  where the narrow peaks are accompanied by the com-  pletely blue-shifted broad emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' When considering  other possible contributions to this blueshifted excess, Si  iii λ1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5 and O v λ1218 lines coincide with the pro-  file at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='707 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='690, respectively (associated  with two offsets identified in S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, neither the  expected narrow width of these lines, nor the expected  low flux can explain the strong broad excess that is best  described by the broad Lyα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Additionally, there appears to be a broad excess of  flux in the red wing of the profile, which could not be  caused by either Si iii, O v, Fe ii or Fe iii lines at any  one of the involved redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Including this component  of the profile improves the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, the widths and  positions of the broad Lyα and the unknown component  become less constrained, as reflected in their uncertain-  ties (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The small bump at rest wavelength  1240 ˚A suggests that the unknown broad component  may be the offset N v emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' With this, the fit of the  whole profile was produced with χ2  ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='77, where the  suspected broad N v line had FWHM = 2907 ± 2862  km s−1 at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6914 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' There is however no  distinct narrow N v λ1240 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The offset of the  broad Lyα and N v components is at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='2σ levels,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The redshift of C iv is measured at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6907±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0008  with χ2  ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='69 per degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Although due to  low spectral resolution and spectral purity the line ap-  pears to be strongly affected by the noise, it is above 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='2σ  noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The spectrum was resampled to improve its  signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' No other distinct emission or ab-  sorption features could be observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' however, the line  can usually be contaminated by some narrow features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  As the left wing of the profile at 1490 − 1520 ˚A (rest  frame) could be affected by the Fe ii and Fe iii and Si  ii at 1531 ˚A emission at the systemic redshift (Vester-  gaard & Wilkes 2001), the width of the C iv fit was  cross-checked by estimating the BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The em-  pirical mass estimator from Dalla Bont`a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2020)  for single epoch measurements yields log(MBH/M⊙) =  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='16, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0σ away from the Mg ii-based  estimate of log(MBH/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='65 from S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The re-  lations based on the C iv line width are less well con-  strained than those made with the Hβ line mainly due  to the lack of observations in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The literature  on the most commonly identified agreements and differ-  ences in the estimators has been summarized in Dalla  Bont`a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Their empirical estimator was cal-  ibrated against identified correlations in the residuals  with the reverberation mapping measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In this  specific case if the Fe emission acted to decrease the  width of the C iv profile, this would only make the dis-  agreement with other estimators worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, the  cross-check above is used as the main argument for the  broad C iv line producing dominant contribution to the  profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It is concluded that the C iv profile is observed  as emission coming entirely from the BLR with the ve-  locity offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, it should be stated that applying  mass estimators to strong outflows makes the estimates  not trustworthy and speculative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' HST IMAGES  As part of the same proposal (15872), ACS/WFC  camera of the HST was used to take 6 dithered images  in F606W and 4 in F850LP filters to produce the com-  bined images totaling 2075 and 2300 second exposures,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The individual exposures were aligned,  background-subtracted and drizzled by grizli pipeline  (Brammer & Matharu 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The final products repre-  sent the combined mosaics of these dithered images for  each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  4  Rusakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Lyα λ1215 and C iv λ1549 spectrum profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Vertical dotted lines indicate the systemic redshift of line centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Lyα profile shows significant flux excess at ∼ 2055 ˚A in the observed frame, fitted here as the offset broad Lyα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  excess of flux in the long-wavelength tail is suggested to arise from the offset N v broad emission, as there is a hint at the narrow  N v λ1240 line, which is unresolved in this spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The spectrum is resampled to ∼ 3 ˚A per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' C iv profile does not appear  affected by strong emission or absorption features when resampled to ∼ 6 ˚A per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Verifying the BH mass using the C iv  estimator suggests the width of the fitted profile is the real C iv width (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The line appears completely offset from the  host quasar redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Measurements of lines in NUV spectrum (this work), optical and NIR (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The offsets are stated with respect to the systemic redshift (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7142) reported in  S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Line  Component  Redshift  FWHM (km s−1)  Offset (km s−1)  N v λ1240?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  BEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6914 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0051  4917 ± 4840  3996 ± 893  C iv λ1549  BEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6907 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0008  12491 ± 953  4107 ± 143  Lyα λ1215  BEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6909 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0026  11511 ± 1781  4079 ± 457  Hα λ6563  BEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='690  ∼7200  4100  Hβ λ4861  BEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='690  ∼7200  4100  Mg ii λ2798  BEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7071 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0006  12800 ± 490  1200  Lyα λ1215  NEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0003  1978 ± 565  123 ± 48  [O iii] λ4959  NEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='714  [O iii] λ5007  NEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='714  [O ii] λ3727  NEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='714  [Ne iii] λ3881  NEL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='714  [A]  Irest-frame  1160  1200  1240  1450  1500  1550  1600  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5  CIV  Lya  6  cm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0  一  [erg  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5  2  NV  X  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0  1950  2000  2050  2100  2150  2500  2600  2700  入[A]5  The reconstruction of the quasar emission was per-  formed by using an effective PSF (ePSF) constructed  with photutils package (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2020) and based  on 6 foreground stars in the images utilised here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It is  an empirical model based on the selection of stars in  the images (Anderson & King 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Anderson 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  It is produced by simply measuring the flux of stellar  sources in the vicinity of the target at individual pixels  and it represents a map of fractional flux produced by a  point source given the optics and the detector sensitiv-  ity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' the instrumental PSF scaled by the pixel sensi-  tivity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This method was shown to be more precise  and numerically efficient than deconvolving the photom-  etry and instrumental effects, assuming and then fitting  an analytical function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The advantages are particularly  justified, when only a single point source is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Figure 2 shows the residuals after fitting and subtract-  ing the best-fit PSF from the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The fit was per-  formed using the 2D image profile fitting code IMFIT  (Erwin 2015) with reduced χ2  r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='83 for F606W and  χ2  r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='36 for F850LP images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  With the pixel scale  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='03′′ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='22 kpc per pixel) the images allowed to re-  solve the host structure around the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The figure  shows both the residuals in terms of the instrumental  read-errors and the residuals in units of Poisson σ de-  fined using the average of the counts in the observed  and model images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' These maps indicate that the resid-  ual host emission around the AGN is most significant in  the F606W band (up to 5σ, Poisson), while the emis-  sion in F850LP (up to 2 − 3σ) is shallower and more  extended outside of the central region than in F606W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  This is roughly in agreement with the relative amount  of flux of the model SED emission of the host galaxy at  the respective wavelengths presented in S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The residuals in either filter do not indicate any dis-  ruption or deformation in the central structure of the  apparently elliptical light profile, which would be ex-  pected from a recent galaxy merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, it was not  possible to perform a detailed study of the host galaxy  profile due to the overall shallow photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The cyan  ‘x’-marks indicate the best-fit centers of the quasar emis-  sion, which are misplaced from the center of the isopho-  tal host emission by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='24 kpc) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='31 kpc)  pixels in F606W and F850LP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Given the  resolution, the offset up to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5 (3σ) pixels is allowed,  which makes the best-fit QSO location consistent with  the center of the host emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' PREVIOUSLY PROPOSED EXPLANATIONS  There exist several mechanisms that are responsible  for creating different distinct velocity components in  BLRs of quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  However, none of them appears to  provide a full explanation for the observed features of  SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Those include outflows, quasar mor-  phology and orientation or quasar motion with respect  to its host that could correspond to one or both of the  velocity components in SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Specifically,  S12 considered multiple objects along the line of sight,  double-peaked emitter profile and a recoiling black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Another considered mechanism is accretion disk winds  that may be responsible for velocity profiles typically  seen in most of the QSOs with outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This section  describes how the previous and new evidence from SDSS  0956+5128 fits within these scenarios and shows that  none of them are capable of explaining the observation  completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Multiple objects along the line of sight  Perhaps the simplest explanation for multiple velocity  components would be multiple objects along the same  line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, this explanation cannot produce  the broad lines observed in SDSS 0956+5128, There is  a unique set of narrow lines, consistent in their red-  shift, and two velocity components, represented by ei-  ther broad Mg ii or Balmer lines, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' With  two objects along the line of sight, two sets of narrow  lines would have to be present, even if there was a spe-  cific combination with a strong Mg ii broad emission  and very weak Balmer lines in one object and a very  weak Mg ii and strong Balmer emission in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In addition, the narrow lines argue against this expla-  nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Narrow Balmer lines at the systemic velocity  indicate that the central region of the presumed host  galaxy is illuminated by the quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The broad Balmer  lines are bluer than the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Thus, if along the same  line of sight, they would need to be in front of the host  and unable to produce these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, a sce-  nario with three distinct components in the same system  has to be considered: emission of the host (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='714)  in the NLR, Mg ii (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='707) and Balmer (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='690)  emission in the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Disk winds  In the context of a single QSO system, one common  mechanism thought to be responsible for outflows in the  BLR is accretion disk winds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=', models by Murray  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Mathews & Blumenthal 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In support  of these models, Gaskell (1985) found evidence that BLR  clouds could be radiatively accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It was proposed  and modeled that radiation pressure could be respon-  sible for accelerating the gas clouds radially outwards  (Emmering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Everett 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In such a scenario, the velocity of the outflow is high-  est for the high-ionization material of the BLR and de-  6  Rusakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' HST/ACS images of SDSS 0956+5128: observed image (first column);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' model PSF (second column);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' positive  residuals after subtracting the PSF in units of instrumental σ-noise (third column);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' residuals after subtracting the PSF in  units of Poisson σ (fourth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The top and bottom rows shows photometry in F606W and F850LP filters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Contours show the residual host emission without strong eccentricity or possible post-merger disruption (smoothed with a  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='3-pixel Gaussian kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The outer-most isophotes extend to the 50th percentile in F606W and F850LP indicating that the  emission is more centrally concentrated in F606W and more extended in F850LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The best-fit centers of the quasar emission  are marked with the cyan crosses, where the PSF centers are offset by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='1 (F606W) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='4 (F850LP) pixels from the isophotal  centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This offset is within the 3σ offset of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The scale is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='03 arcsec per pixel (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='22 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  creases for species with lower-ionization energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Exam-  ples of such profiles can be seen in Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Marziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2010, 1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Broth-  erton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' As the ionization potentials are cor-  related with the radial distances of emission lines, the  lines like N v, C iv and other high-ionization lines ex-  perience some of the largest relative velocities reaching  several thousands km per second (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  intermediate- to low-ionization emission, including hy-  drogen lines and Mg ii is typically seen accelerated to at  most a few hundred km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Often Hα, Hβ and Mg ii  are consistent with the systemic velocity to within ∼ 200  km s−1 (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Hewett & Wild 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Although such a negative velocity gradient could pro-  duce a difference in emission line velocities, the differ-  ence seen between the hydrogen lines and Mg ii in SDSS  0956+5128 is too steep and too high for the lines of  such similar ionization potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' For instance, Meyer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2019) show that the offset velocities of Mg ii and  other low-ionization lines are typically identical in spec-  troscopically observed SDSS quasars, with velocity shifts  well within a few hundred km s−1 at z < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Moreover,  in the rare cases where Mg ii blueshifts of ∼ 103 km s−1  are reported, they are generally connected with other  mechanisms (as in the recoiling black hole candidate,  3C 186, in Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The similarity between ionization potentials of Hβ  and Mg ii alone may not be indicative of their physi-  cal proximity, especially if the lines originate from fully  and partially-ionized regions and their emission is driven  by different excitation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, the corre-  lation between the radius of the species in the BLR and  the accompanying continuum luminosity (at λ = 5100˚A  and 3000 ˚A) indicates that they are likely coming from  the adjacent gas shells in the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Figure 3 shows es-  timates with several of such empirical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' They  may not be applicable to all quasars, because they are  based on different sample selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, any pair  of radius estimates show that Hβ and Mg ii here are  closely spaced and Hβ is closer to the center on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content="  Observed  PSF  Residuals  (instrumental) Residuals  (Poisson  0  20  4060 80  12-  30  12  F606W  51°28'24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5"  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0"  Declination (J2000)  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5"  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0"  F850LP  51028\'24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5″  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0"  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5"  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='0″  1 kpc  gh56m32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='55s  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='45s 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='40s  9h56m32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='55s  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='45s 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='40s  gh56m32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='55s  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='45s 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='40s  gh56m32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='55s  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='45s 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='40s  Right ascention (J2000)7  Thus, if a negative velocity gradient is responsible for  the extreme offsets of the hydrogen and Mg ii lines in  SDSS 0956+5128, there should be an even larger differ-  ence when Mgii is compared with high-ionization lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  For example, such trend was found in the offsets of emis-  sion lines of I Zw 1 (Laor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Vestergaard &  Wilkes 2001) and other quasars (Corbin 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Espey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Wilkes 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, as shown in § 2,  this is inconsistent with the new HST observations pre-  sented in this work, especially with the C iv, possibly N  v and the hydrogen lines being symmetric and part of  the same velocity system at −4100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' QSO jets  Alternatively, QSO jets that provide a way of losing  the angular momentum for the supermassive black hole  (SMBH) can cause QSO outflows (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Even though the jets are highly collimated, they may  cause outflows in the BLR assuming a high covering  factor of the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  It was shown in Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  (1990) that double-peaked Balmer emission can be pro-  duced in such AGN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  However, this does not  explain the symmetric single-peaked lines observed in  SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Double-peaked emitter  In another scenario, large line shifts of over 4000  km s−1 observed in some galaxies can be described by  models with non-axisymmetric accretion disks (Strateva  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2003): flattened, eccentric disks (or other forms of  asymmetries) and a preferred inclination angle (Chen &  Halpern 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Eracleous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The problem in  this case is that BELs in such systems produce double or  asymmetric profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' S12 reported that the Balmer lines  in SDSS 0956+5128 have asymmetric profiles, with a  faint, broad component and that Mg ii, in contrast, is  symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, the asymmetry appeared only in  one of three independent observations (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  If the model of a double-peaked emitter is allowed to  have sufficiently many parameters, it is possible to pro-  duce a reasonable fit to the spectrum in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  However, in this case one double-peaked emitter is not  able to explain scenarios in which different lines show  different offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' At least two eccentric emitter compo-  nents are required to provide an explanation for the two  offsets, but still not sufficient to produce a symmetric  (non-double) Mg ii, Lyα and C iv (and possibly N v),  unless these lines have double profiles with the second  component being very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Recoiling black hole  Finally, S12 suggested that SDSS 0956+5128 may be a  recoiling BH in a post-merger galaxy, in which the BLR  is a combination of eccentric and circular components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' A  SMBH resulting from coalescence of two smaller SMBHs  can receive a kick in some direction, depending on the ro-  tation properties of the system (Campanelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2007b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Schnittman & Buonanno 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Loeb 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  It was shown that such a BH can still exhibit symmet-  ric BELs with the offset of over 1000 km s−1 (Merritt  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Loeb 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, an offset of ∼ 4000  km s−1 would be difficult to produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Some analytical  and numerical considerations limit the maximum recoil  velocity to ∼ 4000 km s−1 (Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Campan-  elli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2007a,b) or even lower (Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2014),  while others can reach up to ∼ 5000 km s−1 for special  configurations (Lousto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In observations, one  potential candidate is the object CID-42 (Civano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  2010, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Blecha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2013), with the offset of up  to 1300 km s−1 detected in the broad Balmer emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Another candidate for a SMBH recoil with more BEL  detections, 3C 186 Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2017), shows the  highest known offset of ∼ 2100 km s−1, which is second  only to SDSS 0956+5128 with the offset nearly twice as  high (∼ 4100 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In addition, the spectroscopic  velocity profile of 3C 186 appears to be constant with  respect to the ionization potential of C iv, C iii] and  Lyα, as well as Mg ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In SDSS 0956+5128, C iv (and possibly N v) and the  hydrogen lines are consistent with this scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' how-  ever, Mg ii is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Given the similarity of the ionization  potentials of hydrogen and Mg ii lines, it seems very  unlikely that the Mg ii region could be moving ∼ 2900  km s−1 slower than the rest of the recoiling BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' One  possibility is to assume that Mg ii is consistent with  the recoil scenario and all of the BLR is offset by at  least the offset of Mg ii (−1200 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Then even  slightly higher-ionization lines, including the hydrogen  lines, could arise due to an eccentric emitter disk that in-  troduces additional offset on top of the recoil and causes  asymmetry of the Balmer lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, in this sce-  nario, the Balmer and Mg ii emission must come from  separate locations, with the latter being further out and  the higher-ionization lines, like C iv or N v, must also  have the same asymmetry, as reported for the Balmer  lines in S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In support of the idea of the recoiling BH, S12 showed  using ground-based photometry that the peak of inten-  sity in SDSS 0956+5128 was possibly offset from the  center of light in the host galaxy, although the data had  low resolution and systematic effects in PSF fitting were  not ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Additionally, photometric decomposition  showed that the host is a dusty galaxy, which can be  consistent with various states of galaxy evolution, in-  cluding a possible post-merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' S12 showed that based  8  Rusakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  on the timescale for the BLR to sustain emission after  the quasar accretion was disrupted, the recoil could have  occurred in the past 140 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This could allow for the  host galaxy to preserve the evidence of a recent merger  on a less than dynamical time, detectable in the follow-  up resolved observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  However, the new evidence presented from HST does  not hold any indications of a BH recoil that could re-  sult from a previous galaxy merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It is shown in § 3  that the QSO, fitted with a PSF, is spatially consis-  tent with location of the isophotal center of the resid-  ual galaxy emission and no strong irregularities in the  structure are seen that could result from a recent galaxy  merger, although the photometry is not deep enough to  accurately fit the host morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  It remains possi-  ble that the strong signatures of the merger could have  been erased to the point of being undetected at the given  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Nevertheless, new observations of the high-  ionization line C iv and low-ionization Lyα appear to be  consistent with the offset of the Balmer lines, they are  symmetric (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The symmetry is contrary to  the expectation from the BH recoil hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It was  also shown by S12 that the system lies on the standard  MBH −Mbulge galaxy mass relation, with the virial mass  of the SMBH of log(MBH/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='65 (S12) lying close  to the empirical relation in the plane of the galaxy lumi-  nosity and the black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, this does not  provide any sharp indications that the SMBH does not  belong to the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In this context, the evidence sug-  gests that the offsets in SDSS 0956+5128 do not appear  to be caused by the motion of the quasar, but are rather  intrinsic to the QSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In § 5, we argue that the radial velocity profile in  the BLR of SDSS 0956+5128 is reminiscent of the post-  shock outflow and use the idea of star formation in the  accretion disks (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=', from Goodman & Tan 2004) to sup-  port this mechanism for causing strong outflows in the  BLR of quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' INTERPRETATION  The observations of SDSS 0956+5128 appear to be in-  consistent with known physical mechanisms and suggest  that an outflow with very distinct velocity signature is  responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The event that produced the outflow must have an  origin at the BH or close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  This appears to be  the case, because the symmetry of the BELs suggests  a spherically uniform outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Additionally, the whole  BLR appears to be affected, where the radial velocity  is constant across a large fraction of the BLR, between  the high-ionization region and low-ionization hydrogen  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Finally, there appears to be a drop-off of the out-  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Compilation of various calculations of the radii  of Hβ and Mg ii obtained using the respective luminosities  λL5100 and λL3000 of the continuum in SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In-  dividual estimates are shown with 1-σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The av-  erage radii are shown as vertical dashed lines with their corre-  sponding shaded uncertainty bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The estimates are based  on the empirical radius-luminosity relations for quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Some of the studies use very different sample selections, such  as different accretion rates  ˙  M in Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2018), to define  the relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Nevertheless, they produce similar radii for  the two species here, where Hβ is on average is closer to the  center than Mg ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  flow velocity starting from the location where Mg ii is  produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, it may be safe to assume any ori-  gin of such event within the inner boundary of the BLR,  with its effects limited to the BLR outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Clearly, some extreme physical conditions must be at  play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The event has to be consistent with the outflow  energetically and with the observed radial velocity pro-  file, as indicated by the offsets of the broad lines and  their expected radial distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The upper limit on the  time of the event can be placed by using the velocity of  the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Below, a shock wave from an energetic explosion is  postulated as a mechanism for causing an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It  is shown that this mechanism can be consistent with  the observed features in SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Then the  plausibility of the extreme physical conditions due to a  supernova explosion is considered, along with a discus-  sion of whether stars can exist in close proximity of a  SMBH and how they can appear there in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Post-shock outflow  The velocity profile of the outflow as a function of  radius in the BLR of SDSS 0956+5128 is strongly rem-  iniscent of that of a “shocked” material (Taylor 1950),  as it shown in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It is expected that the radial  profile in terms of increasing radial distances is as fol-  lows: C iv, Lyα, Hα, Hβ, Mg ii, where Mg ii must be at  least as far out as the Balmer line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Based on  the spectroscopic observations made in Section 2 here  Hβ  Mg IH  Fonseca Alvarez+2020  U+2022  Du+2018 (high M)  Du+2018 (low M)  Bentz+2013  Prince+2022  Trakhtenbrot & Netzer (2012)  Czerny+2019  Vestergaard & Osmer (2009)  Zajacek+2020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='15  R [pc]9  and in S12, the outflow velocity profile is constant at  ∆V ≈ −4100 km s−1 starting at the location of the C  iv line and extending out to the hydrogen lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Then  the radial velocity drops to −1200 km s−1 at the loca-  tion of Mg ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Assuming the velocities estimated from  several spectra across an extended timeline are accurate,  there would be only one explanation for different offset  velocities of the Balmer lines and Mg ii: the latter line  must originate at larger radii due to its lower ionization  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In this case, the overall velocity profile is char-  acteristic of a post-shock outflow, such as the outflows  in the interstellar medium (ISM) produced by supernova  (SN) explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Assuming a uniform density of the medium, the evolu-  tion of shocks can be described as a three-phase process  that starts with a free expansion at high pressure rela-  tive to the surroundings, during which the mass of the  medium swept up by the shock wave along the direction  of travel is insignificant compared to the mass of the  shock material itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Thus, the energy and momentum  of the wave are conserved and the kinetic energy Ek is:  Ek = MejV 2  ej/2,  (1)  where Mej and Vej are the mass and velocity of the  ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  As the shock shell expands in radius, it is opposed  by a larger mass of the ambient medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  When the  two masses become equal, the shock enters the Sedov-  Taylor phase (Taylor 1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' There, the ejecta start to  lose its momentum, while the matter is too heated to  irradiate, so the system remains adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' At this stage,  the expansion of the shock radius RS with time t is found  to depend entirely on the initial kinetic energy Ek and  density of the medium ρ0:  Rs(t) ∝ E1/5  k  ρ−1/5  0  t2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  (2)  Finally, as the temperature of the shock drops, C, N,  O ions start to recombine helping to cool the shock ma-  terial efficiently, which starts to loose its energy radia-  tively until the flow becomes subsonic and merges with  the surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  This evolution can be matched with the velocity com-  ponents observed in SDSS 0956+5128, as illustrated by  the velocity and distance profiles of a shock wave in Fig-  ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In this interpretation, the velocities presented  here sample the first two phases: C iv, Lyα, Hβ and Hα  represent the region in the free expansion phase, while  Mg ii is in the Sedov-Taylor region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Also, in the analysis  above the high-ionization N v line1 was possibly identi-  fied with the offset placing it consistently with the lines  in the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This phase is sampled well with 4 or  5 lines probably spanning a significant fraction of the  BLR radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, the second phase is only  seen with one BEL (Mg ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Fortunately, there have been  two observations of this line 7 years apart to confirm the  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The velocity profile of the shock stages shown in Fig-  ure 4 is annotated with the offset velocities of the corre-  sponding broad lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' While the source energy sets the  initial velocity of the flow, the density of the BLR sets  the distance and time scale of the shock profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Here,  it was assumed that the density is uniform throughout  the region and is equal to the lower limit of ne = 109  cm−3 in the BLR (Osterbrock 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Kwan & Krolik  1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Higher densities act to shrink the distance and  time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The velocity that the ambient BLR material gains  when it crosses over the shock boundary depends on the  ratio between the speed of sound and the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  speed of sound in the BLR gas is cs ≈ 10 km s−1, assum-  ing the BLR is in photoionization equilibrium and has  the uniform temperature of T = 104 K, which is the min-  imum required for photoionization (Osterbrock 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Therefore, the outflows observed in SDSS 0956+5128  (∼ 4100 and 1200 km s−1) are strongly supersonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Hence, the shock wave should be expected even more  so, such that the limit as the Mach number M → ∞  for M = v/cs can be safely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In the reference frame  with the shock at rest, the velocity of the post-shock  material (v2, downstream) can be related to the veloc-  ity of the pre-shock material (v1, upstream) using the  Rankine-Hugoniot jump condition:  v1  v2  =  (γ + 1)M2  (γ + 1) + (γ − 1)(M2 − 1),  (3)  where γ = 5/3 is the adiabatic index for an ideal  monoatomic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The assumption here is that the flow is  adiabatic and the entropy is constant across the shock  boundary, which holds for the first two stages of the  shock wave, before it becomes subsonic and its viscos-  ity cannot be neglected leading to the start of radia-  tive cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' At M → ∞, v2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='25v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, we  would expect the shock to have traversed the locations  of the observed broad emission at v1 ≈ 16400 km s−1  and v1 ≈ 4800 km s−1 to produce the high (v2 ∼ 4100  km s−1) and low (v2 ∼ 1200 km s−1) observed velocity  components in the BLR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  1 N v is the line with the highest ionization potential in our sample,  which places it closest to the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  10  Rusakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Interestingly, under the assumption of the BLR gas  density of ne = 109 cm−3 two key model predictions are  in agreement with the independent expectations from  the empirical radius-luminosity (RL) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  First, the model correctly predicts the location of the  Mg ii gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This prediction is made by using the time  when the calculated downstream flow (black solid line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 4) matches the observed velocity off-  set of Mg ii (orange solid line): t ≈ 27 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Based on  this time the predicted radius of the Mg ii line is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='131  pc (blue arrow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This closely agrees with  the mean of the estimates from various RL relations:  RMgII = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='117 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='005 pc (see Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Prince et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Zajaˇcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Czerny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Trakhten-  brot & Netzer 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Vestergaard & Osmer 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Second, the location of the boundary between the first  two phases (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='057 pc) agrees with the location of Hβ  from RL relations (blue solid line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The  average of several RL estimators yields RHβ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='082 ±  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='006 pc (see Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' U et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Fonseca Alvarez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Bentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This is  consistent with the observation that the hydrogen lines  are part of the high velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  These two  predictions are possible only for a narrow range of BLR  gas densities around ne = 109 cm−3 and for a constant  density as a function of radius, where the time scale  goes as t ∝ n−1/3  e  in the first phase and t ∝ n−1/5  e  in the  second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Predicting the mass and type of the candidate super-  nova star is beyond the scope of this work, but it is  possible to estimate an order of magnitude mass of the  ejecta and their kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The ejecta mass can be  obtained by locating the boundary between the first two  shock regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' This radial distance encloses the mass of  the swept up gas comparable to the mass of the shocked  ejecta, as this is the approximate condition for termina-  tion of the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The boundary must lie between the hydrogen lines  (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=', Hβ) and Mg ii, which are close spatially (see Fig-  ure 3), but correspond to different velocity components  in the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Assuming Hβ is at the boundary (as shown  in the right panel of Figure 4), we can use the estimated  radius of the boundary of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='057 pc and the BLR density  of ne = 109 cm−3 to estimate the ejecta mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  This  produces Mej ∼ 2 × 104 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  With the velocities of the BEL offsets in the first  phase, the kinetic energy of the event is EK = 5 × 1055  erg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Here, it is assumed that most of this mass is con-  2 This energy is 104 times higher than the kinetic energy of the  ejecta of a core-collapse SN explosion of an 8 M⊙ star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  tained in a thin shell at the radius of Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, the  exploded star must have had an initial mass M∗ ≥ Mej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Such objects have not been observed before, although  theoretical considerations of instability growth in the  accretion disks of quasars suggest a possibility for forma-  tion of stars with masses at least as high as 102−107 M⊙  (Goodman & Tan 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Origin of stars in AGN  If a stellar explosion produced the observed features  in the spectrum of SDSS 0956+5128, how could a star  appear in the vicinity of a SMBH in the first place?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  And how could it be nearly as massive as 2 × 104 M⊙?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The star might be formed either in the proximity of the  SMBH or in a host galaxy and then captured into the  accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' One way or another, it probably owes its  large mass to the abundance of material in the accretion  disk of a SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The growth of supermassive stars may stem from the  capture of stars from the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The closest galac-  tic source of stars is the bulge that encircles the quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Stochastic encounters can result in significant changes  of stellar velocities on a relaxation timescale that could  lead to stars being deflected off their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' A “lucky”  star brought to the central object with the mass 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6  M⊙ would not risk being disrupted due to tidal forces,  unless it sinks into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The reason is the tidal  radius of BHs with masses > 108 M⊙ becomes smaller  than the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In fact, sinking into the  BH would be a more likely outcome, as otherwise reach-  ing an exact stable orbit around the BH would require  acquiring a very specific velocity making it very unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Following on the idea of disk-star interactions as a  potential mechanism for removal of angular momentum  from AGNs (Ostriker 1983), Syer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (1991) investi-  gated a possibility for stars to be captured by the ac-  cretion disk for favorable inclinations of stellar orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  This mechanism was characterised by shorter than the  relaxation timescale for delivering stars into the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It  could be possible as the growth of the gaseous disk and  clouds would be expected to extend as far as to the edge  of influence of the BH for the stellar bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Stars would  then be dragged down by the viscous forces to the stable  circular orbit over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' If such stellar migration is real,  then according to the predictions of Artymowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  (1993), after the QSO acquires a disk and turns on, in  the span of 108 yr there can be as many as 104 captured  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' They would be dragged down the disk, enmassed  by the disk material to as high as ∼ 102 M⊙ and evolve  on their respective main-sequence timescale leading to  Type II SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' For the case of SDSS 0956+5128 a single  core collapse supernova with such initial mass would not  11  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Velocity (left) and distance (right) profiles of a shock wave produced by an explosion of a star associated with the  ejecta mass Mej = 2×104M⊙ and kinetic energy EK = 5×1055 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The BLR medium has a uniform density (ne = 109 cm−3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  other main assumptions are stated in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Shadowed regions represent different stages of a shock wave traveling through  the medium: free expansion (momentum and energy conserving);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Sedov-Taylor phase (adiabatic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' energy conserving);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' snowplow  phase (radiative and momentum conserving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Left panel: two observed offset velocities of BELs are indicated by the horizontal  solid lines and correspond to different stages by color, as deduced from their predicted downstream flow velocities (black solid  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The predicted downstream velocity profile is calculated only for the first two phases, before the shock becomes nearly  sonic and valid approximations break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Right panel: predicted radial profile of the shock wave (black dashed line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' the observed  radii of Hβ and Mg ii are obtained using the mean of the estimates from the RL relations and shown by the horizontal lines  with the same colors as before  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' the yellow arrow indicates the Mg ii radius predicted from the time when the shock reaches Mg ii location, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' the time when  the downstream flow coincides with the observed velocity of Mg ii in the left panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' the blue arrow show the location of the  boundary between the first two phases of the shock flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  be able to explain the observed BLR offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Although  ∼ 102 − 103 such SNe simultaneously would, this could  hardly be a plausible scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In addition, it was ar-  gued that the stars might more likely be destroyed by  the release of energy for circularization of their orbit in  the accretion disk, as a very limited set of conditions  for entry must be met to stay bound (Goodman & Tan  2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  On the other hand, Goodman & Tan (2004) consid-  ered in situ star formation and growth of massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The main argument that stars can and possibly should  form in BH accretion disks was that starting from a  distance of ∼ 103 RS, where RS is the Schwarzschild ra-  dius, the disks are dominated by self-gravity and are  expected to fragment via local dynamical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Goodman & Tan (2004) considered fragmentation at the  exact boundary, where the radiation pressure equals the  gas pressure, as at this location, any fragments would be  well-separated and therefore likely to grow via accretion  of the surrounding gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The authors showed that given  the strong accretion in the disk, the star is more likely  to grow than fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The properties of the protostar would be defined by  the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  As mentioned previously, the initial  overdensity masses could reach anywhere in the range  102−107M⊙ and are likely to grow by further accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The mass growth continues until the gap roughly of the  size of the protostar’s Roche lobe is cleared in the accre-  tion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Finally, the material in the Roche lobe would  contract until the star reaches the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Such  stars are likely to be supported by radiation pressure  and experience significant mass loss, although the levels  are not known for the very high masses over 102 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In the theoretical framework of Goodman & Tan  (2004), the mass of a protostar in a disk of 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='6 M⊙  SMBH is expected to be ∼ 105M⊙, if it forms at the  edge of the low self-gravity of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The likely out-  come of the SNe of stars with mass > 102 M⊙ is a com-  plete collapse to a BH (Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2001), although an  instability from e+e− pair production may occur that  101  104  NVCIV  Lyα Hα Hβ  100  Mg II  103  Mg II  S  10-1  Hβ  R  U  10-2  101  10-3  Shock  Downstream BLR  100  2  1  0  1  2  3  4  5  6  7  2  1  0  1  2  3  4  5  6  7  log1o(t [yr])12  Rusakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  would convert all of the stellar mass to ejecta at explo-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore the scenario in which a star with an ini-  tial mass 105 M⊙ evolves into a less massive ∼ 104 M⊙  star by mass loss and causes a pair-instability supernova  that ejects all of the material in an explosion would be  consistent with the observation in SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Following Goodman & Tan (2004), if the star is stable  on the main sequence, it can be expected to migrate  inwards, preserving the disk gap, and reach the tidal  radius on the time (∼ 105 yr) shorter than the main  sequence time (∼ 106 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  It would be of particular  interest here to know whether the star is expected to  disintegrate once on the main sequence or if it survives  for some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Future numerical simulations of  such processes would be of particular interest to answer  these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  If star formation in accretion disks is common, how  frequently would an SDSS 0956+5128-like event occur?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Goodman & Tan (2004) estimate that if star formation  in a QSO happens at a rate of one per viscous time,  then given the approximate number density of QSOs at  redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5 < z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='5, 10−5 massive stars per QSO per  year should form, for a total of a few stars per year in a  survey the size of SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  However, most of those stars will not produce an ob-  servable supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Goodman & Tan (2004) argued that  sinking into the SMBH is the typical outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Fresh,  hydrogen-rich material from the accretion disk can mix  into the convective stellar core, so that the main se-  quence lifetime becomes longer than the timescale for  reaching the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Thus, their deaths would  not be observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Perhaps if the accretion disk becomes sparse towards  either near the Eddington luminosity or towards the end  of a quasar’s lifetime, there might be just a high enough  density to form a massive star, yet not enough additional  material to sustain that star until it reaches the event  horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The resulting supernovae might even provide  a turnoff mechanism, so that quasars die with a bang  rather than with a whimper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' As SDSS 0956+5128 lies  at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='7, below which relatively few quasars are ob-  served, it is likely to turn off in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It is  therefore plausible to associate the extreme features of  SDSS 0956+5128 with a short-lived turnoff mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  If such a process lasts for ∼ 103 − 104 yr (based on the  timeline of the shock wave in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 4), finding one exam-  ple in all of SDSS is plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' DISCUSSION  It is argued here that none of the common and less  so mechanisms, except for the shock wave, are likely to  produce a spectroscopic signature observed in the BLR  of SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The scenarios that involve spe-  cial morphology, alignment of multiple objects, radia-  tion output of the QSO or SMBH mergers each have  at least one sharp inconsistency with the evidence (see  Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Instead, this work proposes the idea of shock  waves to explain the observed spectroscopic offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  toy model requires the tuning of only the BLR den-  sity parameter to match the velocity offsets in SDSS  0956+5128, as well as the location of Mg ii line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  To support or reject the idea of the supernova shock  causing the outflow in the BLR it is necessary to collect  more observational evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Below is the list with a  couple of possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Other offset lines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The shock mechanism predicts  that there must be a continuum of velocity compo-  nents in the second phase of the shocked BLR (see  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='1 and Figure 4), as opposed to only one  that was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Although the most prominent  BEL in the second phase (Mg ii) was already ob-  served, lower-ionization lines in the NUV, like O i,  Si ii or C ii are not detected in the HST spectrum  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Instead, they could be targeted with higher  signal-to-noise spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Additionally, Fe Kα line  in the X-rays would be expected to have the high  velocity offset of 4100 km s−1, as the other lines of  that velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Probably, the direction  of the offset and the line symmetry would depend  on the position relative to the potential SN explo-  sion site in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' If any of these lines  are inconsistent with the prediction, this will rule  out the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Traces of neutrinos and electromagnetic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  From another perspective, SN shocks act as neu-  trino production sites, through pion decays as a  result of proton-proton interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' It was sug-  gested that the diffuse neutrino background may  be substantially contributed to by AGN stellar ex-  plosions (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In order to detect these  events, it was shown (though for much less mas-  sive stars) that short bursts in the diffuse neutrino  background and the ensuing lasting electromag-  netic emission can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, such signa-  ture may be used to search for potential candidates  for follow-up observations or for testing models of  neutrino and electromagnetic emission produced  by energetic stellar explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  After all, the shock hypothesis here is an analytical so-  lution based on the simple assumptions of shock-medium  interaction and lacks a more detailed insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore,  numerical modeling of the involved processes would be  13  required to test the idea more robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Below are the  suggestions for possible experiments and predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Downstream velocity as function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  The  BLR gas with different offsets starts mixing in the  snowplow phase altering the velocity of the BLR  outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' If one can calculate the outflow velocity  of every one of the observed emission lines as a  function of time, it will be possible to predict the  blueshift that the lines should have at a time of a  possible future observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Besides, it will also be  possible to predict how the strength of the broad  line emission will change over time, as the emitting  gas moves away from its ionization zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Addi-  tionally, the density of the BLR assumed here cor-  responds to a rather conservative lower limit and  it has a uniform radial profile, which can be tested  by modeling the broad line emission observed in  SDSS 0956+5128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Numerical simulation of stars in a strong gravi-  tational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' One of the main problems in  the shock hypothesis is what caused it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The only  possibility to create a shock wave of the required  energy known to us is a potential supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' If  the outflow profile in SDSS 0956+5128 is a signa-  ture of an SN (see Section 5), then the mass of the  ejecta is estimated to be ∼ 104 M⊙ and the kinetic  energy is Ek ∼ 1055 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  It is unclear whether  stars can be “safely” delivered to accretion disks  of SMBHs and then grow to such extent by accre-  tion, but there exist theoretical models that pre-  dict formation of such supermassive stars in these  extreme environments in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' To shed  more light on the possibility of the supermassive  stars leading to SNe shocks, numerical simulations  of their formation and evolution are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' In  this regard, the most important questions are: (1)  whether a stellar explosion is close to being spher-  ical in a strong gravitational field (to offset a BLR  entirely rather than in the plane of the disk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (2)  whether a proto-star can remain stable to reach  and stay on the stellar main sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (3) if the  star’s main sequence lifetime is shorter than the  timescale for spiralling down into the SMBH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' (4)  if the “spiralling down” timescale wins, whether  it may be reversed in special cases when the QSO  shuts down accretion and turns off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Solving the problem posed by SDSS 0956+5128 may  shed new light on the evolution of quasars and forma-  tion of massive stars in extreme environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' There  is a potential for studying the physics of star formation  and evolution in QSO disks, as potentially a few stars  per year may form in an SDSS-sized survey of quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  In addition, if stellar explosions are shown to be possible,  this may improve the understanding of extreme stellar  evolution and BLR outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' However, it is less clear  how frequent supernovae may be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' If the offset emission  of the BLR in SDSS 0956+5128 is the result of an SN, it  is likely only one such event in the SDSS, as no similar  candidates have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Although, new multi-  line observations of the evolved quasars with strong out-  flows may reveal more of these objects in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' On  the other hand, it is believed that they are not observ-  able and more likely to cross the event horizon before  they explode, because their lifetimes are prolonged by  the freshly accreted gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Although, this may not be the  case in sparse accretion disks of the old quasars, where  little gas may not be enough to support the stars before  sinking into the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' Therefore, if the occurrence rate  is low because the SN-induced velocity offsets of BLRs  in old quasars are transient on some timescale, then it is  possible that SNe are more frequent and could also play  an important role in turning off QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  We would like to greatly thank Gabriel Brammer for  help with the data processing, Martin Pessah, Darach  Watson, Marianne Vestergaard, Johan Fynbo, Lise  Christensen, Jeremy Goodman, Adam Jermyn, Ersilia  Guarini and Kasper Elm Heintz for helpful and insight-  ful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The Cosmic Dawn Center (DAWN) is  funded by the Danish National Research Foundation un-  der grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  This work made use of data stored at Barbara A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='  Mikulski Archive for Space Telescopes (MAST), NASA  Astrophysics Data System Bibliographic Services and  the NASA/IPAC Extragalactic Database (NED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content=' The  HST photometry and spectroscopy used in this paper  can be found in MAST: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
+page_content='17909/6g1c-9b66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQf0P7C/content/2301.01782v1.pdf'}
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+Stochastic Variable Metric Proximal Gradient with variance
+reduction for non-convex composite optimization
+Gersende Fort1* and Eric Moulines2
+1*Institut de Math´ematiques de Toulouse, CNRS & Universit´e de Toulouse, 118 route de
+Narbonne, Toulouse, 31400, France.
+2CMAP, Ecole Polytechnique, Route de Saclay, Palaiseau, 91128 Cedex, France.
+*Corresponding author(s). E-mail(s): gersende.fort@math.univ-toulouse.fr;
+Contributing authors: eric.moulines@polytechnique.edu;
+Abstract
+This paper introduces a novel algorithm, the Perturbed Proximal Preconditioned SPIDER algo-
+rithm (3P-SPIDER), designed to solve finite sum non-convex composite optimization. It is a
+stochastic Variable Metric Forward-Backward algorithm, which allows approximate preconditioned
+forward operator and uses a variable metric proximity operator as the backward operator; it
+also proposes a mini-batch strategy with variance reduction to address the finite sum setting.
+We show that 3P-SPIDER extends some Stochastic preconditioned Gradient Descent-based algo-
+rithms and some Incremental Expectation Maximization algorithms to composite optimization
+and to the case the forward operator can not be computed in closed form. We also provide
+an explicit control of convergence in expectation of 3P-SPIDER, and study its complexity in
+order to satisfy the epsilon-approximate stationary condition. Our results are the first to com-
+bine the non-convex composite optimization setting, a variance reduction technique to tackle the
+finite sum setting by using a minibatch strategy and, to allow deterministic or random approx-
+imations of the preconditioned forward operator. Finally, through an application to inference
+in a logistic regression model with random effects, we numerically compare 3P-SPIDER to other
+stochastic forward-backward algorithms and discuss the role of some design parameters of 3P-SPIDER.
+Keywords: Stochastic optimization, Variable Metric Forward-Backward splitting, Preconditioned Stochastic
+Gradient Descent, Incremental Expectation Maximization, Proximal methods, Variance reduction,
+Non-asymptotic convergence bounds.
+1 Introduction
+Efficient learning from large data sets require new
+optimization algorithms designed to be robust to
+big data and complex models era. In Statistics and
+Machine Learning, we are often faced with solving
+problems of the form
+argmins∈Rq
+�
+1
+n
+n
+�
+i=1
+Wi(s) + g(s)
+�
+,
+where n is the number of examples in the training
+data set, s is an unknown quantity to be learnt
+from the examples, Wi is a loss function associ-
+ated to the example #i and g is a regularization
+1
+arXiv:2301.00631v1  [cs.LG]  2 Jan 2023
+
+2
+3P-SPIDER
+term encoding a priori knowledge and constraints
+on s; g may also prevent from overfitting. Quite
+often, the regularization term g : Rq → (0, +∞]
+is not differentiable, and the data fidelity term
+n−1 �n
+i=1 Wi is smooth on the domain of g.
+This paper is concerned with stochastic opti-
+mization of a non-convex finite sum composite
+function; more precisely, it addresses the differen-
+tial inclusion problem
+0 ∈ 1
+n
+n
+�
+i=1
+Gi(s) + ∂g(s),
+s ∈ Rq,
+(1)
+where g
+:
+Rq
+→
+(−∞, +∞] is lower semi-
+continuous convex with non-empty domain S and
+for all i, Gi : S → Rq is globally Lipschitz on S.
+The first goal of this paper is to provide a novel
+algorithm. Motivated by applications in Statistics
+and Machine learning, we require this algorithm
+to satisfy the following three constraints. (c1) The
+algorithm uses possibly preconditioned operators
+hi instead of the forward operator Gi:
+∀s ∈ S,
+hi(s, B)
+def
+= −B−1 Gi(s),
+(2)
+where B is a q × q positive definite matrix. Such
+a condition encompasses preconditioned gradient
+methods for example, which also includes gradient
+methods with adaptive step sizes. It also encom-
+passes Expectation Maximization (EM) algo-
+rithms (Dempster et al (1977)) designed for large
+scale learning. (c2) The algorithm may only have
+access to approximations of hi(s, B). Such a con-
+dition addresses the situations when hi(s, B) is
+not explicit, for example when it is defined by an
+intractable integral. This occurs at each E-step
+of EM, when the conditional expectations under
+the a posteriori distributions can not be computed
+exactly. (c3) The algorithm addresses the finite
+sum challenge while keeping the caused variabil-
+ity induced by the algorithmic solution as small as
+possible. For example, when the solution relies on
+a random selection of a mini-batch of examples,
+the algorithm has to propose a variance reduction
+scheme.
+A first class of problems of the form (1)
+are minimizations of regularized loss functions
+through gradient-based algorithms. In that case,
+Gi
+def
+= ∇Wi. (2) allows preconditioned gradients;
+such a variable metric is known to accelerate the
+convergence. Variable Metric Forward-Backward
+(VMFB) algorithms were introduced to solve (1)-
+(2) in the case G is a gradient. Nevertheless, as
+discussed in Section 2.1, to our best knowledge
+none of the variants of VMFB address the three
+constraints c1, c2 and c3 simultaneously.
+A second application of (1)-(2) is the EM
+algorithm, an algorithm originally designed to
+compute the Maximum-Likelihood estimator of
+an unknown parameter θ in latent variable mod-
+els. When the complete data model is from the
+curved exponential family, EM is equivalent to
+an algorithm in the so-called statistic space (see
+e.g. Delyon et al (1999)). This remark is the cor-
+nerstone of stochastic EM algorithms including
+incremental EM ones designed for incremental
+processing of large data sets. EM only supplies
+preconditioned forward operators −B(s)−1Gi(s).
+Therefore, stochastic EM algorithms are naturally
+in the setting (1)-(2) (see Fort et al (2020)). Here
+again, as discussed in Section 2.2, none of the EM
+variants in the literature address the constraints
+(c2) and (c3) simultaneously.
+Our first contribution is the design of a novel
+iterative algorithm, named Perturbed Proximal
+Preconditioned SPIDER
+(3P-SPIDER),
+which
+combines (i) a preconditioned forward operator
+associated to the smooth part n−1 �n
+i=1 Gi(s),
+(ii) a variable metric proximity operator with
+respect to the non-smooth part g, (iii) a stochas-
+tic approach to address the finite sum setting
+induced by n−1 �n
+i=1 Gi(s) combined with a vari-
+ance reduction technique based on the SPIDER
+algorithm (Nguyen et al (2017); Fang et al
+(2018)); it also allows (iv) numerical approxi-
+mations of the preconditioned forward operator
+when it has no closed form. The algorithm is
+introduced in Section 3, together with discussions
+on implementation questions. We also design a
+stochastic VMFB algorithm which answers the
+constraints c1 and c2 but does not contain a
+variance reduction step as required by c3.
+The second contribution is to provide a non-
+asymptotic convergence analysis in expectation
+of 3P-SPIDER in the case the variable metric B
+at iteration #t depends on the current value of
+the iterate, and the Gi’s are gradient operators;
+see Section 4. The proof relies on a Lyapunov
+inequality with an original construction, which
+
+3P-SPIDER
+3
+is a consequence of the non-convex optimization
+setting, and the fact the algorithm uses precon-
+ditioned forward operators and variable metric
+proximity operators (see Section 7.4).
+Theorem 4.1 provides a control of convergence
+in expectation for 3P-SPIDER, which explicitly
+identifies the impact of non-exact preconditioned
+forward operators, and the impact of initializa-
+tion strategies. First, we prove that the learning
+rate of 3P-SPIDER can be chosen constant over
+iterations when the preconditioned forward oper-
+ator is exact or replaced with an unbiased random
+oracle; and is decreasing along iterations when it
+is replaced with a biased oracle (deterministic or
+random). Second, we provide the first convergence
+result for a stochastic VMFB algorithm addressing
+c1, c2 and c3 for non-convex finite sum composite
+optimization. For example, it is the first result for
+incremental EM with a non-smooth penalty term
+(g ̸= 0) and possibly biased Monte Carlo approx-
+imations of the E-step.
+When the forward operator hi(s, B(s)) is exact,
+we study the complexity of 3P-SPIDER (see Corol-
+lary 4.3): in order to satisfy the ϵ-approximate
+stationary condition, the number of calls K¯h to one
+of the operator hi(s, B(s)) is O(√n/ϵ), the number
+of calls Kprox to the backward operator is O(1/ϵ)
+and, the learning rate can be chosen independent
+of the accuracy ϵ. Applied to the Gradient method
+and applied to the EM method when there are
+no constraints (g = 0), these explicit controls of
+convergence retrieve previous results in the litera-
+ture (see e.g. Wang et al (2019); Fort et al (2020))
+which are known to be at the state-of-the-art.
+Finally we show that this complexity analysis
+remains valid when the forward operators are
+approximated. In the difficult case when the
+approximations are biased random oracles based
+on Monte Carlo sums, we show that K¯h and Kprox
+are not impacted by the approximation and are
+the same as with exact operators hi(·, B(·)), by
+choosing an adequate number of terms in the
+Monte Carlo sums. The price to be paid is a Monte
+Carlo complexity of order O(√n/ϵ2).
+In Section 5, 3P-SPIDER is applied to inference
+in a logistic regression model with random effects.
+We show how the problem is of the form (1)-
+(2). In this example, the preconditioned forward
+operators are approximated by a Monte Carlo
+sum computed from a Markov chain Monte Carlo
+sampler. Through numerical analyses in the case
+3P-SPIDER is a stochastic Expectation Maximiza-
+tion algorithm in the statistic space, we discuss the
+choice of design parameters. We also show how the
+SPIDER variance reduction effect can be increased
+by exploiting the Monte Carlo approximations of
+the forward operator.
+The proofs are given in Section 6 and Section 7;
+technical details are also provided in Appendix.
+Notations. We denote by ⟨·, ·⟩ the dot prod-
+uct on Rq, and by ∥ · ∥ the associated norm.
+For a q × q positive definite matrix B, we set
+⟨·, ·⟩B
+def
+= ⟨B·, ·⟩ and ∥ · ∥B the associated norm.
+Iq denotes the q × q identity matrix. For a matrix
+B, B⊤ is its transpose. Pq
++ denotes the set of the
+q × q positive definite matrices.
+N (resp. N⋆) is the set of non negative (resp.
+positive) integers. For n ∈ N⋆, we set [n]
+def
+=
+{0, · · · , n} and [n]⋆ def
+= {1, · · · , n}. R+ is the set
+of the positive real numbers. For q ∈ R, ⌈q⌉ is the
+upper integer part.
+I is the identity function. For a proper function
+g : Rq → (−∞, +∞], ∂g(s) denotes the sub-
+differential at s. For a continuously differentiable
+function W at s ∈ Rq, ∇W(s) is the gradient of
+W at s.
+All the random variables (r.v.) are defined on
+(Ω, A, P); for a r.v. U, σ(U) is the sigma-field
+generated by U.
+2 Motivating examples
+In this section, we show that the Variable Metric
+Proximal-Gradient algorithm and the Expectation
+Maximization algorithm are examples of the gen-
+eral framework described by (1) and (2). In the
+first case, the preconditioning matrices B are cho-
+sen by the user, while in the second case, they are
+supplied by the algorithm.
+2.1 Variable Metric
+Proximal-Gradient algorithms
+Consider the non-convex composite problem with
+finite sum structure
+find s ∈ Rq:
+0 ∈ 1
+n
+n
+�
+i=1
+∇Wi(s) + ∂g(s) ,
+(3)
+
+4
+3P-SPIDER
+where g is a proper lower semicontinuous convex
+function with domain S; and for all i ∈ [n]⋆, Wi is
+continuously differentiable on S. It is of the form
+(1) with Gi
+def
+= ∇Wi being a gradient. (3) is an
+example of the more general problem: finding a
+zero on Rq of the sum of two (set-valued) oper-
+ators 0 ∈ As + Cs. Here, A
+def
+= n−1 �n
+i=1 ∇Wi
+and C
+def
+=
+∂g is a maximally monotone oper-
+ator (see e.g. (Bauschke and Combettes, 2011,
+Theorem 20.25)). (3) can be solved by Forward-
+Backward splitting algorithms (see e.g. Combettes
+and Wajs (2005); Beck (2017)): the forward step
+uses the gradient of some if not all the functions
+Wi’s at each iteration; the backward step uses a
+proximity operator associated to g. This yields
+Proximal-Gradient based algorithms.
+In the case g = 0, which includes uncon-
+strained optimization problems, stochastic gradi-
+ent methods with variance reduction were pro-
+posed in the situation
+n−1
+n
+�
+i=1
+Gi(s) = E [G(Z, s)]
+(4)
+and random oracles G(Z, s) are available; in the
+non-convex setting, let us cite e.g. Ghadimi and
+Lan (2013); Reddi et al (2016); Allen-Zhu and
+Hazan (2016); Nguyen et al (2017); Allen-Zhu
+(2018); Fang et al (2018); Zhou et al (2020).
+These algorithms address the problem (1)-(2) by
+choosing B equal to the identity matrix Iq and
+they use unbiased random oracles G(Z, s) for the
+approximation of the forward operator.
+For non-convex composite optimization (g ̸=
+0), let us cite Ghadimi et al (2016) and Karimi
+et al (2016) for stochastic Proximal-Gradient algo-
+rithms using unbiased oracles G(Z, s) (see (4)).
+Li and Li (2018), Wang et al (2019), Zhang
+and Xiao (2019), Nhan et al (2020) and Metel
+and Takeda (2021) propose stochastic Proximal-
+Gradient methods with unbiased random ora-
+cles and including variance reduction schemes. In
+Metel and Takeda (2021), g may be non-convex
+but admits an efficiently computable proximity
+operator. Atchad´e et al (2017) allow for deter-
+ministic or random approximations of the forward
+operator n−1 �n
+i=1 Gi(s); when the perturbation
+is stochastic, the convergence analysis covers both
+biased and unbiased oracles, includes Nesterov
+acceleration schemes, but is restricted to con-
+vex optimization. Here again, all these algorithms
+address the problem (1)-(2) by choosing B = Iq.
+Forward-Backward suffers from slow conver-
+gence, and Variable Metric Forward-Backward
+(VMFB) methods were proposed by Chen and Rock-
+afellar (1997) in order to accelerate the conver-
+gence (see also refs. 11 to 16 in Chouzenoux et al
+(2014)). VMFB changes the metric at each itera-
+tion by using symmetric positive definite scaling
+matrices multiplying the forward operator. It is
+an alternative to inertial methods such as Heavy
+Ball or Nesterov acceleration which use informa-
+tions from the previous iterates. When solving
+the inclusion (3), VMFB uses preconditioned gradi-
+ents with an iteration-dependent preconditioning
+matrix B−1
+t
+for the forward step at iteration #t,
+and scales the proximal step consequently. Exam-
+ples showing that VMFB is more efficient than
+Forward-Backward and Forward-Backward with
+inertial schemes, are provided in Chouzenoux et al
+(2014) and Repetti et al (2014). Different strate-
+gies exist for the definition of the variable metric
+Bt; for example, it may be a diagonal matrix
+depending on the past history of the algorithm
+(see e.g. Park et al (2019) and references therein
+for variable scalar metrics; see also Chen et al
+(2019) in the case g = 0), or inherited from
+Newton-type methods (see e.g. Becker and Fadili
+(2012); Lee et al (2014); Becker et al (2019) for
+the composite convex case; see also Kolte et al
+(2015); Moritz et al (2016); Gower et al (2016) for
+the smooth convex case with finite sum structure;
+finally, see Zhang et al (2022) for the smooth non-
+convex case with finite sum structure), or defined
+through a Majorize-Minimize strategy to make the
+backward operator explicit (see e.g. Chouzenoux
+et al (2014) and Repetti and Wiaux (2021)). Con-
+vergence results for VMFB exist in the convex case
+(see e.g. Combettes and V˜u (2014) and Bonettini
+et al (2021); and Park et al (2019) for the strongly
+convex case) and in the non-convex case (see e.g.
+Chouzenoux et al (2014) and Repetti and Wiaux
+(2021)). In Yun et al (2021), a stochastic VMFB is
+studied in the non-convex case; the exact gradient
+is approximated by a linear combination of ran-
+dom oracles, with exponential forgetting, and the
+oracles are assumed unbiased and bounded.
+3P-SPIDER addresses non-convex composite
+optimization with a finite sum structure by using
+Proximal-Gradient algorithms accelerated via the
+
+3P-SPIDER
+5
+Variable Metric cunning. It is a stochastic VMFB,
+which contains a variance reduction technique
+in order to overcome the finite sum setting; it
+also allows oracles for the preconditioned forward
+operators, oracles which can be biased or unbi-
+ased when random (see e.g. Atchad´e et al (2017)
+and Fort et al (2018) for examples motivating
+biased random approximations of the gradient).
+The combination of these two sources of pertur-
+bations is an original setting which, to our best
+knowledge, is not addressed in the literature.
+3P-SPIDER uses preconditioning matrices, which
+may depend on the current value of the iterate and
+therefore may be random. The non-asymptotic
+convergence analysis derived in Section 4 will rely
+on weaker minorization assumptions on the spec-
+trum of the scaling matrices (see A 3) than in
+Yun et al (2021); it will not require ordering
+assumptions on the sequence of scaling matrices
+as in Combettes and V˜u (2014) and Bonettini
+et al (2021), and will not assume a Kurdyka-
+�Lojasiewicz condition on the objective function as
+in Chouzenoux et al (2014). As a consequence,
+the construction of the Lyapunov inequality for
+the convergence analysis of 3P-SPIDER differs from
+these previous works.
+3P-SPIDER requires the backward operator to be
+explicit, which may be a strong assumption espe-
+cially for variable metric proximity operators (see
+Section 3.5); extensions of the convergence anal-
+ysis to the case of inexact proximity operators is
+out of the scope of this paper.
+2.2 Expectation Maximization for
+curved exponential families
+Consider the parametric statistical model: the
+observations are independent with density
+y �→
+�
+Z
+p(y, z; θ)µlv(dz) ,
+with respect to (w.r.t.) a σ-finite positive measure
+µo on Rdy. In this model, z acts as a latent vari-
+able taking values in the measurable set (Z, Z)
+endowed with a σ-finite positive measure µlv (see
+e.g. Everitt (1984) for examples of latent vari-
+able models). The goal is to learn the parameter
+θ ∈ Θ ⊆ Rd from n observations Y1, · · · , Yn, by
+minimizing the negative normalized log-likelihood
+F(θ)
+def
+= − 1
+n
+n
+�
+i=1
+log
+�
+Z
+p(Yi, z; θ)µlv(dz)
+(5)
+on Θ. Unfortunately, this is a non-convex problem
+and most often, an optimization algorithm for the
+minimization of (5) can not do better than con-
+verging to a critical point of the objective function
+(see Wu (1983)).
+A popular model is the case when the complete
+data likelihood (y, z) �→ p(y, z; θ) is of the form
+p(y, z; θ)
+def
+= H(y, z) exp (⟨S(y, z), φ(θ)⟩ − ψ(θ))
+where H : Rdy × Z → R+, S : Rdy × Z → Rq,
+φ : Θ → Rq, ψ : Θ → R; it corresponds to the
+so-called curved exponential family assumption. It
+is satisfied by the mixture models as soon as the
+components of the mixture are from the curved
+exponential family. See e.g. Brown (1986) for an
+introduction to curved exponential family of dis-
+tributions; and McLachlan and Krishnan (2008)
+for examples of such latent variable models.
+EM for curved exponential families defines
+iteratively a Θ-valued sequence {θt, t
+≥
+0}
+through the mechanism: given θt,
+• (E-step) Compute ¯s(θt), the expectation of
+the sufficient statistics w.r.t. the a posteriori
+distributions
+¯s(θt)
+def
+= 1
+n
+n
+�
+i=1
+¯si(θt) ,
+¯si(θt)
+def
+=
+�
+Z
+S(Yi, z) p(Yi, z; θt) µlv(dz)
+�
+Z p(Yi, u; θt)µlv(du) .
+• (M-step) Update the parameter
+θt+1
+def
+= argminθ∈Θ (ψ(θ) − ⟨¯s(θt), φ(θ)⟩) .
+The algorithm alternates between a step in the
+parameter space Θ (when computing θt+1 ∈ Rd),
+and a step in the statistic space when computing
+¯s(θt) ∈ Rq. Proposition 2.1 states that the limiting
+points of EM run in the parameter space Θ are the
+fixed points of an operator onto Θ; finding such a
+fixed point is equivalent to find a fixed point of an
+operator onto the statistic space ¯s(Θ) ⊆ Rq.
+
+6
+3P-SPIDER
+Proposition 2.1 Assume that for any s ∈ S ⊇ ¯s(Θ),
+T(s) def
+= argminθ∈Θ (ψ(θ) − ⟨s, φ(θ)⟩)
+exists and is unique. Set Lθ
+def
+= {θ ∈ Θ : T(¯s(θ)) = θ}
+and Ls
+def
+= {s ∈ ¯s(Θ) : ¯s(T(s)) = s}.
+Lθ is the set of the limiting points of EM. If θ⋆ ∈
+Lθ, then s⋆
+def
+= ¯s(θ⋆) is in Ls. Conversely, if s⋆ ∈ Ls
+then θ⋆
+def
+= T(s⋆) is in Lθ.
+See e.g. Delyon et al (1999) for the proof. An
+algorithmic corollary of Proposition 2.1, is that
+EM is equivalent to any algorithm run in the
+statistic space and designed to find the roots of
+the function
+s �→ 1
+n
+n
+�
+i=1
+¯hi(s) ,
+where ¯hi(s)
+def
+= ¯si(T(s)) − s ,
+on the subset ¯s(Θ) of Rq. Under regularity con-
+ditions on the statistical model, it is proved in
+Delyon et al (1999) (see also a statement in (Fort
+et al, 2020, Proposition 1)) that there exists a q×q
+positive definite matrix B(s) such that
+∇ (F ◦ T) (s) = −B(s)
+�
+1
+n
+n
+�
+i=1
+¯hi(s)
+�
+,
+(6)
+where F is the negative normalized log-likelihood
+(see (5)). Therefore, the roots of n−1 �n
+i=1 ¯hi(s)
+on ¯s(Θ) are the roots of n−1 �n
+i=1 Gi(s) on ¯s(Θ),
+where Gi(s)
+def
+= −B(s) ¯hi(s). It also means that
+the roots of n−1 �n
+i=1 ¯hi(s) are the roots of the
+gradient of F◦T, the objective function transferred
+on the statistic space through the map T : Θ →
+Rq.
+As a conclusion, EM in the statistic space is
+an example of problem (1)-(2), where the func-
+tion g collects the constraint on s such as s ∈
+S ⊇ ¯s(Θ): (i) it is designed to find a root of
+n−1 �n
+i=1 Gi(s) = ∇(F ◦ T)(s) under the con-
+straint that s ∈ S; (ii) it uses the quantities ¯hi(s)
+which are preconditioned forward operator since
+there exists B(s) such that ¯hi(s) = −B(s)−1 Gi(s)
+(see (6)); (iii) this preconditioned forward oper-
+ator is intractable for at least two reasons: first,
+due to the inner integrations on the set Z when
+computing ¯si, since p(y, z; θ) and Z are often too
+complex to make the integrals explicit; second,
+due to the outer integration on the n examples
+when computing ¯s, which has a prohibitive com-
+putational cost in large scale learning. However, a
+Monte Carlo approximation of ¯hi(s) is always pos-
+sible, whatever i and s. This remark is the corner-
+stone to understand the stochastic versions of EM
+(see e.g. Celeux and Diebolt (1985); Wei and Tan-
+ner (1990); Delyon et al (1999); Fort and Moulines
+(2003) which address the inner sum intractability;
+and Neal and Hinton (1998); Ng and McLachlan
+(2003); Capp´e and Moulines (2009); Chen et al
+(2018); Karimi et al (2019); Fort et al (2020,
+2021a) for the outer sum intractability). They con-
+sist in running a Stochastic Approximation (SA)
+algorithm with mean field n−1 �n
+i=1 ¯hi(s) (for an
+introduction to SA, see e.g. Benveniste et al (1990)
+or Borkar (2008)); this yields SA within EM pro-
+cedures. They differ through the construction of
+the random field used for the approximation of the
+mean field (see (Fort et al, 2021a, Section 2.2.) for
+a description of some SA within EM algorithms).
+3P-SPIDER is among the SA within EM algo-
+rithms. Compared to previous stochastic EM
+methods, it encompasses the two random approx-
+imations (of the sum in i and of the integrals
+on Z) and a variance reduction step, and it also
+allows a more general penalty term g than the
+{0, +∞}-valued indicator function of a set.
+3 The 3P-SPIDER algorithm
+We introduce a novel algorithm named Perturbed
+Proximal Preconditioned SPIDER (3P-SPIDER),
+solving (1) and satisfying (c1), (c2) and (c3). It
+requires g to satisfy the following assumption
+A 1. g : Rq → (−∞, +∞] is proper, lower semi-
+continuous and convex. Denote by S its domain
+S
+def
+= {s ∈ Rq : g(s) < +∞}.
+Under this assumption, we define a variable
+metric proximity operator. For any γ > 0 and
+B ∈ Pq
++, the proximity operator of the proper
+lower semicontinuous convex function γg : Rq →
+(−∞, +∞] relative to the metric induced by
+B is defined by (see e.g. (Hiriart-Urruty and
+Lemar´echal, 1996, Section XV.4))
+proxB
+γg(s)
+def
+=argminRq
+�
+γg(·) + 1
+2∥ · −s∥2
+B
+�
+. (7)
+
+3P-SPIDER
+7
+When B = Iq, we simply write proxγg(s), which
+is the proximity operator originally defined by
+Moreau (1965). Lemma 3.1 shows that under A1,
+proxB
+γg(s) exists and is unique for all s ∈ Rq, γ > 0
+and B ∈ Pq
++. It also provides characterizations of
+this point. Its proof is in Section 6.1.
+Lemma 3.1 Assume A1.
+1. For any γ > 0, B ∈ Pq
++ and s ∈ Rq, the
+optimization problem (7) has a unique solu-
+tion, characterized as the unique point p ∈ S
+satisfying
+− γ−1 B (p − s) ∈ ∂g(p) .
+2. For any γ > 0, B ∈ Pq
++, s ∈ S and h ∈ Rq,
+s = proxB
+γg(s + γh)
+iff
+Bh ∈ ∂g(s).
+(8)
+For s ∈ Rq and B ∈ Pq
++, set
+hi(s, B)
+def
+= −B−1 Gi(s) ,
+h(s, B)
+def
+= n−1
+n
+�
+i=1
+hi(s, B) .
+(9)
+By Lemma 3.1-item 2, it holds for any B ∈ Pq
++ and
+γ > 0: s = proxB
+γ g (s + γ h(s, B)) iff −Bh(s, B) ∈
+∂g(s). By (9), this yields for any B ∈ Pq
++, and
+γ > 0:
+s⋆ = proxB
+γ g (s⋆ + γ h(s⋆, B))
+iff s⋆ solves (1).
+(10)
+3.1 Variable Metric Proximal and
+Preconditioned Gradient
+(10) shows that when solving the composite opti-
+mization problem (1), as soon as a preconditioned
+version of the operator s �→ n−1 �n
+i=1 Gi(s) is
+used – with preconditioning matrix B−1, a prox-
+imity operator of g relative to a metric induced by
+the matrix B has to be used.
+Based on the characterization (10), a natural
+splitting algorithm to solve (1) under the con-
+dition (c1) is: given σ0 ∈ S, a positive stepsize
+sequence {γk+1, k ≥ 0} and a Pq
++-valued sequence
+{Bk+1, k ≥ 0}, repeat
+σk+1 = proxBk+1
+γk+1 g (σk + γk+1h(σk, Bk+1)) . (11)
+It corresponds to the Variable Metric Forward-
+Backward algorithm (see e.g. Chen and Rockafel-
+lar (1997); Combettes and V˜u (2014)).
+In the large scale learning setting, the full sum
+over the n functions hi (see (9)) can not be com-
+puted at each iteration of (11). In addition, it may
+happen that hi(s) is not explicit (see e.g. the case
+of the incremental EM algorithms, Section 2.2).
+Therefore, a natural idea is to propose the inexact
+version of (11) defined by Algorithm 1: the proxi-
+mal step is unchanged (see line 8); the SA step in
+line 7 uses a random approximation Sk+1 of the
+exact mean field n−1 �n
+i=1 hi(�Sk); this approxi-
+mation, defined by line 6, combines a mini-batch
+approximation of a full sum (see line 3) and
+possibly approximated terms δk+1,i (see line 5).
+Algorithm
+1
+A
+stochastic
+Variable
+Metric
+Forward-Backward
+Require: kout ∈ N⋆, γk > 0 for k ∈ [kout]⋆, b ∈
+N⋆, �Sinit ∈ S.
+Ensure: The sequence {�Sk, k ∈ [kout]}.
+1: �S0 = �Sinit
+2: for k = 0, · · · , kout − 1 do
+3:
+Sample a batch Bk+1 of size b in [n]⋆
+4:
+Choose Bk+1 ∈ Pq
++
+5:
+For i ∈ Bk+1, compute an approximation
+δk+1,i of hi(�Sk, Bk+1).
+6:
+Sk+1 = b−1 �
+i∈Bk+1 δk+1,i
+7:
+�Sk+1/2 = �Sk + γk+1 Sk+1
+8:
+�Sk+1 = proxBk+1
+γk+1 g(�Sk+1/2).
+9: end for
+3.2 The SPIDER variance reduction
+technique
+3P-SPIDER leverages on Algorithm 1 and on
+the
+variance
+reduction
+technique
+SPIDER
+for
+the definition of the field Sk+1 that approxi-
+mates n−1 �n
+i=1 hi(�Sk, Bk+1). SPIDER stands for
+Stochastic Path-Integrated Differential EstimatoR,
+and was originally introduced in the stochastic
+gradient descent literature by Fang et al (2018)
+(see also Nguyen et al (2017); Wang et al (2019)).
+
+8
+3P-SPIDER
+We give the intuition of SPIDER in the SA setting
+which encompasses the stochastic gradient one.
+SA scheme solves a root finding problem ξ(s) =
+0 on Rq by: given an initial value s0 ∈ Rq and
+a stepsize sequence {γk+1, k ≥ 0}, repeat sk+1 =
+sk + γk+1 Ξk+1, where at each iteration #(k + 1),
+Ξk+1 is a random approximation of ξ(sk). Usu-
+ally, it is required that conditionally to the past
+of the algorithm, the expectation of Ξk+1 is ξ(sk);
+in that case, Ξk+1 can be replaced with Sk+1
+def
+=
+Ξk+1 + Vk+1, where conditionally to the past,
+Vk+1 is centered. SPIDER leverages on this remark
+and on the control variate technique: it proposes
+a clever construction of a random variable Vk+1
+approximating zero and correlated to Ξk+1.
+The recipe is as follows: consider that at
+iteration #k, Sk is a random approximation of
+h(�Sk−1, Bk). Then define Sk+1 by Sk+1
+def
+= Hk+1+
+Vk+1 where
+Hk+1
+def
+= b−1
+�
+i∈Bk+1
+hi(�Sk, Bk+1) ,
+Vk+1
+def
+= Sk − b−1
+�
+i∈Bk+1
+hi(�Sk−1, Bk) ,
+and
+Bk+1
+is
+sampled
+at
+random
+in
+[n]⋆.
+The r.v. Vk+1
+approximates zero since both
+b−1 �
+i∈Bk+1 hi(�Sk−1, Bk) and Sk
+approximate
+n−1 �n
+i=1 hi(�Sk−1, Bk); Vk+1 and Hk+1 are corre-
+lated via Bk+1.
+Unfortunately, the r.v. Sk+1 is not an unbi-
+ased approximation of n−1 �n
+i=1 hi(�Sk, Bk+1) (see
+Proposition 7.3 in the case Bk+1 is of the form
+B(�Sk)). In order to remove the bias, SPIDER
+restarts the control variate mechanism regularly:
+every kin iterations, compute a full sum over the n
+terms and set Skin+1 = n−1 �n
+i=1 hi(�Skin, Bkin+1).
+3.3 3P-SPIDER
+3P-SPIDER is given by Algorithm 2. The itera-
+tion index is (t, k) where t is the index of the
+current outer loop and ranges from 1 to kout,
+and k is the index of the current inner loop. At
+outer loop #t, there are kin
+t
+inner iterations. The
+inner iterations are Algorithm 1 (see Lines 8, 9,
+12 and 13 of Algorithm 2) combined with the
+SPIDER variance reduction trick (see Line 11 of
+Algorithm 2) adapted to the case when the quan-
+tities hi(�St,k, Bt,k+1) − hi(�St,k−1, Bt,k) can not be
+computed exactly (see Line 10).
+When Gi is a gradient and B(s) = Iq, different
+strategies were proposed for SPIDER for the choice
+of b′
+t and kin
+t . In Fang et al (2018); Nguyen et al
+(2017); Wang et al (2019), the number of inner
+loops is constant (kin
+t
+= kin for any t ≥ 1) and
+b′
+t = n; Nguyen et al (2017) also considers the case
+when kin
+t
+is adapted based on the history of the
+algorithm while being upper bounded; in Horv´ath
+et al (2022), b′
+t is deterministic and depends on
+t, b depends on t, and kin
+t is a Geometric random
+variable with an expectation depending on t; in Li
+et al (2021), b′
+t does not depend on t and kin
+t
+is
+random.
+For the EM problem (see Section 2.2), Fort et al
+(2020) introduced SPIDER-EM, a variance reduced
+stochastic EM designed for large scale learning,
+in a situation when the computation of ¯hi(s) is
+exact for all s, i. For this algorithm, the benefit
+of an increasing batch size t �→ b′
+t and a geomet-
+ric number of inner loops kin
+t
+with time-varying
+expectation, is discussed in Fort et al (2021b). The
+conclusion is that the best strategy is a determin-
+istic increasing sequence b′
+t in order to have an
+increasing accuracy when refreshing the variable
+S·, and a constant number of inner loops kin
+t = kin.
+This paper allows b′
+t and kin
+t
+to vary with t: they
+may be deterministic functions of t or random ones
+as well.
+The matrices {Bt,k+1, t ∈ [kout]⋆, k ∈ [kin −1]}
+can be deterministic or random. They could be
+chosen prior the run of the algorithm; more effi-
+cient strategies consist in adapting this matrix
+along the run of the algorithm, based on its his-
+tory. In EM (see Section 2.2), Bt,k+1 is of the form
+B(�St,k) where B is defined by the statistic model.
+After kin
+t
+inner iterations, the outer loop
+#(t + 1) starts: the stochastic mean field St+1,0
+is refreshed (see Line 3 to Line 6). Here again,
+two approximations of the original SPIDER algo-
+rithm are allowed: the first one is when computing
+hi(�St,0, Bt,1) and the second one avoids the scan
+of the full data set (one may choose b′
+t < n).
+The input variables of 3P-SPIDER are the num-
+ber of outer loops kout, the number of inner loops
+kin
+t , the stepsize sequence {γt,k, t ≥ 1, k ≥ 1} for
+the SA steps, the size of the mini-batches b and
+
+3P-SPIDER
+9
+Algorithm 2 The Perturbed Proximal Preconditioned SPIDER algorithm (3P-SPIDER)
+Require: kout ∈ N⋆, kin
+t
+∈ N⋆ for t ∈ [kout]⋆, γt,k+1 > 0 for t ∈ [kout]⋆, k ∈ [kin
+t ], b ∈ N⋆, b′
+t ∈ N⋆ for
+t ∈ [kout]⋆, �Sinit ∈ S and Binit ∈ Pq
++
+Ensure: The sequence {�St,k, t ∈ [kout]⋆, k ∈ [kin
+t ]⋆}.
+1: �S0,kin
+0 = �Sinit, B0,kin
+0 = Binit
+2: for t = 1, · · · , kout do
+3:
+�St,0 = �St−1,kin
+t−1,
+�St,−1 = �St−1,kin
+t−1,
+Bt,0 = Bt−1,kin
+t−1
+4:
+Sample a batch Bt,0 of size b′
+t in [n]⋆, with or without replacement.
+5:
+For all i ∈ Bt,0, compute δt,0,i equal to or approximating hi(�St,0, Bt,0).
+6:
+St,0 = (b′
+t)−1 �
+i∈Bt,0 δt,0,i
+7:
+for k = 0, · · · , kin
+t − 1 do
+8:
+Sample a mini batch Bt,k+1 of size b in [n]⋆, with or without replacement.
+9:
+Choose Bt,k+1 ∈ Pq
++.
+10:
+For all i ∈ Bt,k+1, compute δt,k+1,i equal to or approximating hi(�St,k, Bt,k+1)−hi(�St,k−1, Bt,k).
+11:
+St,k+1 = St,k + b−1 �
+i∈Bt,k+1 δt,k+1,i
+12:
+�St,k+1/2 = �St,k + γt,k+1 St,k+1
+13:
+�St,k+1 = proxt,k(�St,k+1/2),
+where proxt,k
+def
+= proxBt,k+1
+γt,k+1 g.
+14:
+end for
+15: end for
+b′
+t, and the initial values of the iterate �Sinit and
+the metric Binit in S and Pq
++ respectively.
+3.4 Monte Carlo approximation of
+hi(s, B)
+Set ϑ
+def
+= (s, i, B) ∈ S ×[n]⋆×Pq
++. In some applica-
+tions, there exist a measurable function Hϑ and a
+probability measure πϑ defined on the measurable
+set (Z, Z) such that
+hi(s, B) =
+�
+Z
+Hϑ(z)πϑ(dz) .
+(12)
+This is the case of EM in the statistic space (see
+Section 2.2) where Hϑ(z) = S(Yi, z) − s and
+πϑ(dz)
+def
+=
+p(Yi, z; T(s))
+�
+Z p(Yi, u; T(s)) µlv(du) µlv(dz) .
+When the integral in (12) is intractable, one can
+resort to Monte Carlo integrations to define the
+approximations δt,k+1,i and δt,0,i (see e.g. Devroye
+(1986) for exact sampling methods, and Robert
+and Casella (2004) for an introduction to Markov
+chain Monte Carlo methods). If {Zϑ
+m, m ≥ 0} are
+independent samples with distribution πϑ(dz) or
+are a path of an ergodic Markov chain with unique
+invariant distribution πϑ(dz), then we can set
+hi(�St,k, Bt,k+1) ≈ 1
+M
+M
+�
+m=1
+Hϑt,k+1,i(Zϑt,k+1,i
+m
+) ,
+where ϑt,k+1,i
+def
+= (�St,k, i, Bt,k+1). We will show
+numerically in Section 5 that when approximating
+the difference hi(�St,k, Bt,k+1) − hi(�St,k−1, Bt,k),
+there is a gain in correlating the two sequences
+{Zϑt,k+1,i
+m
+, m ≥ 0} and {Zϑt,k,i
+m
+m ≥ 0}; this makes
+stronger the effect of the SPIDER control variate
+(see Section 3.2).
+3.5 The computation of proxB
+γg
+When g = 0, proxB
+γg(s) = s. When g ̸= 0, p
+def
+=
+proxB
+γg(s) solves 0 ∈ p − s + γB−1 ∂g(p) and there
+does not always exist an explicit expression of p.
+When B = Iq, (Combettes and Pesquet, 2011,
+Tables 10.1 and 10.2) provide properties of proxγg
+and expressions of proximity operators for many
+functions g.
+When B is the sum of a diagonal matrix and
+of a rank one matrix, (Becker and Fadili, 2012,
+Section 3) presents iterative algorithms for the
+computation of p. For a general positive definite
+matrix B, we have from (Combettes and V˜u, 2014,
+
+10
+3P-SPIDER
+Example 3.9)
+proxB
+γg(s) =
+√
+B
+−1proxγg(
+√
+B
+−1·)(
+√
+Bs) ,
+where
+√
+B is the square root of the matrix B.
+(Becker and Fadili, 2012, Lemma 5) (see also
+Combettes and V˜u (2014)) establishes a Moreau
+identity i.e. an expression of proxB
+γg as a function
+of a proximity operator of the Fenchel conjugate
+of g.
+In the special case g is the {0, +∞}-valued
+indicator function of a closed convex set S, the
+projected Landweber method is an iterative algo-
+rithm for the computation of p (see Eicke (1992),
+see also (Combettes and Pesquet, 2011, Example
+10.10)).
+Finally, for applicatons including a metric
+selection step, metric selection strategies for the
+definition of B can be found in (Park et al,
+2019, Section 3) for diagonal variable metrics; and
+in Repetti and Wiaux (2021) for specific func-
+tions g which circumvent the often challenging
+computation of proxB
+γg.
+4 Non-asymptotic
+convergence analysis
+This section is devoted to explicit non-asymptotic
+bounds for the convergence in expectation of
+3P-SPIDER. We will restrict to the case there exist
+B : S → Pq
++ and
+Bt,k+1
+def
+= B(�St,k) .
+This framework encompasses the EM problem
+(see Section 2.2) and any preconditioned gradient-
+based algorithms (see Section 2.1) when the pre-
+conditioning matrix depends on the past history of
+the algorithm via the current value of the iterate.
+We will also use the notation
+¯hi(s)
+def
+= hi(s, B(s)) ,
+¯h(s)
+def
+= h(s, B(s)) .
+(13)
+3P-SPIDER is designed to solve (1) under the
+constraints c1 to c3. Therefore, based on (10),
+we are interested in a control of the quantities
+proxt,k
+�
+�St,k + γt,k+1 ¯h(�St,k)
+�
+− �St,k where
+proxt,k(s)
+def
+= proxB(�St,k)
+γt,k+1 g(s) .
+Roughly speaking, these quantities evaluate how
+far the algorithm is from the limiting set at
+iteration #(t, k). More precisely, we will con-
+trol the cumulative ”distances to stationary”
+�kout
+t=1
+�kin
+t −1
+k=0
+∆⋆
+t,k+1 where ∆⋆
+t,k+1 is equal to
+∥proxt,k(�St,k + γt,k+1¯h(�St,k)) − �St,k∥2
+B(�St,k)
+γ2
+t,k+1
+; (14)
+h is defined by (9).
+The controls in expectation of the cumulated
+distances are obtained under the assumptions A
+2 to A4. A2 is a smoothness assumption on the
+functions hi, A3 assumes that n−1 �n
+i=1 Gi(s) is
+a gradient operator of some so-called Lyapunov
+function, and the spectrum of the matrices B(s)
+are bounded uniformly in s. A4 are assumptions
+on the approximations δt,k+1,i.
+A 2. For all i ∈ [n]⋆, the function ¯hi is globally
+Lipschitz on S, with constant Li: there exists a
+positive constant Li such that ∀s, s′ ∈ S, ∥¯hi(s) −
+¯hi(s′)∥ ≤ Li ∥s − s′∥. Set L2 def
+= n−1 �n
+i=1 L2
+i .
+A 2 only requires a Lipschitz property on S;
+it is weaker than assuming the Lipschitz property
+on the full space Rq as sometimes assumed in the
+literature (see e.g. Combettes and Wajs (2005)). A
+2 holds for example when S is compact and for all
+i ∈ [n]⋆, the gradient ∇hi exists and is continuous
+on S.
+A3. 1. There exists a function W : Rq → R,
+continuously differentiable on S and such that
+∀s ∈ S,
+∇ W(s) = 1
+n
+n
+�
+i=1
+Gi(s) ;
+in addition, ¯hi(s) = −B(s)−1 Gi(s), where
+B(s) ∈ Pq
++.
+2. ∇ W is globally L ˙W-Lipschitz on S.
+3. There exist 0 < vmin ≤ vmax < +∞ such that
+for any s ∈ S, vmin∥·∥2 ≤ ∥·∥2
+B(s) ≤ vmax∥·∥2.
+Here again, both the Lipschitz property and
+the boundedness condition on the spectrum of the
+matrices B(s) are required on S and not on the
+full space Rq. When B(s) does not depend on
+
+3P-SPIDER
+11
+s (B(s) = B for any s ∈ S), we have L ˙W ≤
+vmaxn−1 �n
+i=1 Li.
+The last assumption is on the fluctuations
+of the errors when approximating ¯hi(�St,k) −
+¯hi(�St,k−1): set ξt,k+1,i
+def
+=
+δt,k+1,i − ¯hi(�St,k) +
+¯hi(�St,k−1) and define its conditional bias and
+variance, conditionally to the σ-field generated
+by Bt,k+1,
+�St,k
+and
+�St,k−1. Set Pt,k+1/2
+def
+=
+σ
+�
+Bt,k+1, �St,k, �St,k−1
+�
+.
+µt,k+1,i
+def
+= E
+�
+ξt,k+1,i|Pt,k+1/2
+�
+σ2
+t,k+1,i
+def
+= E
+�
+∥ξt,k+1,i − µt,k+1,i∥2|Pt,k+1/2
+�
+.
+We assume
+A4. 1. Conditionally to Bt,k+1, �St,k and �St,k−1,
+the approximations {δt,k+1,i, i ∈ Bt,k+1} are
+independent.
+2. There exists a non negative constant Cb and for
+any t ∈ [kout]⋆, there exists a non decreasing
+deterministic sequence {mt,k, k ≥ 1} such that
+for any k ∈ [kin
+t − 1], with probability one,
+∥ 1
+n
+n
+�
+i=1
+µt,k+1,i∥ ≤
+Cb
+mt,k+1
+.
+3. There exist non negative constants Cv and Cvb
+and for any t ∈ [kout]⋆, there exist non decreas-
+ing deterministic sequences {Mt,k, k ≥ 1} and
+{ ¯
+Mt,k, k ≥ 1} such that for any k ∈ [kin
+t − 1],
+with probability one,
+1
+n
+n
+�
+i=1
+σ2
+t,k+1,i ≤
+Cv
+Mt,k+1
+,
+1
+n
+n
+�
+i=1
+∥µt,k+1,i − 1
+n
+n
+�
+j=1
+µt,k+1,j∥2 ≤
+C2
+vb
+¯
+M 2
+t,k+1
+.
+We allow the errors ξt,k+1,i to be deterministic
+or random. When there are no errors (ξt,k+1,i = 0)
+then Cb = Cv = Cvb = 0. When the errors
+are deterministic, we have ξt,k+1,i = µt,k+1,i and
+σ2
+t,k+1,i = 0. When the errors are random and
+unbiased, then µt,k+1,i = 0. Therefore, some of the
+constants Cb, Cv or Cvb can be null as summarized
+in Table 1.
+In Section A, we discuss how A 4 is veri-
+fied in the case ¯hi(s′) − ¯hi(s) is an expectation
+Cb
+Cv
+Cvb
+exact
+0
+0
+0
+deterministic
+≥ 0
+0
+≥ 0
+random, unbiased
+0
+≥ 0
+0
+random, biased
+> 0
+≥ 0
+≥ 0
+Table 1 The sign of the constants Cb, Cv, Cvb when
+there are no approximations on the ¯hi(s)′ (case exact),
+and when there are approximations.
+under a distribution that may depend on (s, s′, i)
+(see Section 3.4), and δt,k+1,i is a Monte Carlo
+approximation.
+Theorem 4.1 provides an explicit upper bound
+of the cumulative distance to stationarity as mea-
+sured by ∆⋆
+t,k+1 (see (14)) along the �kout
+t=1 kin
+t
+iterations of the algorithm. It also provides an
+upper bound on the cumulative errors D⋆
+t,k+1
+defined by
+∥�St,k+1 − proxt,k(�St,k + γt,k+1¯h(�St,k))∥2
+B(�St,k)
+γ2
+t,k+1
+,
+where ¯h is defined by (13). Given the current iter-
+ate �St,k, D⋆
+t,k+1 compares two iterations: the ideal
+one proxt,k(�St,k + γt,k+1¯h(�St,k)) and the available
+one proxt,k(�St,k + γt,k+1St,k+1).
+Theorem 4.1 Assume A 1, A 2, A 3 and A 4. Let
+{kin
+t , t ∈ [kout]⋆} be a deterministic positive sequence.
+For any t ∈ [kout]⋆ and k ∈ [kin
+t − 1], define Λt,k+1 by
+γt,kL ˙W
+vmin
++γ2
+t,kL2 2vmaxkin
+t
+vminb
+�
+1 +
+2 Cvb
+√
+b ¯
+Mt,k+1
+�
+. (15)
+Let {�St,k, t ∈ [kout]⋆, k ∈ [kin
+t ]⋆} be the sequence given
+by Algorithm 2 when the stepsize sequence {γt,k+1, t ∈
+[kout]⋆, k ∈ [kin − 1]} satisfies
+γt,k+1
+�
+1 +
+2Cb
+mt,k+1
+�
+≤ γt,k ,
+Λt,k+1 ∈ (0, 1/2) .
+(16)
+Then,
+kout
+�
+t=1
+kin
+t
+�
+k=1
+γt,k
+�1
+2 − Λt,k+1
+� �
+E
+�
+∆⋆
+t,k
+�
++ E
+�
+D⋆
+t,k
+��
+≤ E
+�
+W(�S1,0) + g(�S1,0)
+�
+− min
+S (W +g)
++ vmax
+kout
+�
+t=1
+γt,0 kin
+t E
+�
+∥Et∥2�
++ vmax
+kout
+�
+t=1
+kin
+t
+�
+k=1
+�
+kin
+t − k + 1
+�
+γt,k Ut,k ,
+
+12
+3P-SPIDER
+where Et
+def
+= St,0 − h(�St,0) and
+Ut,k
+def
+= 2 Cb
+mt,k
++ C2
+b
+m2
+t,k
++
+Cv
+b Mt,k
++
+2 Cvb
+√
+b ¯
+Mt,k
++
+C2
+vb
+b ¯
+M2
+t,k
+.
+The
+proof
+of
+Theorem
+4.1
+is
+given
+in
+Section 7.5. Note that Ut,k+1 = 0 when the algo-
+rithm uses exact preconditioned gradients at each
+iteration: δt,0,i = ¯hi(�St,0) and δt,k+1,i = ¯hi(�St,k) −
+¯hi(�St,k−1) for all i, t, k.
+Random number of inner loops kin
+t . When
+the number of inner loops kin
+t at the outer loop #t
+is a random number, we consider it is drawn prior
+the run of the algorithm. Therefore the expec-
+tations in Theorem 4.1 are conditionally to the
+random sequence {kin
+t , t ∈ [kout]⋆}. The expecta-
+tion w.r.t. the randomness of kin
+t
+can easily be
+obtained from Theorem 4.1; details are omitted.
+The step sizes γt,k. The conditions on the
+sequence {γt,k+1, t ∈ [kout]⋆, k ∈ [kin
+t − 1]} are
+satisfied with
+γt,k+1
+def
+=
+k
+�
+j=0
+�
+1 +
+2 Cb
+mt,j+1
+�−1
+γt,0
+where γt,0 is positive and strictly lower than
+1
+4Lvmaxυ
+b
+kin
+t
+�
+�
+�
+L2
+˙W
+L2 + 4vminvmax
+kin
+t
+b υ − L ˙W
+L
+�
+� ;
+(17)
+υ
+def
+= 1 + 2Cvb/(
+√
+b inft,k ¯
+Mt,k+1) (see the proof in
+Section C.1). First, observe that when Cb = 0, the
+step size can be a constant function of the inner
+loop index k loop (γt,k+1 = γt,0 for any k). On
+the contrary, when Cb > 0 i.e. for a deterministic
+approximation or a biased random approximation
+(see Table 1), the stepsize sequence is a strictly
+decreasing function of the inner loop index k.
+Second, the maximal value of γt,0 is larger
+when Cvb = 0 than when Cvb > 0. Here again,
+deterministic or unbiased random approximations
+requires more aggressive step sizes.
+The initialization of the outer loops. Set
+Nt
+def
+= ∥Et∥2. When Bt,0 = {1, · · · , n} and δt,0,i =
+¯hi(�St,0) for all i, then Nt = 0; otherwise, Nt is
+positive.
+Let us discuss the behavior of Nt when δt,0,i
+is an unbiased random approximation of ¯hi(�St,0)
+with variance denoted by σ2
+t,0,i. When Bt,0 =
+{1, · · · , n}, then
+E [Nt] = 1
+n2
+n
+�
+i=1
+σ2
+t,0,i .
+(18)
+Nevertheless, the strategy Bt,0 = {1, · · · , n} has a
+large computational cost; sampling a subset of size
+b′
+t reduces the computational cost but increases
+the squared norm of the error: we have
+E [Nt] ≤
+1
+b′
+tn
+n
+�
+i=1
+�
+σ2
+t,0,i + ∥¯hi(s) − ¯h(s)∥2�
+,
+(19)
+with an equality if Bt,0 is sampled with replace-
+ment in {1, · · · , n}. See Section C.2 for detailed
+computations. From a numerical point of view,
+an efficient strategy consists in increasing the size
+b′
+t with the outer loop index t (see references in
+Section 3.3 for 3P-SPIDER applied to EM).
+Random
+stopping
+time
+of
+the
+algo-
+rithm. In non-convex optimization, the last iter-
+ate �Skout,kin
+kout is not necessarily the point which
+minimizes, over the sequence {�St,k, t ∈ [kout]⋆, k ∈
+[kin
+t ]⋆}, the distance to the set of solutions of (1).
+The quantity ∆⋆
+· , motivated by (10), can not be
+computed exactly in our framework so that the
+”best” iterate can not be identified thanks to this
+criterion. It is therefore popular to analyze the
+algorithm when stopped at a random time (see
+e.g. (Lan, 2020, Chapter 6)). For sake of simplic-
+ity, we consider the case when kin
+t = kin for any t
+and Cb = 0. We have the following corollary:
+Corollary 4.2 (of Theorem 4.1) Assume that kin
+t
+=
+kin, Cb = 0 and the stepsize sequence is constant
+γt,k = γ⋆. Let (τ, K) be a uniform random variable on
+[kout]⋆ × [kin]⋆, independent of the algorithm. Then
+inf
+(t,k)∈[kout]⋆×[kin]⋆
+�1
+2 − Λt,k
+�
+E
+�
+∆⋆
+τ,K + D⋆
+τ,K
+�
+≤
+E
+�
+W(�S1,0) + g(�S1,0)
+�
+− minS (W +g)
+koutkinγ⋆
++ vmaxE
+�
+∥Eτ∥2�
++ vmax E
+��
+kin − K + 1
+�
+Uτ,K
+�
+.
+
+3P-SPIDER
+13
+An upper bound on Λt,k can easily be obtained
+from (15) as a function of L ˙W, L, vmin, vmax,
+kin, b, γ⋆, Cvb and inft,k ¯
+Mt,k+1.
+Corollary 4.2 shows that, even by stopping
+3P-SPIDER with this simple rule, the first term in
+the RHS is inversely proportional to the maximal
+number of iterations koutkin.
+Complexity analysis when Et = 0, Ut,k =
+0 and kin
+t
+= kin. For smooth first-order opti-
+mization, algorithms are compared through their
+complexity in order to satisfy an ϵ-first order
+stationary condition. In stochastic composite opti-
+mization, this criterion is naturally extended to
+the ϵ-approximate stationary condition defined by
+E
+�
+∆⋆
+τ,K
+�
+≤ ϵ ,
+where (τ, K) is a random variable taking values
+in [kout]⋆ × [kin]⋆; see e.g. (Ghadimi et al, 2016,
+Section 4), (Wang et al, 2019, Section 3) and Fort
+and Moulines (2021)).
+Corollary 4.3 studies the proximal complexity
+Kprox defined as the number of calls to the prox
+operator in order to satisfy the ϵ-approximate sta-
+tionary condition; the stochastic ¯h-complexity K¯h
+defined as the number of calls to one of the ¯hi’s;
+and the total number of iterations kinkout. Again
+for sake of simplicity, and in order to compare our
+results to the literature, we consider a simplified
+setting.
+Corollary 4.3 (of Corollary 4.2) Assume in addi-
+tion
+that
+Et
+=
+0
+and
+Ut,k
+=
+0.
+The
+ϵ-
+approximate stationary condition is satisfied with
+γ⋆
+=
+vmin/(4 L ˙W), kin/b
+=
+L2
+˙W/(vminvmaxL2),
+b
+=
+O(√n√vminvmaxL/L ˙W)
+and
+koutkin
+=
+O(L ˙W/(ϵvmin)). Moreover, Kprox = O(L ˙W/(vmin ϵ))
+and K¯h = O(√vmaxL√n/(ϵ√vmin)).
+The proof is in Section 7.6. This result shows
+that the step size γ⋆ and the number of inner loops
+kin are independent of the accuracy ϵ.
+When applied to Stochastic Gradient Descent,
+3P-SPIDER in the setting of Corollary 4.3 is the
+Prox-SpiderBoost algorithm studied in Wang
+et al (2019): Corollary 4.3 and (Wang et al,
+2019, Theorem 2) state the same complexity
+results. (Wang et al, 2019, Table 1) compares
+Prox-SpiderBoost to other stochastic gradient
+algorithms for composite non-convex finite sum
+optimization. It is shown that the variance reduc-
+tion based on SPIDER order-level outperforms
+other variance reduction strategies such as the
+SVRG one and the SAGA on, introduced respectively
+by Johnson and Zhang (2013) and Defazio et al
+(2014). Hence, 3P-SPIDER reaches the state of the
+art among the proximal stochastic gradient algo-
+rithms designed to solve finite sum non-convex
+composite optimization.
+When applied to EM, 3P-SPIDER in the set-
+ting of Corollary 4.3 is the extension of the
+SPIDER-EM algorithm studied in Fort et al (2020)
+to the case there is a proximal step which man-
+ages the constraint g. Here again, the comparison
+of Corollary 4.3 and (Fort et al, 2020, Theorem
+2) shows that 3P-SPIDER reaches the state of the
+art among the incremental EM algorithms with
+variance reduction, including sEM-VR and FIEM
+introduced respectively in Chen et al (2018) and
+Karimi et al (2019) (see also Fort et al (2021a).
+See the comparison to the literature in Fort et al
+(2020).
+Beyond these two applications, Corollary 4.3
+is - to our best knowledge - the first complexity
+result for an algorithm designed to solve (1) under
+the constraint (2) and for non-convex finite sum
+composite optimization.
+ϵ-approximate stationary condition: the
+cost
+of
+inexact
+preconditioned
+forward
+operators. Let us discuss the cost of inexact
+¯hi(s)’s when the approximation is unbiased and
+random (so that Cb = Cvb = 0, see Table 1):
+does it deteriorate the proximal complexity Kprox
+and the number of calls to an oracle of a precon-
+ditioned forward operator ¯hi (still denoted by K¯h
+below) ? detailed computations of the assertions
+below can be found in Section 7.7.
+If E
+�
+∥Et∥2�
+= O(ϵ1−a′/(√nt)a′) for some a′ ∈
+[0, 1) and
+Mt,k+1 = O
+�n(a−¯a)/2
+ϵ1−a
+ta (k + 1)¯a
+�
+for some a, ¯a ∈ [0, 1), then the ϵ-approximate sta-
+tionary condition is satisfied with kin
+t
+= O(√n),
+b = O(√n) and kout = O(1/(√nϵ)). In addition,
+Kprox = O(1/ϵ) and K¯h = O(√n/ϵ). Therefore,
+the conclusions of Corollary 4.3 remain valid, and
+the approximations of the hi’s do not deteriorate
+
+14
+3P-SPIDER
+the complexity performances of the algorithms, as
+soon as the approximation is small enough.
+Let us now evaluate the computational cost,
+in the case the unbiased random approximation
+is a Monte Carlo approximation computed from
+independent and identically distributed (i.i.d.)
+samples. In this case, Mt,· is the number of terms
+of the Monte Carlo sum (see Section A). The
+Monte Carlo complexity KMC defined as the total
+number of Monte Carlo draws required to satisfy
+the ϵ-approximate stationary condition is: KMC =
+O(√n/ϵ2) for any a, a′, ¯a ∈ [0, 1).
+To our best knowledge, it is the first com-
+plexity analysis with such a Monte Carlo approx-
+imation of the preconditioned forward operators
+¯hi’s.
+5 Application: Penalized
+Logistic Regression with
+random effects
+5.1 The model
+Motivated by applications in classification, we
+consider a logistic regression model with random
+effects.
+Let n pairs of examples {(Xi, Yi), i ∈ [n]⋆}
+where Xi ∈ Rd collects the d explanatory vari-
+ables, and Yi is the binary response variable taking
+values in {−1, 1}. We assume that given {Xi, i ∈
+[n]⋆}, the binary observations {Yi, i ∈ [n]⋆} are
+independent with distribution
+{−1, 1} ∋ yi �→
+�
+Rd(1 + exp(−yi ⟨Xi, zi⟩))−1
+×
+1
+√
+2π
+dσd exp
+�
+−(2σ2)−1∥zi − θ∥2�
+dzi .
+In words, each example #i has an individ-
+ual regression vector Zi in Rd and given Zi,
+the success probability P(Yi
+=
+1
+|
+Zi) is
+(1 + exp(− ⟨Xi, Zi⟩))−1. The regression vectors
+Z1, · · · , Zn are independent with a Gaussian dis-
+tribution N(θ, σ2Id). θ is assumed to be unknown
+and σ2 is known.
+The objective is the estimation of θ by maxi-
+mizing the penalized log-likelihood criterion, with
+a ridge penalty pen(θ)
+def
+= nτ∥θ∥2, where τ > 0. By
+a change of variable, we obtain that the criterion
+to be minimized is (see Lemma B.1)
+F : θ �→ − 1
+n
+n
+�
+i=1
+log
+�
+R
+exp
+�
+x ⟨Xi, θ⟩ /(σ2∥Xi∥)
+�
+1 + exp(−yi∥Xi∥x)
+× exp
+�
+−x2/(2σ2)
+�
+dx + ∥θ∥2
+U ,
+where
+U
+def
+= τId +
+1
+2σ2
+1
+n
+n
+�
+i=1
+XiX⊤
+i
+∥Xi∥2 .
+The following lemma shows that the minimizers
+of F are in a compact set K of Rd thus implying
+that the optimization problem can be constrained
+to K. The proof is given in Section B.2.
+Lemma 5.1 The minimizers of F are in the set K def
+=
+{θ ∈ Rd : ∥θ∥2 ≤ (ln 4)/τ}.
+To solve this optimization problem, we propose
+two approaches: a gradient one, solved in the origi-
+nal space θ ∈ Rd (see Section 2.1); and an EM one,
+solved in the statistic space (see Section 2.2). The
+discussions in Section 5.2 and Section 5.3 show
+that EM is a gradient approach for finding the
+critical points of s �→ F(U −1s/2).
+5.2 A Gradient approach
+We are interested in finding a critical point of F
+in K. Equivalently, we want to solve
+0 ∈ 1
+n
+n
+�
+i=1
+Gi(θ) + ∂g(θ)
+where g is the {0, +∞}-valued indicator function
+of the set K and
+Gi(θ)
+def
+= 2Uθ −
+Xi
+σ2 ∥Xi∥
+�
+R
+z πθ,i(z)dz ;
+πθ,i(z) is the probability density proportional to
+exp
+�
+z ⟨Xi, θ⟩ /(σ2∥Xi∥) − z2/(2σ2)
+�
+1 + exp(−yi∥Xi∥z)
+.
+(20)
+We apply 3P-SPIDER with B
+def
+= Iq and hi(θ, Iq)
+def
+=
+−Gi(θ); note that proxγ g(θ) = argminx∈K∥x −
+
+3P-SPIDER
+15
+θ∥2. hi is the sum of an explicit term and an
+integral with no closed form: it will be approx-
+imated by a Monte Carlo method, based on a
+Markov chain Monte Carlo (MCMC) sampler (see
+Section 5.4 below). Therefore, δt,k,i will be a
+biased random approximation.
+5.3 An EM approach
+The criterion F to be minimized is of the form (5)
+with Z = R, µlv(dz) = dz and p(Yi, z; θ) equal to
+exp
+�
+z ⟨Xi, θ⟩ /(σ2∥Xi∥) − z2/(2σ2) − ∥θ∥2
+U
+�
+1 + exp(−Yi∥Xi∥z)
+.
+The curved exponential family assumption on the
+complete data model is satisfied: p(Yi, z; θ) =
+H(Yi, z) exp (⟨S(Yi, z), φ(θ)⟩ − ψ(θ)) with φ(θ)
+def
+=
+θ, ψ(θ)
+def
+= ∥θ∥2
+U and
+S(Yi, z)
+def
+= z
+Xi
+σ2 ∥Xi∥ .
+From Section 2.2, EM in the statistic space is of
+the form (1)-(2): it solves 0 ∈ n−1 �n
+i=1 ¯Gi(s) +
+∂¯g(s) where ¯Gi(s) = B Gi(Bs), B
+def
+= U −1/2 and
+¯g(s) is the {0, +∞}-valued indicator function of
+the set {s ∈ Rd : T(s) ∈ K} where T(s)
+def
+= Bs; it
+uses
+¯hi(s)
+def
+=
+Xi
+σ2 ∥Xi∥
+�
+R
+z πBs,i(z)dz − s ,
+(21)
+and the metric induced by B(s)
+def
+=
+B. See
+Section B.3 for detailed computations. As in the
+gradient approach, hi requires the expectation
+of the distribution π·,i (see (20)) which has no
+closed form. We will run 3P-SPIDER with B(s) ←
+B and a biased random approximation of the
+¯hi(s)’s (see Section 5.4); note that proxB
+γ ¯g(s) =
+B−1 argminx∈K
+�
+(x − Bs)⊤B−1(x − Bs)
+�
+.
+5.4 The MCMC approximation of ¯hi
+We discuss how to design an efficient MCMC
+sampler for the approximation of
+Ii(θ)
+def
+=
+�
+R
+z πθ,i(z) dz ,
+θ ∈ Rd ,
+where πθ,i is defined, up to a normalizing constant,
+by (20). By using an integration by parts and by
+applying (Polson et al, 2013, Theorem 1), we show
+that a data augmentation scheme is possible to
+approximate integrals w.r.t. πθ,i(z).
+Lemma 5.2 For any i ∈ [n]⋆ and θ ∈ Rd, it holds
+Ii(θ) =
+� Xi
+∥Xi∥, θ
+�
++ yi∥Xi∥σ2
+�
+R
+� +∞
+0
+¯πθ,i(z, ω)
+1 + exp (yi∥Xi∥z) dzdω ,
+where ¯πθ,i(z, ω) is a probability density on R×(0, +∞).
+The conditional distribution of z given ω is a Gaussian
+distribution with parameters
+⟨Xi, θ⟩ /∥Xi∥ + yi∥Xi∥σ2/2
+1 + ωσ2∥Xi∥2
+,
+σ2
+1 + ωσ2∥Xi∥2 ;
+the conditional distribution of ω given z is a Polya-
+Gamma distribution with parameters (1, ∥Xi∥z).
+The proof is given in Section B.4. Therefore,
+a Monte Carlo approximation of integrals w.r.t.
+πθ,i are obtained from a Gibbs sampler targeting
+the distribution ¯πθ,i(z, ω): it produces a sequence
+of pairs {(Zr, Ωr), r ≥ 0} and only the Zr’s are
+retained for the Monte Carlo approximation. For
+example, ¯hi(s) given by (21) can be approximated
+by
+¯hi(s) ≈ −s +
+Xi
+σ2∥Xi∥2 ⟨Xi, Bs⟩
++ yi∥Xi∥ 1
+m
+m
+�
+r=1
+�
+1 + exp(yi∥Xi∥Zs,i
+r )
+�−1 .
+(22)
+This Gibbs sampler is uniformly ergodic (see
+(Choi and Hobert, 2013, Proposition 3.1)); con-
+sequently, upon noting that z �→ Hi(z)
+def
+= (1 +
+exp(yi∥Xi∥z))−1 is bounded by one uniformly in i
+and z, the conditions A5 in Section A are verified
+with U equal to the constant function 1 and with
+a geometric convergence rate ρ(r)
+def
+= υr for some
+υ ∈ (0, 1) (remember that S is a compact set in our
+application); details are provided in Section B.5.
+Therefore, A4 is verified and the rates mt,k+1,
+Mt,k+1 and
+¯
+Mt,k+1 are equal, and equal to the
+number of points in the Monte Carlo sum (see
+Proposition A.1).
+
+16
+3P-SPIDER
+5.5 Numerical illustrations
+Let us run 3P-SPIDER for minimizing the cri-
+terion F; based on previous results comparing
+variance reduced Expectation Maximization algo-
+rithms and variance reduced Gradient algorithms
+(see e.g. (Chen et al, 2018, section 4)), we restrict
+our attention to the EM approach. In this numer-
+ical application, n = 24 989 and d = 21; we choose
+τ = 1 and σ2 = 0.05.
+The data set. The n pairs (yi, Xi) are built
+from the MNIST data set. The 13 007 examples
+labeled yi = −1 are the examples labeled 1 or 7
+in the MNIST training data set; the 11 982 exam-
+ples labeled yi = 1 are the examples labeled 3 or
+8 in the MNIST training data set. The covariates
+Xi are obtained as follows. Let Xim be the 784×n
+matrix collecting the 784 pixels for each image.
+The pixels take values in [0, 1]. Then the rows of
+Xim are centered; by a PCA, each image is reduced
+to a vector in R20. This yields Xred ∈ R20×n.
+Finally, Xred is augmented with a row of ones,
+yielding X ∈ R21×n. The columns of X are the
+Xi’s.
+The
+algorithms. We compare four algo-
+rithms. EM denotes the SAEM algorithm (Delyon
+et al (1999)) combined with a proximal step: each
+iteration processes the full data set so that there
+is one iteration of EM per epoch:
+�SEM
+r+1
+def
+= proxB
+γ g(�SEM
+r + γ
+n
+n
+�
+i=1
+�
+¯hi(�SEM
+r )) .
+Online EM is the algorithm given by Capp´e and
+Moulines (2009) combined with a proximal step;
+each iteration processes b examples and below, we
+will run kin def
+= ⌈n/b⌉ iterations per epoch:
+�SOEM
+r+1
+def
+= proxB
+γ g(�SOEM
+r
++ γ
+b
+�
+i∈Br+1
+�
+¯hi(�S0EM
+r
+)) .
+For
+EM
+and
+Online EM,
+�
+¯hi(�S•
+t )
+is
+a
+Monte
+Carlo approximation of ¯hi(�S•
+t ) computed with mt
+points. 3P-SPIDER is Algorithm 2; we choose kin
+t =
+kin and kin = ⌈n/b⌉ so that one epoch corresponds
+to the kin inner loops; we choose b′
+t = n so that
+the initialization of each outer loop is one epoch;
+the δt,k,i are computed by Monte Carlo sums (see
+(22)) with m0 points for δt,0,i and mt points for
+δt,k+1,i; since �St,0 = �St,−1, we set δt,1,i = 0 for
+all i, so that St,1 = St,0 = n−1 �n
+i=1 δt,0,i. Finally,
+3P-SPIDER and 3P-SPIDER-corr differ as follows:
+the Monte Carlo approximation δt,k+1,i necessi-
+tates a Monte Carlo approximation of ¯hi(�St,k)
+and one of ¯hi(�St,k−1). In 3P-SPIDER, the Monte
+Carlo approximations are based on two indepen-
+dent chains (see (22)) while in 3P-SPIDER-corr
+the chains are correlated.
+All the algorithms are initialized at the null vec-
+tor �Sinit = 0 ∈ Rd. The step size is equal to
+γ = 0.4 during the first six epochs and then equal
+to γ = 0.1. The length of all the paths is 20 epochs.
+On all the figures except Figure 3, we report a
+mean value computed over 25 independent runs of
+each algorithm; the shadowed area is delimited by
+the minimal and maximal value of the displayed
+criterion over these runs.
+Analyses. Most of the comparisons are based
+on the evolution of
+∆t,k+1
+def
+= ∥proxB
+γ g(�St,k + γ St,k+1) − �St,k∥2
+B
+γ2
+as a function of the number of epochs; this crite-
+rion is an approximation of ∆⋆
+t,k+1 (see (14)) which
+can not be computed here since ¯h has no closed
+form in this application. The criterion ∆t,k+1 for
+3P-SPIDER and 3P-SPIDER-corr, is compared to
+∆EM
+r defined by
+∥proxB
+γ g(�SEM
+r + γn−1 �n
+i=1
+�
+¯hi(�SEM
+r )) − �SEM
+r ∥2
+B
+γ2
+;
+and to ∆OEM
+r
+defined by
+∥proxB
+γ g(�SOEM
+r
++ γb−1 �
+i∈Br+1
+�
+¯hi(�SOEM
+r
+)) − �SOEM
+r
+∥2
+B
+γ2
+.
+The best algorithm will have the smallest value of
+∆t,k+1.
+We first study the role of some design param-
+eters of 3P-SPIDER, such as the number of Monte
+Carlo points when computing δt,0,i (denoted by
+m0) and δt,k+1,i (denoted by mt) and the balance
+between kin and b which satisfy kinb ≈ n.
+On Figure 1, two strategies are chosen: first, m0 =
+mt = 2⌈√n⌉; then m0 = mt = 5⌈√n⌉; in all cases,
+kin = ⌈√n/10⌉ and b = ⌈n/kin⌉. For comparison,
+EM and Online EM are also run, with a number of
+Monte Carlo point equal to mt at each iteration.
+
+3P-SPIDER
+17
+Fig. 1 Different strategies for the number of Monte Carlo
+points when approximating ¯hi(s) - see (22). Evolution of
+∆EM
+r
+in green, ∆OEM
+r
+in red, ∆t,k+1 for 3P-SPIDER in blue
+and ∆t,k+1 for 3P-SPIDER corr in black, as a function of
+the number of epochs. [left] m0 = mt = 2⌈√n⌉, [right]
+m0 = mt = 5⌈√n⌉.
+On Figure 2, the case when kin = ⌈√n/10⌉ is
+compared to the case kin = ⌈√n/2⌉; in both cases,
+b = ⌈n/kin⌉ and m0 = mt = 2⌈√n⌉.
+Fig. 2 Number of inner loops per epoch. Evolution of
+∆OEM
+r
+in red, ∆t,k+1 for 3P-SPIDER in blue and ∆t,k+1
+for 3P-SPIDER corr in black, as a function of the number
+of epochs. [left] kin = ⌈√n/10⌉ and b = ⌈n/kin⌉. [right]
+kin = ⌈√n/2⌉ and b = ⌈n/kin⌉.
+Each algorithm returns a sequence of points
+in the s-space, from which a sequence of points
+in the θ-space is deduced through the formula
+θ = T(s) = Bs ∈ Rd. On Figure 3, three compo-
+nents of this θ-sequence are displayed, versus the
+number of epochs.
+Fig. 3 Estimation of three parameters. Evolution of the
+three components of θ by EM in green (top, left), OEM
+in red (top, right), 3P-SPIDER in blue (bottom, left) and
+3P-SPIDER corr in black (bottom, right), as a function of
+the number of epochs.
+Finally, we also display on Figure 4 the evo-
+lution of the squared norm of the iterates ∥�St,k∥2
+obtained by 3P-SPIDER and 3P-SPIDER-corr, and
+∥�SOM
+r ∥2 and ∥�SOEM
+r
+∥2 obtained resp. by EM and
+Online EM. They are plotted as a function of the
+epochs.
+Fig. 4 Squared norm of the iterates. Evolution of ∥�SEM
+r ∥2
+in green (top, left), ∥�SOEM
+r
+∥2 in red (top, right), ∥�St,k∥2 for
+3P-SPIDER in blue (bottom, left) and ∥�St,k∥2 for 3P-SPIDER
+corr in black (bottom, right), as a function of the number
+of epochs.
+Conclusions. EM has a slow convergence rate
+and even fails to converge before 20 epochs con-
+trary to the other algorithms (see e.g. Figure 4):
+one update of the iterate per epoch is not enough
+especially during the first iterations when more
+
+100
+100
+10-2
+102
+104
+10-4
+10-6
+10-6
+国
+w国国
+10-8
+10-8
+01357
+91113151719
+0135
+7
+91113151719100
+100
+10-2
+10°4
+104
+10-4
+10-6
+10-6
+10-8
+10-8
+0135
+791113151719
+01 35791113151719EM
+Online EM
+0.02
+0.02
+0
+-0.02
+-0.02
+-0.04
+-0.04
+0135791113151719
+0135791113151719
+3P SPIDER
+3P SPIDER corr
+0.02
+0.02
+0
+0
+-0.02
+-0.02
+-0.04
+-0.04
+01
+9 1113151719
+013
+5
+9 1113 151719EM
+Online EM
+2
+1.5
+1.9
+1
+1.8
+0.5
+1.7
+5
+7
+91113151719
+5
+6
+1113151719
+3P SPIDER
+3P SPIDER corr
+2
+2
+1.9
+1.9
+1.8
+1.8
+1.7
+1.7
+5
+91113151719
+5 
+7
+6
+111315171918
+3P-SPIDER
+updates even based on part of the data set is a
+better strategy (see e.g. the behavior of Online
+EM, which contains kin updates per epoch).
+Online EM, 3P-SPIDER and 3P-SPIDER-corr pro-
+cess part of the data set at each iteration; com-
+pared to Online EM, the 3P-SPIDER’s contain a
+variance reduction. All the plots illustrate the ben-
+efit of this variance reduction, which reduces the
+variability at convergence.
+The choice of γ impacts this variability: see e.g.
+Figures 1, 2 and Figure 4 where a change occurs at
+epoch #7 (remember that from epoch 2ℓ to 2ℓ+1,
+Online EM runs kin updates of the iterates while
+the 3P-SPIDER’s do not update the iterate since
+they compute St,0).
+3P-SPIDER-corr improves on 3P-SPIDER. The
+control variate has a larger impact when the cor-
+relation is increased, as illustrated by all plots. It
+decreases the variability introduced by the mini-
+batches (b < n) and the variability introduced by
+the Monte Carlo approximation δt,k+1,i.
+Given the budget of n examples processed per
+outer loops, Figure 2 shows that at convergence,
+the accuracy is improved by larger mini batch sizes
+and therefore a smaller number of inner loops.
+Not surprisingly, a larger number of Monte Carlo
+points decreases the variability at convergence (see
+Figure 1).
+6 Proof of Section 3
+6.1 Proof of Lemma 3.1
+Lemma
+6.1
+collects
+the
+two
+statements
+of
+Lemma 3.1 and a third property.
+Lemma 6.1 Assume A1.
+1. For any γ > 0, B ∈ Pq
++ and s ∈ Rq, the
+optimization problem (7) has a unique solu-
+tion, characterized as the unique point p ∈ S
+satisfying −γ−1 B(p − s) ∈ ∂g(p).
+2. For any γ > 0, B ∈ Pq
++, s ∈ S and h ∈ Rq,
+s = proxB
+γg(s + γh)
+iff
+Bh ∈ ∂g(s).
+(23)
+3. Let γ > 0 and B ∈ Pq
++. The operator proxB
+γg is
+firmly nonexpansive; this implies that for any
+s, s′ ∈ Rq,
+∥proxB
+γg(s′) − proxB
+γg(s)∥2
+B
+≤
+�
+proxB
+γg(s′) − proxB
+γg(s), s′ − s
+�
+B .
+Proof Existence, uniqueness and characterization are
+established in Hiriart-Urruty and Lemar´echal (1996,
+Chapter XV, Lemma 4.1.1). The statement (23) fol-
+lows from the characterization; note that proxB
+γg(s) ∈
+S for any s ∈ Rq. The firmly nonexpansive property
+is a consequence of Hiriart-Urruty and Lemar´echal
+(1996, Chapter XV, Theorem 4.1.4).
+□
+7 Proof of Section 4
+7.1 Notations
+Define for any s ∈ S,
+¯h(s)
+def
+= 1
+n
+n
+�
+i=1
+¯hi(s) ,
+¯hB
+def
+= 1
+b
+�
+i∈B
+¯hi ,
+where B is an n-tuple of elements of [n]⋆ (with or
+without multiplicity) of cardinal b.
+All the random variables are defined on a prob-
+ability space (Ω, A, P). It is endowed with the
+following filtrations for t ≥ 0 and k ≥ 0,
+F0,kin
+0
+def
+= σ(�Sinit),
+Ft,0
+def
+= Ft−1,kin
+t−1
+�
+σ (Bt,0, δt,0,i for all i) ,
+Ft,k+ 1
+2
+def
+= Ft,k
+�
+σ(Bt,k+1),
+Ft,k+1
+def
+= Ft,k+ 1
+2
+�
+σ (δt,k+1,i for all i ∈ Bt,k+1) .
+For any t ∈ [kout]⋆, set
+Et
+def
+= St,0 − ¯h(�St,0) = 1
+b′
+t
+�
+i∈Bt,0
+δt,0,i − ¯h(�St,0) .
+Et is the error when replacing the full sum using
+exact terms ¯hi(�St,0), with a possibly subsum of
+size b′
+t < n using approximations of ¯hi(�St,0).
+Remember that
+ξt,k+1,i
+def
+= δt,k+1,i − ¯hi(�St,k) + ¯hi(�St,k−1) ,
+and
+µt,k+1,i
+def
+= E
+�
+ξt,k+1,i|Ft,k+1/2
+�
+,
+σ2
+t,k+1,i
+def
+= E
+�
+∥ξt,k+1,i − µt,k+1,i∥2|Ft,k+1/2
+�
+.
+
+3P-SPIDER
+19
+Finally, set
+ηt,k+1
+def
+= 1
+b
+�
+i∈Bt,k+1
+ξt,k+1,i.
+Throughout the proof, we will use the shorthand
+notation
+Bt,k
+def
+= B(�St,k) .
+7.2 Preliminary lemmas
+Lemma 7.1 Let B be a batch of [n]⋆ of size b, sampled
+at random (with or without replacement).
+1. For
+any
+family
+{f1, · · · , fn},
+E
+�
+b−1 �
+i∈B fi
+�
+= n−1 �n
+i=1 fi.
+2. For any family {f1, · · · , fn},
+E
+����1
+b
+�
+i∈B
+fi − 1
+n
+n
+�
+i=1
+fi
+���
+2
+�
+≤ 1
+b n
+n
+�
+i=1
+∥fi − 1
+n
+n
+�
+j=1
+fj∥2 .
+3. Assume A2. For any s, s′ ∈ S, it holds
+E
+����
+�¯hB(s) − ¯hB(s′)
+�
+−
+�¯h(s) − ¯h(s′)
+� ���
+2�
+≤ 1
+b
+�
+L2∥s − s′∥2 − ∥¯h(s) − ¯h(s′)∥2�
+.
+Proof The proof is along the same lines as the proof
+of (Fort et al, 2020, Lemma 4). A detailed proof is
+provided in Section C.3.
+□
+Lemma 7.2 Assume A4-item 1 and A4-item 3. For
+any t ∈ [kout]⋆ and k ∈ [kin
+t − 1], it holds
+E
+�
+ηt,k+1|Ft,k+1/2
+�
+= 1
+b
+�
+i∈Bt,k+1
+µt,k+1,i ,
+E
+�
+ηt,k+1|Ft,k
+�
+= 1
+n
+n
+�
+i=1
+µt,k+1,i ,
+E
+�
+∥ηt,k+1 − E
+�
+ηt,k+1|Ft,k
+�
+∥2|Ft,k
+�
+≤ 1
+b
+�
+Cv
+Mt,k+1
++
+C2
+vb
+¯
+M2
+t,k+1
+�
+.
+Proof Let t ∈ [kout]⋆ and k ∈ [kin
+t − 1]. We have
+E
+�
+ηt,k+1|Ft,k+1/2
+�
+= 1
+b
+�
+i∈Bt,k+1
+µt,k+1,i ,
+since Bt,k+1 ∈ Ft,k+1/2; and by Lemma 7.1,
+E
+�
+ηt,k+1|Ft,k
+�
+= 1
+n
+n
+�
+i=1
+µt,k+1,i .
+We write
+ηt,k+1 − E
+�
+ηt,k+1|Ft,k
+�
+= 1
+b
+�
+i∈Bt,k+1
+ξt,k+1,i − 1
+n
+n
+�
+i=1
+µt,k+1,i
+= 1
+b
+�
+i∈Bt,k+1
+�
+ξt,k+1,i − µt,k+1,i
+�
++ 1
+b
+�
+i∈Bt,k+1
+µt,k+1,i − 1
+n
+n
+�
+i=1
+µt,k+1,i .
+The RHS is of the form U + V
+and we write
+∥U + V ∥2 = ∥U∥2 + ∥V ∥2 + 2 ⟨U, V ⟩ with U
+←
+b−1 �
+i∈Bt,k+1
+�
+ξt,k+1,i − µt,k+1,i
+�
+. By conditioning
+and by definition of σ2
+t,k+1,i, we have
+E
+�
+∥U∥2|Ft,k
+�
+= 1
+b2 E
+�
+�
+�
+i∈Bt,k+1
+σ2
+t,k+1,i|Ft,k
+�
+� .
+Under A4-item 1, we have by Lemma 7.1
+E
+�
+∥U∥2|Ft,k
+�
+=
+1
+b n
+n
+�
+i=1
+σ2
+t,k+1,i ≤
+Cv
+b Mt,k+1
+.
+By Lemma 7.1 again, it holds
+E
+�
+∥V ∥2|Ft,k
+�
+≤
+1
+b n
+n
+�
+i=1
+∥µt,k+1,i − 1
+n
+n
+�
+j=1
+µt,k+1,j∥2 ,
+which yields
+E
+�
+∥V ∥2|Ft,k
+�
+≤
+C2
+vb
+b ¯
+M2
+t,k+1
+.
+Finally, upon noting that E
+�
+U|Ft,k+1/2
+�
+= 0 and V ∈
+Ft,k+1/2, we have
+E
+�
+⟨U, V ⟩ |Ft,k
+�
+= E
+��
+E
+�
+U|Ft,k+1/2
+�
+, V
+�
+|Ft,k
+�
+= 0 .
+This concludes the proof.
+□
+7.3 Results on the variables St,k
+Proposition 7.3 studies the bias of the variables
+St,k+1. It shows that St,k+1 is a biased approxima-
+tion of ¯h(�St,k):
+E [St,k+1|Ft,k] ̸= ¯h(�St,k).
+
+20
+3P-SPIDER
+When k = 0, we may have E [St,1|Ft,0] = ¯h(�St,0)
+if δt,0,i = hi(�St,0) and Bt,0 = [n]⋆. The choice
+Bt,0 = [n]⋆ is the strategy proposed in Wang et al
+(2019) for SpiderBoost; it has an important com-
+putational cost but has the advantage to cancel
+the bias of the variable S· at the beginning of each
+outer loop. Along the inner loops, a (signed) bias
+appears.
+Proposition 7.3 For any t ∈ [kout]⋆ and k ∈ [kin
+t −
+1], it holds
+E
+�
+St,k+1|Ft,k
+�
+− ¯h(�St,k)
+= St,k − ¯h(�St,k−1) + E
+�
+ηt,k+1|Ft,k
+�
+,
+and
+E
+�
+St,k+1 − ¯h(�St,k)|Ft,0
+�
+= Et +
+k+1
+�
+j=1
+E
+�
+ηt,j|Ft,0
+�
+.
+Proof Let t ∈ [kout]⋆ and k ∈ [kin
+t
+− 1]. We write
+St,k+1 = St,k + hBt,k+1(�St,k) − hBt,k+1(�St,k−1) +
+ηt,k+1. By Lemma 7.1,
+E
+�
+St,k+1|Ft,k
+�
+= St,k + ¯h(�St,k) − ¯h(�St,k−1)
++ E
+�
+ηt,k+1|Ft,k
+�
+.
+Since Ft,0 ⊆ Ft,k, we have
+E
+�
+St,k+1 − ¯h(�St,k)|Ft,0
+�
+= E
+�
+St,k − ¯h(�St,k−1)|Ft,0
+�
++ E
+�
+ηt,k+1|Ft,0
+�
+.
+Summing from j = 0 to j = k yields
+E
+�
+St,k+1 − ¯h(�St,k)|Ft,0
+�
+= St,0 − ¯h(�St,−1)
++
+k
+�
+j=0
+E
+�
+ηt,j+1|Ft,0
+�
+.
+The proof is concluded by using �St,0 = �St,−1 and the
+definition of Et; note that Et ∈ Ft,0.
+□
+Proposition 7.4 provides a control of the con-
+ditional variance of St,k.
+Proposition 7.4 Assume A 2, A 4-item 1 and A 4-
+item 3. For any t ∈ [kout]⋆ and k ∈ [kin
+t − 1], it holds
+E
+����St,k+1 − E
+�
+St,k+1|Ft,k
+� ���
+2
+|Ft,k
+�
+≤ L2
+b
+�
+1 +
+2Cvb
+√
+b ¯
+Mt,k+1
+�
+∥�St,k − �St,k−1∥2
++
+Cv
+b Mt,k+1
++
+C2
+vb
+b ¯
+M2
+t,k+1
++
+2Cvb
+√
+b ¯
+Mt,k+1
+.
+Proof Let t ∈ [kout]⋆, k ∈ [kin
+t − 1]. By Lemma 7.1,
+Proposition 7.3, the definitions of St,k+1 and of the
+filtration Ft,k,
+St,k+1 − E
+�
+St,k+1|Ft,k
+�
+= St,k+1 − ¯h(�St,k) − St,k + ¯h(�St,k−1) − E
+�
+ηt,k+1|Ft,k
+�
+= ηt,k+1 − E
+�
+ηt,k+1|Ft,k
+�
++ ¯hBt,k+1(�St,k) − ¯hBt,k+1(�St,k−1) − ¯h(�St,k) + ¯h(�St,k−1) .
+The RHS is of the form U + V with U ← ηt,k+1 −
+E
+�
+ηt,k+1|Ft,k
+�
+and V
+∈ Ft,k+1/2. Then, we write
+E
+�
+∥U + V ∥2|Ft,k
+�
+= E
+�
+∥U∥2|Ft,k
+�
++E
+�
+∥V ∥2|Ft,k
+�
++
+2E
+��
+E
+�
+U|Ft,k+1/2
+�
+, V
+�
+|Ft,k
+�
+.
+The
+term
+E
+�
+∥V ∥2|Ft,k
+�
+is
+controlled
+by
+Lemma 7.1: an upper bound is L2b−1∥�St,k− �St,k−1∥2.
+The term E
+�
+∥U∥2|Ft,k
+�
+is controlled by Lemma 7.2:
+an upper bound is Cv/(b Mt,k+1) + C2
+vb/(b ¯
+M2
+t,k+1).
+Upon noting that V ∈ Ft,k+1/2, and using Lemma 7.2
+and Lemma 7.1, the scalar product is upper bounded
+by
+2 E
+�
+∥V ∥
+���E
+�
+U|Ft,k+1/2
+� ���
+���Ft,k
+�
+≤ 2
+Cvb
+√
+b ¯
+Mt,k+1
+�
+E
+�
+∥V ∥2���Ft,k
+��1/2
+≤ 2
+Cvb
+√
+b ¯
+Mt,k+1
+�
+1 + E
+�
+∥V ∥2|Ft,k
+��
+,
+where we used that a ≤ 1 + a2.
+□
+Proposition 7.5 establishes an upper bound on
+the conditional expectation of the quadratic error
+∥St,k+1 − ¯h(�St,k)∥2.
+Proposition 7.5 Assume A2 and A4. For any t ∈
+[kout]⋆ and k ∈ [kin
+t − 1], it holds
+E
+�
+∥St,k+1 − ¯h(�St,k)∥2|Ft,k
+�
+≤
+�
+1 +
+2Cb
+mt,k+1
+�
+∥St,k − ¯h(�St,k−1)∥2
++ L2
+b
+�
+1 +
+2Cvb
+√
+b ¯
+Mt,k+1
+�
+∥�St,k − �St,k−1∥2
++ Ut,k+1 ,
+where
+Ut,k
+def
+=
+2Cb
+mt,k
++ C2
+b
+m2
+t,k
++
+Cv
+b Mt,k
++
+2Cvb
+√
+b ¯
+Mt,k
++
+C2
+vb
+b ¯
+M2
+t,k
+.
+
+3P-SPIDER
+21
+Proof Let t ∈ [kout]⋆ and k ∈ [kin
+t − 1]. By definition
+of the conditional expectation, we have for any r.v.
+U, V and any σ-field F such that V ∈ F:
+E
+�
+∥U − V ∥2|F
+�
+= E
+�
+∥U − E [U|F] ∥2|F
+�
++ ∥E [U|F] − V ∥2 .
+We apply this equality with U ← �St,k+1, V ← ¯h(�St,k)
+and F ← Ft,k. Proposition 7.4 controls the first term.
+For the second one, by Proposition 7.3, Lemma 7.2
+and A4-item 2 we have
+∥E
+�
+St,k+1|Ft,k
+�
+− ¯h(�St,k)∥2
+= ∥St,k − ¯h(�St,k−1) + E
+�
+ηt,k+1|Ft,k
+�
+∥2
+≤ ∥St,k − ¯h(�St,k−1)∥2 +
+C2
+b
+m2
+t,k+1
++ 2
+Cb
+mt,k+1
+∥St,k − ¯h(�St,k−1)∥ .
+We conclude by using ∥a∥ ≤ 1+∥a∥2 with a ← ∥St,k−
+¯h(�St,k−1)∥.
+□
+Corollary 7.6 (of Proposition 7.5) Assume also A3-
+item 3. For t ∈ [kout]⋆ and k ∈ [kin
+t −1], define Dt,k+1
+by
+∥�St,k+1 − proxBt,k
+γt,k+1g(�St,k + γt,k+1¯h(�St,k))∥2
+Bt,k ,
+and Dt,0
+def
+= 0. For t ∈ [kout]⋆ and k ∈ [kin
+t −], it holds
+E
+�
+∥St,k+1 − ¯h(�St,k)∥2|Ft,k
+�
+≤
+�
+1 +
+2Cb
+mt,k+1
+�
+∥St,k − ¯h(�St,k−1)∥2
++ γ2
+t,k
+2
+vmin
+L2
+b
+�
+1 +
+2Cvb
+√
+b ¯
+Mt,k+1
+�
+∆⋆
+t,k
++
+2
+vmin
+L2
+b
+�
+1 +
+2Cvb
+√
+b ¯
+Mt,k+1
+�
+Dt,k
++ Ut,k+1 .
+By convention, ∆⋆
+t,0
+def
+= 0.
+Proof The proof consists in an upper bound for ∥�St,k−
+�St,k−1∥2. Let s ∈ S, H, h ∈ Rq, γ > 0 and B be a
+q × q positive definite matrix. For any β > 0, it holds
+∥proxB
+γg(s+γH)−s∥2
+B ≤ (1+ 1
+β ) ∥proxB
+γg(s+γh)−s∥2
+B
++ (1 + β) ∥proxB
+γg(s + γH) − proxB
+γg(s + γh)∥2
+B .
+We apply these inequalities with γ ← γt,k, B ←
+Bt,k−1, s ← �St,k−1, H ← St,k and h ← ¯h(�St,k−1).
+Then, for any k > 0,
+∥�St,k − �St,k−1∥2
+Bt,k−1
+≤ (1 + β−1) γ2
+t,k∆⋆
+t,k + (1 + β)Dt,k .
+(24)
+We choose β = 1 and conclude by A3-item 3: ∥ · ∥2 ≤
+v−1
+min∥ · ∥2
+Bt,k−1.
+When k = 0, ∥�St,k − �St,k−1∥2
+Bt,k−1 = 0 since by
+definition, �St,0 = �St,−1. Therefore, (24) remains valid
+since Dt,0 = 0 and ∆⋆
+t,0 = 0 by convention. This con-
+cludes the proof.
+□
+Corollary 7.7 (of Corollary 7.6) Let {ρt,k, t ≥ 1, k ≥
+0} be a positive sequence satisfying
+ρt,k+1
+�
+1 +
+2Cb
+mt,k+1
+�
+≤ ρt,k .
+(25)
+For any t ∈ [kout]⋆, k ∈ [kin
+t − 1], it holds
+ρt,k+1E
+�
+∥St,k+1 − ¯h(�St,k)∥2|Ft,0
+�
+≤ ρt,0∥Et∥2 +
+k+1
+�
+ℓ=1
+ρt,ℓ Ut,ℓ +
+2
+vmin
+L2
+b · · ·
+×
+� k
+�
+ℓ=1
+γ2
+t,ℓρt,ℓ+1
+�
+1 +
+2Cvb
+√
+b ¯
+Mt,ℓ+1
+�
+E
+�
+∆⋆
+t,ℓ|Ft,0
+�
++
+k
+�
+ℓ=1
+ρt,ℓ+1
+�
+1 +
+2Cvb
+√
+b ¯
+Mt,ℓ+1
+�
+E
+�
+Dt,ℓ|Ft,0
+�
+�
+.
+Proof In Corollary 7.6, the claim is of the form
+E
+�
+∥St,k+1 − ¯h(�St,k)∥2|Ft,k
+�
+≤
+�
+1 +
+2Cb
+mt,k+1
+�
+∥St,k − ¯h(�St,k−1)∥2 + Ak .
+This yields, by using the condition (25),
+ρt,k+1E
+�
+∥St,k+1 − ¯h(�St,k)∥2|Ft,k
+�
+≤ ρt,k ∥St,k − ¯h(�St,k−1)∥2 + ρt,k+1 Ak .
+Using E[U|Ft,0] = E
+�
+E[U|Ft,k]|Ft,0
+�
+and summing
+from ℓ = 0 to ℓ = k yields
+ρt,k+1 E
+�
+∥St,k+1 − ¯h(�St,k)∥2|Ft,0
+�
+≤
+k
+�
+ℓ=0
+ρt,ℓ+1E
+�
+Aℓ|Ft,0
+�
++ ρt,0∥St,0 − ¯h(�St,−1)∥2 ;
+we then conclude by using the equality �St,−1 = �St,0
+and the definition of Et. Note also that ∆⋆
+t,0 = 0 and
+Dt,0 = 0.
+□
+
+22
+3P-SPIDER
+7.4 Lyapunov inequalities for W, g
+and W +g
+Lemma 7.8, while being classical in smooth opti-
+mization, is provided for a self-content purpose.
+Lemma 7.8 Assume A3. For any s, s′ ∈ S and γ > 0,
+W(s′) ≤ W(s) −
+�¯h(s), s′ − s
+�
+B(s) + L ˙W
+2 ∥s′ − s∥2 .
+Proof S is convex since it is the domain of a convex
+function. By A3, W is continuously differentiable on S
+with L ˙W-Lipschitz gradient. Then for any s, s′ ∈ S,
+W(s′) − W(s) ≤
+�
+∇ W(s), s′ − s
+�
++ L ˙W
+2 ∥s′ − s∥2 .
+We use that ∇ W(s) = −B(s)¯h(s), so that
+�
+∇ W(s), s′ − s
+�
+= −
+�¯h(s), s′ − s
+�
+B(s) .
+□
+Lemma 7.9 Assume A1. Let B be a q × q positive
+definite matrix. For any s ∈ S, γ > 0, H, h ∈ Rq and
+β > 0,
+g
+�
+proxB
+γg(s + γH)
+�
+≤ g(s)
+− 1
+γ
+�
+1 − β
+4
+�
+∥proxB
+γg(s + γh) − s∥2
+B
+− 1
+γ (1 − 1
+β )∥proxB
+γg(s + γh) − proxB
+γg(s + γH)∥2
+B
+−
+�
+h, s − proxB
+γg(s + γh)
+�
+B
++
+�
+H, proxB
+γg(s + γH) − proxB
+γg(s + γh)
+�
+B .
+Proof In this proof, we use the shorthand notation
+pH
+def
+= proxB
+γg(s + γH) and ph
+def
+= proxB
+γg(s + γh). By
+Lemma 3.1 and the definition of the subdifferential at
+a point, it holds
+g(ph) ≥ g(pH) − γ−1 ⟨pH − s − γH, ph − pH⟩B
+g(s) ≥ g(ph) − γ−1 ⟨ph − s − γh, s − ph⟩B .
+This yields
+g(pH) ≤ g(s) − γ−1∥ph − s∥2
+B − ⟨h, s − ph⟩B
+− ⟨H, ph − pH⟩B + γ−1 ⟨pH − s, ph − pH⟩B .
+For the last term, we write for any β > 0,
+γ−1 ⟨pH − s, ph − pH⟩B + γ−1∥ph − pH∥2
+B
+= γ−1 ⟨ph − s, ph − pH⟩B
+≤ 2
+�
+(ph − s)
+√β
+2√γ , (ph − pH)
+1
+√βγ
+�
+B
+≤ β
+4γ ∥ph − s∥2
+B + 1
+βγ ∥ph − pH∥2
+B .
+This concludes the proof.
+□
+Proposition 7.10 Assume A1, A2 and A3. For any
+t ∈ [kout]⋆, k ∈ [kin
+t − 1] and β > 0,
+E
+�
+W(�St,k+1) + g(�St,k+1)|Ft,0
+�
+≤ E
+�
+W(�St,k) + g(�St,k)|Ft,0
+�
+− γt,k+1
+�
+1 − β
+4 − L ˙Wγt,k+1
+vmin
+�
+E
+�
+∆⋆
+t,k+1|Ft,0
+�
+−
+1
+γt,k+1
+�
+1 − 1
+β − L ˙W
+vmin
+γt,k+1
+�
+E
+�
+Dt,k+1|Ft,0
+�
++ γt,k+1E
+�
+∥St,k+1 − ¯h(�St,k)∥2
+Bt,k|Ft,0
+�
+,
+where Dt,k+1 is defined in Corollary 7.6.
+Proof Let γ > 0, s ∈ S and H ∈ Rq. Apply Lemma 7.8
+with s′ ← proxB(s)
+γg
+(s+γH) ∈ S; and Lemma 7.9 with
+h ← ¯h(s). This yields for any β > 0,
+W(proxB
+γg(s + γH)) + g(proxB
+γg(s + γH))
+≤ W(s) + g(s) − 1
+γ (1 − β
+4 )∥proxB
+γg(s + γ¯h(s)) − s∥2
+B
+− 1
+γ
+�
+1 − 1
+β
+�
+∥proxB
+γg(s+γH)−proxB
+γg(s+γ¯h(s))∥2
+B
+−
+�
+¯h(s) − H, proxB
+γg(s + γH) − proxB
+γg(s + γ¯h(s))
+�
+B
++ L ˙W
+2 ∥proxB
+γg(s + γH) − s∥2 .
+Since proxB
+γg is firmly nonexpansive (see Lemma 6.1),
+the scalar product is upper bounded by γ∥H −¯h(s)∥2
+B.
+By A3-item 3, we write
+∥proxB
+γg(s+γH)−s∥2 ≤
+1
+vmin
+∥proxB
+γg(s+γH)−s∥2
+B ;
+then we use ∥a + b∥2
+B ≤ 2∥a∥2
+B + 2∥b∥2
+B with a ←
+proxB
+γg(s + γH) − proxB
+γg(s + γ¯h(s)). This yields
+L ˙W
+2 ∥proxB
+γg(s + γH) − s∥2
+≤ L ˙W
+vmin
+∥proxB
+γg(s + γH) − proxB
+γg(s + γ¯h(s))∥2
+B
++ L ˙W
+vmin
+∥proxB
+γg(s + γ¯h(s)) − s∥2
+B .
+We apply these inequalities with s ←
+�St,k, γ ←
+γt,k+1, H ← St,k+1, s′ ← �St,k+1 and B ← Bt,k. Note
+that proxB
+γg(s+γH) = �St,k+1. The proof is concluded.
+□
+
+3P-SPIDER
+23
+7.5 Proof of Theorem 4.1
+Let t ∈ [kout]⋆. Let µ ∈ (0, 1). Throughout the
+proof, set
+At,k+1
+def
+=
+�
+1 +
+2Cvb
+√
+b ¯
+Mt,k+1
+�
+.
+From Corollary 7.7 applied with ρt,k+1 ← γt,k+1
+and Proposition 7.10 applied with β ← 4µ, it holds
+for any k ∈ [kin
+t − 1],
+E
+�
+W(�St,k+1) + g(�St,k+1)|Ft,0
+�
+≤ E
+�
+W(�St,k) + g(�St,k)|Ft,0
+�
+− γt,k+1
+�
+1 − µ − L ˙W
+vmin
+γt,k+1
+�
+E
+�
+∆⋆
+t,k+1|Ft,0
+�
+−
+1
+γt,k+1
+�
+1 − 1
+4µ − L ˙W
+vmin
+γt,k+1
+�
+E [Dt,k+1|Ft,0]
++ γt,0vmax∥Et∥2 + vmax
+k+1
+�
+ℓ=1
+γt,ℓ Ut,ℓ
++ 2vmax
+vmin
+L2
+b
+k
+�
+ℓ=1
+γ3
+t,ℓAt,ℓ+1E
+�
+∆⋆
+t,ℓ|Ft,0
+�
++ 2vmax
+vmin
+L2
+b
+k
+�
+ℓ=1
+γt,ℓ+1At,ℓ+1E [Dt,ℓ|Ft,0] .
+Above, we used that γt,k+1 ≤ γt,ℓ for any ℓ ∈
+[k + 1]. We now sum from k = 0 to k = kin
+t − 1.
+This yields,
+E
+�
+W(�St,kin
+t ) + g(�St,kin
+t )|Ft,0
+�
+≤ E
+�
+W(�St,0) + g(�St,0)|Ft,0
+�
+−
+kin
+t
+�
+k=1
+γt,k
+�
+1 − µ − L ˙W
+vmin
+γt,k
+�
+E
+�
+∆⋆
+t,k|Ft,0
+�
+−
+kin
+t
+�
+k=1
+1
+γt,k
+�
+1 − 1
+4µ − L ˙W
+vmin
+γt,k
+�
+E [Dt,k|Ft,0]
++ γt,0vmaxkin
+t ∥Et∥2
++ vmax
+kin
+t
+�
+ℓ=1
+(kin
+t − ℓ + 1)γt,ℓ Ut,ℓ
++ 2vmax
+vmin
+kin
+t
+kin
+t −1
+�
+k=1
+γ3
+t,kAt,k+1E
+�
+∆⋆
+t,k|Ft,0
+�
++ 2vmax
+vmin
+kin
+t
+kin
+t −1
+�
+k=1
+γt,k+1At,k+1E [Dt,k|Ft,0] .
+Observe
+that
+the
+coefficient
+in
+front
+of
+E
+�
+∆⋆
+t,k|Ft,0
+�
+is γt,k(1 − µ − Λt,k+1); and the term
+in front of E [Dt,k|Ft,0] is γ−1
+t,k (1−1/(4µ)−Λt,k+1).
+By symmetry, we choose µ
+=
+1/2 so that
+µ = 1/(4µ). This yields
+E
+�
+W(�St,kin
+t ) + g(�St,kin
+t )|Ft,0
+�
+≤ E
+�
+W(�St,0) + g(�St,0)|Ft,0
+�
+−
+kin
+t
+�
+k=1
+γt,k
+�1
+2 − Λt,k+1
+�
+E
+�
+∆⋆
+t,k|Ft,0
+�
+−
+kin
+t
+�
+k=1
+1
+γt,k
+�1
+2 − Λt,k+1
+�
+E [Dt,k|Ft,0]
++ γt,0vmaxkin
+t ∥Et∥2
++ vmax
+kin
+t
+�
+ℓ=1
+(kin
+t − ℓ + 1)γt,ℓ Ut,ℓ .
+We now sum for t = 1 to t = kout and compute the
+expectation. This yields, by using that �St+1,0 =
+�St,kin,
+kout
+�
+t=1
+kin
+t
+�
+k=1
+γt,k
+�1
+2 − Λt,k+1
+�
+E
+�
+∆⋆
+t,k
+�
++
+kout
+�
+t=1
+kin
+t
+�
+k=1
+1
+γt,k
+�1
+2 − Λt,k+1
+�
+E [Dt,k]
+≤
+E
+�
+W(�St,0) + g(�St,0)
+�
+− E
+�
+W(�Skout,kin
+kout ) + g(�Skout,kin
+kout )
+�
++ vmax
+kout
+�
+t=1
+γt,0kin
+t E
+�
+∥Et∥2�
++ vmax
+kout
+�
+t=1
+kin
+t
+�
+ℓ=1
+�
+kin
+t − ℓ + 1
+�
+γt,ℓ Ut,ℓ .
+The
+proof
+is
+concluded
+upon
+noting
+that
+E
+�
+W(�Skout,kin
+kout ) + g(�Skout,kin
+kout )
+�
+≥ minS(W +
+g).
+
+24
+3P-SPIDER
+7.6 Proof of Corollary 4.3
+Since Ut,k = 0, we have Cb = Cvb = 0. In addition,
+kin
+t
+= kin for any t. Therefore, we can consider
+a constant stepsize sequence γt,k = γ⋆ where γ⋆
+satisfies (see (15) and (16))
+γ⋆
+L ˙W
+vmin
++ γ2
+⋆
+2vmax
+vmin
+L2 kin
+b ∈ (0, 1/2) .
+This condition is satisfied by choosing
+kin
+b
+def
+=
+1
+vminvmax
+L2
+˙W
+L2 ,
+γ⋆
+def
+=
+1
+4vmax
+L ˙W
+L2
+b
+kin = vmin
+4L ˙W
+.
+Such a choice implies that inft,k(1/2 − Λt,k) =
+1/2 − 3/8 = 1/8. Since Et = Ut,k = 0, we obtain
+from Corollary 4.2 that
+E
+�
+∆⋆
+τ,K + D⋆
+τ,K
+�
+≤ 32 L ˙W
+vmin
+�
+E
+�
+W(�S1,0) + g(�S1,0)
+�
+− minS (W +g)
+�
+koutkin
+.
+The ϵ-approximate stationary condition is sat-
+isfied by choosing koutkin
+=
+O(L ˙W/(vmin ϵ)).
+The number of calls to the proximal operator is
+koutkin so that Kprox = O(L ˙W/(vmin ϵ)). Finally,
+we have b′
+t = n so that the number of calls
+to one of the ¯hi’s is kout n + 2koutkinb. We
+can choose b = O
+�√n√vminvmaxL/L ˙W
+�
+. This
+yields kout = O(L√vmax/(√vminϵ√n)), and K¯h =
+O(√vmaxL√n/(ϵ√vmin)).
+7.7 Cost of the approximation on
+the ¯hi’s
+Following the rates obtained in Corollary 4.3, let
+us set kin
+t
+= O(√n), b = O(√n) and kout =
+O(1/(√nϵ)) and let us show that we can define
+random approximations δt,0,i and δt,k+1,i such
+that the ϵ-approximate stationarity condition is
+satisfied.
+On
+the
+term
+E
+�
+∥Eτ∥2�
+.
+We
+write
+E
+�
+∥Eτ∥2�
+= (kout)−1 �kout
+t=1 E
+�
+∥Et∥2�
+and
+1
+kout
+kout
+�
+t=1
+ϵ1−a′
+√na′ta′ = ϵ O(1) .
+Let us compute the associated Monte Carlo
+complexity in the case Et = n−1 �n
+i=1{δt,0,i −
+¯hi(�St,0)} and δt,0,i is equal to a Monte Carlo
+sum with mt,0 i.i.d. samples. Then E
+�
+∥Et∥2�
+=
+n−1m−1
+t,0O(1). It is equal to O(ϵ1−a′/(√nt)a′)
+when mt,0 = O(na′/2−1ta′/ϵ1−a′). Therefore, the
+Monte Carlo cost is
+kout
+�
+t=1
+nmt,0 = O
+�
+1
+√nϵ2
+�
+.
+On the term E
+��
+kin − K + 1
+�
+Uτ,K
+�
+. This
+term is upper bounded by kin E [Uτ,K] and we
+write
+kin E [Uτ,K] ≤
+1
+kout
+kout
+�
+t=1
+kin
+�
+k=1
+O
+�
+1
+b Mt,k+1
+�
+.
+The RHS is O(ϵ). The associated Monte Carlo
+complexity is
+2b
+kout
+�
+t=1
+kin
+�
+k=1
+Mt,k = O
+�√n
+ϵ2
+�
+,
+whatever a, ¯a ∈ [0, 1).
+Acknowledgments.
+This work was partly sup-
+ported by the Fondation Simone et Cino del Duca,
+under the program OpSiMorE; and by the french
+Agence Nationale de la Recherche (ANR) under
+the program ANR-19-CE23 MASDOL.
+Appendix A
+The condition A
+4 in the Monte
+Carlo case
+Following the framework detailed in Section 3.4,
+let us assume that (i) the intractable quantities
+hi(�St,k, Bt,k+1) and hi(�St,k−1, Bt,k) are of the form
+hi(s, B) =
+�
+Z
+Hϑ(z)πϑ(dz) ,
+(A1)
+where ϑ
+def
+=
+(s, i, B); and (ii) these integrals
+are approximated by a Monte Carlo sum: set
+ϑt,ℓ+1,i
+def
+= (�St,ℓ, i, Bt,ℓ+1) and
+
+3P-SPIDER
+25
+δt,k+1,i
+def
+=
+1
+mt,k+1
+mt,k+1
+�
+r=1
+�
+Hϑt,k+1,i(Zϑt,k+1,i
+r
+)
+−Hϑt,k,i(Zϑt,k,i
+r
+)
+�
+,
+(A2)
+where, conditionally to �St,k−1,
+�St,k, Bt,k and
+Bt,k+1, the samples {Zϑt,ℓ,i
+r
+, r
+≥
+1} are a
+Markov chain with unique stationary distribution
+πϑt,ℓ,i(dz); ℓ ∈ {k, k + 1}. Below, we show that
+A4 is verified when the Markov chain is ergodic
+enough. Let us start with introducing few nota-
+tions from the Markov chain theory (see e.g. Meyn
+and Tweedie (1993)).
+Let P be a transition kernel onto the measur-
+able set (Z, Z) and λ, π be probability measures on
+(Z, Z). For a measurable function ξ : Z → [0, +∞),
+define
+π(ξ)
+def
+=
+�
+Z
+ξ(z) π(dz) .
+For any r ∈ N, the r-iterated transition kernel P r
+is defined by induction:
+P r+1(z, A)
+def
+=
+�
+Z
+P r(z, dy) P(y, A)
+=
+�
+Z
+P(z, dy) P r(y, A) ,
+for all z ∈ Z, A ∈ Z; by convention, P 0(z, A)
+def
+=
+χA(z) the {0, 1}-valued indicator function and
+P 0(z, A) = δz(A), the Dirac mass at zero. Given
+a probability measure λ on (Z, Z), λP stands for
+the probability measure on (Z, Z) given by
+λP(A)
+def
+=
+�
+Z
+λ(dy)P(y, A) ,
+∀A ∈ Z .
+For a function U
+: Z → [1, +∞) such that
+λP r(U) + π(U) < +∞, define the U-norm of a
+measurable function ξ : Z → Rq
+∥ξ∥U
+def
+= sup
+Z
+∥ξ∥
+U
+;
+and the U-norm of the signed measure λP r −π by
+∥λP r − π∥U
+def
+=
+sup
+ξ:∥ξ∥U≤1
+∥λP r(ξ) − π(ξ)∥ .
+Let us go back to sufficient conditions for verifying
+A4. Denote by Pϑ a Markov transition kernel with
+invariant distribution πϑ(dz): at iteration (t, k+1),
+conditionally to (�St,k−1, �St,k, Bt,k, Bt,k+1), the
+chains {Zϑt,k
+r
+, r ≥ 0} and {Zϑt,k+1
+r
+, r ≥ 0} are
+Markov chains with transition kernels Pϑt,k and
+Pϑt,k+1 respectively. They have the same initial
+value λ. Assume
+A5. 1. There exists a measurable function U :
+Z → [1, +∞) such that
+H⋆
+def
+=
+sup
+(s,i,B)∈S×[n]⋆×Pq
++
+∥Hϑ∥U < +∞ ,
+where Hϑ is defined by (A1).
+2. There exist a function ρ : N → [0, 1] and a
+positive constant CMC such that for any r ∈ N,
+sup
+ϑ∈S×[n]⋆×Pq
++
+∥λP r
+ϑ − πϑ∥U ≤ CMC ρ(r) .
+In addition, �
+r≥1 ρ(r) < +∞.
+3. Let ϑ ∈ S × [n]⋆ × Pq
++. Let {Zϑ
+r , r ≥ 1} be
+a Markov chain with transition kernel Pϑ and
+initial distribution λ. There exists a positive
+constant C′
+MC such that for any ϑ ∈ S × [n]⋆ ×
+Pq
++ and m′ ∈ N⋆,
+E
+�
+�∥
+m′
+�
+r=1
+{Hϑ(Zϑ
+r ) − πϑ(Hϑ)}∥2
+�
+�
+≤ H2
+⋆ C′
+MC m′ .
+A5-item 2 is a uniform-in-s ergodicity condi-
+tion. Sufficient conditions for it are provided in
+(Fort et al, 2011, Lemma 2.3.) in the case of a
+geometric rate ρ(r) = κr for some κ ∈ (0, 1). By
+adapting (Andrieu et al, 2015, Theorem 1), sim-
+ilar conditions can be obtained in the case of a
+subgeometric rate ρ(r). Sufficient conditions for
+A5-item 3 can be obtained from a trivial adap-
+tation of (Fort and Moulines, 2003, Proposition
+12).
+We prove the following result.
+Proposition
+A.1 Assume
+A
+5.
+Let
+δt,k+1,i
+be
+given
+by
+(A2),
+where
+conditionally
+to
+(�St,k−1, �St,k, Bt,k, Bt,k+1), {Zϑt,ℓ,i
+r
+, r
+≥
+0} is a
+Markov chain with transition kernel Pϑt,ℓ,i and ini-
+tial distribution λ, for ℓ ∈ {k, k + 1}. Then A 4 is
+verified with mt,k+1 = Mt,k+1 = ¯
+Mt,k+1 ← mt,k+1,
+
+26
+3P-SPIDER
+Cb
+=
+Cvb
+def
+=
+2 H⋆ CMC
+�
+r≥1 ρ(r)
+and
+Cv
+def
+= 2H2⋆ C′
+MC.
+Proof We will use the notations
+Pℓ,i
+def
+= Pϑt,ℓ+1,i, πℓ,i
+def
+= πϑt,ℓ+1,i, Hℓ,i
+def
+= Hϑt,ℓ+1,i .
+• Expression of µt,k+1. We have
+µt,k+1,i
+def
+=
+1
+mt,k+1
+mt,k+1
+�
+r=1
+�
+λP r
+k,iHk,i − πk,i(Hk,i)
+�
+−
+1
+mt,k+1
+mt,k+1
+�
+r=1
+�
+λP r
+k−1,iHk−1,i − πk−1,i(Hk−1,i)
+�
+.
+• The condition A4-Item 2. By A5 and since
+�S• ∈ S, we write
+sup
+k,i
+∥λP r
+k,iHk,i − πk,i(Hk,i)∥ ≤ H⋆ CMC ρ(r) .
+This implies that
+∥µt,k+1,i∥ ≤ 2 H⋆ CMC
+1
+mt,k+1
+mt,k+1
+�
+r=1
+ρ(r) .
+Since �
+r ρ(r) < ∞, the RHS is of the form Cb/mt,k+1
+with Cb
+def
+= 2 H⋆ CMC
+�
+r ρ(r).
+• The condition A4-Item 3. We write
+σ2
+t,k+1,i ≤ E
+�
+∥ξt,k+1,i∥2|Pt,k+1/2
+�
+.
+Then, we have
+E
+�
+∥ξt,k+1,i∥2|Pt,k+1/2
+�
+≤ 2 sup
+s,i
+E
+�
+∥
+1
+mt,k+1
+mt,k+1
+�
+r=1
+Hs,i(Zs,i
+r ) − πs,i(Hs,i)∥2
+�
+and the RHS is upper bounded by 2H2⋆ C′
+MC/mt,k+1
+by A5-Item 3.
+We also have
+1
+n
+n
+�
+i=1
+∥µt,k+1,i− 1
+n
+n
+�
+j=1
+µt,k+1,j∥2 ≤ 1
+n
+n
+�
+i=1
+∥µt,k+1,i∥2.
+From the upper bound on ∥µt,k+1,i∥ above, we have
+∥µt,k+1,i∥2 ≤
+C2
+b
+m2
+t,k+1
+.
+This concludes the proof.
+□
+Appendix B
+Supplementary
+materials for
+Section 5
+B.1
+The penalized log-likelihood
+criterion
+The observations are assumed independent, so the
+log-likelihood is given by
+θ �→
+n
+�
+i=1
+log
+�
+Rd(1 + exp(−yi ⟨Xi, zi⟩))−1
+×
+1
+√
+2π
+dσd exp
+�
+−(2σ2)−1∥zi − θ∥2�
+dzi .
+The penalty term is −nτ∥θ∥2.
+Lemma B.1 The sum of the log-likelihood and the
+penalty term is equal to
+− n
+2 log(2πσ2) −
+1
+2σ2 θ⊤
+n
+�
+i=1
+XiX⊤
+i
+∥Xi∥2 θ − nτ∥θ∥2
++
+n
+�
+i=1
+log
+�
+R
+exp
+�
+x ⟨Xi, θ⟩ /(σ2∥Xi∥)
+�
+1 + exp(−yi∥Xi∥x)
+exp(− x2
+2σ2 )dx .
+Proof Let i ∈ [n]⋆. Define an orthogonal d × d matrix
+Q with columns denoted by (Q1, · · · , Qd), such that
+Q1
+def
+= Xi/∥Xi∥. We have ⟨zi, Xi⟩ = ∥Xi∥ ⟨Q1, zi⟩,
+⟨zi, θ⟩ =
+�
+Q⊤zi, Q⊤θ
+�
+and ∥zi∥2 = ∥Q⊤zi∥2. This
+implies that
+− yi ⟨zi, Xi⟩ = −yi∥Xi∥ ⟨Q1, zi⟩
+∥zi − θ∥2 = ∥θ∥2 + ∥QT zi∥2 − 2
+�
+Q⊤zi, Q⊤θ
+�
+so that the log-likelihood of the observation Yi is (up
+to the additive constant C1
+def
+= −d ln σ − (d/2) ln(2π))
+yi �→ −∥θ∥2
+2σ2 +log
+�
+Rd(1+exp(−yi∥Xi∥ ⟨Q1, zi⟩))−1
+×exp
+�
+−(2σ2)−1 �
+∥QT zi∥2 − 2
+�
+Q⊤zi, Q⊤θ
+���
+dzi .
+By a change of variable v = (v1, · · · , vq) ← Q⊤zi, the
+logarithm of the integral is equal to
+log
+�
+Rd
+exp
+�
+−(2σ2)−1 �
+∥v∥2 − 2
+�
+v, Q⊤θ
+���
+1 + exp(−yi∥Xi∥v1)
+dv
+= log
+�
+R
+exp
+�
+−(2σ2)−1 �
+v2
+1 − 2v1 ⟨Q1, θ⟩
+��
+1 + exp(−yi∥Xi∥v1)
+dv1
+
+3P-SPIDER
+27
++
+d
+�
+u=2
+log
+�
+R
+exp
+�
+− 1
+2σ2 {v2
+u − 2vu ⟨Qu, θ⟩}
+�
+dvu .
+The last (d − 1) integrals have a closed form. Observe
+indeed that v2u − 2vu ⟨Qu, θ⟩ = (vu − ⟨Qu, θ⟩)2 −
+(⟨Qu, θ⟩)2 so that up to the additive constant C2
+def
+=
+(d − 1){log(2π)/2 + log σ}
+d
+�
+u=2
+log
+�
+R
+exp
+�
+− 1
+2σ2 {v2
+u − 2vu ⟨Qu, θ⟩}
+�
+dvu
+=
+d
+�
+u=2
+(⟨Qu, θ⟩)2
+2σ2
+= ∥θ∥2 − (⟨Q1, θ⟩)2
+2σ2
+= ∥θ∥2 − (⟨Xi, θ⟩)2/∥Xi∥2
+2σ2
+.
+This concludes the proof; the constant (w.r.t. to θ) is
+equal to C1 + C2.
+□
+B.2
+Proof of Lemma 5.1
+The criterion F is equal to −L(θ) − log(2πσ2)/2,
+where L(θ) is the normalized penalized log-
+likelihood.
+The likelihood is the product of probabili-
+ties, taking values in (0, 1); therefore, its loga-
+rithm is negative. The penalized log-likelihood is
+upper bounded −pen(θ) = −nτ∥θ∥2. The nor-
+malized penalized log-likelihood is upper bounded
+−τ∥θ∥2. Therefore the criterion is lower bounded
+by τ∥θ∥2 − ln(2πσ2)/2.
+On the other hand, the minimum of the crite-
+rion is smaller than the value of the criterion at
+θ = 0. Let us show that this value is (ln 4)/n −
+ln(2πσ2)/2. This will imply that the minimizers of
+the criterion are in the set {θ ∈ Rd : τ∥θ∥2 ≤ ln 4}
+and conclude the proof.
+We have pen(0) = 0. Let us lower bound the
+likelihood of an observation Yi = +1 at θ = 0. The
+likelihood is equal to
+1
+√
+2π
+dσd
+�
+Rd
+exp
+�
+−(2σ2)−1∥zi∥2�
+1 + exp(− ⟨Xi, zi⟩) dzi .
+By using the same change of variable than in the
+proof of Lemma B.1, it is equal to
+1
+√
+2πσ
+�
+R
+exp
+�
+−x2/(2σ2)
+�
+1 + exp(−∥Xi∥x)dx
+�
+1
+√
+2πσ
+�
+R
+exp
+�
+−x2/(2σ2)
+�
+dx
+�d−1
+,
+and is lower bounded by (note that the (d − 1)
+identical integrals are equal to one)
+1
+√
+2πσ
+�
+R+
+exp
+�
+−x2/(2σ2)
+�
+1 + exp(−∥Xi∥x)dx ,
+which is in turn lower bounded by 1/4 since 1 +
+exp(−∥Xi∥x) ≤ 2 for all x ≥ 0.
+The proof for the case Yi = −1 is on the same
+lines and is omitted.
+This implies that the likelihood of the n vari-
+ables is lower bounded by 1/4n; the normalized
+log-likelihood is lower bounded by − ln 4; the
+criterion is upper bounded by ln 4 − ln(2πσ2)/2.
+B.3
+The optimization problem
+seen as an EM
+We established that for any θ ∈ Rd, ∇F(θ) =
+n−1 �n
+i=1 Gi(θ) where
+Gi(θ)
+def
+= 2Uθ −
+Xi
+σ2 ∥Xi∥
+�
+R
+z πθ,i(z)dz ,
+and πθ,i(z) is the probability density proportional
+to (20).
+From the expressions of φ, ψ and S(Yi, z), we
+obtain that T, defined by Proposition 2.1, is given
+by T(s)
+def
+= U −1s/2 for any s ∈ Rd. This implies
+that for any s ∈ Rd,
+hi(s, B)
+def
+=
+�
+R
+S(Yi, z) πT(s),i(z) dz − s
+=
+Xi
+σ2 ∥Xi∥
+�
+R
+z πBs,i(z) dz − s .
+For any s ∈ Rd, let us find the matrix B(s)
+satisfying ∇(F ◦ T)(s) = −n−1 �n
+i=1 B(s)hi(s)
+(see (6)). We have ∇(F ◦ T)
+=
+∇F(B·)
+=
+B⊤ (∇F)(B·) = B (∇F)(B·). This yields
+∇(F ◦ T)(s) = B 1
+n
+n
+�
+i=1
+Gi(Bs)
+= B 1
+n
+n
+�
+i=1
+�
+2Uθ −
+Xi
+σ2 ∥Xi∥
+�
+R
+z πBs,i(z)dz
+�
+= Bs − B 1
+n
+n
+�
+i=1
+Xi
+σ2 ∥Xi∥
+�
+R
+z πBs,i(z)dz
+
+28
+3P-SPIDER
+= −B 1
+n
+n
+�
+i=1
+hi(s) .
+This yields B(s)
+def
+= B for any s ∈ Rd.
+B.4
+Proof of Lemma 5.2
+Let i ∈ [n]⋆ and θ ∈ Rd. Step 1. By using
+− z2/(2σ2) + z ⟨Xi, θ⟩ /(σ2∥Xi∥)
+= −(z − ⟨Xi, θ⟩ /∥Xi∥)2/(2σ2)
++ (⟨Xi, θ⟩)2/(2σ2∥Xi∥2) ,
+we write
+πθ,i(z) =
+exp
+�
+− (z − ⟨Xi, θ⟩ /∥Xi∥)2 /(2σ2)
+�
+Zθ,i 1 + exp(−yi∥Xi∥z)
+where Zθ,i is the normalizing constant. Second,
+we use z πθ,i(z) = (z − ai) πθ,i(z) + ai πθ,i(z) with
+ai ← ⟨Xi, θ⟩ /∥Xi∥ and since
+�
+R πθ,i(z)dz = 1, we
+obtain
+Ii(θ) = ⟨Xi, θ⟩
+∥Xi∥ +
+�
+R
+�
+z − ⟨Xi, θ⟩
+∥Xi∥
+�
+πθ,i(z) dz .
+Finally, the integral in the RHS being of the form
+σ2
+�
+R
+f ′(z)
+Zθ,i 1 + exp(−yi∥Xi∥z)dz
+with
+f(z)
+def
+= − exp
+�
+− (z − ⟨Xi, θ⟩ /∥Xi∥)2 /(2σ2)
+�
+,
+we use an integration by parts. Upon noting that
+the derivative of z �→ 1/(1 + exp(−yi∥Xi∥z)) is
+yi∥Xi∥
+exp(−yi∥Xi∥z)
+(1 + exp(−yi∥Xi∥z))2 ,
+we write
+�
+R
+f ′(z)
+1 + exp(−yi∥Xi∥z)dz
+= −yi∥Xi∥
+�
+f(z)
+exp(−yi∥Xi∥z)
+(1 + exp(−yi∥Xi∥z))2 dz .
+Therefore, the conclusion of this first step is
+Ii(θ) =
+� Xi
+∥Xi∥, θ
+�
++ yi∥Xi∥σ2
+�
+R
+πθ,i(z)
+1 + exp (yi∥Xi∥z) dz .
+Step 2. This step is classical in the MCMC
+literature (see e.g. Choi and Hobert (2013) and
+references therein). We prove that for any z ∈ R,
+πθ,i(z) =
+� +∞
+0
+¯πθ,i(z, ω)dω .
+By (Polson et al, 2013, Theorem 1), it holds
+1
+1 + exp (−yi∥Xi∥z) = 1
+2 exp (yi∥Xi∥z/2)
+×
+� +∞
+0
+exp(−ω∥Xi∥2z2/2)p(ω; 1)dω ,
+where p(ω; b)dω is a Polya-Gamma distribution
+with parameter b. This implies that πθ,i(z) is equal
+to
+exp
+�yi∥Xi∥z
+2
+− (z − ⟨Xi, θ⟩ /∥Xi∥)2
+2σ2
+�
+×
+1
+2Zθ,i
+� +∞
+0
+exp(−ω∥Xi∥2z2/2)p(ω; 1)dω .
+This concludes the proof.
+B.5
+The assumption A4 is verified.
+Define the Markov kernel with density
+Ps,i(z; z′)
+def
+=
+�� ∞
+0
+π2(ω|z; i) π1(z′|ω; s, i) dω
+�
+w.r.t.
+the
+Lebesgue
+measure
+on
+R;
+here,
+π1(z′|ω; s, i) is the density of a Gaussian distribu-
+tion with expectation ms,i(ω) and variance vi(ω)
+given by
+vi(ω)
+def
+=
+σ2
+1 + ωσ2∥Xi∥2 ,
+ms,i(ω)
+def
+= vi(ω)
+� 1
+σ2
+� Xi
+∥Xi∥, Bs
+�
++ 1
+2yi∥Xi∥
+�
+;
+and π2(ω|z;
+i) is a Polya-Gamma distribution
+with parameter (1, ∥Xi∥z). The Gibbs kernel
+described by Lemma 5.2 and targeting the density
+
+3P-SPIDER
+29
+distribution ¯πBs,i(z, ω)dzdω, produces a Markov
+chain {(Zs,i
+r , Ωs,i
+r ), r ≥ 0} such that the marginal
+{Zs,i
+r , r ≥ 0} is a Markov chain with transition
+kernel Ps,i(z;
+z′) dz′. We apply the results of
+(Choi and Hobert, 2013, Proposition 3.1) with
+y ∈ {0, 1}
+yi ∈ {−1, 1}
+n
+1
+Ω(ω)
+ω
+X
+∥Xi∥
+B
+σ2
+b
+⟨Xi, Bs⟩ /∥Xi∥
+Table B1 [left] The notations of Choi and Hobert
+(2013). [right] the notations in this paper.
+This yields
+�
+A
+Ps,i(z; z′)dz′
+≥ ε
+�
+A
+exp(−0.5(x − m⋆)2/v2
+⋆) dx
+(B3)
+where
+ε
+def
+=
+inf
+s∈S,i∈[n]⋆
+exp
+�
+− 1
+4 − {ms,i(1/2)}2 σ2∥Xi∥2
+4 vi(1/2)
+�
+2
+�
+1 + σ2∥Xi∥2/2
+and (m⋆, v⋆) satisfy for any x ∈ R,
+inf
+s∈S,i∈[n]⋆ exp(−0.5 (x − ms,i(1/2))2/vi(1/2))
+≥ exp(−0.5(x − m⋆)2/v2
+⋆) .
+Lemma B.2 Since S is bounded, then ε > 0 and
+m⋆, v⋆ exist in R × (0, +∞).
+The minorization condition (B3) implies that
+the kernel Ps,i(z, z′)dz′ is uniformly ergodic, uni-
+formly in s, i and z. By (Meyn and Tweedie, 1993,
+Theorem 16.0.2.) and (Fort and Moulines, 2003,
+Proposition 1), A 5-Item 2 and A 5-Item 3 are
+satisfied.
+Appendix C
+Detailed proofs
+C.1
+Proof of (17)
+Let t ∈ [kout]⋆. The sequence given by γt,k+1
+def
+=
+�k
+j=0
+�
+1 +
+2 Cb
+mt,j+1
+�−1
+γt,0 for any k ≥ 0, satisfies
+γt,k+1
+�
+1 +
+2Cb
+mt,k+1
+�
+≤ γt,k .
+A sufficient condition for the property Λt,k+1 ∈
+(0, 1/2) to hold is aγ2
+t,0 + ¯aγt,0 − 1/2 < 0 where
+¯a
+def
+= L ˙W
+vmin
+,
+a
+def
+= L2 2vmaxkin
+t
+vminb
+�
+1 +
+2 Cvb
+√
+b ¯
+Mt,k+1
+�
+.
+The function x �→ ax2 + ¯ax − 1/2 possesses two
+roots: one is positive and one is negative. The pos-
+itive one is given by (−¯a +
+√
+¯a2 + 2a)/(2a); it is
+equal to (17).
+C.2
+Proof of (18) and (19)
+We write St,0 − h(�St,0) = U + V where
+U
+def
+= 1
+b′
+t
+�
+i∈Bt,0
+�
+δt,0,i − E
+�
+δt,0,i|�St,0, Bt,0
+��
+,
+V
+def
+= 1
+b′
+t
+�
+i∈Bt,0
+E
+�
+δt,0,i|�St,0, Bt,0
+�
+− h(�St,0) .
+We have E
+�
+∥U + V ∥2�
+= E
+�
+∥U∥2�
++ E
+�
+∥V ∥2�
+by definition of the conditional expectation. Since
+δt,0,i is an unbiased random approximation of
+hi(�St,0), we have E
+�
+δt,0,i|�St,0, Bt,0
+�
+= hi(�St,0).
+In the case Bt,0 = {1, · · · , n}, then V = 0.
+Therefore,
+E
+�
+∥St,0 − h(�St,0)∥2�
+= 1
+n2 E
+�
+∥
+n
+�
+i=1
+{δt,0,i − E
+�
+δt,0,i|�St,0
+�
+}∥2
+�
+= 1
+n2
+n
+�
+i=1
+σ2
+t,0,i ,
+
+30
+3P-SPIDER
+where we used that the variables {δt,0,i, i ∈ [n]⋆}
+are independent conditionally to �St,0, and with
+variance σ2
+t,0,i.
+In the case Bt,0 is a subset of [n]⋆ of cardinality
+b′
+t, then we write
+E
+�
+∥U∥2�
+=
+1
+(b′
+t)2 E
+�
+E
+�
+∥U∥2|Bt,0, �St,0
+��
+= 1
+b′
+t
+E
+�
+� 1
+b′
+t
+�
+i∈Bt,0
+σ2
+t,0,i
+�
+�
+and we conclude by Lemma 7.1. Again from
+Lemma 7.1, we have
+E
+�
+∥V ∥2�
+≤
+1
+b′
+t n
+n
+�
+i=1
+∥hi(�St,0) − h(�St,0)∥2 ,
+with an equality when Bt,0 is sampled with
+replacement. This concludes the proof.
+C.3
+Proof of Lemma 7.1
+Set, for ease of notations,
+B
+def
+= Bt,k,
+hB
+def
+= 1
+b
+�
+i∈B
+hi .
+C.3.1
+Case with replacement
+We write B = {I1, · · · , Ib} where the r.v. Ii’s are
+independent, and uniformly distributed on [n]⋆.
+• Then
+E [fB] = 1
+b
+b
+�
+ℓ=1
+E [fIℓ] = E [fI1] = n−1
+n
+�
+i=1
+fi(u) .
+• Set ¯f
+def
+= n−1 �n
+i=1 fi. We have, by using that
+the r.v. {I1, · · · , Ib} are independent,
+E
+�
+∥fB − ¯f∥2�
+= E
+�
+∥1
+b
+b
+�
+ℓ=1
+�
+fIℓ − ¯f
+�
+∥2
+�
+= 1
+b2
+b
+�
+ℓ=1
+E
+�
+∥fIℓ − ¯f∥2�
+= 1
+bE
+�
+∥fI1 − ¯f∥2�
+= 1
+b n
+n
+�
+i=1
+∥fi − ¯f∥2 .
+• Since the variance of the sum is the sum of
+the variance for independent r.v.
+E
+�
+∥hB(u) − hB(u′) − {h(u) − h(u′)}∥2�
+= 1
+b2
+b
+�
+ℓ=1
+E
+�
+∥hIℓ(u) − hIℓ(u′) − h(u) + h(u′)∥2�
+.
+Then, since Iℓ is uniformly distributed on [n]⋆,
+E
+�
+∥hIℓ(u) − hIℓ(u′) − h(u) + h(u′)∥2�
+= 1
+n
+n
+�
+i=1
+E
+�
+∥hi(u) − hi(u′)∥2�
+− ∥h(u) − h(u′)∥2
+≤ ∥u − u′∥2 1
+n
+n
+�
+i=1
+L2
+i − ∥h(u) − h(u′)∥2 .
+(C4)
+C.3.2
+Case without replacement
+Set B = {I1, · · · , Ib} and ¯f
+def
+= n−1 �n
+i=1 fi. I1 is a
+uniform random variable on [n]⋆ so that E [fI1] =
+¯f.
+• Conditionally to I1, I2 is a uniform random
+variable on [n]⋆ \ {I1}. Therefore
+E [fI2] =
+1
+n − 1
+� n
+�
+j=1
+fj − E [fI1]
+�
+=
+n
+n − 1
+¯f −
+1
+n − 1
+¯f = ¯f .
+By induction, for any ℓ ≥ 2,
+E [fIℓ]
+=
+1
+n − ℓ + 1
+� n
+�
+j=1
+fj −
+ℓ−1
+�
+r=1
+E [fIr]
+�
+=
+n
+n − ℓ + 1
+¯f −
+ℓ − 1
+n − ℓ + 1
+¯f = ¯f .
+As a conclusion, b−1 �b
+ℓ=1 E [fIℓ] = ¯f.
+• Let u, u′ ∈ S; set φ(Iℓ)
+def
+= hIℓ(u) − h(u) −
+hIℓ(u′)+h(u′). Then E [φ(Iℓ)] = 0. First, we prove
+by induction that E
+�
+∥φ(Iℓ)∥2�
+= E
+�
+∥φ(I1)∥2�
+.
+Upon noting that I1 is a uniform random variable
+on [n]⋆ and by using the induction assumption,
+(n − ℓ + 1)E
+�
+∥φ(Iℓ)∥2�
+
+3P-SPIDER
+31
+=
+� n
+�
+i=1
+∥φ(i)∥2 − E
+�ℓ−1
+�
+r=1
+∥φ(Ir)∥2
+��
+= nE
+�
+∥φ(I1)∥2�
+− (ℓ − 1)E
+�
+∥φ(I1)∥2�
+,
+which concludes the induction. Second, let us
+prove that for any ℓ ≥ 0,
+E
+�
+∥
+ℓ+1
+�
+r=1
+φ(Ir)∥2
+�
+≤ (ℓ + 1)E
+�
+∥φ(I1)∥2�
+.
+(C5)
+Since �n
+i=1 φ(i) = nE [φ(I1)] = 0,
+E
+�� ℓ
+�
+p=1
+φ(Ip), φ(Iℓ+1)
+��
+= E
+�� ℓ
+�
+p=1
+φ(Ip), E
+�
+φ(Iℓ+1)
+���I1, · · · , Iℓ
+���
+=
+1
+n − ℓE
+�� ℓ
+�
+p=1
+φ(Ip),
+n
+�
+i=1
+φ(i) −
+ℓ
+�
+p=1
+φ(Ip)
+��
+= −
+1
+n − ℓE
+�
+∥
+ℓ
+�
+p=1
+φ(Ip)∥2
+�
+,
+so that
+E
+�
+∥
+ℓ+1
+�
+p=1
+φ(Ip)∥2
+�
+=
+�
+1 −
+2
+n − ℓ
+�
+E
+�
+∥
+ℓ
+�
+p=1
+φ(Ip)∥2
+�
++ E
+�
+∥φ(Iℓ+1)∥2�
+≤ (ℓ + 1)E
+�
+∥φ(I1)∥2�
+.
+The proof of the first bound follows from (C5) and
+(C4) since here again, I1 is uniformly distributed
+on [n]⋆.
+• The proof of the second bound is similar
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf,len=1421
+page_content='Stochastic Variable Metric Proximal Gradient with variance  reduction for non-convex composite optimization  Gersende Fort1* and Eric Moulines2  1*Institut de Math´ematiques de Toulouse, CNRS & Universit´e de Toulouse, 118 route de  Narbonne, Toulouse, 31400, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2CMAP, Ecole Polytechnique, Route de Saclay, Palaiseau, 91128 Cedex, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' E-mail(s): gersende.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='fort@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='univ-toulouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Contributing authors: eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='moulines@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Abstract  This paper introduces a novel algorithm, the Perturbed Proximal Preconditioned SPIDER algo-  rithm (3P-SPIDER), designed to solve finite sum non-convex composite optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is a  stochastic Variable Metric Forward-Backward algorithm, which allows approximate preconditioned  forward operator and uses a variable metric proximity operator as the backward operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it  also proposes a mini-batch strategy with variance reduction to address the finite sum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We show that 3P-SPIDER extends some Stochastic preconditioned Gradient Descent-based algo-  rithms and some Incremental Expectation Maximization algorithms to composite optimization  and to the case the forward operator can not be computed in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We also provide  an explicit control of convergence in expectation of 3P-SPIDER, and study its complexity in  order to satisfy the epsilon-approximate stationary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Our results are the first to com-  bine the non-convex composite optimization setting, a variance reduction technique to tackle the  finite sum setting by using a minibatch strategy and, to allow deterministic or random approx-  imations of the preconditioned forward operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Finally, through an application to inference  in a logistic regression model with random effects, we numerically compare 3P-SPIDER to other  stochastic forward-backward algorithms and discuss the role of some design parameters of 3P-SPIDER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Keywords: Stochastic optimization, Variable Metric Forward-Backward splitting, Preconditioned Stochastic  Gradient Descent, Incremental Expectation Maximization, Proximal methods, Variance reduction,  Non-asymptotic convergence bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  1 Introduction  Efficient learning from large data sets require new  optimization algorithms designed to be robust to  big data and complex models era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In Statistics and  Machine Learning, we are often faced with solving  problems of the form  argmins∈Rq  �  1  n  n  �  i=1  Wi(s) + g(s)  �  ,  where n is the number of examples in the training  data set, s is an unknown quantity to be learnt  from the examples, Wi is a loss function associ-  ated to the example #i and g is a regularization  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='00631v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='LG]  2 Jan 2023  2  3P-SPIDER  term encoding a priori knowledge and constraints  on s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' g may also prevent from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Quite  often, the regularization term g : Rq → (0, +∞]  is not differentiable, and the data fidelity term  n−1 �n  i=1 Wi is smooth on the domain of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This paper is concerned with stochastic opti-  mization of a non-convex finite sum composite  function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' more precisely, it addresses the differen-  tial inclusion problem  0 ∈ 1  n  n  �  i=1  Gi(s) + ∂g(s),  s ∈ Rq,  (1)  where g  :  Rq  →  (−∞, +∞] is lower semi-  continuous convex with non-empty domain S and  for all i, Gi : S → Rq is globally Lipschitz on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The first goal of this paper is to provide a novel  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Motivated by applications in Statistics  and Machine learning, we require this algorithm  to satisfy the following three constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (c1) The  algorithm uses possibly preconditioned operators  hi instead of the forward operator Gi:  ∀s ∈ S,  hi(s, B)  def  = −B−1 Gi(s),  (2)  where B is a q × q positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Such  a condition encompasses preconditioned gradient  methods for example, which also includes gradient  methods with adaptive step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It also encom-  passes Expectation Maximization (EM) algo-  rithms (Dempster et al (1977)) designed for large  scale learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (c2) The algorithm may only have  access to approximations of hi(s, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Such a con-  dition addresses the situations when hi(s, B) is  not explicit, for example when it is defined by an  intractable integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This occurs at each E-step  of EM, when the conditional expectations under  the a posteriori distributions can not be computed  exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (c3) The algorithm addresses the finite  sum challenge while keeping the caused variabil-  ity induced by the algorithmic solution as small as  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For example, when the solution relies on  a random selection of a mini-batch of examples,  the algorithm has to propose a variance reduction  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A first class of problems of the form (1)  are minimizations of regularized loss functions  through gradient-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In that case,  Gi  def  = ∇Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (2) allows preconditioned gradients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  such a variable metric is known to accelerate the  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Variable Metric Forward-Backward  (VMFB) algorithms were introduced to solve (1)-  (2) in the case G is a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Nevertheless, as  discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1, to our best knowledge  none of the variants of VMFB address the three  constraints c1, c2 and c3 simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A second application of (1)-(2) is the EM  algorithm, an algorithm originally designed to  compute the Maximum-Likelihood estimator of  an unknown parameter θ in latent variable mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When the complete data model is from the  curved exponential family, EM is equivalent to  an algorithm in the so-called statistic space (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Delyon et al (1999)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This remark is the cor-  nerstone of stochastic EM algorithms including  incremental EM ones designed for incremental  processing of large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' EM only supplies  preconditioned forward operators −B(s)−1Gi(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Therefore, stochastic EM algorithms are naturally  in the setting (1)-(2) (see Fort et al (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Here  again, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2, none of the EM  variants in the literature address the constraints  (c2) and (c3) simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Our first contribution is the design of a novel  iterative algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' named Perturbed Proximal  Preconditioned SPIDER  (3P-SPIDER),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  which  combines (i) a preconditioned forward operator  associated to the smooth part n−1 �n  i=1 Gi(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (ii) a variable metric proximity operator with  respect to the non-smooth part g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (iii) a stochas-  tic approach to address the finite sum setting  induced by n−1 �n  i=1 Gi(s) combined with a vari-  ance reduction technique based on the SPIDER  algorithm (Nguyen et al (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Fang et al  (2018));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it also allows (iv) numerical approxi-  mations of the preconditioned forward operator  when it has no closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The algorithm is  introduced in Section 3, together with discussions  on implementation questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We also design a  stochastic VMFB algorithm which answers the  constraints c1 and c2 but does not contain a  variance reduction step as required by c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The second contribution is to provide a non-  asymptotic convergence analysis in expectation  of 3P-SPIDER in the case the variable metric B  at iteration #t depends on the current value of  the iterate, and the Gi’s are gradient operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The proof relies on a Lyapunov  inequality with an original construction, which  3P-SPIDER  3  is a consequence of the non-convex optimization  setting, and the fact the algorithm uses precon-  ditioned forward operators and variable metric  proximity operators (see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 provides a control of convergence  in expectation for 3P-SPIDER, which explicitly  identifies the impact of non-exact preconditioned  forward operators, and the impact of initializa-  tion strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' First, we prove that the learning  rate of 3P-SPIDER can be chosen constant over  iterations when the preconditioned forward oper-  ator is exact or replaced with an unbiased random  oracle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and is decreasing along iterations when it  is replaced with a biased oracle (deterministic or  random).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Second, we provide the first convergence  result for a stochastic VMFB algorithm addressing  c1, c2 and c3 for non-convex finite sum composite  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For example, it is the first result for  incremental EM with a non-smooth penalty term  (g ̸= 0) and possibly biased Monte Carlo approx-  imations of the E-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When the forward operator hi(s, B(s)) is exact,  we study the complexity of 3P-SPIDER (see Corol-  lary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3): in order to satisfy the ϵ-approximate  stationary condition, the number of calls K¯h to one  of the operator hi(s, B(s)) is O(√n/ϵ), the number  of calls Kprox to the backward operator is O(1/ϵ)  and, the learning rate can be chosen independent  of the accuracy ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Applied to the Gradient method  and applied to the EM method when there are  no constraints (g = 0), these explicit controls of  convergence retrieve previous results in the litera-  ture (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Wang et al (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Fort et al (2020))  which are known to be at the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Finally we show that this complexity analysis  remains valid when the forward operators are  approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In the difficult case when the  approximations are biased random oracles based  on Monte Carlo sums, we show that K¯h and Kprox  are not impacted by the approximation and are  the same as with exact operators hi(·, B(·)), by  choosing an adequate number of terms in the  Monte Carlo sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The price to be paid is a Monte  Carlo complexity of order O(√n/ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In Section 5, 3P-SPIDER is applied to inference  in a logistic regression model with random effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We show how the problem is of the form (1)-  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In this example, the preconditioned forward  operators are approximated by a Monte Carlo  sum computed from a Markov chain Monte Carlo  sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Through numerical analyses in the case  3P-SPIDER is a stochastic Expectation Maximiza-  tion algorithm in the statistic space, we discuss the  choice of design parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We also show how the  SPIDER variance reduction effect can be increased  by exploiting the Monte Carlo approximations of  the forward operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The proofs are given in Section 6 and Section 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  technical details are also provided in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We denote by ⟨·, ·⟩ the dot prod-  uct on Rq, and by ∥ · ∥ the associated norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For a q × q positive definite matrix B, we set  ⟨·, ·⟩B  def  = ⟨B·, ·⟩ and ∥ · ∥B the associated norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Iq denotes the q × q identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For a matrix  B, B⊤ is its transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Pq  + denotes the set of the  q × q positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  N (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' N⋆) is the set of non negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  positive) integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For n ∈ N⋆, we set [n]  def  =  {0, · · · , n} and [n]⋆ def  = {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' R+ is the set  of the positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For q ∈ R, ⌈q⌉ is the  upper integer part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  I is the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For a proper function  g : Rq → (−∞, +∞], ∂g(s) denotes the sub-  differential at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For a continuously differentiable  function W at s ∈ Rq, ∇W(s) is the gradient of  W at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  All the random variables (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=') are defined on  (Ω, A, P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' for a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' U, σ(U) is the sigma-field  generated by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2 Motivating examples  In this section, we show that the Variable Metric  Proximal-Gradient algorithm and the Expectation  Maximization algorithm are examples of the gen-  eral framework described by (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In the  first case, the preconditioning matrices B are cho-  sen by the user, while in the second case, they are  supplied by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Variable Metric  Proximal-Gradient algorithms  Consider the non-convex composite problem with  finite sum structure  find s ∈ Rq:  0 ∈ 1  n  n  �  i=1  ∇Wi(s) + ∂g(s) ,  (3)  4  3P-SPIDER  where g is a proper lower semicontinuous convex  function with domain S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and for all i ∈ [n]⋆, Wi is  continuously differentiable on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is of the form  (1) with Gi  def  = ∇Wi being a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (3) is an  example of the more general problem: finding a  zero on Rq of the sum of two (set-valued) oper-  ators 0 ∈ As + Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Here, A  def  = n−1 �n  i=1 ∇Wi  and C  def  =  ∂g is a maximally monotone oper-  ator (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (Bauschke and Combettes, 2011,  Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='25)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (3) can be solved by Forward-  Backward splitting algorithms (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Combettes  and Wajs (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Beck (2017)): the forward step  uses the gradient of some if not all the functions  Wi’s at each iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the backward step uses a  proximity operator associated to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This yields  Proximal-Gradient based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In the case g = 0, which includes uncon-  strained optimization problems, stochastic gradi-  ent methods with variance reduction were pro-  posed in the situation  n−1  n  �  i=1  Gi(s) = E [G(Z, s)]  (4)  and random oracles G(Z, s) are available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' in the  non-convex setting, let us cite e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Ghadimi and  Lan (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Reddi et al (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Allen-Zhu and  Hazan (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Nguyen et al (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Allen-Zhu  (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Fang et al (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Zhou et al (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  These algorithms address the problem (1)-(2) by  choosing B equal to the identity matrix Iq and  they use unbiased random oracles G(Z, s) for the  approximation of the forward operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For non-convex composite optimization (g ̸=  0), let us cite Ghadimi et al (2016) and Karimi  et al (2016) for stochastic Proximal-Gradient algo-  rithms using unbiased oracles G(Z, s) (see (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Li and Li (2018), Wang et al (2019), Zhang  and Xiao (2019), Nhan et al (2020) and Metel  and Takeda (2021) propose stochastic Proximal-  Gradient methods with unbiased random ora-  cles and including variance reduction schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In  Metel and Takeda (2021), g may be non-convex  but admits an efficiently computable proximity  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Atchad´e et al (2017) allow for deter-  ministic or random approximations of the forward  operator n−1 �n  i=1 Gi(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' when the perturbation  is stochastic, the convergence analysis covers both  biased and unbiased oracles, includes Nesterov  acceleration schemes, but is restricted to con-  vex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Here again, all these algorithms  address the problem (1)-(2) by choosing B = Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Forward-Backward suffers from slow conver-  gence, and Variable Metric Forward-Backward  (VMFB) methods were proposed by Chen and Rock-  afellar (1997) in order to accelerate the conver-  gence (see also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 11 to 16 in Chouzenoux et al  (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' VMFB changes the metric at each itera-  tion by using symmetric positive definite scaling  matrices multiplying the forward operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is  an alternative to inertial methods such as Heavy  Ball or Nesterov acceleration which use informa-  tions from the previous iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When solving  the inclusion (3), VMFB uses preconditioned gradi-  ents with an iteration-dependent preconditioning  matrix B−1  t  for the forward step at iteration #t,  and scales the proximal step consequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Exam-  ples showing that VMFB is more efficient than  Forward-Backward and Forward-Backward with  inertial schemes, are provided in Chouzenoux et al  (2014) and Repetti et al (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Different strate-  gies exist for the definition of the variable metric  Bt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' for example, it may be a diagonal matrix  depending on the past history of the algorithm  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Park et al (2019) and references therein  for variable scalar metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' see also Chen et al  (2019) in the case g = 0), or inherited from  Newton-type methods (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Becker and Fadili  (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Lee et al (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Becker et al (2019) for  the composite convex case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' see also Kolte et al  (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Moritz et al (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Gower et al (2016) for  the smooth convex case with finite sum structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  finally, see Zhang et al (2022) for the smooth non-  convex case with finite sum structure), or defined  through a Majorize-Minimize strategy to make the  backward operator explicit (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Chouzenoux  et al (2014) and Repetti and Wiaux (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Con-  vergence results for VMFB exist in the convex case  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Combettes and V˜u (2014) and Bonettini  et al (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and Park et al (2019) for the strongly  convex case) and in the non-convex case (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Chouzenoux et al (2014) and Repetti and Wiaux  (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In Yun et al (2021), a stochastic VMFB is  studied in the non-convex case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the exact gradient  is approximated by a linear combination of ran-  dom oracles, with exponential forgetting, and the  oracles are assumed unbiased and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER addresses non-convex composite  optimization with a finite sum structure by using  Proximal-Gradient algorithms accelerated via the  3P-SPIDER  5  Variable Metric cunning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is a stochastic VMFB,  which contains a variance reduction technique  in order to overcome the finite sum setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it  also allows oracles for the preconditioned forward  operators, oracles which can be biased or unbi-  ased when random (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Atchad´e et al (2017)  and Fort et al (2018) for examples motivating  biased random approximations of the gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The combination of these two sources of pertur-  bations is an original setting which, to our best  knowledge, is not addressed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER uses preconditioning matrices, which  may depend on the current value of the iterate and  therefore may be random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The non-asymptotic  convergence analysis derived in Section 4 will rely  on weaker minorization assumptions on the spec-  trum of the scaling matrices (see A 3) than in  Yun et al (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it will not require ordering  assumptions on the sequence of scaling matrices  as in Combettes and V˜u (2014) and Bonettini  et al (2021), and will not assume a Kurdyka-  �Lojasiewicz condition on the objective function as  in Chouzenoux et al (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' As a consequence,  the construction of the Lyapunov inequality for  the convergence analysis of 3P-SPIDER differs from  these previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER requires the backward operator to be  explicit, which may be a strong assumption espe-  cially for variable metric proximity operators (see  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' extensions of the convergence anal-  ysis to the case of inexact proximity operators is  out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 Expectation Maximization for  curved exponential families  Consider the parametric statistical model: the  observations are independent with density  y �→  �  Z  p(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ)µlv(dz) ,  with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=') a σ-finite positive measure  µo on Rdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In this model, z acts as a latent vari-  able taking values in the measurable set (Z, Z)  endowed with a σ-finite positive measure µlv (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Everitt (1984) for examples of latent vari-  able models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The goal is to learn the parameter  θ ∈ Θ ⊆ Rd from n observations Y1, · · · , Yn, by  minimizing the negative normalized log-likelihood  F(θ)  def  = − 1  n  n  �  i=1  log  �  Z  p(Yi, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ)µlv(dz)  (5)  on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Unfortunately, this is a non-convex problem  and most often, an optimization algorithm for the  minimization of (5) can not do better than con-  verging to a critical point of the objective function  (see Wu (1983)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A popular model is the case when the complete  data likelihood (y, z) �→ p(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ) is of the form  p(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ)  def  = H(y, z) exp (⟨S(y, z), φ(θ)⟩ − ψ(θ))  where H : Rdy × Z → R+, S : Rdy × Z → Rq,  φ : Θ → Rq, ψ : Θ → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it corresponds to the  so-called curved exponential family assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It  is satisfied by the mixture models as soon as the  components of the mixture are from the curved  exponential family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Brown (1986) for an  introduction to curved exponential family of dis-  tributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and McLachlan and Krishnan (2008)  for examples of such latent variable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  EM for curved exponential families defines  iteratively a Θ-valued sequence {θt, t  ≥  0}  through the mechanism: given θt,  (E-step) Compute ¯s(θt), the expectation of  the sufficient statistics w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the a posteriori  distributions  ¯s(θt)  def  = 1  n  n  �  i=1  ¯si(θt) ,  ¯si(θt)  def  =  �  Z  S(Yi, z) p(Yi, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θt) µlv(dz)  �  Z p(Yi, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θt)µlv(du) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (M-step) Update the parameter  θt+1  def  = argminθ∈Θ (ψ(θ) − ⟨¯s(θt), φ(θ)⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The algorithm alternates between a step in the  parameter space Θ (when computing θt+1 ∈ Rd),  and a step in the statistic space when computing  ¯s(θt) ∈ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 states that the limiting  points of EM run in the parameter space Θ are the  fixed points of an operator onto Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' finding such a  fixed point is equivalent to find a fixed point of an  operator onto the statistic space ¯s(Θ) ⊆ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  6  3P-SPIDER  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Assume that for any s ∈ S ⊇ ¯s(Θ),  T(s) def  = argminθ∈Θ (ψ(θ) − ⟨s, φ(θ)⟩)  exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Set Lθ  def  = {θ ∈ Θ : T(¯s(θ)) = θ}  and Ls  def  = {s ∈ ¯s(Θ) : ¯s(T(s)) = s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lθ is the set of the limiting points of EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' If θ⋆ ∈  Lθ, then s⋆  def  = ¯s(θ⋆) is in Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Conversely, if s⋆ ∈ Ls  then θ⋆  def  = T(s⋆) is in Lθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Delyon et al (1999) for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' An  algorithmic corollary of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1, is that  EM is equivalent to any algorithm run in the  statistic space and designed to find the roots of  the function  s �→ 1  n  n  �  i=1  ¯hi(s) ,  where ¯hi(s)  def  = ¯si(T(s)) − s ,  on the subset ¯s(Θ) of Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Under regularity con-  ditions on the statistical model, it is proved in  Delyon et al (1999) (see also a statement in (Fort  et al, 2020, Proposition 1)) that there exists a q×q  positive definite matrix B(s) such that  ∇ (F ◦ T) (s) = −B(s)  �  1  n  n  �  i=1  ¯hi(s)  �  ,  (6)  where F is the negative normalized log-likelihood  (see (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore, the roots of n−1 �n  i=1 ¯hi(s)  on ¯s(Θ) are the roots of n−1 �n  i=1 Gi(s) on ¯s(Θ),  where Gi(s)  def  = −B(s) ¯hi(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It also means that  the roots of n−1 �n  i=1 ¯hi(s) are the roots of the  gradient of F◦T, the objective function transferred  on the statistic space through the map T : Θ →  Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  As a conclusion, EM in the statistic space is  an example of problem (1)-(2), where the func-  tion g collects the constraint on s such as s ∈  S ⊇ ¯s(Θ): (i) it is designed to find a root of  n−1 �n  i=1 Gi(s) = ∇(F ◦ T)(s) under the con-  straint that s ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (ii) it uses the quantities ¯hi(s)  which are preconditioned forward operator since  there exists B(s) such that ¯hi(s) = −B(s)−1 Gi(s)  (see (6));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (iii) this preconditioned forward oper-  ator is intractable for at least two reasons: first,  due to the inner integrations on the set Z when  computing ¯si, since p(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ) and Z are often too  complex to make the integrals explicit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' second,  due to the outer integration on the n examples  when computing ¯s, which has a prohibitive com-  putational cost in large scale learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' However, a  Monte Carlo approximation of ¯hi(s) is always pos-  sible, whatever i and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This remark is the corner-  stone to understand the stochastic versions of EM  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Celeux and Diebolt (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Wei and Tan-  ner (1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Delyon et al (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Fort and Moulines  (2003) which address the inner sum intractability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  and Neal and Hinton (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Ng and McLachlan  (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Capp´e and Moulines (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Chen et al  (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Karimi et al (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Fort et al (2020,  2021a) for the outer sum intractability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' They con-  sist in running a Stochastic Approximation (SA)  algorithm with mean field n−1 �n  i=1 ¯hi(s) (for an  introduction to SA, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Benveniste et al (1990)  or Borkar (2008));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' this yields SA within EM pro-  cedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' They differ through the construction of  the random field used for the approximation of the  mean field (see (Fort et al, 2021a, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=') for  a description of some SA within EM algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER is among the SA within EM algo-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Compared to previous stochastic EM  methods, it encompasses the two random approx-  imations (of the sum in i and of the integrals  on Z) and a variance reduction step, and it also  allows a more general penalty term g than the  {0, +∞}-valued indicator function of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3 The 3P-SPIDER algorithm  We introduce a novel algorithm named Perturbed  Proximal Preconditioned SPIDER (3P-SPIDER),  solving (1) and satisfying (c1), (c2) and (c3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It  requires g to satisfy the following assumption  A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' g : Rq → (−∞, +∞] is proper, lower semi-  continuous and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Denote by S its domain  S  def  = {s ∈ Rq : g(s) < +∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Under this assumption, we define a variable  metric proximity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any γ > 0 and  B ∈ Pq  +, the proximity operator of the proper  lower semicontinuous convex function γg : Rq →  (−∞, +∞] relative to the metric induced by  B is defined by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (Hiriart-Urruty and  Lemar´echal, 1996, Section XV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4))  proxB  γg(s)  def  =argminRq  �  γg(·) + 1  2∥ · −s∥2  B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (7)  3P-SPIDER  7  When B = Iq, we simply write proxγg(s), which  is the proximity operator originally defined by  Moreau (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 shows that under A1,  proxB  γg(s) exists and is unique for all s ∈ Rq, γ > 0  and B ∈ Pq  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It also provides characterizations of  this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Its proof is in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Assume A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any γ > 0, B ∈ Pq  + and s ∈ Rq, the  optimization problem (7) has a unique solu-  tion, characterized as the unique point p ∈ S  satisfying  − γ−1 B (p − s) ∈ ∂g(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any γ > 0, B ∈ Pq  +, s ∈ S and h ∈ Rq,  s = proxB  γg(s + γh)  iff  Bh ∈ ∂g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (8)  For s ∈ Rq and B ∈ Pq  +, set  hi(s, B)  def  = −B−1 Gi(s) ,  h(s, B)  def  = n−1  n  �  i=1  hi(s, B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (9)  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1-item 2, it holds for any B ∈ Pq  + and  γ > 0: s = proxB  γ g (s + γ h(s, B)) iff −Bh(s, B) ∈  ∂g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By (9), this yields for any B ∈ Pq  +, and  γ > 0:  s⋆ = proxB  γ g (s⋆ + γ h(s⋆, B))  iff s⋆ solves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (10)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Variable Metric Proximal and  Preconditioned Gradient  (10) shows that when solving the composite opti-  mization problem (1), as soon as a preconditioned  version of the operator s �→ n−1 �n  i=1 Gi(s) is  used – with preconditioning matrix B−1, a prox-  imity operator of g relative to a metric induced by  the matrix B has to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Based on the characterization (10), a natural  splitting algorithm to solve (1) under the con-  dition (c1) is: given σ0 ∈ S, a positive stepsize  sequence {γk+1, k ≥ 0} and a Pq  +-valued sequence  {Bk+1, k ≥ 0}, repeat  σk+1 = proxBk+1  γk+1 g (σk + γk+1h(σk, Bk+1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (11)  It corresponds to the Variable Metric Forward-  Backward algorithm (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Chen and Rockafel-  lar (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Combettes and V˜u (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In the large scale learning setting, the full sum  over the n functions hi (see (9)) can not be com-  puted at each iteration of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In addition, it may  happen that hi(s) is not explicit (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the case  of the incremental EM algorithms, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Therefore, a natural idea is to propose the inexact  version of (11) defined by Algorithm 1: the proxi-  mal step is unchanged (see line 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the SA step in  line 7 uses a random approximation Sk+1 of the  exact mean field n−1 �n  i=1 hi(�Sk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' this approxi-  mation, defined by line 6, combines a mini-batch  approximation of a full sum (see line 3) and  possibly approximated terms δk+1,i (see line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Algorithm  1  A  stochastic  Variable  Metric  Forward-Backward  Require: kout ∈ N⋆, γk > 0 for k ∈ [kout]⋆, b ∈  N⋆, �Sinit ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Ensure: The sequence {�Sk, k ∈ [kout]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  1: �S0 = �Sinit  2: for k = 0, · · · , kout − 1 do  3:  Sample a batch Bk+1 of size b in [n]⋆  4:  Choose Bk+1 ∈ Pq  +  5:  For i ∈ Bk+1, compute an approximation  δk+1,i of hi(�Sk, Bk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  6:  Sk+1 = b−1 �  i∈Bk+1 δk+1,i  7:  �Sk+1/2 = �Sk + γk+1 Sk+1  8:  �Sk+1 = proxBk+1  γk+1 g(�Sk+1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  9: end for  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 The SPIDER variance reduction  technique  3P-SPIDER leverages on Algorithm 1 and on  the  variance  reduction  technique  SPIDER  for  the definition of the field Sk+1 that approxi-  mates n−1 �n  i=1 hi(�Sk, Bk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' SPIDER stands for  Stochastic Path-Integrated Differential EstimatoR,  and was originally introduced in the stochastic  gradient descent literature by Fang et al (2018)  (see also Nguyen et al (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Wang et al (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  8  3P-SPIDER  We give the intuition of SPIDER in the SA setting  which encompasses the stochastic gradient one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  SA scheme solves a root finding problem ξ(s) =  0 on Rq by: given an initial value s0 ∈ Rq and  a stepsize sequence {γk+1, k ≥ 0}, repeat sk+1 =  sk + γk+1 Ξk+1, where at each iteration #(k + 1),  Ξk+1 is a random approximation of ξ(sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Usu-  ally, it is required that conditionally to the past  of the algorithm, the expectation of Ξk+1 is ξ(sk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  in that case, Ξk+1 can be replaced with Sk+1  def  =  Ξk+1 + Vk+1, where conditionally to the past,  Vk+1 is centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' SPIDER leverages on this remark  and on the control variate technique: it proposes  a clever construction of a random variable Vk+1  approximating zero and correlated to Ξk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The recipe is as follows: consider that at  iteration #k, Sk is a random approximation of  h(�Sk−1, Bk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then define Sk+1 by Sk+1  def  = Hk+1+  Vk+1 where  Hk+1  def  = b−1  �  i∈Bk+1  hi(�Sk, Bk+1) ,  Vk+1  def  = Sk − b−1  �  i∈Bk+1  hi(�Sk−1, Bk) ,  and  Bk+1  is  sampled  at  random  in  [n]⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Vk+1  approximates zero since both  b−1 �  i∈Bk+1 hi(�Sk−1, Bk) and Sk  approximate  n−1 �n  i=1 hi(�Sk−1, Bk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Vk+1 and Hk+1 are corre-  lated via Bk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Unfortunately, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Sk+1 is not an unbi-  ased approximation of n−1 �n  i=1 hi(�Sk, Bk+1) (see  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 in the case Bk+1 is of the form  B(�Sk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In order to remove the bias, SPIDER  restarts the control variate mechanism regularly:  every kin iterations, compute a full sum over the n  terms and set Skin+1 = n−1 �n  i=1 hi(�Skin, Bkin+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 3P-SPIDER  3P-SPIDER is given by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The itera-  tion index is (t, k) where t is the index of the  current outer loop and ranges from 1 to kout,  and k is the index of the current inner loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' At  outer loop #t, there are kin  t  inner iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The  inner iterations are Algorithm 1 (see Lines 8, 9,  12 and 13 of Algorithm 2) combined with the  SPIDER variance reduction trick (see Line 11 of  Algorithm 2) adapted to the case when the quan-  tities hi(�St,k, Bt,k+1) − hi(�St,k−1, Bt,k) can not be  computed exactly (see Line 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When Gi is a gradient and B(s) = Iq, different  strategies were proposed for SPIDER for the choice  of b′  t and kin  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In Fang et al (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Nguyen et al  (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Wang et al (2019), the number of inner  loops is constant (kin  t  = kin for any t ≥ 1) and  b′  t = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Nguyen et al (2017) also considers the case  when kin  t  is adapted based on the history of the  algorithm while being upper bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' in Horv´ath  et al (2022), b′  t is deterministic and depends on  t, b depends on t, and kin  t is a Geometric random  variable with an expectation depending on t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' in Li  et al (2021), b′  t does not depend on t and kin  t  is  random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For the EM problem (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2), Fort et al  (2020) introduced SPIDER-EM, a variance reduced  stochastic EM designed for large scale learning,  in a situation when the computation of ¯hi(s) is  exact for all s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For this algorithm, the benefit  of an increasing batch size t �→ b′  t and a geomet-  ric number of inner loops kin  t  with time-varying  expectation, is discussed in Fort et al (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The  conclusion is that the best strategy is a determin-  istic increasing sequence b′  t in order to have an  increasing accuracy when refreshing the variable  S·, and a constant number of inner loops kin  t = kin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This paper allows b′  t and kin  t  to vary with t: they  may be deterministic functions of t or random ones  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The matrices {Bt,k+1, t ∈ [kout]⋆, k ∈ [kin −1]}  can be deterministic or random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' They could be  chosen prior the run of the algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' more effi-  cient strategies consist in adapting this matrix  along the run of the algorithm, based on its his-  tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In EM (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2), Bt,k+1 is of the form  B(�St,k) where B is defined by the statistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  After kin  t  inner iterations, the outer loop  #(t + 1) starts: the stochastic mean field St+1,0  is refreshed (see Line 3 to Line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Here again,  two approximations of the original SPIDER algo-  rithm are allowed: the first one is when computing  hi(�St,0, Bt,1) and the second one avoids the scan  of the full data set (one may choose b′  t < n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The input variables of 3P-SPIDER are the num-  ber of outer loops kout,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the number of inner loops  kin  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the stepsize sequence {γt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' t ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' k ≥ 1} for  the SA steps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the size of the mini-batches b and  3P-SPIDER  9  Algorithm 2 The Perturbed Proximal Preconditioned SPIDER algorithm (3P-SPIDER)  Require: kout ∈ N⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' kin  t  ∈ N⋆ for t ∈ [kout]⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' γt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='k+1 > 0 for t ∈ [kout]⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' k ∈ [kin  t ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' b ∈ N⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' b′  t ∈ N⋆ for  t ∈ [kout]⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' �Sinit ∈ S and Binit ∈ Pq  +  Ensure: The sequence {�St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' t ∈ [kout]⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' k ∈ [kin  t ]⋆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  1: �S0,kin  0 = �Sinit, B0,kin  0 = Binit  2: for t = 1, · · · , kout do  3:  �St,0 = �St−1,kin  t−1,  �St,−1 = �St−1,kin  t−1,  Bt,0 = Bt−1,kin  t−1  4:  Sample a batch Bt,0 of size b′  t in [n]⋆, with or without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  5:  For all i ∈ Bt,0, compute δt,0,i equal to or approximating hi(�St,0, Bt,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  6:  St,0 = (b′  t)−1 �  i∈Bt,0 δt,0,i  7:  for k = 0, · · · , kin  t − 1 do  8:  Sample a mini batch Bt,k+1 of size b in [n]⋆, with or without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  9:  Choose Bt,k+1 ∈ Pq  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  10:  For all i ∈ Bt,k+1, compute δt,k+1,i equal to or approximating hi(�St,k, Bt,k+1)−hi(�St,k−1, Bt,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  11:  St,k+1 = St,k + b−1 �  i∈Bt,k+1 δt,k+1,i  12:  �St,k+1/2 = �St,k + γt,k+1 St,k+1  13:  �St,k+1 = proxt,k(�St,k+1/2),  where proxt,k  def  = proxBt,k+1  γt,k+1 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  14:  end for  15: end for  b′  t, and the initial values of the iterate �Sinit and  the metric Binit in S and Pq  + respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 Monte Carlo approximation of  hi(s, B)  Set ϑ  def  = (s, i, B) ∈ S ×[n]⋆×Pq  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In some applica-  tions, there exist a measurable function Hϑ and a  probability measure πϑ defined on the measurable  set (Z, Z) such that  hi(s, B) =  �  Z  Hϑ(z)πϑ(dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (12)  This is the case of EM in the statistic space (see  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2) where Hϑ(z) = S(Yi, z) − s and  πϑ(dz)  def  =  p(Yi, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' T(s))  �  Z p(Yi, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' T(s)) µlv(du) µlv(dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When the integral in (12) is intractable, one can  resort to Monte Carlo integrations to define the  approximations δt,k+1,i and δt,0,i (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Devroye  (1986) for exact sampling methods, and Robert  and Casella (2004) for an introduction to Markov  chain Monte Carlo methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' If {Zϑ  m, m ≥ 0} are  independent samples with distribution πϑ(dz) or  are a path of an ergodic Markov chain with unique  invariant distribution πϑ(dz), then we can set  hi(�St,k, Bt,k+1) ≈ 1  M  M  �  m=1  Hϑt,k+1,i(Zϑt,k+1,i  m  ) ,  where ϑt,k+1,i  def  = (�St,k, i, Bt,k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We will show  numerically in Section 5 that when approximating  the difference hi(�St,k, Bt,k+1) − hi(�St,k−1, Bt,k),  there is a gain in correlating the two sequences  {Zϑt,k+1,i  m  , m ≥ 0} and {Zϑt,k,i  m  m ≥ 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' this makes  stronger the effect of the SPIDER control variate  (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5 The computation of proxB  γg  When g = 0, proxB  γg(s) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When g ̸= 0, p  def  =  proxB  γg(s) solves 0 ∈ p − s + γB−1 ∂g(p) and there  does not always exist an explicit expression of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When B = Iq, (Combettes and Pesquet, 2011,  Tables 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2) provide properties of proxγg  and expressions of proximity operators for many  functions g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When B is the sum of a diagonal matrix and  of a rank one matrix, (Becker and Fadili, 2012,  Section 3) presents iterative algorithms for the  computation of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For a general positive definite  matrix B, we have from (Combettes and V˜u, 2014,  10  3P-SPIDER  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='9)  proxB  γg(s) =  √  B  −1proxγg(  √  B  −1·)(  √  Bs) ,  where  √  B is the square root of the matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (Becker and Fadili, 2012, Lemma 5) (see also  Combettes and V˜u (2014)) establishes a Moreau  identity i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' an expression of proxB  γg as a function  of a proximity operator of the Fenchel conjugate  of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In the special case g is the {0, +∞}-valued  indicator function of a closed convex set S, the  projected Landweber method is an iterative algo-  rithm for the computation of p (see Eicke (1992),  see also (Combettes and Pesquet, 2011, Example  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Finally, for applicatons including a metric  selection step, metric selection strategies for the  definition of B can be found in (Park et al,  2019, Section 3) for diagonal variable metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and  in Repetti and Wiaux (2021) for specific func-  tions g which circumvent the often challenging  computation of proxB  γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  4 Non-asymptotic  convergence analysis  This section is devoted to explicit non-asymptotic  bounds for the convergence in expectation of  3P-SPIDER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We will restrict to the case there exist  B : S → Pq  + and  Bt,k+1  def  = B(�St,k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This framework encompasses the EM problem  (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2) and any preconditioned gradient-  based algorithms (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1) when the pre-  conditioning matrix depends on the past history of  the algorithm via the current value of the iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We will also use the notation  ¯hi(s)  def  = hi(s, B(s)) ,  ¯h(s)  def  = h(s, B(s)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (13)  3P-SPIDER is designed to solve (1) under the  constraints c1 to c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore, based on (10),  we are interested in a control of the quantities  proxt,k  �  �St,k + γt,k+1 ¯h(�St,k)  �  − �St,k where  proxt,k(s)  def  = proxB(�St,k)  γt,k+1 g(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Roughly speaking, these quantities evaluate how  far the algorithm is from the limiting set at  iteration #(t, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' More precisely, we will con-  trol the cumulative ”distances to stationary”  �kout  t=1  �kin  t −1  k=0  ∆⋆  t,k+1 where ∆⋆  t,k+1 is equal to  ∥proxt,k(�St,k + γt,k+1¯h(�St,k)) − �St,k∥2  B(�St,k)  γ2  t,k+1  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (14)  h is defined by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The controls in expectation of the cumulated  distances are obtained under the assumptions A  2 to A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' A2 is a smoothness assumption on the  functions hi, A3 assumes that n−1 �n  i=1 Gi(s) is  a gradient operator of some so-called Lyapunov  function, and the spectrum of the matrices B(s)  are bounded uniformly in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' A4 are assumptions  on the approximations δt,k+1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For all i ∈ [n]⋆, the function ¯hi is globally  Lipschitz on S, with constant Li: there exists a  positive constant Li such that ∀s, s′ ∈ S, ∥¯hi(s) −  ¯hi(s′)∥ ≤ Li ∥s − s′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Set L2 def  = n−1 �n  i=1 L2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A 2 only requires a Lipschitz property on S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  it is weaker than assuming the Lipschitz property  on the full space Rq as sometimes assumed in the  literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Combettes and Wajs (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' A  2 holds for example when S is compact and for all  i ∈ [n]⋆, the gradient ∇hi exists and is continuous  on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' There exists a function W : Rq → R,  continuously differentiable on S and such that  ∀s ∈ S,  ∇ W(s) = 1  n  n  �  i=1  Gi(s) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  in addition, ¯hi(s) = −B(s)−1 Gi(s), where  B(s) ∈ Pq  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' ∇ W is globally L ˙W-Lipschitz on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' There exist 0 < vmin ≤ vmax < +∞ such that  for any s ∈ S, vmin∥·∥2 ≤ ∥·∥2  B(s) ≤ vmax∥·∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Here again, both the Lipschitz property and  the boundedness condition on the spectrum of the  matrices B(s) are required on S and not on the  full space Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When B(s) does not depend on  3P-SPIDER  11  s (B(s) = B for any s ∈ S), we have L ˙W ≤  vmaxn−1 �n  i=1 Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The last assumption is on the fluctuations  of the errors when approximating ¯hi(�St,k) −  ¯hi(�St,k−1): set ξt,k+1,i  def  =  δt,k+1,i − ¯hi(�St,k) +  ¯hi(�St,k−1) and define its conditional bias and  variance, conditionally to the σ-field generated  by Bt,k+1,  �St,k  and  �St,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Set Pt,k+1/2  def  =  σ  �  Bt,k+1, �St,k, �St,k−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  µt,k+1,i  def  = E  �  ξt,k+1,i|Pt,k+1/2  �  σ2  t,k+1,i  def  = E  �  ∥ξt,k+1,i − µt,k+1,i∥2|Pt,k+1/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We assume  A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Conditionally to Bt,k+1, �St,k and �St,k−1,  the approximations {δt,k+1,i, i ∈ Bt,k+1} are  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' There exists a non negative constant Cb and for  any t ∈ [kout]⋆, there exists a non decreasing  deterministic sequence {mt,k, k ≥ 1} such that  for any k ∈ [kin  t − 1], with probability one,  ∥ 1  n  n  �  i=1  µt,k+1,i∥ ≤  Cb  mt,k+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' There exist non negative constants Cv and Cvb  and for any t ∈ [kout]⋆, there exist non decreas-  ing deterministic sequences {Mt,k, k ≥ 1} and  { ¯  Mt,k, k ≥ 1} such that for any k ∈ [kin  t − 1],  with probability one,  1  n  n  �  i=1  σ2  t,k+1,i ≤  Cv  Mt,k+1  ,  1  n  n  �  i=1  ∥µt,k+1,i − 1  n  n  �  j=1  µt,k+1,j∥2 ≤  C2  vb  ¯  M 2  t,k+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We allow the errors ξt,k+1,i to be deterministic  or random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When there are no errors (ξt,k+1,i = 0)  then Cb = Cv = Cvb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When the errors  are deterministic, we have ξt,k+1,i = µt,k+1,i and  σ2  t,k+1,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When the errors are random and  unbiased, then µt,k+1,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore, some of the  constants Cb, Cv or Cvb can be null as summarized  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In Section A, we discuss how A 4 is veri-  fied in the case ¯hi(s′) − ¯hi(s) is an expectation  Cb  Cv  Cvb  exact  0  0  0  deterministic  ≥ 0  0  ≥ 0  random, unbiased  0  ≥ 0  0  random, biased  > 0  ≥ 0  ≥ 0  Table 1 The sign of the constants Cb, Cv, Cvb when  there are no approximations on the ¯hi(s)′ (case exact),  and when there are approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  under a distribution that may depend on (s, s′, i)  (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4), and δt,k+1,i is a Monte Carlo  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 provides an explicit upper bound  of the cumulative distance to stationarity as mea-  sured by ∆⋆  t,k+1 (see (14)) along the �kout  t=1 kin  t  iterations of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It also provides an  upper bound on the cumulative errors D⋆  t,k+1  defined by  ∥�St,k+1 − proxt,k(�St,k + γt,k+1¯h(�St,k))∥2  B(�St,k)  γ2  t,k+1  ,  where ¯h is defined by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Given the current iter-  ate �St,k, D⋆  t,k+1 compares two iterations: the ideal  one proxt,k(�St,k + γt,k+1¯h(�St,k)) and the available  one proxt,k(�St,k + γt,k+1St,k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Assume A 1, A 2, A 3 and A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let  {kin  t , t ∈ [kout]⋆} be a deterministic positive sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For any t ∈ [kout]⋆ and k ∈ [kin  t − 1], define Λt,k+1 by  γt,kL ˙W  vmin  +γ2  t,kL2 2vmaxkin  t  vminb  �  1 +  2 Cvb  √  b ¯  Mt,k+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (15)  Let {�St,k, t ∈ [kout]⋆, k ∈ [kin  t ]⋆} be the sequence given  by Algorithm 2 when the stepsize sequence {γt,k+1, t ∈  [kout]⋆, k ∈ [kin − 1]} satisfies  γt,k+1  �  1 +  2Cb  mt,k+1  �  ≤ γt,k ,  Λt,k+1 ∈ (0, 1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (16)  Then,  kout  �  t=1  kin  t  �  k=1  γt,k  �1  2 − Λt,k+1  � �  E  �  ∆⋆  t,k  �  + E  �  D⋆  t,k  ��  ≤ E  �  W(�S1,0) + g(�S1,0)  �  − min  S (W +g)  + vmax  kout  �  t=1  γt,0 kin  t E  �  ∥Et∥2�  + vmax  kout  �  t=1  kin  t  �  k=1  �  kin  t − k + 1  �  γt,k Ut,k ,  12  3P-SPIDER  where Et  def  = St,0 − h(�St,0) and  Ut,k  def  = 2 Cb  mt,k  + C2  b  m2  t,k  +  Cv  b Mt,k  +  2 Cvb  √  b ¯  Mt,k  +  C2  vb  b ¯  M2  t,k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The  proof  of  Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  is  given  in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Note that Ut,k+1 = 0 when the algo-  rithm uses exact preconditioned gradients at each  iteration: δt,0,i = ¯hi(�St,0) and δt,k+1,i = ¯hi(�St,k) −  ¯hi(�St,k−1) for all i, t, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Random number of inner loops kin  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When  the number of inner loops kin  t at the outer loop #t  is a random number, we consider it is drawn prior  the run of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore the expec-  tations in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 are conditionally to the  random sequence {kin  t , t ∈ [kout]⋆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The expecta-  tion w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the randomness of kin  t  can easily be  obtained from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' details are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The step sizes γt,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The conditions on the  sequence {γt,k+1, t ∈ [kout]⋆, k ∈ [kin  t − 1]} are  satisfied with  γt,k+1  def  =  k  �  j=0  �  1 +  2 Cb  mt,j+1  �−1  γt,0  where γt,0 is positive and strictly lower than  1  4Lvmaxυ  b  kin  t  �  �  �  L2  ˙W  L2 + 4vminvmax  kin  t  b υ − L ˙W  L  �  � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (17)  υ  def  = 1 + 2Cvb/(  √  b inft,k ¯  Mt,k+1) (see the proof in  Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' First, observe that when Cb = 0, the  step size can be a constant function of the inner  loop index k loop (γt,k+1 = γt,0 for any k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' On  the contrary, when Cb > 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' for a deterministic  approximation or a biased random approximation  (see Table 1), the stepsize sequence is a strictly  decreasing function of the inner loop index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Second, the maximal value of γt,0 is larger  when Cvb = 0 than when Cvb > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Here again,  deterministic or unbiased random approximations  requires more aggressive step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The initialization of the outer loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Set  Nt  def  = ∥Et∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When Bt,0 = {1, · · · , n} and δt,0,i =  ¯hi(�St,0) for all i, then Nt = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' otherwise, Nt is  positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let us discuss the behavior of Nt when δt,0,i  is an unbiased random approximation of ¯hi(�St,0)  with variance denoted by σ2  t,0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' When Bt,0 =  {1, · · · , n}, then  E [Nt] = 1  n2  n  �  i=1  σ2  t,0,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (18)  Nevertheless, the strategy Bt,0 = {1, · · · , n} has a  large computational cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' sampling a subset of size  b′  t reduces the computational cost but increases  the squared norm of the error: we have  E [Nt] ≤  1  b′  tn  n  �  i=1  �  σ2  t,0,i + ∥¯hi(s) − ¯h(s)∥2�  ,  (19)  with an equality if Bt,0 is sampled with replace-  ment in {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' See Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 for detailed  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' From a numerical point of view,  an efficient strategy consists in increasing the size  b′  t with the outer loop index t (see references in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 for 3P-SPIDER applied to EM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Random  stopping  time  of  the  algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In non-convex optimization, the last iter-  ate �Skout,kin  kout is not necessarily the point which  minimizes, over the sequence {�St,k, t ∈ [kout]⋆, k ∈  [kin  t ]⋆}, the distance to the set of solutions of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The quantity ∆⋆  , motivated by (10), can not be  computed exactly in our framework so that the  ”best” iterate can not be identified thanks to this  criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is therefore popular to analyze the  algorithm when stopped at a random time (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (Lan, 2020, Chapter 6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For sake of simplic-  ity, we consider the case when kin  t = kin for any t  and Cb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We have the following corollary:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 (of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1) Assume that kin  t  =  kin, Cb = 0 and the stepsize sequence is constant  γt,k = γ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let (τ, K) be a uniform random variable on  [kout]⋆ × [kin]⋆, independent of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then  inf  (t,k)∈[kout]⋆×[kin]⋆  �1  2 − Λt,k  �  E  �  ∆⋆  τ,K + D⋆  τ,K  �  ≤  E  �  W(�S1,0) + g(�S1,0)  �  − minS (W +g)  koutkinγ⋆  + vmaxE  �  ∥Eτ∥2�  + vmax E  ��  kin − K + 1  �  Uτ,K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER  13  An upper bound on Λt,k can easily be obtained  from (15) as a function of L ˙W, L, vmin, vmax,  kin, b, γ⋆, Cvb and inft,k ¯  Mt,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 shows that, even by stopping  3P-SPIDER with this simple rule, the first term in  the RHS is inversely proportional to the maximal  number of iterations koutkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Complexity analysis when Et = 0, Ut,k =  0 and kin  t  = kin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For smooth first-order opti-  mization, algorithms are compared through their  complexity in order to satisfy an ϵ-first order  stationary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In stochastic composite opti-  mization, this criterion is naturally extended to  the ϵ-approximate stationary condition defined by  E  �  ∆⋆  τ,K  �  ≤ ϵ ,  where (τ, K) is a random variable taking values  in [kout]⋆ × [kin]⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (Ghadimi et al, 2016,  Section 4), (Wang et al, 2019, Section 3) and Fort  and Moulines (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 studies the proximal complexity  Kprox defined as the number of calls to the prox  operator in order to satisfy the ϵ-approximate sta-  tionary condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the stochastic ¯h-complexity K¯h  defined as the number of calls to one of the ¯hi’s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  and the total number of iterations kinkout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Again  for sake of simplicity, and in order to compare our  results to the literature, we consider a simplified  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 (of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2) Assume in addi-  tion  that  Et  =  0  and  Ut,k  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The  ϵ-  approximate stationary condition is satisfied with  γ⋆  =  vmin/(4 L ˙W), kin/b  =  L2  ˙W/(vminvmaxL2),  b  =  O(√n√vminvmaxL/L ˙W)  and  koutkin  =  O(L ˙W/(ϵvmin)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Moreover, Kprox = O(L ˙W/(vmin ϵ))  and K¯h = O(√vmaxL√n/(ϵ√vmin)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The proof is in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This result shows  that the step size γ⋆ and the number of inner loops  kin are independent of the accuracy ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When applied to Stochastic Gradient Descent,  3P-SPIDER in the setting of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 is the  Prox-SpiderBoost algorithm studied in Wang  et al (2019): Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 and (Wang et al,  2019, Theorem 2) state the same complexity  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (Wang et al, 2019, Table 1) compares  Prox-SpiderBoost to other stochastic gradient  algorithms for composite non-convex finite sum  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is shown that the variance reduc-  tion based on SPIDER order-level outperforms  other variance reduction strategies such as the  SVRG one and the SAGA on, introduced respectively  by Johnson and Zhang (2013) and Defazio et al  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Hence, 3P-SPIDER reaches the state of the  art among the proximal stochastic gradient algo-  rithms designed to solve finite sum non-convex  composite optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When applied to EM, 3P-SPIDER in the set-  ting of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 is the extension of the  SPIDER-EM algorithm studied in Fort et al (2020)  to the case there is a proximal step which man-  ages the constraint g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Here again, the comparison  of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 and (Fort et al, 2020, Theorem  2) shows that 3P-SPIDER reaches the state of the  art among the incremental EM algorithms with  variance reduction, including sEM-VR and FIEM  introduced respectively in Chen et al (2018) and  Karimi et al (2019) (see also Fort et al (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  See the comparison to the literature in Fort et al  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Beyond these two applications, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3  is - to our best knowledge - the first complexity  result for an algorithm designed to solve (1) under  the constraint (2) and for non-convex finite sum  composite optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  ϵ-approximate stationary condition: the  cost  of  inexact  preconditioned  forward  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let us discuss the cost of inexact  ¯hi(s)’s when the approximation is unbiased and  random (so that Cb = Cvb = 0, see Table 1):  does it deteriorate the proximal complexity Kprox  and the number of calls to an oracle of a precon-  ditioned forward operator ¯hi (still denoted by K¯h  below) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' detailed computations of the assertions  below can be found in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  If E  �  ∥Et∥2�  = O(ϵ1−a′/(√nt)a′) for some a′ ∈  [0, 1) and  Mt,k+1 = O  �n(a−¯a)/2  ϵ1−a  ta (k + 1)¯a  �  for some a, ¯a ∈ [0, 1), then the ϵ-approximate sta-  tionary condition is satisfied with kin  t  = O(√n),  b = O(√n) and kout = O(1/(√nϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In addition,  Kprox = O(1/ϵ) and K¯h = O(√n/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore,  the conclusions of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 remain valid, and  the approximations of the hi’s do not deteriorate  14  3P-SPIDER  the complexity performances of the algorithms, as  soon as the approximation is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let us now evaluate the computational cost,  in the case the unbiased random approximation  is a Monte Carlo approximation computed from  independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=')  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In this case, Mt,· is the number of terms  of the Monte Carlo sum (see Section A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The  Monte Carlo complexity KMC defined as the total  number of Monte Carlo draws required to satisfy  the ϵ-approximate stationary condition is: KMC =  O(√n/ϵ2) for any a, a′, ¯a ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  To our best knowledge, it is the first com-  plexity analysis with such a Monte Carlo approx-  imation of the preconditioned forward operators  ¯hi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  5 Application: Penalized  Logistic Regression with  random effects  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 The model  Motivated by applications in classification, we  consider a logistic regression model with random  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let n pairs of examples {(Xi, Yi), i ∈ [n]⋆}  where Xi ∈ Rd collects the d explanatory vari-  ables, and Yi is the binary response variable taking  values in {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We assume that given {Xi, i ∈  [n]⋆}, the binary observations {Yi, i ∈ [n]⋆} are  independent with distribution  {−1, 1} ∋ yi �→  �  Rd(1 + exp(−yi ⟨Xi, zi⟩))−1  ×  1  √  2π  dσd exp  �  −(2σ2)−1∥zi − θ∥2�  dzi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In words, each example #i has an individ-  ual regression vector Zi in Rd and given Zi,  the success probability P(Yi  =  1  |  Zi) is  (1 + exp(− ⟨Xi, Zi⟩))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The regression vectors  Z1, · · · , Zn are independent with a Gaussian dis-  tribution N(θ, σ2Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ is assumed to be unknown  and σ2 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The objective is the estimation of θ by maxi-  mizing the penalized log-likelihood criterion, with  a ridge penalty pen(θ)  def  = nτ∥θ∥2, where τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By  a change of variable, we obtain that the criterion  to be minimized is (see Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1)  F : θ �→ − 1  n  n  �  i=1  log  �  R  exp  �  x ⟨Xi, θ⟩ /(σ2∥Xi∥)  �  1 + exp(−yi∥Xi∥x)  × exp  �  −x2/(2σ2)  �  dx + ∥θ∥2  U ,  where  U  def  = τId +  1  2σ2  1  n  n  �  i=1  XiX⊤  i  ∥Xi∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The following lemma shows that the minimizers  of F are in a compact set K of Rd thus implying  that the optimization problem can be constrained  to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The proof is given in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 The minimizers of F are in the set K def  =  {θ ∈ Rd : ∥θ∥2 ≤ (ln 4)/τ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  To solve this optimization problem, we propose  two approaches: a gradient one, solved in the origi-  nal space θ ∈ Rd (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and an EM one,  solved in the statistic space (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The  discussions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 show  that EM is a gradient approach for finding the  critical points of s �→ F(U −1s/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 A Gradient approach  We are interested in finding a critical point of F  in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Equivalently, we want to solve  0 ∈ 1  n  n  �  i=1  Gi(θ) + ∂g(θ)  where g is the {0, +∞}-valued indicator function  of the set K and  Gi(θ)  def  = 2Uθ −  Xi  σ2 ∥Xi∥  �  R  z πθ,i(z)dz ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  πθ,i(z) is the probability density proportional to  exp  �  z ⟨Xi, θ⟩ /(σ2∥Xi∥) − z2/(2σ2)  �  1 + exp(−yi∥Xi∥z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (20)  We apply 3P-SPIDER with B  def  = Iq and hi(θ, Iq)  def  =  −Gi(θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' note that proxγ g(θ) = argminx∈K∥x −  3P-SPIDER  15  θ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' hi is the sum of an explicit term and an  integral with no closed form: it will be approx-  imated by a Monte Carlo method, based on a  Markov chain Monte Carlo (MCMC) sampler (see  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore, δt,k,i will be a  biased random approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 An EM approach  The criterion F to be minimized is of the form (5)  with Z = R, µlv(dz) = dz and p(Yi, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ) equal to  exp  �  z ⟨Xi, θ⟩ /(σ2∥Xi∥) − z2/(2σ2) − ∥θ∥2  U  �  1 + exp(−Yi∥Xi∥z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The curved exponential family assumption on the  complete data model is satisfied: p(Yi, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' θ) =  H(Yi, z) exp (⟨S(Yi, z), φ(θ)⟩ − ψ(θ)) with φ(θ)  def  =  θ, ψ(θ)  def  = ∥θ∥2  U and  S(Yi, z)  def  = z  Xi  σ2 ∥Xi∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  From Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2, EM in the statistic space is of  the form (1)-(2): it solves 0 ∈ n−1 �n  i=1 ¯Gi(s) +  ∂¯g(s) where ¯Gi(s) = B Gi(Bs), B  def  = U −1/2 and  ¯g(s) is the {0, +∞}-valued indicator function of  the set {s ∈ Rd : T(s) ∈ K} where T(s)  def  = Bs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it  uses  ¯hi(s)  def  =  Xi  σ2 ∥Xi∥  �  R  z πBs,i(z)dz − s ,  (21)  and the metric induced by B(s)  def  =  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' See  Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 for detailed computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' As in the  gradient approach, hi requires the expectation  of the distribution π·,i (see (20)) which has no  closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We will run 3P-SPIDER with B(s) ←  B and a biased random approximation of the  ¯hi(s)’s (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' note that proxB  γ ¯g(s) =  B−1 argminx∈K  �  (x − Bs)⊤B−1(x − Bs)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 The MCMC approximation of ¯hi  We discuss how to design an efficient MCMC  sampler for the approximation of  Ii(θ)  def  =  �  R  z πθ,i(z) dz ,  θ ∈ Rd ,  where πθ,i is defined, up to a normalizing constant,  by (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By using an integration by parts and by  applying (Polson et al, 2013, Theorem 1), we show  that a data augmentation scheme is possible to  approximate integrals w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' πθ,i(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 For any i ∈ [n]⋆ and θ ∈ Rd, it holds  Ii(θ) =  � Xi  ∥Xi∥, θ  �  + yi∥Xi∥σ2  �  R  � +∞  0  ¯πθ,i(z, ω)  1 + exp (yi∥Xi∥z) dzdω ,  where ¯πθ,i(z, ω) is a probability density on R×(0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The conditional distribution of z given ω is a Gaussian  distribution with parameters  ⟨Xi, θ⟩ /∥Xi∥ + yi∥Xi∥σ2/2  1 + ωσ2∥Xi∥2  ,  σ2  1 + ωσ2∥Xi∥2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  the conditional distribution of ω given z is a Polya-  Gamma distribution with parameters (1, ∥Xi∥z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The proof is given in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore,  a Monte Carlo approximation of integrals w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  πθ,i are obtained from a Gibbs sampler targeting  the distribution ¯πθ,i(z, ω): it produces a sequence  of pairs {(Zr, Ωr), r ≥ 0} and only the Zr’s are  retained for the Monte Carlo approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For  example, ¯hi(s) given by (21) can be approximated  by  ¯hi(s) ≈ −s +  Xi  σ2∥Xi∥2 ⟨Xi, Bs⟩  + yi∥Xi∥ 1  m  m  �  r=1  �  1 + exp(yi∥Xi∥Zs,i  r )  �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (22)  This Gibbs sampler is uniformly ergodic (see  (Choi and Hobert, 2013, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' con-  sequently, upon noting that z �→ Hi(z)  def  = (1 +  exp(yi∥Xi∥z))−1 is bounded by one uniformly in i  and z, the conditions A5 in Section A are verified  with U equal to the constant function 1 and with  a geometric convergence rate ρ(r)  def  = υr for some  υ ∈ (0, 1) (remember that S is a compact set in our  application);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' details are provided in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Therefore, A4 is verified and the rates mt,k+1,  Mt,k+1 and  ¯  Mt,k+1 are equal, and equal to the  number of points in the Monte Carlo sum (see  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  16  3P-SPIDER  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5 Numerical illustrations  Let us run 3P-SPIDER for minimizing the cri-  terion F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' based on previous results comparing  variance reduced Expectation Maximization algo-  rithms and variance reduced Gradient algorithms  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' (Chen et al, 2018, section 4)), we restrict  our attention to the EM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In this numer-  ical application, n = 24 989 and d = 21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' we choose  τ = 1 and σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The n pairs (yi, Xi) are built  from the MNIST data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The 13 007 examples  labeled yi = −1 are the examples labeled 1 or 7  in the MNIST training data set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the 11 982 exam-  ples labeled yi = 1 are the examples labeled 3 or  8 in the MNIST training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The covariates  Xi are obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let Xim be the 784×n  matrix collecting the 784 pixels for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The pixels take values in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then the rows of  Xim are centered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' by a PCA, each image is reduced  to a vector in R20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This yields Xred ∈ R20×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Finally, Xred is augmented with a row of ones,  yielding X ∈ R21×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The columns of X are the  Xi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We compare four algo-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' EM denotes the SAEM algorithm (Delyon  et al (1999)) combined with a proximal step: each  iteration processes the full data set so that there  is one iteration of EM per epoch:  �SEM  r+1  def  = proxB  γ g(�SEM  r + γ  n  n  �  i=1  �  ¯hi(�SEM  r )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Online EM is the algorithm given by Capp´e and  Moulines (2009) combined with a proximal step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  each iteration processes b examples and below, we  will run kin def  = ⌈n/b⌉ iterations per epoch:  �SOEM  r+1  def  = proxB  γ g(�SOEM  r  + γ  b  �  i∈Br+1  �  ¯hi(�S0EM  r  )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For  EM  and  Online EM,  �  ¯hi(�S•  t )  is  a  Monte  Carlo approximation of ¯hi(�S•  t ) computed with mt  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 3P-SPIDER is Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' we choose kin  t =  kin and kin = ⌈n/b⌉ so that one epoch corresponds  to the kin inner loops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' we choose b′  t = n so that  the initialization of each outer loop is one epoch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  the δt,k,i are computed by Monte Carlo sums (see  (22)) with m0 points for δt,0,i and mt points for  δt,k+1,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' since �St,0 = �St,−1, we set δt,1,i = 0 for  all i, so that St,1 = St,0 = n−1 �n  i=1 δt,0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Finally,  3P-SPIDER and 3P-SPIDER-corr differ as follows:  the Monte Carlo approximation δt,k+1,i necessi-  tates a Monte Carlo approximation of ¯hi(�St,k)  and one of ¯hi(�St,k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In 3P-SPIDER, the Monte  Carlo approximations are based on two indepen-  dent chains (see (22)) while in 3P-SPIDER-corr  the chains are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  All the algorithms are initialized at the null vec-  tor �Sinit = 0 ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The step size is equal to  γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 during the first six epochs and then equal  to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The length of all the paths is 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  On all the figures except Figure 3, we report a  mean value computed over 25 independent runs of  each algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the shadowed area is delimited by  the minimal and maximal value of the displayed  criterion over these runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Most of the comparisons are based  on the evolution of  ∆t,k+1  def  = ∥proxB  γ g(�St,k + γ St,k+1) − �St,k∥2  B  γ2  as a function of the number of epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' this crite-  rion is an approximation of ∆⋆  t,k+1 (see (14)) which  can not be computed here since ¯h has no closed  form in this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The criterion ∆t,k+1 for  3P-SPIDER and 3P-SPIDER-corr, is compared to  ∆EM  r defined by  ∥proxB  γ g(�SEM  r + γn−1 �n  i=1  �  ¯hi(�SEM  r )) − �SEM  r ∥2  B  γ2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  and to ∆OEM  r  defined by  ∥proxB  γ g(�SOEM  r  + γb−1 �  i∈Br+1  �  ¯hi(�SOEM  r  )) − �SOEM  r  ∥2  B  γ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The best algorithm will have the smallest value of  ∆t,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We first study the role of some design param-  eters of 3P-SPIDER, such as the number of Monte  Carlo points when computing δt,0,i (denoted by  m0) and δt,k+1,i (denoted by mt) and the balance  between kin and b which satisfy kinb ≈ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  On Figure 1, two strategies are chosen: first, m0 =  mt = 2⌈√n⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' then m0 = mt = 5⌈√n⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' in all cases,  kin = ⌈√n/10⌉ and b = ⌈n/kin⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For comparison,  EM and Online EM are also run, with a number of  Monte Carlo point equal to mt at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER  17  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 1 Different strategies for the number of Monte Carlo  points when approximating ¯hi(s) - see (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Evolution of  ∆EM  r  in green, ∆OEM  r  in red, ∆t,k+1 for 3P-SPIDER in blue  and ∆t,k+1 for 3P-SPIDER corr in black, as a function of  the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' [left] m0 = mt = 2⌈√n⌉, [right]  m0 = mt = 5⌈√n⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  On Figure 2, the case when kin = ⌈√n/10⌉ is  compared to the case kin = ⌈√n/2⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' in both cases,  b = ⌈n/kin⌉ and m0 = mt = 2⌈√n⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 2 Number of inner loops per epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Evolution of  ∆OEM  r  in red, ∆t,k+1 for 3P-SPIDER in blue and ∆t,k+1  for 3P-SPIDER corr in black, as a function of the number  of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' [left] kin = ⌈√n/10⌉ and b = ⌈n/kin⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' [right]  kin = ⌈√n/2⌉ and b = ⌈n/kin⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Each algorithm returns a sequence of points  in the s-space, from which a sequence of points  in the θ-space is deduced through the formula  θ = T(s) = Bs ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' On Figure 3, three compo-  nents of this θ-sequence are displayed, versus the  number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 3 Estimation of three parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Evolution of the  three components of θ by EM in green (top, left), OEM  in red (top, right), 3P-SPIDER in blue (bottom, left) and  3P-SPIDER corr in black (bottom, right), as a function of  the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Finally, we also display on Figure 4 the evo-  lution of the squared norm of the iterates ∥�St,k∥2  obtained by 3P-SPIDER and 3P-SPIDER-corr, and  ∥�SOM  r ∥2 and ∥�SOEM  r  ∥2 obtained resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' by EM and  Online EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' They are plotted as a function of the  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 4 Squared norm of the iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Evolution of ∥�SEM  r ∥2  in green (top, left), ∥�SOEM  r  ∥2 in red (top, right), ∥�St,k∥2 for  3P-SPIDER in blue (bottom, left) and ∥�St,k∥2 for 3P-SPIDER  corr in black (bottom, right), as a function of the number  of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' EM has a slow convergence rate  and even fails to converge before 20 epochs con-  trary to the other algorithms (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Figure 4):  one update of the iterate per epoch is not enough  especially during the first iterations when more  100  100  10-2  102  104  10-4  10-6  10-6  国  w国国  10-8  10-8  01357  91113151719  0135  7  91113151719100  100  10-2  10°4  104  10-4  10-6  10-6  10-8  10-8  0135  791113151719  01 35791113151719EM  Online EM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='04  0135791113151719  0135791113151719  3P SPIDER  3P SPIDER corr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='04  01  9 1113151719  013  5  9 1113 151719EM  Online EM  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='7  5  7  91113151719  5  6  1113151719  3P SPIDER  3P SPIDER corr  2  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='7  5  91113151719  5  7  6  111315171918  3P-SPIDER  updates even based on part of the data set is a  better strategy (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the behavior of Online  EM, which contains kin updates per epoch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Online EM, 3P-SPIDER and 3P-SPIDER-corr pro-  cess part of the data set at each iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' com-  pared to Online EM, the 3P-SPIDER’s contain a  variance reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' All the plots illustrate the ben-  efit of this variance reduction, which reduces the  variability at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The choice of γ impacts this variability: see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Figures 1, 2 and Figure 4 where a change occurs at  epoch #7 (remember that from epoch 2ℓ to 2ℓ+1,  Online EM runs kin updates of the iterates while  the 3P-SPIDER’s do not update the iterate since  they compute St,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER-corr improves on 3P-SPIDER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The  control variate has a larger impact when the cor-  relation is increased, as illustrated by all plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It  decreases the variability introduced by the mini-  batches (b < n) and the variability introduced by  the Monte Carlo approximation δt,k+1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Given the budget of n examples processed per  outer loops, Figure 2 shows that at convergence,  the accuracy is improved by larger mini batch sizes  and therefore a smaller number of inner loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Not surprisingly, a larger number of Monte Carlo  points decreases the variability at convergence (see  Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  6 Proof of Section 3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  collects  the  two  statements  of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 and a third property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Assume A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any γ > 0, B ∈ Pq  + and s ∈ Rq, the  optimization problem (7) has a unique solu-  tion, characterized as the unique point p ∈ S  satisfying −γ−1 B(p − s) ∈ ∂g(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any γ > 0, B ∈ Pq  +, s ∈ S and h ∈ Rq,  s = proxB  γg(s + γh)  iff  Bh ∈ ∂g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (23)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let γ > 0 and B ∈ Pq  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The operator proxB  γg is  firmly nonexpansive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' this implies that for any  s, s′ ∈ Rq,  ∥proxB  γg(s′) − proxB  γg(s)∥2  B  ≤  �  proxB  γg(s′) − proxB  γg(s), s′ − s  �  B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof Existence, uniqueness and characterization are  established in Hiriart-Urruty and Lemar´echal (1996,  Chapter XV, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The statement (23) fol-  lows from the characterization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' note that proxB  γg(s) ∈  S for any s ∈ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The firmly nonexpansive property  is a consequence of Hiriart-Urruty and Lemar´echal  (1996, Chapter XV, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  7 Proof of Section 4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Notations  Define for any s ∈ S,  ¯h(s)  def  = 1  n  n  �  i=1  ¯hi(s) ,  ¯hB  def  = 1  b  �  i∈B  ¯hi ,  where B is an n-tuple of elements of [n]⋆ (with or  without multiplicity) of cardinal b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  All the random variables are defined on a prob-  ability space (Ω, A, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is endowed with the  following filtrations for t ≥ 0 and k ≥ 0,  F0,kin  0  def  = σ(�Sinit),  Ft,0  def  = Ft−1,kin  t−1  �  σ (Bt,0, δt,0,i for all i) ,  Ft,k+ 1  2  def  = Ft,k  �  σ(Bt,k+1),  Ft,k+1  def  = Ft,k+ 1  2  �  σ (δt,k+1,i for all i ∈ Bt,k+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For any t ∈ [kout]⋆, set  Et  def  = St,0 − ¯h(�St,0) = 1  b′  t  �  i∈Bt,0  δt,0,i − ¯h(�St,0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Et is the error when replacing the full sum using  exact terms ¯hi(�St,0), with a possibly subsum of  size b′  t < n using approximations of ¯hi(�St,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Remember that  ξt,k+1,i  def  = δt,k+1,i − ¯hi(�St,k) + ¯hi(�St,k−1) ,  and  µt,k+1,i  def  = E  �  ξt,k+1,i|Ft,k+1/2  �  ,  σ2  t,k+1,i  def  = E  �  ∥ξt,k+1,i − µt,k+1,i∥2|Ft,k+1/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER  19  Finally, set  ηt,k+1  def  = 1  b  �  i∈Bt,k+1  ξt,k+1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Throughout the proof, we will use the shorthand  notation  Bt,k  def  = B(�St,k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 Preliminary lemmas  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Let B be a batch of [n]⋆ of size b, sampled  at random (with or without replacement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For  any  family  {f1, · · · , fn},  E  �  b−1 �  i∈B fi  �  = n−1 �n  i=1 fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any family {f1, · · · , fn},  E  ����1  b  �  i∈B  fi − 1  n  n  �  i=1  fi  ���  2  �  ≤ 1  b n  n  �  i=1  ∥fi − 1  n  n  �  j=1  fj∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Assume A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any s, s′ ∈ S, it holds  E  ����  �¯hB(s) − ¯hB(s′)  �  −  �¯h(s) − ¯h(s′)  � ���  2�  ≤ 1  b  �  L2∥s − s′∥2 − ∥¯h(s) − ¯h(s′)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof The proof is along the same lines as the proof  of (Fort et al, 2020, Lemma 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' A detailed proof is  provided in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 Assume A4-item 1 and A4-item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For  any t ∈ [kout]⋆ and k ∈ [kin  t − 1], it holds  E  �  ηt,k+1|Ft,k+1/2  �  = 1  b  �  i∈Bt,k+1  µt,k+1,i ,  E  �  ηt,k+1|Ft,k  �  = 1  n  n  �  i=1  µt,k+1,i ,  E  �  ∥ηt,k+1 − E  �  ηt,k+1|Ft,k  �  ∥2|Ft,k  �  ≤ 1  b  �  Cv  Mt,k+1  +  C2  vb  ¯  M2  t,k+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof Let t ∈ [kout]⋆ and k ∈ [kin  t − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We have  E  �  ηt,k+1|Ft,k+1/2  �  = 1  b  �  i∈Bt,k+1  µt,k+1,i ,  since Bt,k+1 ∈ Ft,k+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1,  E  �  ηt,k+1|Ft,k  �  = 1  n  n  �  i=1  µt,k+1,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We write  ηt,k+1 − E  �  ηt,k+1|Ft,k  �  = 1  b  �  i∈Bt,k+1  ξt,k+1,i − 1  n  n  �  i=1  µt,k+1,i  = 1  b  �  i∈Bt,k+1  �  ξt,k+1,i − µt,k+1,i  �  + 1  b  �  i∈Bt,k+1  µt,k+1,i − 1  n  n  �  i=1  µt,k+1,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The RHS is of the form U + V  and we write  ∥U + V ∥2 = ∥U∥2 + ∥V ∥2 + 2 ⟨U, V ⟩ with U  ←  b−1 �  i∈Bt,k+1  �  ξt,k+1,i − µt,k+1,i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By conditioning  and by definition of σ2  t,k+1,i, we have  E  �  ∥U∥2|Ft,k  �  = 1  b2 E  �  �  �  i∈Bt,k+1  σ2  t,k+1,i|Ft,k  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Under A4-item 1, we have by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  E  �  ∥U∥2|Ft,k  �  =  1  b n  n  �  i=1  σ2  t,k+1,i ≤  Cv  b Mt,k+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 again, it holds  E  �  ∥V ∥2|Ft,k  �  ≤  1  b n  n  �  i=1  ∥µt,k+1,i − 1  n  n  �  j=1  µt,k+1,j∥2 ,  which yields  E  �  ∥V ∥2|Ft,k  �  ≤  C2  vb  b ¯  M2  t,k+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Finally, upon noting that E  �  U|Ft,k+1/2  �  = 0 and V ∈  Ft,k+1/2, we have  E  �  ⟨U, V ⟩ |Ft,k  �  = E  ��  E  �  U|Ft,k+1/2  �  , V  �  |Ft,k  �  = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 Results on the variables St,k  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 studies the bias of the variables  St,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It shows that St,k+1 is a biased approxima-  tion of ¯h(�St,k):  E [St,k+1|Ft,k] ̸= ¯h(�St,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  20  3P-SPIDER  When k = 0, we may have E [St,1|Ft,0] = ¯h(�St,0)  if δt,0,i = hi(�St,0) and Bt,0 = [n]⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The choice  Bt,0 = [n]⋆ is the strategy proposed in Wang et al  (2019) for SpiderBoost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it has an important com-  putational cost but has the advantage to cancel  the bias of the variable S· at the beginning of each  outer loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Along the inner loops, a (signed) bias  appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3 For any t ∈ [kout]⋆ and k ∈ [kin  t −  1], it holds  E  �  St,k+1|Ft,k  �  − ¯h(�St,k)  = St,k − ¯h(�St,k−1) + E  �  ηt,k+1|Ft,k  �  ,  and  E  �  St,k+1 − ¯h(�St,k)|Ft,0  �  = Et +  k+1  �  j=1  E  �  ηt,j|Ft,0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof Let t ∈ [kout]⋆ and k ∈ [kin  t  − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We write  St,k+1 = St,k + hBt,k+1(�St,k) − hBt,k+1(�St,k−1) +  ηt,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1,  E  �  St,k+1|Ft,k  �  = St,k + ¯h(�St,k) − ¯h(�St,k−1)  + E  �  ηt,k+1|Ft,k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Since Ft,0 ⊆ Ft,k, we have  E  �  St,k+1 − ¯h(�St,k)|Ft,0  �  = E  �  St,k − ¯h(�St,k−1)|Ft,0  �  + E  �  ηt,k+1|Ft,0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Summing from j = 0 to j = k yields  E  �  St,k+1 − ¯h(�St,k)|Ft,0  �  = St,0 − ¯h(�St,−1)  +  k  �  j=0  E  �  ηt,j+1|Ft,0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The proof is concluded by using �St,0 = �St,−1 and the  definition of Et;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' note that Et ∈ Ft,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 provides a control of the con-  ditional variance of St,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 Assume A 2, A 4-item 1 and A 4-  item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any t ∈ [kout]⋆ and k ∈ [kin  t − 1], it holds  E  ����St,k+1 − E  �  St,k+1|Ft,k  � ���  2  |Ft,k  �  ≤ L2  b  �  1 +  2Cvb  √  b ¯  Mt,k+1  �  ∥�St,k − �St,k−1∥2  +  Cv  b Mt,k+1  +  C2  vb  b ¯  M2  t,k+1  +  2Cvb  √  b ¯  Mt,k+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof Let t ∈ [kout]⋆, k ∈ [kin  t − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1,  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3, the definitions of St,k+1 and of the  filtration Ft,k,  St,k+1 − E  �  St,k+1|Ft,k  �  = St,k+1 − ¯h(�St,k) − St,k + ¯h(�St,k−1) − E  �  ηt,k+1|Ft,k  �  = ηt,k+1 − E  �  ηt,k+1|Ft,k  �  + ¯hBt,k+1(�St,k) − ¯hBt,k+1(�St,k−1) − ¯h(�St,k) + ¯h(�St,k−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The RHS is of the form U + V with U ← ηt,k+1 −  E  �  ηt,k+1|Ft,k  �  and V  ∈ Ft,k+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then, we write  E  �  ∥U + V ∥2|Ft,k  �  = E  �  ∥U∥2|Ft,k  �  +E  �  ∥V ∥2|Ft,k  �  +  2E  ��  E  �  U|Ft,k+1/2  �  , V  �  |Ft,k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The  term  E  �  ∥V ∥2|Ft,k  �  is  controlled  by  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1: an upper bound is L2b−1∥�St,k− �St,k−1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The term E  �  ∥U∥2|Ft,k  �  is controlled by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2:  an upper bound is Cv/(b Mt,k+1) + C2  vb/(b ¯  M2  t,k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Upon noting that V ∈ Ft,k+1/2, and using Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2  and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1, the scalar product is upper bounded  by  2 E  �  ∥V ∥  ���E  �  U|Ft,k+1/2  � ���  ���Ft,k  �  ≤ 2  Cvb  √  b ¯  Mt,k+1  �  E  �  ∥V ∥2���Ft,k  ��1/2  ≤ 2  Cvb  √  b ¯  Mt,k+1  �  1 + E  �  ∥V ∥2|Ft,k  ��  ,  where we used that a ≤ 1 + a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5 establishes an upper bound on  the conditional expectation of the quadratic error  ∥St,k+1 − ¯h(�St,k)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5 Assume A2 and A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any t ∈  [kout]⋆ and k ∈ [kin  t − 1], it holds  E  �  ∥St,k+1 − ¯h(�St,k)∥2|Ft,k  �  ≤  �  1 +  2Cb  mt,k+1  �  ∥St,k − ¯h(�St,k−1)∥2  + L2  b  �  1 +  2Cvb  √  b ¯  Mt,k+1  �  ∥�St,k − �St,k−1∥2  + Ut,k+1 ,  where  Ut,k  def  =  2Cb  mt,k  + C2  b  m2  t,k  +  Cv  b Mt,k  +  2Cvb  √  b ¯  Mt,k  +  C2  vb  b ¯  M2  t,k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3P-SPIDER  21  Proof Let t ∈ [kout]⋆ and k ∈ [kin  t − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By definition  of the conditional expectation, we have for any r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  U, V and any σ-field F such that V ∈ F:  E  �  ∥U − V ∥2|F  �  = E  �  ∥U − E [U|F] ∥2|F  �  + ∥E [U|F] − V ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We apply this equality with U ← �St,k+1, V ← ¯h(�St,k)  and F ← Ft,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 controls the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For the second one, by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2  and A4-item 2 we have  ∥E  �  St,k+1|Ft,k  �  − ¯h(�St,k)∥2  = ∥St,k − ¯h(�St,k−1) + E  �  ηt,k+1|Ft,k  �  ∥2  ≤ ∥St,k − ¯h(�St,k−1)∥2 +  C2  b  m2  t,k+1  + 2  Cb  mt,k+1  ∥St,k − ¯h(�St,k−1)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We conclude by using ∥a∥ ≤ 1+∥a∥2 with a ← ∥St,k−  ¯h(�St,k−1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='6 (of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5) Assume also A3-  item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For t ∈ [kout]⋆ and k ∈ [kin  t −1], define Dt,k+1  by  ∥�St,k+1 − proxBt,k  γt,k+1g(�St,k + γt,k+1¯h(�St,k))∥2  Bt,k ,  and Dt,0  def  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For t ∈ [kout]⋆ and k ∈ [kin  t −], it holds  E  �  ∥St,k+1 − ¯h(�St,k)∥2|Ft,k  �  ≤  �  1 +  2Cb  mt,k+1  �  ∥St,k − ¯h(�St,k−1)∥2  + γ2  t,k  2  vmin  L2  b  �  1 +  2Cvb  √  b ¯  Mt,k+1  �  ∆⋆  t,k  +  2  vmin  L2  b  �  1 +  2Cvb  √  b ¯  Mt,k+1  �  Dt,k  + Ut,k+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By convention, ∆⋆  t,0  def  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof The proof consists in an upper bound for ∥�St,k−  �St,k−1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let s ∈ S, H, h ∈ Rq, γ > 0 and B be a  q × q positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any β > 0, it holds  ∥proxB  γg(s+γH)−s∥2  B ≤ (1+ 1  β ) ∥proxB  γg(s+γh)−s∥2  B  + (1 + β) ∥proxB  γg(s + γH) − proxB  γg(s + γh)∥2  B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We apply these inequalities with γ ← γt,k, B ←  Bt,k−1, s ← �St,k−1, H ← St,k and h ← ¯h(�St,k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Then, for any k > 0,  ∥�St,k − �St,k−1∥2  Bt,k−1  ≤ (1 + β−1) γ2  t,k∆⋆  t,k + (1 + β)Dt,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (24)  We choose β = 1 and conclude by A3-item 3: ∥ · ∥2 ≤  v−1  min∥ · ∥2  Bt,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  When k = 0, ∥�St,k − �St,k−1∥2  Bt,k−1 = 0 since by  definition, �St,0 = �St,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore, (24) remains valid  since Dt,0 = 0 and ∆⋆  t,0 = 0 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This con-  cludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='7 (of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='6) Let {ρt,k, t ≥ 1, k ≥  0} be a positive sequence satisfying  ρt,k+1  �  1 +  2Cb  mt,k+1  �  ≤ ρt,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (25)  For any t ∈ [kout]⋆, k ∈ [kin  t − 1], it holds  ρt,k+1E  �  ∥St,k+1 − ¯h(�St,k)∥2|Ft,0  �  ≤ ρt,0∥Et∥2 +  k+1  �  ℓ=1  ρt,ℓ Ut,ℓ +  2  vmin  L2  b · · ·  ×  � k  �  ℓ=1  γ2  t,ℓρt,ℓ+1  �  1 +  2Cvb  √  b ¯  Mt,ℓ+1  �  E  �  ∆⋆  t,ℓ|Ft,0  �  +  k  �  ℓ=1  ρt,ℓ+1  �  1 +  2Cvb  √  b ¯  Mt,ℓ+1  �  E  �  Dt,ℓ|Ft,0  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof In Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='6, the claim is of the form  E  �  ∥St,k+1 − ¯h(�St,k)∥2|Ft,k  �  ≤  �  1 +  2Cb  mt,k+1  �  ∥St,k − ¯h(�St,k−1)∥2 + Ak .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This yields, by using the condition (25),  ρt,k+1E  �  ∥St,k+1 − ¯h(�St,k)∥2|Ft,k  �  ≤ ρt,k ∥St,k − ¯h(�St,k−1)∥2 + ρt,k+1 Ak .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Using E[U|Ft,0] = E  �  E[U|Ft,k]|Ft,0  �  and summing  from ℓ = 0 to ℓ = k yields  ρt,k+1 E  �  ∥St,k+1 − ¯h(�St,k)∥2|Ft,0  �  ≤  k  �  ℓ=0  ρt,ℓ+1E  �  Aℓ|Ft,0  �  + ρt,0∥St,0 − ¯h(�St,−1)∥2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  we then conclude by using the equality �St,−1 = �St,0  and the definition of Et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Note also that ∆⋆  t,0 = 0 and  Dt,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  22  3P-SPIDER  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4 Lyapunov inequalities for W, g  and W +g  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='8, while being classical in smooth opti-  mization, is provided for a self-content purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='8 Assume A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any s, s′ ∈ S and γ > 0,  W(s′) ≤ W(s) −  �¯h(s), s′ − s  �  B(s) + L ˙W  2 ∥s′ − s∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof S is convex since it is the domain of a convex  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By A3, W is continuously differentiable on S  with L ˙W-Lipschitz gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then for any s, s′ ∈ S,  W(s′) − W(s) ≤  �  ∇ W(s), s′ − s  �  + L ˙W  2 ∥s′ − s∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We use that ∇ W(s) = −B(s)¯h(s), so that  �  ∇ W(s), s′ − s  �  = −  �¯h(s), s′ − s  �  B(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='9 Assume A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let B be a q × q positive  definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any s ∈ S, γ > 0, H, h ∈ Rq and  β > 0,  g  �  proxB  γg(s + γH)  �  ≤ g(s)  − 1  γ  �  1 − β  4  �  ∥proxB  γg(s + γh) − s∥2  B  − 1  γ (1 − 1  β )∥proxB  γg(s + γh) − proxB  γg(s + γH)∥2  B  −  �  h, s − proxB  γg(s + γh)  �  B  +  �  H, proxB  γg(s + γH) − proxB  γg(s + γh)  �  B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof In this proof, we use the shorthand notation  pH  def  = proxB  γg(s + γH) and ph  def  = proxB  γg(s + γh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 and the definition of the subdifferential at  a point, it holds  g(ph) ≥ g(pH) − γ−1 ⟨pH − s − γH, ph − pH⟩B  g(s) ≥ g(ph) − γ−1 ⟨ph − s − γh, s − ph⟩B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This yields  g(pH) ≤ g(s) − γ−1∥ph − s∥2  B − ⟨h, s − ph⟩B  − ⟨H, ph − pH⟩B + γ−1 ⟨pH − s, ph − pH⟩B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For the last term, we write for any β > 0,  γ−1 ⟨pH − s, ph − pH⟩B + γ−1∥ph − pH∥2  B  = γ−1 ⟨ph − s, ph − pH⟩B  ≤ 2  �  (ph − s)  √β  2√γ , (ph − pH)  1  √βγ  �  B  ≤ β  4γ ∥ph − s∥2  B + 1  βγ ∥ph − pH∥2  B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='10 Assume A1, A2 and A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For any  t ∈ [kout]⋆, k ∈ [kin  t − 1] and β > 0,  E  �  W(�St,k+1) + g(�St,k+1)|Ft,0  �  ≤ E  �  W(�St,k) + g(�St,k)|Ft,0  �  − γt,k+1  �  1 − β  4 − L ˙Wγt,k+1  vmin  �  E  �  ∆⋆  t,k+1|Ft,0  �  −  1  γt,k+1  �  1 − 1  β − L ˙W  vmin  γt,k+1  �  E  �  Dt,k+1|Ft,0  �  + γt,k+1E  �  ∥St,k+1 − ¯h(�St,k)∥2  Bt,k|Ft,0  �  ,  where Dt,k+1 is defined in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof Let γ > 0, s ∈ S and H ∈ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='8  with s′ ← proxB(s)  γg  (s+γH) ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='9 with  h ← ¯h(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This yields for any β > 0,  W(proxB  γg(s + γH)) + g(proxB  γg(s + γH))  ≤ W(s) + g(s) − 1  γ (1 − β  4 )∥proxB  γg(s + γ¯h(s)) − s∥2  B  − 1  γ  �  1 − 1  β  �  ∥proxB  γg(s+γH)−proxB  γg(s+γ¯h(s))∥2  B  −  �  ¯h(s) − H, proxB  γg(s + γH) − proxB  γg(s + γ¯h(s))  �  B  + L ˙W  2 ∥proxB  γg(s + γH) − s∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Since proxB  γg is firmly nonexpansive (see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1),  the scalar product is upper bounded by γ∥H −¯h(s)∥2  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By A3-item 3, we write  ∥proxB  γg(s+γH)−s∥2 ≤  1  vmin  ∥proxB  γg(s+γH)−s∥2  B ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  then we use ∥a + b∥2  B ≤ 2∥a∥2  B + 2∥b∥2  B with a ←  proxB  γg(s + γH) − proxB  γg(s + γ¯h(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This yields  L ˙W  2 ∥proxB  γg(s + γH) − s∥2  ≤ L ˙W  vmin  ∥proxB  γg(s + γH) − proxB  γg(s + γ¯h(s))∥2  B  + L ˙W  vmin  ∥proxB  γg(s + γ¯h(s)) − s∥2  B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We apply these inequalities with s ←  �St,k, γ ←  γt,k+1, H ← St,k+1, s′ ← �St,k+1 and B ← Bt,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Note  that proxB  γg(s+γH) = �St,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The proof is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  3P-SPIDER  23  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  Let t ∈ [kout]⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let µ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Throughout the  proof, set  At,k+1  def  =  �  1 +  2Cvb  √  b ¯  Mt,k+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  From Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='7 applied with ρt,k+1 ← γt,k+1  and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='10 applied with β ← 4µ, it holds  for any k ∈ [kin  t − 1],  E  �  W(�St,k+1) + g(�St,k+1)|Ft,0  �  ≤ E  �  W(�St,k) + g(�St,k)|Ft,0  �  − γt,k+1  �  1 − µ − L ˙W  vmin  γt,k+1  �  E  �  ∆⋆  t,k+1|Ft,0  �  −  1  γt,k+1  �  1 − 1  4µ − L ˙W  vmin  γt,k+1  �  E [Dt,k+1|Ft,0]  + γt,0vmax∥Et∥2 + vmax  k+1  �  ℓ=1  γt,ℓ Ut,ℓ  + 2vmax  vmin  L2  b  k  �  ℓ=1  γ3  t,ℓAt,ℓ+1E  �  ∆⋆  t,ℓ|Ft,0  �  + 2vmax  vmin  L2  b  k  �  ℓ=1  γt,ℓ+1At,ℓ+1E [Dt,ℓ|Ft,0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Above, we used that γt,k+1 ≤ γt,ℓ for any ℓ ∈  [k + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We now sum from k = 0 to k = kin  t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This yields,  E  �  W(�St,kin  t ) + g(�St,kin  t )|Ft,0  �  ≤ E  �  W(�St,0) + g(�St,0)|Ft,0  �  −  kin  t  �  k=1  γt,k  �  1 − µ − L ˙W  vmin  γt,k  �  E  �  ∆⋆  t,k|Ft,0  �  −  kin  t  �  k=1  1  γt,k  �  1 − 1  4µ − L ˙W  vmin  γt,k  �  E [Dt,k|Ft,0]  + γt,0vmaxkin  t ∥Et∥2  + vmax  kin  t  �  ℓ=1  (kin  t − ℓ + 1)γt,ℓ Ut,ℓ  + 2vmax  vmin  kin  t  kin  t −1  �  k=1  γ3  t,kAt,k+1E  �  ∆⋆  t,k|Ft,0  �  + 2vmax  vmin  kin  t  kin  t −1  �  k=1  γt,k+1At,k+1E [Dt,k|Ft,0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Observe  that  the  coefficient  in  front  of  E  �  ∆⋆  t,k|Ft,0  �  is γt,k(1 − µ − Λt,k+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and the term  in front of E [Dt,k|Ft,0] is γ−1  t,k (1−1/(4µ)−Λt,k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By symmetry, we choose µ  =  1/2 so that  µ = 1/(4µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This yields  E  �  W(�St,kin  t ) + g(�St,kin  t )|Ft,0  �  ≤ E  �  W(�St,0) + g(�St,0)|Ft,0  �  −  kin  t  �  k=1  γt,k  �1  2 − Λt,k+1  �  E  �  ∆⋆  t,k|Ft,0  �  −  kin  t  �  k=1  1  γt,k  �1  2 − Λt,k+1  �  E [Dt,k|Ft,0]  + γt,0vmaxkin  t ∥Et∥2  + vmax  kin  t  �  ℓ=1  (kin  t − ℓ + 1)γt,ℓ Ut,ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We now sum for t = 1 to t = kout and compute the  expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This yields, by using that �St+1,0 =  �St,kin,  kout  �  t=1  kin  t  �  k=1  γt,k  �1  2 − Λt,k+1  �  E  �  ∆⋆  t,k  �  +  kout  �  t=1  kin  t  �  k=1  1  γt,k  �1  2 − Λt,k+1  �  E [Dt,k]  ≤  E  �  W(�St,0) + g(�St,0)  �  − E  �  W(�Skout,kin  kout ) + g(�Skout,kin  kout )  �  + vmax  kout  �  t=1  γt,0kin  t E  �  ∥Et∥2�  + vmax  kout  �  t=1  kin  t  �  ℓ=1  �  kin  t − ℓ + 1  �  γt,ℓ Ut,ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The  proof  is  concluded  upon  noting  that  E  �  W(�Skout,kin  kout ) + g(�Skout,kin  kout )  �  ≥ minS(W +  g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  24  3P-SPIDER  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='6 Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3  Since Ut,k = 0, we have Cb = Cvb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' In addition,  kin  t  = kin for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore, we can consider  a constant stepsize sequence γt,k = γ⋆ where γ⋆  satisfies (see (15) and (16))  γ⋆  L ˙W  vmin  + γ2  ⋆  2vmax  vmin  L2 kin  b ∈ (0, 1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This condition is satisfied by choosing  kin  b  def  =  1  vminvmax  L2  ˙W  L2 ,  γ⋆  def  =  1  4vmax  L ˙W  L2  b  kin = vmin  4L ˙W  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Such a choice implies that inft,k(1/2 − Λt,k) =  1/2 − 3/8 = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Since Et = Ut,k = 0, we obtain  from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 that  E  �  ∆⋆  τ,K + D⋆  τ,K  �  ≤ 32 L ˙W  vmin  �  E  �  W(�S1,0) + g(�S1,0)  �  − minS (W +g)  �  koutkin  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The ϵ-approximate stationary condition is sat-  isfied by choosing koutkin  =  O(L ˙W/(vmin ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The number of calls to the proximal operator is  koutkin so that Kprox = O(L ˙W/(vmin ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Finally,  we have b′  t = n so that the number of calls  to one of the ¯hi’s is kout n + 2koutkinb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We  can choose b = O  �√n√vminvmaxL/L ˙W  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This  yields kout = O(L√vmax/(√vminϵ√n)), and K¯h =  O(√vmaxL√n/(ϵ√vmin)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='7 Cost of the approximation on  the ¯hi’s  Following the rates obtained in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3, let  us set kin  t  = O(√n), b = O(√n) and kout =  O(1/(√nϵ)) and let us show that we can define  random approximations δt,0,i and δt,k+1,i such  that the ϵ-approximate stationarity condition is  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  On  the  term  E  �  ∥Eτ∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We  write  E  �  ∥Eτ∥2�  = (kout)−1 �kout  t=1 E  �  ∥Et∥2�  and  1  kout  kout  �  t=1  ϵ1−a′  √na′ta′ = ϵ O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let us compute the associated Monte Carlo  complexity in the case Et = n−1 �n  i=1{δt,0,i −  ¯hi(�St,0)} and δt,0,i is equal to a Monte Carlo  sum with mt,0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then E  �  ∥Et∥2�  =  n−1m−1  t,0O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' It is equal to O(ϵ1−a′/(√nt)a′)  when mt,0 = O(na′/2−1ta′/ϵ1−a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore, the  Monte Carlo cost is  kout  �  t=1  nmt,0 = O  �  1  √nϵ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  On the term E  ��  kin − K + 1  �  Uτ,K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This  term is upper bounded by kin E [Uτ,K] and we  write  kin E [Uτ,K] ≤  1  kout  kout  �  t=1  kin  �  k=1  O  �  1  b Mt,k+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The RHS is O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The associated Monte Carlo  complexity is  2b  kout  �  t=1  kin  �  k=1  Mt,k = O  �√n  ϵ2  �  ,  whatever a, ¯a ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This work was partly sup-  ported by the Fondation Simone et Cino del Duca,  under the program OpSiMorE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and by the french  Agence Nationale de la Recherche (ANR) under  the program ANR-19-CE23 MASDOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Appendix A  The condition A  4 in the Monte  Carlo case  Following the framework detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4,  let us assume that (i) the intractable quantities  hi(�St,k, Bt,k+1) and hi(�St,k−1, Bt,k) are of the form  hi(s, B) =  �  Z  Hϑ(z)πϑ(dz) ,  (A1)  where ϑ  def  =  (s, i, B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' and (ii) these integrals  are approximated by a Monte Carlo sum: set  ϑt,ℓ+1,i  def  = (�St,ℓ, i, Bt,ℓ+1) and  3P-SPIDER  25  δt,k+1,i  def  =  1  mt,k+1  mt,k+1  �  r=1  �  Hϑt,k+1,i(Zϑt,k+1,i  r  )  −Hϑt,k,i(Zϑt,k,i  r  )  �  ,  (A2)  where, conditionally to �St,k−1,  �St,k, Bt,k and  Bt,k+1, the samples {Zϑt,ℓ,i  r  , r  ≥  1} are a  Markov chain with unique stationary distribution  πϑt,ℓ,i(dz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' ℓ ∈ {k, k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Below, we show that  A4 is verified when the Markov chain is ergodic  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let us start with introducing few nota-  tions from the Markov chain theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Meyn  and Tweedie (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let P be a transition kernel onto the measur-  able set (Z, Z) and λ, π be probability measures on  (Z, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' For a measurable function ξ : Z → [0, +∞),  define  π(ξ)  def  =  �  Z  ξ(z) π(dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For any r ∈ N, the r-iterated transition kernel P r  is defined by induction:  P r+1(z, A)  def  =  �  Z  P r(z, dy) P(y, A)  =  �  Z  P(z, dy) P r(y, A) ,  for all z ∈ Z, A ∈ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' by convention, P 0(z, A)  def  =  χA(z) the {0, 1}-valued indicator function and  P 0(z, A) = δz(A), the Dirac mass at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Given  a probability measure λ on (Z, Z), λP stands for  the probability measure on (Z, Z) given by  λP(A)  def  =  �  Z  λ(dy)P(y, A) ,  ∀A ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For a function U  : Z → [1, +∞) such that  λP r(U) + π(U) < +∞, define the U-norm of a  measurable function ξ : Z → Rq  ∥ξ∥U  def  = sup  Z  ∥ξ∥  U  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  and the U-norm of the signed measure λP r −π by  ∥λP r − π∥U  def  =  sup  ξ:∥ξ∥U≤1  ∥λP r(ξ) − π(ξ)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let us go back to sufficient conditions for verifying  A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Denote by Pϑ a Markov transition kernel with  invariant distribution πϑ(dz): at iteration (t, k+1),  conditionally to (�St,k−1, �St,k, Bt,k, Bt,k+1), the  chains {Zϑt,k  r  , r ≥ 0} and {Zϑt,k+1  r  , r ≥ 0} are  Markov chains with transition kernels Pϑt,k and  Pϑt,k+1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' They have the same initial  value λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Assume  A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' There exists a measurable function U :  Z → [1, +∞) such that  H⋆  def  =  sup  (s,i,B)∈S×[n]⋆×Pq  +  ∥Hϑ∥U < +∞ ,  where Hϑ is defined by (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' There exist a function ρ : N → [0, 1] and a  positive constant CMC such that for any r ∈ N,  sup  ϑ∈S×[n]⋆×Pq  +  ∥λP r  ϑ − πϑ∥U ≤ CMC ρ(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In addition, �  r≥1 ρ(r) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let ϑ ∈ S × [n]⋆ × Pq  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let {Zϑ  r , r ≥ 1} be  a Markov chain with transition kernel Pϑ and  initial distribution λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' There exists a positive  constant C′  MC such that for any ϑ ∈ S × [n]⋆ ×  Pq  + and m′ ∈ N⋆,  E  �  �∥  m′  �  r=1  {Hϑ(Zϑ  r ) − πϑ(Hϑ)}∥2  �  �  ≤ H2  ⋆ C′  MC m′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A5-item 2 is a uniform-in-s ergodicity condi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Sufficient conditions for it are provided in  (Fort et al, 2011, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=') in the case of a  geometric rate ρ(r) = κr for some κ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By  adapting (Andrieu et al, 2015, Theorem 1), sim-  ilar conditions can be obtained in the case of a  subgeometric rate ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Sufficient conditions for  A5-item 3 can be obtained from a trivial adap-  tation of (Fort and Moulines, 2003, Proposition  12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proposition  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 Assume  A  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let  δt,k+1,i  be  given  by  (A2),  where  conditionally  to  (�St,k−1, �St,k, Bt,k, Bt,k+1), {Zϑt,ℓ,i  r  , r  ≥  0} is a  Markov chain with transition kernel Pϑt,ℓ,i and ini-  tial distribution λ, for ℓ ∈ {k, k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then A 4 is  verified with mt,k+1 = Mt,k+1 = ¯  Mt,k+1 ← mt,k+1,  26  3P-SPIDER  Cb  =  Cvb  def  =  2 H⋆ CMC  �  r≥1 ρ(r)  and  Cv  def  = 2H2⋆ C′  MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof We will use the notations  Pℓ,i  def  = Pϑt,ℓ+1,i, πℓ,i  def  = πϑt,ℓ+1,i, Hℓ,i  def  = Hϑt,ℓ+1,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Expression of µt,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We have  µt,k+1,i  def  =  1  mt,k+1  mt,k+1  �  r=1  �  λP r  k,iHk,i − πk,i(Hk,i)  �  −  1  mt,k+1  mt,k+1  �  r=1  �  λP r  k−1,iHk−1,i − πk−1,i(Hk−1,i)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The condition A4-Item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By A5 and since  �S• ∈ S, we write  sup  k,i  ∥λP r  k,iHk,i − πk,i(Hk,i)∥ ≤ H⋆ CMC ρ(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This implies that  ∥µt,k+1,i∥ ≤ 2 H⋆ CMC  1  mt,k+1  mt,k+1  �  r=1  ρ(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Since �  r ρ(r) < ∞, the RHS is of the form Cb/mt,k+1  with Cb  def  = 2 H⋆ CMC  �  r ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The condition A4-Item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We write  σ2  t,k+1,i ≤ E  �  ∥ξt,k+1,i∥2|Pt,k+1/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Then, we have  E  �  ∥ξt,k+1,i∥2|Pt,k+1/2  �  ≤ 2 sup  s,i  E  �  ∥  1  mt,k+1  mt,k+1  �  r=1  Hs,i(Zs,i  r ) − πs,i(Hs,i)∥2  �  and the RHS is upper bounded by 2H2⋆ C′  MC/mt,k+1  by A5-Item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We also have  1  n  n  �  i=1  ∥µt,k+1,i− 1  n  n  �  j=1  µt,k+1,j∥2 ≤ 1  n  n  �  i=1  ∥µt,k+1,i∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  From the upper bound on ∥µt,k+1,i∥ above, we have  ∥µt,k+1,i∥2 ≤  C2  b  m2  t,k+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  Appendix B  Supplementary  materials for  Section 5  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  The penalized log-likelihood  criterion  The observations are assumed independent, so the  log-likelihood is given by  θ �→  n  �  i=1  log  �  Rd(1 + exp(−yi ⟨Xi, zi⟩))−1  ×  1  √  2π  dσd exp  �  −(2σ2)−1∥zi − θ∥2�  dzi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The penalty term is −nτ∥θ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1 The sum of the log-likelihood and the  penalty term is equal to  − n  2 log(2πσ2) −  1  2σ2 θ⊤  n  �  i=1  XiX⊤  i  ∥Xi∥2 θ − nτ∥θ∥2  +  n  �  i=1  log  �  R  exp  �  x ⟨Xi, θ⟩ /(σ2∥Xi∥)  �  1 + exp(−yi∥Xi∥x)  exp(− x2  2σ2 )dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Proof Let i ∈ [n]⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Define an orthogonal d × d matrix  Q with columns denoted by (Q1, · · · , Qd), such that  Q1  def  = Xi/∥Xi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We have ⟨zi, Xi⟩ = ∥Xi∥ ⟨Q1, zi⟩,  ⟨zi, θ⟩ =  �  Q⊤zi, Q⊤θ  �  and ∥zi∥2 = ∥Q⊤zi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This  implies that  − yi ⟨zi, Xi⟩ = −yi∥Xi∥ ⟨Q1, zi⟩  ∥zi − θ∥2 = ∥θ∥2 + ∥QT zi∥2 − 2  �  Q⊤zi, Q⊤θ  �  so that the log-likelihood of the observation Yi is (up  to the additive constant C1  def  = −d ln σ − (d/2) ln(2π))  yi �→ −∥θ∥2  2σ2 +log  �  Rd(1+exp(−yi∥Xi∥ ⟨Q1, zi⟩))−1  ×exp  �  −(2σ2)−1 �  ∥QT zi∥2 − 2  �  Q⊤zi, Q⊤θ  ���  dzi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By a change of variable v = (v1, · · · , vq) ← Q⊤zi, the  logarithm of the integral is equal to  log  �  Rd  exp  �  −(2σ2)−1 �  ∥v∥2 − 2  �  v, Q⊤θ  ���  1 + exp(−yi∥Xi∥v1)  dv  = log  �  R  exp  �  −(2σ2)−1 �  v2  1 − 2v1 ⟨Q1, θ⟩  ��  1 + exp(−yi∥Xi∥v1)  dv1  3P-SPIDER  27  +  d  �  u=2  log  �  R  exp  �  − 1  2σ2 {v2  u − 2vu ⟨Qu, θ⟩}  �  dvu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The last (d − 1) integrals have a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Observe  indeed that v2u − 2vu ⟨Qu, θ⟩ = (vu − ⟨Qu, θ⟩)2 −  (⟨Qu, θ⟩)2 so that up to the additive constant C2  def  =  (d − 1){log(2π)/2 + log σ}  d  �  u=2  log  �  R  exp  �  − 1  2σ2 {v2  u − 2vu ⟨Qu, θ⟩}  �  dvu  =  d  �  u=2  (⟨Qu, θ⟩)2  2σ2  = ∥θ∥2 − (⟨Q1, θ⟩)2  2σ2  = ∥θ∥2 − (⟨Xi, θ⟩)2/∥Xi∥2  2σ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This concludes the proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the constant (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' to θ) is  equal to C1 + C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  □  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  The criterion F is equal to −L(θ) − log(2πσ2)/2,  where L(θ) is the normalized penalized log-  likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The likelihood is the product of probabili-  ties, taking values in (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' therefore, its loga-  rithm is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The penalized log-likelihood is  upper bounded −pen(θ) = −nτ∥θ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The nor-  malized penalized log-likelihood is upper bounded  −τ∥θ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore the criterion is lower bounded  by τ∥θ∥2 − ln(2πσ2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  On the other hand, the minimum of the crite-  rion is smaller than the value of the criterion at  θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let us show that this value is (ln 4)/n −  ln(2πσ2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This will imply that the minimizers of  the criterion are in the set {θ ∈ Rd : τ∥θ∥2 ≤ ln 4}  and conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We have pen(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Let us lower bound the  likelihood of an observation Yi = +1 at θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The  likelihood is equal to  1  √  2π  dσd  �  Rd  exp  �  −(2σ2)−1∥zi∥2�  1 + exp(− ⟨Xi, zi⟩) dzi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By using the same change of variable than in the  proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1, it is equal to  1  √  2πσ  �  R  exp  �  −x2/(2σ2)  �  1 + exp(−∥Xi∥x)dx  �  1  √  2πσ  �  R  exp  �  −x2/(2σ2)  �  dx  �d−1  ,  and is lower bounded by (note that the (d − 1)  identical integrals are equal to one)  1  √  2πσ  �  R+  exp  �  −x2/(2σ2)  �  1 + exp(−∥Xi∥x)dx ,  which is in turn lower bounded by 1/4 since 1 +  exp(−∥Xi∥x) ≤ 2 for all x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The proof for the case Yi = −1 is on the same  lines and is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This implies that the likelihood of the n vari-  ables is lower bounded by 1/4n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the normalized  log-likelihood is lower bounded by − ln 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' the  criterion is upper bounded by ln 4 − ln(2πσ2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3  The optimization problem  seen as an EM  We established that for any θ ∈ Rd, ∇F(θ) =  n−1 �n  i=1 Gi(θ) where  Gi(θ)  def  = 2Uθ −  Xi  σ2 ∥Xi∥  �  R  z πθ,i(z)dz ,  and πθ,i(z) is the probability density proportional  to (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  From the expressions of φ, ψ and S(Yi, z), we  obtain that T, defined by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1, is given  by T(s)  def  = U −1s/2 for any s ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This implies  that for any s ∈ Rd,  hi(s, B)  def  =  �  R  S(Yi, z) πT(s),i(z) dz − s  =  Xi  σ2 ∥Xi∥  �  R  z πBs,i(z) dz − s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  For any s ∈ Rd, let us find the matrix B(s)  satisfying ∇(F ◦ T)(s) = −n−1 �n  i=1 B(s)hi(s)  (see (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We have ∇(F ◦ T)  =  ∇F(B·)  =  B⊤ (∇F)(B·) = B (∇F)(B·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This yields  ∇(F ◦ T)(s) = B 1  n  n  �  i=1  Gi(Bs)  = B 1  n  n  �  i=1  �  2Uθ −  Xi  σ2 ∥Xi∥  �  R  z πBs,i(z)dz  �  = Bs − B 1  n  n  �  i=1  Xi  σ2 ∥Xi∥  �  R  z πBs,i(z)dz  28  3P-SPIDER  = −B 1  n  n  �  i=1  hi(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This yields B(s)  def  = B for any s ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='4  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2  Let i ∈ [n]⋆ and θ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By using  − z2/(2σ2) + z ⟨Xi, θ⟩ /(σ2∥Xi∥)  = −(z − ⟨Xi, θ⟩ /∥Xi∥)2/(2σ2)  + (⟨Xi, θ⟩)2/(2σ2∥Xi∥2) ,  we write  πθ,i(z) =  exp  �  − (z − ⟨Xi, θ⟩ /∥Xi∥)2 /(2σ2)  �  Zθ,i 1 + exp(−yi∥Xi∥z)  where Zθ,i is the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Second,  we use z πθ,i(z) = (z − ai) πθ,i(z) + ai πθ,i(z) with  ai ← ⟨Xi, θ⟩ /∥Xi∥ and since  �  R πθ,i(z)dz = 1, we  obtain  Ii(θ) = ⟨Xi, θ⟩  ∥Xi∥ +  �  R  �  z − ⟨Xi, θ⟩  ∥Xi∥  �  πθ,i(z) dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Finally, the integral in the RHS being of the form  σ2  �  R  f ′(z)  Zθ,i 1 + exp(−yi∥Xi∥z)dz  with  f(z)  def  = − exp  �  − (z − ⟨Xi, θ⟩ /∥Xi∥)2 /(2σ2)  �  ,  we use an integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Upon noting that  the derivative of z �→ 1/(1 + exp(−yi∥Xi∥z)) is  yi∥Xi∥  exp(−yi∥Xi∥z)  (1 + exp(−yi∥Xi∥z))2 ,  we write  �  R  f ′(z)  1 + exp(−yi∥Xi∥z)dz  = −yi∥Xi∥  �  f(z)  exp(−yi∥Xi∥z)  (1 + exp(−yi∥Xi∥z))2 dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Therefore, the conclusion of this first step is  Ii(θ) =  � Xi  ∥Xi∥, θ  �  + yi∥Xi∥σ2  �  R  πθ,i(z)  1 + exp (yi∥Xi∥z) dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This step is classical in the MCMC  literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Choi and Hobert (2013) and  references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We prove that for any z ∈ R,  πθ,i(z) =  � +∞  0  ¯πθ,i(z, ω)dω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By (Polson et al, 2013, Theorem 1), it holds  1  1 + exp (−yi∥Xi∥z) = 1  2 exp (yi∥Xi∥z/2)  ×  � +∞  0  exp(−ω∥Xi∥2z2/2)p(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 1)dω ,  where p(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' b)dω is a Polya-Gamma distribution  with parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This implies that πθ,i(z) is equal  to  exp  �yi∥Xi∥z  2  − (z − ⟨Xi, θ⟩ /∥Xi∥)2  2σ2  �  ×  1  2Zθ,i  � +∞  0  exp(−ω∥Xi∥2z2/2)p(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' 1)dω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5  The assumption A4 is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Define the Markov kernel with density  Ps,i(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' z′)  def  =  �� ∞  0  π2(ω|z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' i) π1(z′|ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' s, i) dω  �  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  the  Lebesgue  measure  on  R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  here,  π1(z′|ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' s, i) is the density of a Gaussian distribu-  tion with expectation ms,i(ω) and variance vi(ω)  given by  vi(ω)  def  =  σ2  1 + ωσ2∥Xi∥2 ,  ms,i(ω)  def  = vi(ω)  � 1  σ2  � Xi  ∥Xi∥, Bs  �  + 1  2yi∥Xi∥  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  and π2(ω|z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  i) is a Polya-Gamma distribution  with parameter (1, ∥Xi∥z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The Gibbs kernel  described by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 and targeting the density  3P-SPIDER  29  distribution ¯πBs,i(z, ω)dzdω, produces a Markov  chain {(Zs,i  r , Ωs,i  r ), r ≥ 0} such that the marginal  {Zs,i  r , r ≥ 0} is a Markov chain with transition  kernel Ps,i(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  z′) dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We apply the results of  (Choi and Hobert, 2013, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1) with  y ∈ {0, 1}  yi ∈ {−1, 1}  n  1  Ω(ω)  ω  X  ∥Xi∥  B  σ2  b  ⟨Xi, Bs⟩ /∥Xi∥  Table B1 [left] The notations of Choi and Hobert  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' [right] the notations in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  This yields  �  A  Ps,i(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' z′)dz′  ≥ ε  �  A  exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5(x − m⋆)2/v2  ⋆) dx  (B3)  where  ε  def  =  inf  s∈S,i∈[n]⋆  exp  �  − 1  4 − {ms,i(1/2)}2 σ2∥Xi∥2  4 vi(1/2)  �  2  �  1 + σ2∥Xi∥2/2  and (m⋆, v⋆) satisfy for any x ∈ R,  inf  s∈S,i∈[n]⋆ exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5 (x − ms,i(1/2))2/vi(1/2))  ≥ exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='5(x − m⋆)2/v2  ⋆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2 Since S is bounded, then ε > 0 and  m⋆, v⋆ exist in R × (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The minorization condition (B3) implies that  the kernel Ps,i(z, z′)dz′ is uniformly ergodic, uni-  formly in s, i and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' By (Meyn and Tweedie, 1993,  Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=') and (Fort and Moulines, 2003,  Proposition 1), A 5-Item 2 and A 5-Item 3 are  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Appendix C  Detailed proofs  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  Proof of (17)  Let t ∈ [kout]⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The sequence given by γt,k+1  def  =  �k  j=0  �  1 +  2 Cb  mt,j+1  �−1  γt,0 for any k ≥ 0, satisfies  γt,k+1  �  1 +  2Cb  mt,k+1  �  ≤ γt,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  A sufficient condition for the property Λt,k+1 ∈  (0, 1/2) to hold is aγ2  t,0 + ¯aγt,0 − 1/2 < 0 where  ¯a  def  = L ˙W  vmin  ,  a  def  = L2 2vmaxkin  t  vminb  �  1 +  2 Cvb  √  b ¯  Mt,k+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The function x �→ ax2 + ¯ax − 1/2 possesses two  roots: one is positive and one is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' The pos-  itive one is given by (−¯a +  √  ¯a2 + 2a)/(2a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it is  equal to (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2  Proof of (18) and (19)  We write St,0 − h(�St,0) = U + V where  U  def  = 1  b′  t  �  i∈Bt,0  �  δt,0,i − E  �  δt,0,i|�St,0, Bt,0  ��  ,  V  def  = 1  b′  t  �  i∈Bt,0  E  �  δt,0,i|�St,0, Bt,0  �  − h(�St,0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  We have E  �  ∥U + V ∥2�  = E  �  ∥U∥2�  + E  �  ∥V ∥2�  by definition of the conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Since  δt,0,i is an unbiased random approximation of  hi(�St,0), we have E  �  δt,0,i|�St,0, Bt,0  �  = hi(�St,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In the case Bt,0 = {1, · · · , n}, then V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Therefore,  E  �  ∥St,0 − h(�St,0)∥2�  = 1  n2 E  �  ∥  n  �  i=1  {δt,0,i − E  �  δt,0,i|�St,0  �  }∥2  �  = 1  n2  n  �  i=1  σ2  t,0,i ,  30  3P-SPIDER  where we used that the variables {δt,0,i, i ∈ [n]⋆}  are independent conditionally to �St,0, and with  variance σ2  t,0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  In the case Bt,0 is a subset of [n]⋆ of cardinality  b′  t, then we write  E  �  ∥U∥2�  =  1  (b′  t)2 E  �  E  �  ∥U∥2|Bt,0, �St,0  ��  = 1  b′  t  E  �  � 1  b′  t  �  i∈Bt,0  σ2  t,0,i  �  �  and we conclude by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Again from  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1, we have  E  �  ∥V ∥2�  ≤  1  b′  t n  n  �  i=1  ∥hi(�St,0) − h(�St,0)∥2 ,  with an equality when Bt,0 is sampled with  replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3  Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  Set, for ease of notations,  B  def  = Bt,k,  hB  def  = 1  b  �  i∈B  hi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='1  Case with replacement  We write B = {I1, · · · , Ib} where the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Ii’s are  independent, and uniformly distributed on [n]⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Then  E [fB] = 1  b  b  �  ℓ=1  E [fIℓ] = E [fI1] = n−1  n  �  i=1  fi(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Set ¯f  def  = n−1 �n  i=1 fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' We have, by using that  the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' {I1, · · · , Ib} are independent,  E  �  ∥fB − ¯f∥2�  = E  �  ∥1  b  b  �  ℓ=1  �  fIℓ − ¯f  �  ∥2  �  = 1  b2  b  �  ℓ=1  E  �  ∥fIℓ − ¯f∥2�  = 1  bE  �  ∥fI1 − ¯f∥2�  = 1  b n  n  �  i=1  ∥fi − ¯f∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Since the variance of the sum is the sum of  the variance for independent r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  E  �  ∥hB(u) − hB(u′) − {h(u) − h(u′)}∥2�  = 1  b2  b  �  ℓ=1  E  �  ∥hIℓ(u) − hIℓ(u′) − h(u) + h(u′)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Then, since Iℓ is uniformly distributed on [n]⋆,  E  �  ∥hIℓ(u) − hIℓ(u′) − h(u) + h(u′)∥2�  = 1  n  n  �  i=1  E  �  ∥hi(u) − hi(u′)∥2�  − ∥h(u) − h(u′)∥2  ≤ ∥u − u′∥2 1  n  n  �  i=1  L2  i − ∥h(u) − h(u′)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (C4)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='2  Case without replacement  Set B = {I1, · · · , Ib} and ¯f  def  = n−1 �n  i=1 fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' I1 is a  uniform random variable on [n]⋆ so that E [fI1] =  ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Conditionally to I1, I2 is a uniform random  variable on [n]⋆ \\ {I1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Therefore  E [fI2] =  1  n − 1  � n  �  j=1  fj − E [fI1]  �  =  n  n − 1  ¯f −  1  n − 1  ¯f = ¯f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  By induction, for any ℓ ≥ 2,  E [fIℓ]  =  1  n − ℓ + 1  � n  �  j=1  fj −  ℓ−1  �  r=1  E [fIr]  �  =  n  n − ℓ + 1  ¯f −  ℓ − 1  n − ℓ + 1  ¯f = ¯f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  As a conclusion, b−1 �b  ℓ=1 E [fIℓ] = ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Let u, u′ ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' set φ(Iℓ)  def  = hIℓ(u) − h(u) −  hIℓ(u′)+h(u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Then E [φ(Iℓ)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' First, we prove  by induction that E  �  ∥φ(Iℓ)∥2�  = E  �  ∥φ(I1)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  Upon noting that I1 is a uniform random variable  on [n]⋆ and by using the induction assumption,  (n − ℓ + 1)E  �  ∥φ(Iℓ)∥2�  3P-SPIDER  31  =  � n  �  i=1  ∥φ(i)∥2 − E  �ℓ−1  �  r=1  ∥φ(Ir)∥2  ��  = nE  �  ∥φ(I1)∥2�  − (ℓ − 1)E  �  ∥φ(I1)∥2�  ,  which concludes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' Second, let us  prove that for any ℓ ≥ 0,  E  �  ∥  ℓ+1  �  r=1  φ(Ir)∥2  �  ≤ (ℓ + 1)E  �  ∥φ(I1)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  (C5)  Since �n  i=1 φ(i) = nE [φ(I1)] = 0,  E  �� ℓ  �  p=1  φ(Ip), φ(Iℓ+1)  ��  = E  �� ℓ  �  p=1  φ(Ip), E  �  φ(Iℓ+1)  ���I1, · · · , Iℓ  ���  =  1  n − ℓE  �� ℓ  �  p=1  φ(Ip),  n  �  i=1  φ(i) −  ℓ  �  p=1  φ(Ip)  ��  = −  1  n − ℓE  �  ∥  ℓ  �  p=1  φ(Ip)∥2  �  ,  so that  E  �  ∥  ℓ+1  �  p=1  φ(Ip)∥2  �  =  �  1 −  2  n − ℓ  �  E  �  ∥  ℓ  �  p=1  φ(Ip)∥2  �  + E  �  ∥φ(Iℓ+1)∥2�  ≤ (ℓ + 1)E  �  ∥φ(I1)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The proof of the first bound follows from (C5) and  (C4) since here again, I1 is uniformly distributed  on [n]⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content='  The proof of the second bound is similar  (change the definition of φ(Iℓ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
+page_content=' it is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf'}
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+Denoising Convolution Algorithms and Applications
+to SAR Signal Processing
+Alina Chertock∗, Chris Leonard†, Semyon Tsynkov‡, Sergey Utyuzhnikov§
+February 1, 2023
+Abstract
+Convolutions are one of the most important operations in signal processing. They often involve large
+arrays and require significant computing time. Moreover, in practice, the signal data to be processed by
+convolution may be corrupted by noise. In this paper, we introduce a new method for computing the
+convolutions in the quantized tensor train (QTT) format and removing noise from data using the QTT
+decomposition. We demonstrate the performance of our method using a common mathematical model
+for synthetic aperture radar (SAR) processing that involves a sinc kernel and present the entire cost of
+decomposing the original data array, computing the convolutions, and then reformatting the data back
+into full arrays.
+1
+Introduction
+Convolution operations are used in different practical applications. They often involve large arrays of data
+and require optimization with respect to memory and computational cost. While input data are usually
+available only in a discrete form, the standard realization based on a vector-matrix representation is not
+often efficient since it leads to using sparse matrices. On the other hand, a tensor decomposition looks
+very attractive because it might reduce the volume of data very drastically, minimizing the number of zero
+elements. In addition, arithmetic operations between tensors can be implemented efficiently.
+There are different forms of tensor decomposition. The most popular approach is based on the canonical
+decomposition [12] where a multidimensional array is represented (might be approximate) via a sum of outer
+products of vectors. For matrices, such decomposition is reduced to skeleton decomposition. However, it
+is known to be unstable in the cases of multiple tensor dimensions, also referred to as tensor modes. The
+Tucker decomposition [27] represents a natural stable generalization of the canonical decomposition and
+can provide a high compression rate. The main drawback of the Tucker decomposition is related to the
+so-called curse of dimensionality; that is, the algorithm’s complexity grows exponentially with the number
+of tensor modes. A way to overcome these difficulties is to use the Tensor Train (TT) decomposition, which
+was originally introduced in [23, 24]. Effectively, the TT decomposition represents a generalization of the
+classical SVD decomposition to the case of multiple modes. It can also be interpreted as a hierarchical
+Tucker decomposition [10].
+∗Department of Mathematics, North Carolina State University, Raleigh, NC, USA; chertock@math.ncsu.edu
+†Department of Mathematics, North Carolina State University, Raleigh, NC, USA; cleonar@ncsu.edu
+‡Corresponding
+author.
+Department
+of
+Mathematics,
+North
+Carolina
+State
+University,
+Raleigh,
+NC,
+USA;
+tsynkov@math.ncsu.edu
+§Department
+of
+Mechanical,
+Aerospace
+&
+Civil
+Engineering,
+University
+of
+Manchester,
+Manchester,
+UK;
+S.Utyuzhnikov@manchester.ac.uk
+1
+arXiv:2301.13339v1  [math.NA]  31 Jan 2023
+
+2
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+Computing the TT decomposition fully can be very expensive if we use the standard TT-SVD algorithm
+given, e.g., by Algorithm 1 below. Therefore, many modifications to this algorithm were proposed in the
+literature to help speed it up. One such improvement was presented in [18], where results comparable to
+those obtained by the TT-SVD algorithm were produced in a fraction of the time for sparse tensor data.
+Another algorithm that uses the column space of the unfolding tensors was designed to compute the TT cores
+in parallel; see [26]. The most popular approach to efficiently compute the TT decomposition is based on
+using a randomized algorithm; see, e.g., [1,5,13].
+Maximal compression with the TT decomposition can be reached with matrices whose dimensions are
+powers of two, as proposed in the so-called Quantized TT (QTT) algorithm [20]. As shown in [15], the
+convolution realized for multilevel Toeplitz matrices via QTT has a logarithmic complexity with respect to
+the number of elements in each mode, N, and is proportional to the number of modes. It is proven that
+the result cannot be asymptotically improved. However, this algorithm is improved for finite and practically
+important N ∼ 104 in [25] thanks to the cross-convolution in the Fourier (image) space. The improvement is
+demonstrated for convolutions with three modes with Newton’s potential. It is to be noted that QTT can also
+be applied to the Fast Fourier Transform (FTT) to decrease its complexity, as shown in [3]. This super-fast
+FFT (QTT-FFT) beats the standard FFT for extremely large N such as N ∼ 260 for one mode tensors and
+N ∼ 220 for tensors with three modes.
+For practical applications, a critical issue is denoising. Real-life data, such as radar signals, are typically
+contaminated with noise. Denoising is not addressed in the papers we have cited previously. However,
+TT decomposition itself potentially has the property of denoising, owing to the SVD incorporated in the
+algorithm [4,8]. In the current work, we propose and implement the low-rank modifications for the previously
+developed TT-SVD algorithm of [21]. These modifications speed up the computations. We also demonstrate
+the denoising capacity of numerical convolutions computed using the QTT decomposition. Specifically,
+we employ a common model for synthetic aperture radar (SAR) signal processing based on the convolution
+with a sinc imaging kernel (called the generalized ambiguity function) [6, Chapter 2] and show that when
+a convolution with this kernel is evaluated in the QTT format, the noise level in the resulting image is
+substantially reduced compared to that in the original data.
+It should be observed that most papers on tensor convolution only consider the run time cost of the
+convolution after the tensor decomposition has been applied to the objective function and the kernel function
+and either ignores the cost of the actual tensor decompositions or puts it as a side note. In this paper, we
+consider every step of computing the convolution using the QTT-FFT algorithm, including the decomposition
+of the arrays into the QTT format using the TT-SVD algorithm (see Algorithm 1), computation of the QTT-
+FFT algorithm once in that format, and then extracting the data back after the computation is conducted
+(Section 5). As the QTT decomposition is computationally expensive, we consider several approaches to
+speed up the decomposition run time. Without these modifications to the TT-SVD decomposition algorithm,
+the convolution can take a long to compute and is not a practical approach. We provide more detail in section
+7.1.
+The methods we use to speed up our TT decompositions are based on truncating SVD ranks in the
+decomposition algorithm (Algorithm 1) and lead to a significant noise reduction in the data (Section 6).
+Thus, in Chapter 6, we present algorithms to compute convolutions in a reasonable time while significantly
+reducing the noise in the data at the same time. Our contribution includes developing and analyzing new
+approaches to speeding up the tensor train decomposition, see Section 5. In Section 6, we consider the
+effects of convolutions on removing noise in data. Finally, in Section 7, we show numerical examples and
+compare our results with other approaches to computing convolutions.
+
+Denoising Convolution Algorithms
+3
+2
+Convolution
+The convolution operation is widely used in different applications in signal processing, data imaging,
+physics, and probability, to name a few. This operation is a way to combine two signals, usually represented
+as functions, and produce a third signal with meaningful information. The D-dimensional convolution
+between two functions f and g is defined as
+I(x) = [f ∗ g](x) =
+�
+RD f(y)g(x − y) dy,
+∀x ∈ RD.
+(2.1)
+Often to compute the convolution numerically, we assume the support of f and g, denoted supp(f) and
+supp(g) respectively, are compact. For simplicity, in this paper, we assume supp(f) = supp(g) = [−L, L]D
+for some L ∈ R. Next, we discretize the domain [−L, L]D uniformly into ND points such that
+xj = (xj1, . . . , xjD),
+xjd = −L + ∆x
+2 + jd∆x,
+jd = 0, . . . , N − 1,
+d = 1, . . . , D,
+where ∆x = 2L
+N and j = (j1, . . . , jD). We then let f and g be D-dimensional arrays such that
+fj = f(xj),
+gj = g(xj)
+for all j. This leads to the discrete convolution I such that
+Ij := (∆x)D �
+i
+figj−i+( N
+2 −1)1 ≈ I(xj),
+(2.2)
+where 1 = (1, . . . , 1) and the sums are over all indices i = (i1, . . . , iD) that lead to legal subscripts. This
+Riemann sum approximation (2.2) to the integral (2.1) uses the midpoint rule, thus having O(∆x2) accuracy.
+Remark 2.1. The convolution defined in (2.2) is equivalent to Matlab’s convn function with the optional
+shape input set to ’same’ and then multiplied by (∆x)D.
+To compute this convolution directly takes O(N2D) operations, but it can be reduced to O(ND log(ND))
+by using the fast Fourier transform (FFT) and the discrete convolution theorem. The FFT algorithm is an
+efficient algorithm used to compute the D-dimensional discrete Fourier transforms (DFT) of V ∈ RN×...×N,
+ˆVα := DFT(V) =
+N−1
+�
+j=0
+Vjωj·α
+N
+where the sum is over the multi-indexed array j,
+α = (α1, . . . , αD),
+αd = 0, . . . , N − 1,
+d = 1, . . . , D,
+N = (N, . . . , N),
+0 = (0, . . . , 0),
+and ωN = e− 2πˆı
+N , where ˆı = √−1 is the imaginary unit. Similarly, the D-dimensional inverse discrete
+Fourier transform (IDFT), such that
+V = IDFT(DFT(V)),
+of the array ˆV ∈ RN×...×N is given by
+Vj =
+1
+ND
+N−1
+�
+α=0
+ˆVαω−j·α
+N
+.
+
+4
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+Using the discrete Fourier transform, we can compute the circular convolution Ic = (V ⊛ W) defined
+as
+Ic
+j =
+N−1
+�
+i=0
+Vi ¯
+Wj−i
+¯
+Wi1,...,iD = Wj1,...,jD,
+id ≡ jd mod(N),
+d = 1, . . . , D,
+by taking the DFT of W and V, multiplying the results together, and then taking the IDFT of the given
+result. Thus, we have
+Ic = IDFT(DFT(W) ⊙ DFT(V))
+where ⊙ is Hadamard product (element-wise product) of D-dimensional arrays. The circular convolution is
+the same as the convolution of two periodic functions (up to a constant scaling), thus to obtain the convolution
+given in (2.2) (also known as a linear convolution), we need to pad the vectors f and g with at least N − 1
+zeros in each dimension. For example, given the vectors f 0, g0 ∈ R2N−1 with
+f 0
+j =
+�
+fj
+0 ≤ j ≤ N − 1
+0
+j > N − 1
+,
+and
+g0
+j =
+�
+gj
+0 ≤ j ≤ N − 1
+0
+j > N − 1
+,
+and Ic = (f 0 ⊛ g0) as the circular convolution between them, the linear convolution I in (2.2) is given by
+Ij = ∆xIc
+j+ N−1
+2 ,
+j = 0, . . . , N − 1.
+In this paper, we let g be a predefined kernel, such as the SAR generalized ambiguity function (GAF) (see
+Section 3 and [6, Chapter 2] for detail) and f be a smooth gradually varying function contaminated with white
+noise. To compute the convolution, we use the QTT decomposition [16] and the QTT-FFT algorithm [3].
+The QTT decomposition is a particular case of the more general TT decomposition (see Section 4 and [21]
+for detail).
+3
+Synthetic aperture radar (SAR)
+SAR is a coherent remote sensing technology capable of producing two-dimensional images of the Earth’s
+surface from overhead platforms (airborne or spaceborne). SAR illuminates the chosen area on the surface of
+the Earth with microwaves (specially modulated pulses) and generates the image by digitally processing the
+returns (i.e., reflected signals). SAR processing involves the application of the matched filter and summation
+along the synthetic array, which is a collection of successive locations of the SAR antenna along the flight
+path. Matched filtering yields the image in the direction normal to the platform flight trajectory or orbit
+(called cross-track or range), while summation along the array yields the image in the direction parallel to
+the trajectory or orbit (along-the-track or azimuth).
+Mathematically, each of the two signal processing stages can be interpreted as the convolution of the signal
+received by the SAR antenna with a known function. Equivalently, it can be represented as a convolution of
+the ground reflectivity function, which is the unknown quantity that SAR aims to reconstruct the imaging
+kernel or generalized ambiguity function. The advantage of this equivalent representation is that it leads to
+a very convenient partition: the GAF depends on the imaging system’s characteristics, whereas the target’s
+properties determine the ground reflectivity function. Moreover, image representation via GAF allows one
+to see clearly how signal compression (a property that pertains to SAR interrogating waveforms) enables
+SAR resolution, i.e., the capacity of the sensor to distinguish between closely located targets.
+
+Denoising Convolution Algorithms
+5
+In the simplest possible imaging scenario, when the propagation of radar signals between the antenna
+and the target is assumed unobstructed, and several additional assumptions also hold; the GAF in either range
+or azimuthal direction is given by the sinc (or spherical Bessel) function:
+g(x) = A sinc
+�
+π x
+∆x
+�
+≡ A
+sin
+�
+π x
+∆x
+�
+π x
+∆x
+,
+(3.1)
+where the constant A is determined by normalization, x denotes a given direction, and the quantity ∆x is the
+resolution in this direction. From the formula (3.1), we see that the resolution is defined as half-width of the
+sinc main lobe, i.e., the distance from is central maximum to the first zero. When x is the range direction
+(cross-track), the resolution ∆x is inversely proportional to the SAR signal bandwidth, see [6, Section 2.4.4].
+When x is the azimuthal direction (along-the-track), the resolution is inversely proportional to the length of
+the synthetic array, i.e., synthetic aperture, see [6, Section 2.4.3]. Note that lower values of ∆x correspond to
+better resolution because SAR can tell between the targets located closer to one another. It can also be shown
+that as ∆x → 0 the GAF given by (3.1) converges to the δ-function in the sense of distributions [7, Section
+3.3]. In this case, the image, which is a convolution of the ground reflectivity with the GAF, coincides
+with ground reflectivity. This would be ideal because the image would reconstruct the unknown ground
+reflectivity exactly. This situation, however, is never realized in practice because having ∆x → 0 requires
+either the SAR bandwidth (range direction) or synthetic aperture (azimuthal direction) to become infinitely
+large, which is not possible.
+The literature on SAR imaging is vast. Among the more mathematical sources, we mention the mono-
+graphs [2], and [6].
+4
+Tensor Train Decomposition
+Consider the K-mode, tensor A ∈ CM1×...×MK such that
+A = a(i1, . . . , iK),
+ik = 0, . . . , Mk − 1,
+k = 1, . . . , K,
+where Mk is the size of each mode, and a(i1, . . . , iK) ∈ C are the elements of the tensor A for all
+ik = 0, . . . , Mk − 1 and k = 1, . . . , K. The tensor train format of A decomposes the tensor into K cores
+A(k) ∈ Crk−1×Mk×rk such that
+a(i1, . . . , iK) = A(1)
+i1 A(2)
+i2 · · · A(K)
+iK ,
+where the matrices A(k)(:, ik, :) = A(k)
+ik
+∈ Crk−1×rk, for all ik = 0, . . . , Mk − 1, k = 1, . . . , K (In
+Matlab notation, A(k)
+ik
+= squeeze(A(k)(:, ik, :)), where squeez() is used to convert the Crk−1×1×rk tensor
+into a Crk−1×rk matrix). The matrix dimensions rk, k = 1, . . . , K, are referred to as the TT-ranks of
+the tensor decomposition, and the 3−mode tensors A(k) are the TT-cores.
+Since we are interested in
+the case when a(i1, . . . , iK) ∈ C, we impose the condition r0 = rK = 1. Let M = max1≤k≤K Mk
+and r = max1≤k≤K−1 rk, then the the tensor A, which has O(MK) elements, can be represented with
+O(MKr2) elements in the TT format.
+We can also represent the TT decomposition as the product of tensor contraction operators. Define
+the tensor contraction between the tensors A ∈ CM1×...×MK and B ∈ CMK×...×M ˜
+K (note that the first
+dimension size of B equals the last dimension size of A) as C = A ◦ B ∈ CM1×...×MK−1×MK+1×...×M ˜
+K
+where
+C(i1, . . . , iK−1, iK+1, . . . , i ˜
+K) =
+MK−1
+�
+p=0
+A(i1, . . . , p)B(p, . . . , i ˜
+K).
+
+6
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+Then the TT format of A can be represented as
+A = A(1) ◦ . . . ◦ A(K).
+Before we show how to find the TT-cores, we first need to define a few properties of tensors. First, let
+the matrix A{k} be the k-th unfolding of the tensor A such that
+A{k}(α, β) = a(i1, . . . , iK),
+α = i1 + i2M1 + . . . + ikΠk−1
+l=1 Ml,
+β = ik+1 + ik+2Mk+1 + . . . + iKΠK−1
+l=k+1Ml.
+Thus, we have that A{k} ∈ CM1M2...Mk×Mk+1Mk+2...MK which we write as
+A{k} = a(i1 . . . ik, ik+1 . . . iK).
+We denote the process of unfolding a tensor A into a matrix A{k} ∈ CM1M2...Mk×Mk+1Mk+2...MK as
+A{k} = reshape(A, [M1M2 . . . Mk, Mk+1Mk+2 . . . MK])
+and folding a matrix into a tensor A ∈ CM1×...×MK as
+A = reshape(A{k}, [M1, M2, . . . , MK]).
+(Note this is to be consistent with the Matlab function reshape()).
+From [21] it can be shown that there exist a TT-decomposition of A such that
+rk = rank(A{k}),
+k = 1, . . . , K.
+Denote the Frobenius norm of a tensor A ∈ CM1×...×MK as
+∥A∥F =
+�
+�
+�
+�
+M1−1
+�
+i1=0
+. . .
+MK−1
+�
+iK=0
+|a(i1, . . . , iK)|2,
+and the εk-rank of the matrix A{k} as
+rankεk(A{k}) := min{rank(B) : ∥A{k} − B∥F ≤ εk}.
+Given a set {εk}K
+k=1, we can approximate the tensor A with a tensor ˜A in the TT format such that it has TT-
+ranks ˜rk ≤ rankεk(A{k}) and
+∥A − ˜A∥F ≤ ε,
+ε2 = ε2
+1 + . . . + ε2
+K−1.
+In Algorithm 1, we present the TT-SVD algorithm [21], which computes a TT-decomposition of a
+tensor A with a prescribed accuracy ε. In Section 5, we present some modifications to this algorithm that
+relax the prescribed tolerance and allow us to compute an approximate decomposition faster. For a tensor
+A ∈ CM1×...×MK, define
+|A| = number of elements in A = M1M2 . . . MK.
+
+Denoising Convolution Algorithms
+7
+Algorithm 1: TT-SVD
+input : A, ε
+output : TT-Cores: A(1), A(2), ..., A(K)
+τ :=
+ε
+√M−1∥A∥F
+r0 := 1;
+for k=1,...,K-1 do
+A{k} := reshape(A, [Mkrk−1,
+|A|
+Mkrk−1 ])
+Compute truncated SVD: UΣV ∗ + E = A{k} such that ∥E∥F ≤ τ
+rk := rank(Σ) = rankτ(A{k})
+A(k) := reshape(U, [rk−1, Mk, rk])
+A := ΣV ∗
+end
+A(K) := A
+The TT-decomposition can also be applied to tensors with a small number of modes by using the
+quantized tensor train decomposition (QTT). For instance, let v ∈ C2K be a vector (1-mode tensor). To
+apply the QTT-decomposition of v, we reshape it into the K-mode tensor V ∈ C2×...×2 such that
+V(i1, i2, . . . , iK) = v(i),
+where
+i =
+K
+�
+k=1
+ik2k−1,
+ik = 0, 1,
+then compute the TT-decomposition of the tensor V (you can think of iK . . . i1 as the binary representation
+of i). Extending the QTT-decomposition to matrices (2-mode tensors) V ∈ C2K×2K can be done similarly
+by reshaping them into 2K-mode tensors V ∈ C2×...×2, then computing the TT-decomposition of V.
+We can approximate the discrete Fourier transform of a vector v ∈ R2K (or 2D discrete Fourier
+transform of a matrix V ∈ R2K×2K) in the QTT format using what is known as the QTT-FFT approximation
+algorithm [3]. Let ˆv = DFT(v) be the discrete Fourier transform of v and let V and ˆV be the tensors
+in the QTT-format that represent the vectors v and ˆv respectively. Given V, the QTT- FFT approximation
+algorithm can approximate ˆV with a tensor ˜V such that
+∥ ˜V − ˆV∥F ≤ ε
+(4.1)
+for some given tolerance ε. Similarly, we could prescribe some maximum TT-rank, ˆRmax, for the QTT-FFT
+algorithm such that ˜rk ≤ ˆRmax for all TT-ranks of ˜V, {˜rk}K
+k=0. The QTT-FFT algorithm can easily be
+modified to the inverse Fourier transform of a vector (or matrix) in the QTT format, which we denote as the
+QTT-iFFT algorithm.
+5
+Computing the convolution with QTT decomposition
+In practice, we often need to compute the convolution (2.1), where f is the function of interest and g is a
+given kernel, but f is not given explicitly. Instead, we are given noisy data
+(fξ)j = f(xj) + ξj
+(5.1)
+at discrete points xj, j = (j1, . . . , jD). In particular, representing the ground reflectivity function for SAR
+reconstruction in the form (5.1) helps one model the noise in the received data. We assume that ξj is white
+noise from a normal distribution with the standard deviation σ, i.e., ξj ∼ N(0, σ2).
+
+8
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+Since the kernel function g is known, we can discretize it as
+gj = g(xj),
+for the same xj values as in (5.1). We assume the D-dimensional spatial domain is uniformly discretized
+into ND points where N = 2K−1 − 1, see (2). To compute the discrete convolution (2.2), we propose using
+the quantized tensor train (QTT) decomposition. To represent the arrays in the QTT format, we pad them
+with zeros such that the new arrays are D-mode tensors in R2K×...×2K. We can relax the condition on the
+size N, but to compute the convolution with an FFT algorithm, we need to zero-pad each dimension with at
+least N − 1 extra zeros (see Section 2). Also, for the QTT decomposition, we need each dimension to be of
+size 2K for some K ∈ N. Let Fξ, G be the zero-padded tensors representing fξ and g respectively in the
+QTT format. Here, we assume that the discretization of f, f, has a low, but not exactly known, TT-rank in
+the QTT-format. This is motivated by the fact that many standard piecewise smooth functions naturally have
+a low TT-rank, see [9,16,22].
+To find approximations of these tensors in the TT-format, we modify the original TT-SVD algorithm.
+This is because with the full TT-SVD algorithm, if the tolerance ε is small, see equation (4.1), the TT-
+decomposition has close to full rank. Not only does it take a very long time to compute these decompositions,
+but most of the noise is still present. However, if ε is too large, the TT-SVD algorithm loses too much
+information about the true function f. For these reasons, we present slight modifications to the TT-SVD
+algorithm. They are needed to significantly reduce the computing time, as illustrated by the example in
+Section 7.1.
+We consider three different modifications to the TT-SVD algorithm. These modifications are as follows:
+(1) Set some max rank Rmax and truncate the SVD in Algorithm 1 with ranks less than or equal to this
+threshold. Denote this method as the max rank TT-SVD algorithm.
+(2) Set some max rank Rmax and replace the SVD in Algorithm 1 with a randomized SVD (RSVD) given
+in [11] with max ranks set to Rmax (see Appendix A). Denote this method as the max rank TT-RSVD
+algorithm. Note that for this algorithm, we also need to prescribe an oversampling parameter p. We
+could choose from several randomized SVD algorithms, but due to simplicity and effectiveness, we
+use the approach described in Appendix A. This algorithm implements the direct SVD.
+(3) Truncate the SVD in Algorithm 1 based on when there is a relative drop in singular values, i.e., if
+σk+1
+σk
+< δ (0 < δ < 1) for a given threshold δ, then truncate the singular values less than σk. Denote
+this method as the SV drop off TT-SVD algorithm.
+For the max rank TT-RSVD, if the unfolding matrices A{k} ∈ Rmk×nk, where min(mk, nk) ≤ Rmax + p,
+then we revert to the max rank TT-SVD algorithm (without the randomized SVD).
+We can modify the QTT-FFT and QTT-iFFT algorithms similarly to our modifications of the TT-SVD
+algorithms to get a low-rank approximation to the discrete Fourier transform representations of Fξ and G.
+For this, we replace the SVD in the QTT-FFT algorithm (QTT-iFFT) with the truncated SVD algorithms
+(1)-(3) given above, but with possibly a different max rank which we denote ˆRmax for (1) and (2), or different
+threshold ˆδ for (3). For the examples in Section 7, we distinguish between Rmax and ˆRmax. However, we use
+the same threshold for δ in the TT-SVD algorithm and the QTT-FFT algorithm. Thus, we do not distinguish
+between the two. Note that using the threshold (1) in the QTT-FFT algorithm is not new and is mentioned
+in [3].
+With these above modifications to the TT-SVD algorithm and QTT-FFT (QTT-iFFT) algorithms, we
+propose the following algorithm (Algorithm 2) to approximate the convolution between the D-dimensional
+arrays f and g. For this algorithm, we denote
+
+Denoising Convolution Algorithms
+9
+• QFFT ˆRmax(ˆδ): QTT-FFT algorithm with a max rank of ˆRmax (or threshold ˆδ),
+• QiFFT ˆRmax(ˆδ): QTT-iFFT algorithm with a max rank of ˆRmax (or threshold ˆδ).
+Algorithm 2: QTT convolution
+input : fξ, g
+output : I
+Step 1: F ξ = reshape(fξ, [2, . . . , 2]), G = reshape(g, [2, . . . , 2])
+Step 2: Decompose F ξ and G into the QTT format using one of the modified TT-SVD algorithms.
+Step 3: I = QiFFT ˆ
+Rmax(ˆδ)(QFFT ˆ
+Rmax(ˆδ)(F ξ) ⊙ QFFT ˆ
+Rmax(ˆδ)(G)).
+Step 4: Retrieve I from I. (see Algorithm 3)
+In Theorem 5.2, we show the asymptotic run time behavior of computing a convolution in one spatial
+dimension (D = 1) with the max rank TT-SVD algorithm. First, we prove an auxiliary result about the size
+of the unfolding matrices for this algorithm; see Lemma 5.1. For Theorem 5.2, we consider the whole process
+of converting the vector into the QTT-format, computing the convolution, then converting the convolution
+in the QTT format back into a vector, as is demonstrated in Algorithm 2. For the last step, to convert a
+tensor in the TT-format back into the standard format, we use the ’full’ algorithm from the Matlab toolbox
+oseledets/TT-Toolbox. This is given in Algorithm 3. We then reshape this tensor into a vector with a bit of
+run time.
+Algorithm 3: Full
+input : A(1), A(2), ..., A(K), and size of output tensor [M1, . . . , Mk]
+output : A ∈ CM1×...×Mk
+Let A = A(1)
+for k=2,...,K do
+A = reshape(A, [ (|A|)
+rk−1 , rk−1])
+B = reshape(A(k), [rk−1, 2rk])
+A = AB
+end
+A = reshape(A, [M1, . . . , Mk])
+Lemma 5.1. Let A ∈ R2×...×2 be a K-mode tensor. Let {A{k}}K−1
+k=1 be the unfolding matrices of A in the
+max rank TT-SVD algorithm with a max rank of Rmax and with each A{k} ∈ Cmk×nk. Then
+mk = 2rk−1 ≤ 2Rmax
+and
+nk = 2K−k.
+Proof. Since Mk = 2 for all k, the proof for mk = 2rk−1 ≤ 2Rmax is trivial by the first line inside the for
+loop in Algorithm 1. For nk, we do a proof by induction. First, note that |A{1}| = 2K and r0 = 1, thus
+n1 = |A{1}|
+2r0
+= 2K
+2 = 2K−1.
+Assume nℓ = 2K−ℓ for all 1 ≤ ℓ ≤ k − 1. Then,
+nk = |A{k}|
+2rk−1
+= |Σk−1V ∗
+k−1|
+2rk−1
+= rk−1nk−1
+2rk−1
+= nk−1
+2
+= 2K−(k−1)
+2
+= 2K−k.
+Thus, we get
+mk = 2rk−1 ≤ 2Rmax
+and
+nk = 2K−k.
+
+10
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+Theorem 5.2. Let fξ, g ∈ R2K−1−1 for some positive integer K. Then the computational complexity,
+CQTT-conv, of approximating the convolution fξ ∗ g with the max rank TT-SVD and max rank QTT-SVD
+algorithms described above is
+CQTT-conv ≤ O(R2
+max2K),
+where Rmax is the prescribed max rank for both the TT-SVD algorithms and the QTT-FFT algorithm.
+Proof. We show that the computational complexity is dominated asymptotically by the max rank TT-SVD
+algorithms and the full tensor algorithm. First, let Csvd be the computational cost of the SVD in big O
+notation. Then, for a matrix A ∈ Cm×n, Csvd(A) = O(mn min(m, n)). Note that, in Algorithm 1 (as
+well as in our max rank modifications), the computational complexity is dominated by the SVD algorithm.
+Denote the unfolding matrices at the kth iterations as A{k} ∈ Cmk×nk. Hence, the computational cost of
+the max rank TT-SVD algorithm is
+K−1
+�
+k=1
+Csvd(A{k}) =
+K−1
+�
+k=1
+O(mknk min(mk, nk))
+≤
+K−1
+�
+k=1
+O((2Rmax)22K−k)
+= 4R2
+max
+K−1
+�
+k=1
+O(2k)
+= 4R2
+maxO(2K − 2)
+= O(R2
+max2K).
+From [3], we have that for the QTT-FFT and QTT-iFFT algorithms, the computational complexity is
+O(K2R3
+max). In Algorithm 3, the computational complexity comes from the multiplication AB in every
+loop. For the kth loop, A ∈ C2k−1×rk−1 and B ∈ Rrk−1×2rk for k = 2, . . . , K, thus the computational
+complexity is proportional to the cost of multiplying A by B, i.e.,
+Cfull =
+K
+�
+k=2
+O(2k−1rk−12rk)
+≤ R2
+max
+K
+�
+k=2
+O(2k)
+= R2
+maxO(2K+1 − 4)
+= O(R2
+max2K).
+Hence, the total computational complexity is
+O(R2
+max2K) + O(K2R3
+max) + O(R2
+max2K) = O(R2
+max2K).
+For the randomized SVD, we have the computational complexity Crsvd(A{k}) = O(mknk(Rmax + p)) =
+O(2K−kRmax(Rmax + p)). Thus, the run time for the convolution with a max rank TT-RSVD is similar
+when p is small. In D spatial dimensions, we can obtain a similar result but by replacing K with DK in the
+max rank TT-SVD algorithm and the full tensor algorithm, and the QTT-FFT algorithm is O(DK2R3
+max).
+Hence, the total run time complexity in D spatial dimensions is O(R2
+max2DK).
+
+Denoising Convolution Algorithms
+11
+6
+Denoising
+It is well known that the SVD can remove noise from matrix data, as seen in [4,14], but little research has
+been done in denoising with tensor decompositions. In [17] and [19], the Tucker decomposition was used to
+help remove noise from point cloud data and electron holograms, respectively. In [8], it was shown that the
+TT-decomposition might have some advantages to denoising as opposed to the Tucker decomposition. This
+is because a low-rank Tucker matrix guarantees a low TT-rank for the data. However, the converse statement
+is not always true.
+Let F be the low TT-rank tensor representing f in the QTT format.
+Then for some core tensors
+F(k) ∈ Rrk−1×2×rk with tensor slices F(k)(:, ik, :) = f (k)
+ik
+∈ Rrk−1×rk, ik = 0, 1. Each element of F can
+be represented in the TT format as
+F(i1, . . . , ik) = f (1)
+i1 . . . f (K)
+iK ,
+ik = 0, 1,
+k = 1, . . . , K,
+where each f (k)
+ik
+is a low rank matrix. In practice, it is unlikely the data collected has a low-rank TT
+decomposition since almost all real radar data has noise due to hardware limitations or other signals interfering
+with the data. Instead, we have the noisy data fξ whose tensor representation is
+Fξ = F + ξ,
+where ξ is the realization of the random noise in the TT format. The tensor Fξ almost surely has full TT-rank
+when represented exactly in the QTT format. Ideally, we would like to be able to find an approximate TT
+decomposition ˜F with TT-cores ˜F
+(k), k = 1, . . . , K, using the noisy data such that ˜F
+(k) ≈ F(k). However,
+it is hard to guarantee any bound on this. We argue, though, that by using our proposed methods when given
+the noisy data Fξ, we can find a TT decomposition ˜F with low rank such that ˜F ≈ F.
+Consider the first iteration of the for loop of algorithm 1, with A = A0 + Aξ as the sum of a smooth
+tensor (A0) and a noisy tensor (Aξ). Then, after it is reshaped, we obtain the matrix
+A{1} = A{1}
+0
++ A{1}
+ξ
+,
+where A{1}
+0
+is a low rank matrix and A{1}
+ξ
+is added noise. Let A{1} = UΣV ∗ + E be the truncated
+SVD of A{1} and A{1}
+0
+= U0Σ0V ∗
+0 be the SVD of A{1}
+0
+. Note that UΣV ∗ ≈ A{1}
+0
+does not imply that
+U ≈ U0, and thus the TT-core A(1) is not guaranteed to be approximately equal to A(1)
+0 , where A(1)
+0
+is the
+first TT-core of A0. However, if we let A2 = A on the second iteration of the loop in Algorithm 1 (and
+similarly for A0), we do get that the elements of the tensor contraction A(1) ◦ A2 ≈ A(1)
+0
+◦ A2
+0. Similarly, if
+we can approximate the noise-free component on every iteration of the for loop, we obtain an approximation
+for the tensor A0. While we do not have a theoretical bound on this error, our experiments in Section 7
+show that this method works well at removing the noise. Since our method computes multiple SVDs, it can
+reduce a lot more noise than if we just did a single SVD and can do so without excessive smearing.
+7
+Numerical simulations
+This section presents some examples in one and two spatial dimensions. The original code for the TT-
+decompositions and the QTT-FFT algorithms comes from the Matlab toolbox oseledets/TT-Toolbox. We
+have modified it accordingly for the max rank TT-SVD, max rank TT-RSVD, and SV drop off TT-SVD
+algorithm, as discussed in Section 5. For all our examples, we compare the run time and errors of computing
+the convolution (2.1) using several methods. The error for every example is the l2 relative error
+E2(I) = ∥I − Iref∥2
+∥Iref∥2
+,
+(7.1)
+
+12
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+where in D spatial dimensions
+∥I∥2 =
+�
+�
+�
+� 1
+ND
+N−1
+�
+j=0
+|Ij|2.
+The reference solution, Iref, is the discrete convolution (2.2) computed without any noise. In all of the
+examples, we compare our methods against computing the convolution with the randomized TT-SVD
+algorithm from [13], as well as computing the true noisy convolution with FFT. In two space dimensions,
+we also approximate the convolution using a low matrix rank approximation to the noisy data fξ, where the
+truncated rank is determined by the actual matrix rank of f.
+For all of these examples, we use the normalized sinc imaging kernel that corresponds to the GAF (3.1)
+truncated to a sufficiently large interval [−L, L]:
+g(x) =
+sinc(π x
+∆x )
+� L
+−L sinc(π x
+∆x ) dx
+,
+x ∈ [−L, L]
+(7.2)
+for D = 1, and
+g(x, y) =
+sinc(π x
+∆x )sinc(π y
+∆y )
+�� L
+−L sinc(π x
+∆x )sinc(π y
+∆y ) dxdy
+,
+(x, y) ∈ [−L, L] × [−L, L]
+(7.3)
+for D = 2, where the resolution ∆x in (7.2) and ∆x = ∆y in (7.3) is a given parameter. The one- dimensional
+kernel (7.2) for ∆x = 0.04π is shown in Figure 7.1.
+In Table 7.1, we present the relative error for each example for K = 20 when D = 1, and K = 10 when
+D = 2. In this table, the convolution fξ ∗g is denoted by Iξ and computed using the FFT algorithm, the QTT-
+convolution computed with the max rank TT-SVD algorithm is denoted by IQTT0, the QTT-convolution
+computed with the max rank TT-RSVD is denoted by IQTTr, and the convolution computed using the SV
+drop off TT-SVD algorithm is denoted by Iδ. In turn, the convolutions computed using the randomized
+TT-decomposition are denoted by IRTT , and in two dimensions, the convolution computed using low-rank
+approximations of f is denoted by Ilr. For Iδ and Ilr, we also denote what parameter δ and truncation
+matrix rank R are used, respectively, for each example using a subscript of the error.
+Figure 7.1: Kernel function (7.2) with ∆x = 0.04π.
+For each example, we show the TT-ranks of the original function without noise, f, in the QTT format
+given by the tensor F. This QTT approximation is computed with Algorithm 1 with the tolerance ε = 10−10.
+
+g(x)
+8
+6
+4
+2
+0
+-2
+-10
+-5
+0
+5
+10
+XDenoising Convolution Algorithms
+13
+We compute these TT-ranks for K = 20 when D = 1 and K = 10 when D = 2. However, it is worth noting
+that these TT-ranks do not change much for any data size. Notice that the max TT-ranks we choose for our
+algorithms are less than the TT-ranks of f from Algorithm 1, yet still provide a reasonable estimate.
+Table 7.1: l2-norm relative error for K = 20 for examples 1 and 2, and K = 10 for example 3.
+example
+Iξ
+IQTT0
+IQTTr
+Iδ
+IRTT
+Ilr
+1
+0.0383
+0.0028
+0.0102
+0.0280δ=0.02
+0.0430
+-
+2
+0.0131
+0.0011
+0.0075
+0.0068δ=0.01
+0.0201
+-
+3
+0.1142
+0.0151
+0.0447
+0.1650δ=0.09
+0.1534
+0.0470rank=23
+7.1
+Example 1
+For this example, let
+f(x) = e−( 3x
+10 )2(0.4 sin(8πx) − 0.7 cos(6πx)),
+x ∈ [−10, 10],
+and
+(fξ)j = f(xj) + ξj,
+xj = −10 + ∆x
+2 + j∆x,
+j = 0, . . . , N − 1,
+∆x = 20
+N ,
+N = 2K−1 − 1,
+with ξj ∼ N(0, 0.02). We also set the resolution ∆x = 4∆x in (7.2), where ∆x is the size of the spatial
+discretization. Thus, the width of the main lobe of the sinc is 8∆x on the x-axis.
+As we can see in Figure 7.2, the FFT-QTT algorithm removed much of the noise in the data compared
+to the true convolution. For K = 20, we also tried computing the convolution using the original TT-SVD
+algorithm given in Algorithm 1 with multiple values of ε. The smallest error, as defined in (7.1), occurred
+when ε = 0.01 and gave the relative error of E2(I) = 0.03202. This is close to the error of the true
+convolution of the noisy data and took over 100 seconds to compute. However, as we can see in Table 7.2,
+the run times for all of our methods on the same grid took less than a second. This indicates that the original
+TT-SVD algorithm is practically unsuitable for removing data noise.
+The max TT-rank of the discretization of f(x) in the QTT format, F, is 17, yet we were able to achieve
+our approximation using a max rank of Rmax = 10 for the max rank TT-SVD and max rank TT-RSVD
+algorithms and ˆRmax = 15 for the QTT-FFT algorithm. Thus, even if we do not know the exact TT-rank,
+we can still compute a good approximation.
+Table 7.2 shows run times for different grid sizes for each method. We can see that computing the
+convolution with FFT is faster than our methods for these values of K. However, the convolution with our
+QTT methods gets closer to the FFT run time as K increases. This is shown in the last column of Table
+7.2 where we see the ratio of the max rank TT-SVD convolution method to the FFT convolution method
+is getting smaller as K grows. This helps verify our theoretical result that for some constant max rank
+Rmax (and ˆRmax), the max rank TT-SVD convolution method is asymptotically faster than computing the
+convolution with FFT. The amount of data needed for our method to outperform the FFT method may be
+impractical for most real-world applications in 1-2 spatial dimensions.
+Table 7.3 shows the number of elements to represent the data fξ fully versus how many elements are
+required to store the data in the QTT-format with a prescribed max rank of Rmax = 10, Fξ, in Example 1.
+As we can see, storing all the elements takes a lot of data and grows exponentially in K, while storing the
+elements in the QTT format takes a lot less data and only grows linearly in K. These values for the QTT-data
+
+14
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+Figure 7.2: Top Left: True convolution of data without noise, I. Top Right: Function data with noise, fξ.
+Middle Left: True convolution of data with noise, Iξ. Middle Right: Convolution using the max rank
+TT-SVD algorithm, IQTT0. Bottom Left: Absolute error of Iξ. Bottom Right: Absolute error of IQTT0.
+storage can be found by looking at the size of the core tensors. For the tensor Fξ in the QTT format and
+
+Iref
+1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+-10
+-5
+0
+5
+10
+Xf(α) +s
+1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+-10
+-5
+0
+5
+10
+XI:
+1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+-10
+-5
+0
+5
+10
+XIQTT
+1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+-10
+-5
+0
+5
+10
+X[I-Irefl
+0.05
+0.04
+0.03
+0.02
+0.01
+0
+-10
+-5
+0
+5
+10
+XIQTT -Irefl
+0.05
+0.04
+0.03
+0.02
+0.01
+0
+-10
+-5
+0
+5
+10
+XDenoising Convolution Algorithms
+15
+Table 7.2: Run time (seconds): Example 1 convolutions.
+K
+Iξ
+IQTT0
+IQTTr
+Iδ
+IRTT
+IQTT0/Iξ
+16
+0.005
+0.325
+0.369
+1.485δ=0.02
+0.414
+65
+20
+0.067
+0.653
+0.694
+0.479δ=0.02
+0.609
+9.7463
+24
+1.21
+4.41
+4.86
+3.10δ=0.02
+2.80
+3.6446
+26
+5.77
+17.75
+19.68
+13.93δ=0.02
+10.97
+3.0763
+28
+66.6
+95.5
+108.7
+79.0δ=0.02
+62.49
+1.4339
+Table 7.3: Data storage for Example 1.
+K
+fξ
+Fξ
+16
+65,536
+2088
+20
+1,048,576
+2888
+24
+16,777,216
+3688
+26
+67,108,864
+4088
+28
+268,435,456
+4488
+with a max TT-rank of Rmax = 10, we have the TT-cores
+F(1)
+ξ , F(K)
+ξ
+∈ R1×2×2,
+F(2)
+ξ , F(K−1)
+ξ
+∈ R2×2×4,
+F(3)
+ξ , F(K−2)
+ξ
+∈ R4×2×8,
+F(4)
+ξ , F(K−3)
+ξ
+∈ R8×2×10,
+F(k)
+ξ
+∈ R10×2×10,
+k = 5, . . . , K − 4.
+Thus, the number of elements, Nel, that make up this QTT tensors is:
+Nel = 2(1 × 2 × 2) + 2(2 × 2 × 4) + 2(4 × 2 × 8) + 2(8 × 2 × 10) + (K − 8)(10 × 2 × 10).
+The max rank TT-RSVD algorithm is not able to produce results as good as the max rank TT-SVD
+(see Table 7.1 for relative error comparison and Table 7.2 for a run time comparison) but is still able to
+produce a reasonably low error. While the run time for the max rank TT-SVD is faster than the max rank
+TT-RSVD for all of our methods, the max rank TT-RSVD can be faster for tensors with larger mode sizes.
+This is due to the SVD in max rank TT-SVD algorithm with mode sizes, Mk, may be computed on a matrix
+with mk = MkRmax rows. In contrast, for the max rank TT-RSVD algorithm, the SVD is computed on a
+matrix with mk = Rmax + p rows when Mk = 2 (such as for the QTT decomposition). The difference in
+the sizes of mk does not make up for the extra amount of work the RSVD algorithm does. Although this
+paper focuses on the QTT-decomposition and thus Mk = 2, we believe this is important to note as the max
+rank TT-RSVD algorithm can speed up the TT-decomposition for higher mode tensor data and still produce
+accurate approximations. We verify this by computing the max rank TT-SVD algorithm and the max rank
+TT-RSVD algorithm on a tensor with K = 8 modes with each mode of size Mk = 10, k = 1, . . . , K. Each
+element of this tensor is taken from the uniform distribution U[0, 1). The max rank TT-SVD algorithm took
+9.57 seconds, and the max rank TT-RSVD algorithm only took 5.12 seconds, almost half the time of the
+max rank TT-SVD algorithm.
+
+16
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+7.2
+Example 2
+If we were to choose a coarser resolution for the example of Section 7.1 (i.e., a wider sinc function), we could
+reduce the noise using the standard convolution at the cost of smoothing out the solution’s peaks. Doing this
+gives similar results for the true convolution and with our methods (Section 5). In this section, we show an
+example where the ground reflectivity is very oscillatory. Here, the resolution ∆x determined by the GAF
+must be small (i.e., the sinc function must be “skinny”). Otherwise, if the sinc window is close to or larger
+than the characteristic scale of variation of the ground reflectivity, then the convolution can smooth out the
+actual oscillations instead of just the noise, losing most of the information in f.
+We choose the ground reflectivity as
+f(x) = e−(3x)2(0.9 sin( 2xπ
+5∆x) + 1.4 cos( xπ
+3∆x)),
+x ∈ [−1, 1],
+and
+(fξ)j = f(xj) + ξj,
+xj = −1 + ∆x
+2 + j∆x,
+j = 0, . . . , N − 1,
+∆x = 2
+N ,
+N = 2K−1 − 1,
+with ξj ∼ N(0, .01). We use ∆x = 2∆x and the max TT-rank of the discretization of the smooth function
+f(x) in the QTT-format is 26.
+Again, we use the max ranks of Rmax = 10 and ˆRmax = 15 for the max rank TT-SVD and QTT-SVD
+algorithms, respectively. As the function is too oscillatory to see a lot of helpful information in the full graph
+(see top left of Figure 7.3), we show a zoomed-in plot of the graph of Iξ, IQTT0, and Iref (see top right of
+Figure 7.3). While there is some error, the QTT-FFT convolution agrees with the true convolution Iref very
+well, whereas Iξ has a more considerable noticeable difference. This is verified by the graphs of the absolute
+error given in Figure 7.3, where the bottom left shows the error for Iξ, and the bottom right shows the error
+for IQTT0.
+7.3
+Example 3
+Let
+f(x, y) = e−((2x)2+(2y)2)(sin(2πx) − cos(7πy) + cos(4πxy) − sin(3πxy),
+(x, y) ∈ [−1, 1] × [−1, 1]
+and
+(fξ)j,k = f(xj, yk) + ξj,k,
+xj = −1 + ∆x
+2 + j∆x,
+yk = −1 + ∆y
+2 + k∆y,
+j, k = 0, . . . , N − 1,
+∆x = ∆y = 2
+N ,
+N = 2K−1 − 1,
+with ξj,k ∼ N(0, 0.1). We have ∆x = ∆y = 2∆x. Thus, the diameter of the main lobe of the sinc is 4∆x
+on the xy-plane. Here, we show a 2D example whose discretization of a smooth function f has a matrix
+rank of 23 and a TT-rank of 26 when represented in the QTT format. We still use the ranks Rmax = 10 and
+ˆRmax = 15 for our max rank TT-SVD (max rank TT-RSVD) and max rank QTT-FFT. Thus, our TT-ranks
+are much smaller than the true TT-ranks. In Figure 7.4 and Table 7.1, notice that our method can still capture
+the shape of the original function with an error that is an order of magnitude smaller than the error from the
+
+Denoising Convolution Algorithms
+17
+Figure 7.3: Top Left: True convolution of data without noise, I. Top Right: Zoomed in graph of Iξ, IQTT0,
+and Iref. Bottom Left: Absolute error of Iξ. Bottom Right: Absolute error of IQTT0.
+true convolution using FFT. The plots on the bottom of Figure 7.4 are a side view of the error graphs, as it
+is easier to compare the errors in this view. The 2D examples are similar to the previous test case. Thus, it
+is reasonable to assume our method works about the same regardless of the spatial dimension.
+In Table 7.4, we compare the run times for the different methods of computing the convolution in two
+spatial dimensions. We get similar results as the one-dimensional case, where the fastest run time is from
+the convolution with FFT, but with the max rank TT-SVD and max rank TT-RSVD methods approaching
+its run time asymptotically. In 2D, when there is the same amount of data as in the 1D case (for example,
+214×14 in 2D compared to 228 in 1D), the 2D examples do not run as fast as the 1D example. This is due to
+the extra work in the 2D QTT-FFT algorithm from [3].
+Again we compare the amount of data stored in the full format versus in the QTT-format. Note that the
+spatial dimension of the original function does not matter in how much storage it takes, just the dimensionality
+of the data. For example, it takes just as much data to store a vector in R220 as it does to store a matrix in
+R210×210 in the QTT format with a max rank of Rmax.
+
+[I -Irefl
+0.035
+0.03
+0.025
+0.02
+0.015
+0.01
+0.005
+0
+-1
+-0.5
+0
+0.5
+XIQTT -Irefl
+0.035
+0.03
+0.025
+0.02
+0.015
+0.01
+0.005
+0
+-1
+-0.5
+0
+0.5
+XIref
+3
+2
+1
+0
+-1
+-2 
+-3
+-1
+-0.5
+0
+0.5
+1
+X0.2
+M
+-0.2
+IQTTo
+Iref
+0.534
+0.5341
+0.5342
+0.5343
+0.5344
+X18
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+Figure 7.4: Top Left: True convolution of data without noise, I. Top Right: Function data with noise, fξ.
+Middle Left: True convolution of data with noise, Iξ. Middle Right: Convolution using the max rank
+TT-SVD algorithm, IQTT0. Bottom Left: Absolute error of Iξ. Bottom Right: Absolute error of IQTT0.
+
+Iref
+2 
+1
+0
+-1
+1
+0
+0
+-1
+-1f(α) +$
+2 
+1
+0
+7
+1
+0
+0
+-1
+-12 
+1
+0
+-1
+1
+0
+0
+-1
+-12 
+1
+0
+.1
+1
+0
+0
+-1
+-1[Is -Irefl
+0.2
+0.15
+0.1
+0.05
+0
+1
+0.5
+0
+-0.5
+-1IIQTT。 - Irefl
+0.2
+0.15
+0.1
+0.05
+0
+1
+0.5
+0
+-0.5
+1Denoising Convolution Algorithms
+19
+Table 7.4: Run times (seconds): Example 3 convolutions.
+K
+Iξ
+IQTT0
+IQTTr
+Iδ
+IRTT
+Ilr
+IQTT0/Iξ
+8
+0.0034
+0.2213
+0.310
+7.102δ=0.04
+0.297
+0.0090rank=2
+65.088
+10
+0.0629
+0.9209
+1.428
+0.670δ=0.04
+1.321
+0.154rank=2
+14.651
+12
+0.948
+8.6462
+11.258
+22.79δ=0.04
+10.27
+5.15rank=2
+9.121
+14
+58.67
+147.8
+151.05
+1087δ=0.04
+164.5
+286.2rank=2
+2.519
+Table 7.5: Data storage for Example 3.
+K
+fξ
+Fξ
+8
+65,536
+2088
+10
+1,048,576
+2888
+12
+16,777,216
+3688
+14
+268,435,456
+4488
+8
+Conclusions
+In this paper, we have shown that the QTT decomposition, along with the QTT-FFT algorithm, can effectively
+remove noise from signals with full TT-ranks when the true signal is of low rank. As we have seen in the
+numerical examples, we could drastically remove the amount of noise from the signal compared to if we
+took the convolution in the traditional way of using the FFT algorithm. This comes at the cost of run time,
+but our methods still run at a reasonable speed which got closer to the FFT run time as the dimensionality
+of the data increased. This is demonstrated by three different examples, two in one spatial dimension and
+one in two spatial dimensions. We are even able to show that our method works on very oscillatory data
+where it is required to have a sinc kernel with a narrow main lobe. Using approximate TT-ranks smaller than
+the TT-ranks of the actual signal data, we are able to recover most of the signal. This indicates that as long
+as the signal is reasonably smooth, the QTT decomposition can effectively be used for noise reduction for
+high-dimensional data, even if the true TT-rank is unknown.
+From our three new approaches, the max rank TT-SVD convolution algorithm works much better than
+the max rank TT-RSVD and the SV drop off TT-SVD convolution algorithms. As was stated before, the
+max rank TT-RSVD can outperform the max rank TT-SVD algorithm when there are larger mode sizes
+that are used in this paper. For this reason, we present this algorithm, as we have not seen it in the literature
+elsewhere. The SV drop off TT-SVD convolution algorithms do not produce as accurate of a method as the
+max rank TT-SVD or the max rank TT-RSVD algorithm; however, in some cases, it does run faster, and
+this method may give a higher degree of confidence that the truncated singular values are of little importance.
+Unfortunately, this method can also lead to long run times, as is seen in Table 7.4 when K = 14.
+A
+Randomized SVD
+Here, we give a brief overview of the randomized SVD (RSVD) decomposition from [11]. To compute the
+RSVD of the matrix A ∈ Rm×n, the first step is to find Q ∈ Rm×(k+p) such that
+A ≈ QQ∗A
+where Q has orthonormal columns and whose columns are approximations for the range of A. Here, k is
+the number of singular values that we want in our approximation to be close to the singular values of A, and
+
+20
+A. Chertock, C. Leonard, S. Tsunkov, S. Utyuzhnikov
+p is what is known as an oversampling parameter. To find Q, we use the following Algorithm 4.
+Algorithm 4: Solving the Fixed-Rank Problem
+input : A, k, p
+output : Q
+Draw random matrix Ω ∈ Rn×(k+p) such that Ωi,j ∼ N(0, 1).
+Let Y = AΩ.
+Compute QR factorization QR = Y .
+Once we have obtained Q, we can compute the low-rank RSVD using Algorithm 5 (Algorithm 5.1
+in [11]).
+Algorithm 5: RSVD
+input : A, Q, k
+output : UΣV ∗
+1. Let B = Q∗A.
+2. Compute SVD: ˜UΣV ∗ = B.
+3. Let U = Q ˜U.
+With these algorithms, we obtain an approximation ˜
+A to A such that
+∥A − ˜
+A∥ ≤ (1 + 11
+�
+k + p
+�
+min(m, n))σk+1,
+(A.1)
+with probability 1 − 6p−p. If we truncate the SVD to only the leading k singular values in Algorithm 5, then
+the error on the left-hand side of (A.1) only increases by at most σk+1. The computational complexity for
+each step of this algorithm is given as
+• O(mn(k + p))
+• O((k + p)2n)
+• O((k + p)2m),
+Thus, for k + p < min(m, n), the overall algorithm requires O(mn(k + p)) operations.
+Acknowledgments
+The work of A. Chertock was supported in part by NSF grants DMS-1818684 and DMS-2208438. The work
+of C. Leonard was supported in part by NSF grant DMS-1818684. The work of S. Tsynkov was supported
+in part by US Air Force Office of Scientific Research (AFOSR) under grant # FA9550-21-1-0086.
+References
+[1] Maolin Che and Yimin Wei. Randomized algorithms for the approximations of Tucker and the tensor
+train decompositions. Adv Comput Math, 45:395–428, 2019.
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+Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 2009.
+[3] Sergey Dolgov, Boris Khoromskij, and Dmitry Savostyanov. Superfast Fourier transform using QTT
+approximation. J. Fourier Anal. Appl., 18(5):915–953, 2012.
+
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+Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham, Switzerland, 2017.
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+Technical Report 308, Institut für Geometrie und Praktische Mathematik RWTH Aachen, April 2010.
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+multimodal factor analysis. UCLA Working Papers in Phonetics, 16:1–84, December 1970.
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+261–290. Birkhäuser/Springer, Cham, 2017.
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+electronic nose data processing. IEEE Sensors Journal, 11(1):35–44, 2011.
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+generated by tensor-structured vectors and convolution with logarithmic complexity. SIAM J. Sci.
+Comput., 35(3):A1511–A1536, 2013.
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+2019.
+[18] Lingjie Li, Wenjian Yu, and Kim Batselier. Faster tensor train decomposition for sparse data. Journal
+of Computational and Applied Mathematics, 405:113972, 2022.
+[19] Yuki Nomura, Kazuo Yamamoto, Satoshi Anada, Tsukasa Hirayama, Emiko Igaki, and Koh Saitoh.
+Denoising of series electron holograms using tensor decomposition. Microscopy, 70(3):255–264, 09
+2020.
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+Appl., 31(4):2130–2145, 2009/10.
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+[21] I. V. Oseledets. Tensor-train decomposition. SIAM Journal on Scientific Computing, 33(5):2295–2317,
+2011.
+[22] I. V. Oseledets. Constructive representation of functions in low-rank tensor formats. Constr. Approx.,
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+many dimensions. SIAM J. Sci. Comput., 31(5):3744–3759, 2009.
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+decomposition, 2021.
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+1966.
+
diff --git a/Z9FQT4oBgHgl3EQffTZu/content/tmp_files/load_file.txt b/Z9FQT4oBgHgl3EQffTZu/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/Z9FQT4oBgHgl3EQffTZu/content/tmp_files/load_file.txt
@@ -0,0 +1,1053 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf,len=1052
+page_content='Denoising Convolution Algorithms and Applications  to SAR Signal Processing  Alina Chertock∗, Chris Leonard†, Semyon Tsynkov‡, Sergey Utyuzhnikov§  February 1, 2023  Abstract  Convolutions are one of the most important operations in signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' They often involve large  arrays and require significant computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Moreover, in practice, the signal data to be processed by  convolution may be corrupted by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In this paper, we introduce a new method for computing the  convolutions in the quantized tensor train (QTT) format and removing noise from data using the QTT  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We demonstrate the performance of our method using a common mathematical model  for synthetic aperture radar (SAR) processing that involves a sinc kernel and present the entire cost of  decomposing the original data array, computing the convolutions, and then reformatting the data back  into full arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  1  Introduction  Convolution operations are used in different practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' They often involve large arrays of data  and require optimization with respect to memory and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' While input data are usually  available only in a discrete form, the standard realization based on a vector-matrix representation is not  often efficient since it leads to using sparse matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' On the other hand, a tensor decomposition looks  very attractive because it might reduce the volume of data very drastically, minimizing the number of zero  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In addition, arithmetic operations between tensors can be implemented efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  There are different forms of tensor decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The most popular approach is based on the canonical  decomposition [12] where a multidimensional array is represented (might be approximate) via a sum of outer  products of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For matrices, such decomposition is reduced to skeleton decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, it  is known to be unstable in the cases of multiple tensor dimensions, also referred to as tensor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The  Tucker decomposition [27] represents a natural stable generalization of the canonical decomposition and  can provide a high compression rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The main drawback of the Tucker decomposition is related to the  so-called curse of dimensionality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' that is, the algorithm’s complexity grows exponentially with the number  of tensor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' A way to overcome these difficulties is to use the Tensor Train (TT) decomposition, which  was originally introduced in [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Effectively, the TT decomposition represents a generalization of the  classical SVD decomposition to the case of multiple modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' It can also be interpreted as a hierarchical  Tucker decomposition [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  ∗Department of Mathematics, North Carolina State University, Raleigh, NC, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' chertock@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='edu  †Department of Mathematics, North Carolina State University, Raleigh, NC, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' cleonar@ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='edu  ‡Corresponding  author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Department  of  Mathematics,  North  Carolina  State  University,  Raleigh,  NC,  USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  tsynkov@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='edu  §Department  of  Mechanical,  Aerospace  &  Civil  Engineering,  University  of  Manchester,  Manchester,  UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='Utyuzhnikov@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='uk  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='13339v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='NA]  31 Jan 2023  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  Computing the TT decomposition fully can be very expensive if we use the standard TT-SVD algorithm  given, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', by Algorithm 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Therefore, many modifications to this algorithm were proposed in the  literature to help speed it up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' One such improvement was presented in [18], where results comparable to  those obtained by the TT-SVD algorithm were produced in a fraction of the time for sparse tensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Another algorithm that uses the column space of the unfolding tensors was designed to compute the TT cores  in parallel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' see [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The most popular approach to efficiently compute the TT decomposition is based on  using a randomized algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', [1,5,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Maximal compression with the TT decomposition can be reached with matrices whose dimensions are  powers of two, as proposed in the so-called Quantized TT (QTT) algorithm [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' As shown in [15], the  convolution realized for multilevel Toeplitz matrices via QTT has a logarithmic complexity with respect to  the number of elements in each mode, N, and is proportional to the number of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' It is proven that  the result cannot be asymptotically improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, this algorithm is improved for finite and practically  important N ∼ 104 in [25] thanks to the cross-convolution in the Fourier (image) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The improvement is  demonstrated for convolutions with three modes with Newton’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' It is to be noted that QTT can also  be applied to the Fast Fourier Transform (FTT) to decrease its complexity, as shown in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This super-fast  FFT (QTT-FFT) beats the standard FFT for extremely large N such as N ∼ 260 for one mode tensors and  N ∼ 220 for tensors with three modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  For practical applications, a critical issue is denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Real-life data, such as radar signals, are typically  contaminated with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Denoising is not addressed in the papers we have cited previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However,  TT decomposition itself potentially has the property of denoising, owing to the SVD incorporated in the  algorithm [4,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In the current work, we propose and implement the low-rank modifications for the previously  developed TT-SVD algorithm of [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' These modifications speed up the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We also demonstrate  the denoising capacity of numerical convolutions computed using the QTT decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Specifically,  we employ a common model for synthetic aperture radar (SAR) signal processing based on the convolution  with a sinc imaging kernel (called the generalized ambiguity function) [6, Chapter 2] and show that when  a convolution with this kernel is evaluated in the QTT format, the noise level in the resulting image is  substantially reduced compared to that in the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  It should be observed that most papers on tensor convolution only consider the run time cost of the  convolution after the tensor decomposition has been applied to the objective function and the kernel function  and either ignores the cost of the actual tensor decompositions or puts it as a side note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In this paper, we  consider every step of computing the convolution using the QTT-FFT algorithm, including the decomposition  of the arrays into the QTT format using the TT-SVD algorithm (see Algorithm 1), computation of the QTT-  FFT algorithm once in that format, and then extracting the data back after the computation is conducted  (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' As the QTT decomposition is computationally expensive, we consider several approaches to  speed up the decomposition run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Without these modifications to the TT-SVD decomposition algorithm,  the convolution can take a long to compute and is not a practical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We provide more detail in section  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  The methods we use to speed up our TT decompositions are based on truncating SVD ranks in the  decomposition algorithm (Algorithm 1) and lead to a significant noise reduction in the data (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Thus, in Chapter 6, we present algorithms to compute convolutions in a reasonable time while significantly  reducing the noise in the data at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Our contribution includes developing and analyzing new  approaches to speeding up the tensor train decomposition, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In Section 6, we consider the  effects of convolutions on removing noise in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Finally, in Section 7, we show numerical examples and  compare our results with other approaches to computing convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Denoising Convolution Algorithms  3  2  Convolution  The convolution operation is widely used in different applications in signal processing, data imaging,  physics, and probability, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This operation is a way to combine two signals, usually represented  as functions, and produce a third signal with meaningful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The D-dimensional convolution  between two functions f and g is defined as  I(x) = [f ∗ g](x) =  �  RD f(y)g(x − y) dy,  ∀x ∈ RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1)  Often to compute the convolution numerically, we assume the support of f and g, denoted supp(f) and  supp(g) respectively, are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For simplicity, in this paper, we assume supp(f) = supp(g) = [−L, L]D  for some L ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Next, we discretize the domain [−L, L]D uniformly into ND points such that  xj = (xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , xjD),  xjd = −L + ∆x  2 + jd∆x,  jd = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , N − 1,  d = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , D,  where ∆x = 2L  N and j = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , jD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We then let f and g be D-dimensional arrays such that  fj = f(xj),  gj = g(xj)  for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This leads to the discrete convolution I such that  Ij := (∆x)D �  i  figj−i+( N  2 −1)1 ≈ I(xj),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2)  where 1 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , 1) and the sums are over all indices i = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iD) that lead to legal subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This  Riemann sum approximation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) to the integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1) uses the midpoint rule, thus having O(∆x2) accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The convolution defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) is equivalent to Matlab’s convn function with the optional  shape input set to ’same’ and then multiplied by (∆x)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  To compute this convolution directly takes O(N2D) operations, but it can be reduced to O(ND log(ND))  by using the fast Fourier transform (FFT) and the discrete convolution theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The FFT algorithm is an  efficient algorithm used to compute the D-dimensional discrete Fourier transforms (DFT) of V ∈ RN×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×N,  ˆVα := DFT(V) =  N−1  �  j=0  Vjωj·α  N  where the sum is over the multi-indexed array j,  α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , αD),  αd = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , N − 1,  d = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , D,  N = (N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , N),  0 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , 0),  and ωN = e− 2πˆı  N , where ˆı = √−1 is the imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Similarly, the D-dimensional inverse discrete  Fourier transform (IDFT), such that  V = IDFT(DFT(V)),  of the array ˆV ∈ RN×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×N is given by  Vj =  1  ND  N−1  �  α=0  ˆVαω−j·α  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  Using the discrete Fourier transform, we can compute the circular convolution Ic = (V ⊛ W) defined  as  Ic  j =  N−1  �  i=0  Vi ¯  Wj−i  ¯  Wi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=',iD = Wj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=',jD,  id ≡ jd mod(N),  d = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , D,  by taking the DFT of W and V, multiplying the results together, and then taking the IDFT of the given  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, we have  Ic = IDFT(DFT(W) ⊙ DFT(V))  where ⊙ is Hadamard product (element-wise product) of D-dimensional arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The circular convolution is  the same as the convolution of two periodic functions (up to a constant scaling), thus to obtain the convolution  given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) (also known as a linear convolution), we need to pad the vectors f and g with at least N − 1  zeros in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For example, given the vectors f 0, g0 ∈ R2N−1 with  f 0  j =  �  fj  0 ≤ j ≤ N − 1  0  j > N − 1  ,  and  g0  j =  �  gj  0 ≤ j ≤ N − 1  0  j > N − 1  ,  and Ic = (f 0 ⊛ g0) as the circular convolution between them, the linear convolution I in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) is given by  Ij = ∆xIc  j+ N−1  2 ,  j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  In this paper, we let g be a predefined kernel, such as the SAR generalized ambiguity function (GAF) (see  Section 3 and [6, Chapter 2] for detail) and f be a smooth gradually varying function contaminated with white  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' To compute the convolution, we use the QTT decomposition [16] and the QTT-FFT algorithm [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  The QTT decomposition is a particular case of the more general TT decomposition (see Section 4 and [21]  for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  3  Synthetic aperture radar (SAR)  SAR is a coherent remote sensing technology capable of producing two-dimensional images of the Earth’s  surface from overhead platforms (airborne or spaceborne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' SAR illuminates the chosen area on the surface of  the Earth with microwaves (specially modulated pulses) and generates the image by digitally processing the  returns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', reflected signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' SAR processing involves the application of the matched filter and summation  along the synthetic array, which is a collection of successive locations of the SAR antenna along the flight  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Matched filtering yields the image in the direction normal to the platform flight trajectory or orbit  (called cross-track or range), while summation along the array yields the image in the direction parallel to  the trajectory or orbit (along-the-track or azimuth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Mathematically, each of the two signal processing stages can be interpreted as the convolution of the signal  received by the SAR antenna with a known function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Equivalently, it can be represented as a convolution of  the ground reflectivity function, which is the unknown quantity that SAR aims to reconstruct the imaging  kernel or generalized ambiguity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The advantage of this equivalent representation is that it leads to  a very convenient partition: the GAF depends on the imaging system’s characteristics, whereas the target’s  properties determine the ground reflectivity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Moreover, image representation via GAF allows one  to see clearly how signal compression (a property that pertains to SAR interrogating waveforms) enables  SAR resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', the capacity of the sensor to distinguish between closely located targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Denoising Convolution Algorithms  5  In the simplest possible imaging scenario, when the propagation of radar signals between the antenna  and the target is assumed unobstructed, and several additional assumptions also hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' the GAF in either range  or azimuthal direction is given by the sinc (or spherical Bessel) function:  g(x) = A sinc  �  π x  ∆x  �  ≡ A  sin  �  π x  ∆x  �  π x  ∆x  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1)  where the constant A is determined by normalization, x denotes a given direction, and the quantity ∆x is the  resolution in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' From the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1), we see that the resolution is defined as half-width of the  sinc main lobe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', the distance from is central maximum to the first zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' When x is the range direction  (cross-track), the resolution ∆x is inversely proportional to the SAR signal bandwidth, see [6, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  When x is the azimuthal direction (along-the-track), the resolution is inversely proportional to the length of  the synthetic array, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', synthetic aperture, see [6, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Note that lower values of ∆x correspond to  better resolution because SAR can tell between the targets located closer to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' It can also be shown  that as ∆x → 0 the GAF given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1) converges to the δ-function in the sense of distributions [7, Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In this case, the image, which is a convolution of the ground reflectivity with the GAF, coincides  with ground reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This would be ideal because the image would reconstruct the unknown ground  reflectivity exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This situation, however, is never realized in practice because having ∆x → 0 requires  either the SAR bandwidth (range direction) or synthetic aperture (azimuthal direction) to become infinitely  large, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  The literature on SAR imaging is vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Among the more mathematical sources, we mention the mono-  graphs [2], and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  4  Tensor Train Decomposition  Consider the K-mode, tensor A ∈ CM1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×MK such that  A = a(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK),  ik = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , Mk − 1,  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K,  where Mk is the size of each mode, and a(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK) ∈ C are the elements of the tensor A for all  ik = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , Mk − 1 and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The tensor train format of A decomposes the tensor into K cores  A(k) ∈ Crk−1×Mk×rk such that  a(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK) = A(1)  i1 A(2)  i2 · · · A(K)  iK ,  where the matrices A(k)(:, ik, :) = A(k)  ik  ∈ Crk−1×rk, for all ik = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , Mk − 1, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K (In  Matlab notation, A(k)  ik  = squeeze(A(k)(:, ik, :)), where squeez() is used to convert the Crk−1×1×rk tensor  into a Crk−1×rk matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The matrix dimensions rk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K, are referred to as the TT-ranks of  the tensor decomposition, and the 3−mode tensors A(k) are the TT-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Since we are interested in  the case when a(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK) ∈ C, we impose the condition r0 = rK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let M = max1≤k≤K Mk  and r = max1≤k≤K−1 rk, then the the tensor A, which has O(MK) elements, can be represented with  O(MKr2) elements in the TT format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  We can also represent the TT decomposition as the product of tensor contraction operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Define  the tensor contraction between the tensors A ∈ CM1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×MK and B ∈ CMK×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×M ˜  K (note that the first  dimension size of B equals the last dimension size of A) as C = A ◦ B ∈ CM1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×MK−1×MK+1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×M ˜  K  where  C(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK−1, iK+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , i ˜  K) =  MK−1  �  p=0  A(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , p)B(p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , i ˜  K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  Then the TT format of A can be represented as  A = A(1) ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' ◦ A(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Before we show how to find the TT-cores, we first need to define a few properties of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' First, let  the matrix A{k} be the k-th unfolding of the tensor A such that  A{k}(α, β) = a(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK),  α = i1 + i2M1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' + ikΠk−1  l=1 Ml,  β = ik+1 + ik+2Mk+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' + iKΠK−1  l=k+1Ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Thus, we have that A{k} ∈ CM1M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='Mk×Mk+1Mk+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='MK which we write as  A{k} = a(i1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' ik, ik+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' iK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  We denote the process of unfolding a tensor A into a matrix A{k} ∈ CM1M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='Mk×Mk+1Mk+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='MK as  A{k} = reshape(A, [M1M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Mk, Mk+1Mk+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' MK])  and folding a matrix into a tensor A ∈ CM1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×MK as  A = reshape(A{k}, [M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , MK]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  (Note this is to be consistent with the Matlab function reshape()).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  From [21] it can be shown that there exist a TT-decomposition of A such that  rk = rank(A{k}),  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Denote the Frobenius norm of a tensor A ∈ CM1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×MK as  ∥A∥F =  �  �  �  �  M1−1  �  i1=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  MK−1  �  iK=0  |a(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK)|2,  and the εk-rank of the matrix A{k} as  rankεk(A{k}) := min{rank(B) : ∥A{k} − B∥F ≤ εk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Given a set {εk}K  k=1, we can approximate the tensor A with a tensor ˜A in the TT format such that it has TT-  ranks ˜rk ≤ rankεk(A{k}) and  ∥A − ˜A∥F ≤ ε,  ε2 = ε2  1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' + ε2  K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  In Algorithm 1, we present the TT-SVD algorithm [21], which computes a TT-decomposition of a  tensor A with a prescribed accuracy ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In Section 5, we present some modifications to this algorithm that  relax the prescribed tolerance and allow us to compute an approximate decomposition faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For a tensor  A ∈ CM1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×MK, define  |A| = number of elements in A = M1M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Denoising Convolution Algorithms  7  Algorithm 1: TT-SVD  input : A, ε  output : TT-Cores: A(1), A(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', A(K)  τ :=  ε  √M−1∥A∥F  r0 := 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  for k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=',K-1 do  A{k} := reshape(A, [Mkrk−1,  |A|  Mkrk−1 ])  Compute truncated SVD: UΣV ∗ + E = A{k} such that ∥E∥F ≤ τ  rk := rank(Σ) = rankτ(A{k})  A(k) := reshape(U, [rk−1, Mk, rk])  A := ΣV ∗  end  A(K) := A  The TT-decomposition can also be applied to tensors with a small number of modes by using the  quantized tensor train decomposition (QTT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For instance, let v ∈ C2K be a vector (1-mode tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' To  apply the QTT-decomposition of v, we reshape it into the K-mode tensor V ∈ C2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×2 such that  V(i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , iK) = v(i),  where  i =  K  �  k=1  ik2k−1,  ik = 0, 1,  then compute the TT-decomposition of the tensor V (you can think of iK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' i1 as the binary representation  of i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Extending the QTT-decomposition to matrices (2-mode tensors) V ∈ C2K×2K can be done similarly  by reshaping them into 2K-mode tensors V ∈ C2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×2, then computing the TT-decomposition of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  We can approximate the discrete Fourier transform of a vector v ∈ R2K (or 2D discrete Fourier  transform of a matrix V ∈ R2K×2K) in the QTT format using what is known as the QTT-FFT approximation  algorithm [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let ˆv = DFT(v) be the discrete Fourier transform of v and let V and ˆV be the tensors  in the QTT-format that represent the vectors v and ˆv respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Given V, the QTT- FFT approximation  algorithm can approximate ˆV with a tensor ˜V such that  ∥ ˜V − ˆV∥F ≤ ε  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1)  for some given tolerance ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Similarly, we could prescribe some maximum TT-rank, ˆRmax, for the QTT-FFT  algorithm such that ˜rk ≤ ˆRmax for all TT-ranks of ˜V, {˜rk}K  k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The QTT-FFT algorithm can easily be  modified to the inverse Fourier transform of a vector (or matrix) in the QTT format, which we denote as the  QTT-iFFT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  5  Computing the convolution with QTT decomposition  In practice, we often need to compute the convolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1), where f is the function of interest and g is a  given kernel, but f is not given explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Instead, we are given noisy data  (fξ)j = f(xj) + ξj  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1)  at discrete points xj, j = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , jD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In particular, representing the ground reflectivity function for SAR  reconstruction in the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1) helps one model the noise in the received data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We assume that ξj is white  noise from a normal distribution with the standard deviation σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', ξj ∼ N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  Since the kernel function g is known, we can discretize it as  gj = g(xj),  for the same xj values as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We assume the D-dimensional spatial domain is uniformly discretized  into ND points where N = 2K−1 − 1, see (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' To compute the discrete convolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2), we propose using  the quantized tensor train (QTT) decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' To represent the arrays in the QTT format, we pad them  with zeros such that the new arrays are D-mode tensors in R2K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We can relax the condition on the  size N, but to compute the convolution with an FFT algorithm, we need to zero-pad each dimension with at  least N − 1 extra zeros (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Also, for the QTT decomposition, we need each dimension to be of  size 2K for some K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let Fξ, G be the zero-padded tensors representing fξ and g respectively in the  QTT format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Here, we assume that the discretization of f, f, has a low, but not exactly known, TT-rank in  the QTT-format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This is motivated by the fact that many standard piecewise smooth functions naturally have  a low TT-rank, see [9,16,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  To find approximations of these tensors in the TT-format, we modify the original TT-SVD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  This is because with the full TT-SVD algorithm, if the tolerance ε is small, see equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1), the TT-  decomposition has close to full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Not only does it take a very long time to compute these decompositions,  but most of the noise is still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, if ε is too large, the TT-SVD algorithm loses too much  information about the true function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For these reasons, we present slight modifications to the TT-SVD  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' They are needed to significantly reduce the computing time, as illustrated by the example in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  We consider three different modifications to the TT-SVD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' These modifications are as follows:  (1) Set some max rank Rmax and truncate the SVD in Algorithm 1 with ranks less than or equal to this  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Denote this method as the max rank TT-SVD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  (2) Set some max rank Rmax and replace the SVD in Algorithm 1 with a randomized SVD (RSVD) given  in [11] with max ranks set to Rmax (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Denote this method as the max rank TT-RSVD  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Note that for this algorithm, we also need to prescribe an oversampling parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We  could choose from several randomized SVD algorithms, but due to simplicity and effectiveness, we  use the approach described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This algorithm implements the direct SVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  (3) Truncate the SVD in Algorithm 1 based on when there is a relative drop in singular values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', if  σk+1  σk  < δ (0 < δ < 1) for a given threshold δ, then truncate the singular values less than σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Denote  this method as the SV drop off TT-SVD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  For the max rank TT-RSVD, if the unfolding matrices A{k} ∈ Rmk×nk, where min(mk, nk) ≤ Rmax + p,  then we revert to the max rank TT-SVD algorithm (without the randomized SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  We can modify the QTT-FFT and QTT-iFFT algorithms similarly to our modifications of the TT-SVD  algorithms to get a low-rank approximation to the discrete Fourier transform representations of Fξ and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  For this, we replace the SVD in the QTT-FFT algorithm (QTT-iFFT) with the truncated SVD algorithms  (1)-(3) given above, but with possibly a different max rank which we denote ˆRmax for (1) and (2), or different  threshold ˆδ for (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For the examples in Section 7, we distinguish between Rmax and ˆRmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, we use  the same threshold for δ in the TT-SVD algorithm and the QTT-FFT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, we do not distinguish  between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Note that using the threshold (1) in the QTT-FFT algorithm is not new and is mentioned  in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  With these above modifications to the TT-SVD algorithm and QTT-FFT (QTT-iFFT) algorithms, we  propose the following algorithm (Algorithm 2) to approximate the convolution between the D-dimensional  arrays f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For this algorithm, we denote  Denoising Convolution Algorithms  9  QFFT ˆRmax(ˆδ): QTT-FFT algorithm with a max rank of ˆRmax (or threshold ˆδ),  QiFFT ˆRmax(ˆδ): QTT-iFFT algorithm with a max rank of ˆRmax (or threshold ˆδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Algorithm 2: QTT convolution  input : fξ, g  output : I  Step 1: F ξ = reshape(fξ, [2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , 2]), G = reshape(g, [2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , 2])  Step 2: Decompose F ξ and G into the QTT format using one of the modified TT-SVD algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Step 3: I = QiFFT ˆ  Rmax(ˆδ)(QFFT ˆ  Rmax(ˆδ)(F ξ) ⊙ QFFT ˆ  Rmax(ˆδ)(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Step 4: Retrieve I from I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' (see Algorithm 3)  In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2, we show the asymptotic run time behavior of computing a convolution in one spatial  dimension (D = 1) with the max rank TT-SVD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' First, we prove an auxiliary result about the size  of the unfolding matrices for this algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2, we consider the whole process  of converting the vector into the QTT-format, computing the convolution, then converting the convolution  in the QTT format back into a vector, as is demonstrated in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For the last step, to convert a  tensor in the TT-format back into the standard format, we use the ’full’ algorithm from the Matlab toolbox  oseledets/TT-Toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This is given in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We then reshape this tensor into a vector with a bit of  run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Algorithm 3: Full  input : A(1), A(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', A(K), and size of output tensor [M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , Mk]  output : A ∈ CM1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×Mk  Let A = A(1)  for k=2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=',K do  A = reshape(A, [ (|A|)  rk−1 , rk−1])  B = reshape(A(k), [rk−1, 2rk])  A = AB  end  A = reshape(A, [M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , Mk])  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let A ∈ R2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='×2 be a K-mode tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let {A{k}}K−1  k=1 be the unfolding matrices of A in the  max rank TT-SVD algorithm with a max rank of Rmax and with each A{k} ∈ Cmk×nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Then  mk = 2rk−1 ≤ 2Rmax  and  nk = 2K−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Since Mk = 2 for all k, the proof for mk = 2rk−1 ≤ 2Rmax is trivial by the first line inside the for  loop in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For nk, we do a proof by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' First, note that |A{1}| = 2K and r0 = 1, thus  n1 = |A{1}|  2r0  = 2K  2 = 2K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Assume nℓ = 2K−ℓ for all 1 ≤ ℓ ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Then,  nk = |A{k}|  2rk−1  = |Σk−1V ∗  k−1|  2rk−1  = rk−1nk−1  2rk−1  = nk−1  2  = 2K−(k−1)  2  = 2K−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Thus, we get  mk = 2rk−1 ≤ 2Rmax  and  nk = 2K−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let fξ, g ∈ R2K−1−1 for some positive integer K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Then the computational complexity,  CQTT-conv, of approximating the convolution fξ ∗ g with the max rank TT-SVD and max rank QTT-SVD  algorithms described above is  CQTT-conv ≤ O(R2  max2K),  where Rmax is the prescribed max rank for both the TT-SVD algorithms and the QTT-FFT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We show that the computational complexity is dominated asymptotically by the max rank TT-SVD  algorithms and the full tensor algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' First, let Csvd be the computational cost of the SVD in big O  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Then, for a matrix A ∈ Cm×n, Csvd(A) = O(mn min(m, n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Note that, in Algorithm 1 (as  well as in our max rank modifications), the computational complexity is dominated by the SVD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Denote the unfolding matrices at the kth iterations as A{k} ∈ Cmk×nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Hence, the computational cost of  the max rank TT-SVD algorithm is  K−1  �  k=1  Csvd(A{k}) =  K−1  �  k=1  O(mknk min(mk, nk))  ≤  K−1  �  k=1  O((2Rmax)22K−k)  = 4R2  max  K−1  �  k=1  O(2k)  = 4R2  maxO(2K − 2)  = O(R2  max2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  From [3], we have that for the QTT-FFT and QTT-iFFT algorithms, the computational complexity is  O(K2R3  max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In Algorithm 3, the computational complexity comes from the multiplication AB in every  loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For the kth loop, A ∈ C2k−1×rk−1 and B ∈ Rrk−1×2rk for k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K, thus the computational  complexity is proportional to the cost of multiplying A by B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=',  Cfull =  K  �  k=2  O(2k−1rk−12rk)  ≤ R2  max  K  �  k=2  O(2k)  = R2  maxO(2K+1 − 4)  = O(R2  max2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Hence, the total computational complexity is  O(R2  max2K) + O(K2R3  max) + O(R2  max2K) = O(R2  max2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  For the randomized SVD, we have the computational complexity Crsvd(A{k}) = O(mknk(Rmax + p)) =  O(2K−kRmax(Rmax + p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, the run time for the convolution with a max rank TT-RSVD is similar  when p is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In D spatial dimensions, we can obtain a similar result but by replacing K with DK in the  max rank TT-SVD algorithm and the full tensor algorithm, and the QTT-FFT algorithm is O(DK2R3  max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Hence, the total run time complexity in D spatial dimensions is O(R2  max2DK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Denoising Convolution Algorithms  11  6  Denoising  It is well known that the SVD can remove noise from matrix data, as seen in [4,14], but little research has  been done in denoising with tensor decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In [17] and [19], the Tucker decomposition was used to  help remove noise from point cloud data and electron holograms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In [8], it was shown that the  TT-decomposition might have some advantages to denoising as opposed to the Tucker decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This  is because a low-rank Tucker matrix guarantees a low TT-rank for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, the converse statement  is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Let F be the low TT-rank tensor representing f in the QTT format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Then for some core tensors  F(k) ∈ Rrk−1×2×rk with tensor slices F(k)(:, ik, :) = f (k)  ik  ∈ Rrk−1×rk, ik = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Each element of F can  be represented in the TT format as  F(i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , ik) = f (1)  i1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' f (K)  iK ,  ik = 0, 1,  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K,  where each f (k)  ik  is a low rank matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In practice, it is unlikely the data collected has a low-rank TT  decomposition since almost all real radar data has noise due to hardware limitations or other signals interfering  with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Instead, we have the noisy data fξ whose tensor representation is  Fξ = F + ξ,  where ξ is the realization of the random noise in the TT format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The tensor Fξ almost surely has full TT-rank  when represented exactly in the QTT format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Ideally, we would like to be able to find an approximate TT  decomposition ˜F with TT-cores ˜F  (k), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K, using the noisy data such that ˜F  (k) ≈ F(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However,  it is hard to guarantee any bound on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We argue, though, that by using our proposed methods when given  the noisy data Fξ, we can find a TT decomposition ˜F with low rank such that ˜F ≈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Consider the first iteration of the for loop of algorithm 1, with A = A0 + Aξ as the sum of a smooth  tensor (A0) and a noisy tensor (Aξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Then, after it is reshaped, we obtain the matrix  A{1} = A{1}  0  + A{1}  ξ  ,  where A{1}  0  is a low rank matrix and A{1}  ξ  is added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let A{1} = UΣV ∗ + E be the truncated  SVD of A{1} and A{1}  0  = U0Σ0V ∗  0 be the SVD of A{1}  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Note that UΣV ∗ ≈ A{1}  0  does not imply that  U ≈ U0, and thus the TT-core A(1) is not guaranteed to be approximately equal to A(1)  0 , where A(1)  0  is the  first TT-core of A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, if we let A2 = A on the second iteration of the loop in Algorithm 1 (and  similarly for A0), we do get that the elements of the tensor contraction A(1) ◦ A2 ≈ A(1)  0  A2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Similarly, if  we can approximate the noise-free component on every iteration of the for loop, we obtain an approximation  for the tensor A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' While we do not have a theoretical bound on this error, our experiments in Section 7  show that this method works well at removing the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Since our method computes multiple SVDs, it can  reduce a lot more noise than if we just did a single SVD and can do so without excessive smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  7  Numerical simulations  This section presents some examples in one and two spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The original code for the TT-  decompositions and the QTT-FFT algorithms comes from the Matlab toolbox oseledets/TT-Toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We  have modified it accordingly for the max rank TT-SVD, max rank TT-RSVD, and SV drop off TT-SVD  algorithm, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For all our examples, we compare the run time and errors of computing  the convolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1) using several methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The error for every example is the l2 relative error  E2(I) = ∥I − Iref∥2  ∥Iref∥2  ,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1)  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  where in D spatial dimensions  ∥I∥2 =  �  �  �  � 1  ND  N−1  �  j=0  |Ij|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  The reference solution, Iref, is the discrete convolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) computed without any noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In all of the  examples, we compare our methods against computing the convolution with the randomized TT-SVD  algorithm from [13], as well as computing the true noisy convolution with FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In two space dimensions,  we also approximate the convolution using a low matrix rank approximation to the noisy data fξ, where the  truncated rank is determined by the actual matrix rank of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  For all of these examples, we use the normalized sinc imaging kernel that corresponds to the GAF (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1)  truncated to a sufficiently large interval [−L, L]:  g(x) =  sinc(π x  ∆x )  � L  −L sinc(π x  ∆x ) dx  ,  x ∈ [−L, L]  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2)  for D = 1, and  g(x, y) =  sinc(π x  ∆x )sinc(π y  ∆y )  �� L  −L sinc(π x  ∆x )sinc(π y  ∆y ) dxdy  ,  (x, y) ∈ [−L, L] × [−L, L]  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3)  for D = 2, where the resolution ∆x in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) and ∆x = ∆y in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3) is a given parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The one- dimensional  kernel (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) for ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='04π is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  In Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1, we present the relative error for each example for K = 20 when D = 1, and K = 10 when  D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In this table, the convolution fξ ∗g is denoted by Iξ and computed using the FFT algorithm, the QTT-  convolution computed with the max rank TT-SVD algorithm is denoted by IQTT0, the QTT-convolution  computed with the max rank TT-RSVD is denoted by IQTTr, and the convolution computed using the SV  drop off TT-SVD algorithm is denoted by Iδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In turn, the convolutions computed using the randomized  TT-decomposition are denoted by IRTT , and in two dimensions, the convolution computed using low-rank  approximations of f is denoted by Ilr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For Iδ and Ilr, we also denote what parameter δ and truncation  matrix rank R are used, respectively, for each example using a subscript of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1: Kernel function (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2) with ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='04π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  For each example, we show the TT-ranks of the original function without noise, f, in the QTT format  given by the tensor F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This QTT approximation is computed with Algorithm 1 with the tolerance ε = 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  g(x)  8  6  4  2  0  2  10  5  0  5  10  XDenoising Convolution Algorithms  13  We compute these TT-ranks for K = 20 when D = 1 and K = 10 when D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, it is worth noting  that these TT-ranks do not change much for any data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Notice that the max TT-ranks we choose for our  algorithms are less than the TT-ranks of f from Algorithm 1, yet still provide a reasonable estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1: l2-norm relative error for K = 20 for examples 1 and 2, and K = 10 for example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  example  Iξ  IQTT0  IQTTr  Iδ  IRTT  Ilr  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0383  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0280δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0430 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0131  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0068δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0201 3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1142  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0151  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0447  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1650δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1534  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0470rank=23  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1  Example 1  For this example, let  f(x) = e−( 3x  10 )2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4 sin(8πx) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='7 cos(6πx)),  x ∈ [−10, 10],  and  (fξ)j = f(xj) + ξj,  xj = −10 + ∆x  2 + j∆x,  j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , N − 1,  ∆x = 20  N ,  N = 2K−1 − 1,  with ξj ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We also set the resolution ∆x = 4∆x in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2), where ∆x is the size of the spatial  discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, the width of the main lobe of the sinc is 8∆x on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  As we can see in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2, the FFT-QTT algorithm removed much of the noise in the data compared  to the true convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For K = 20, we also tried computing the convolution using the original TT-SVD  algorithm given in Algorithm 1 with multiple values of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The smallest error, as defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1), occurred  when ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='01 and gave the relative error of E2(I) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='03202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This is close to the error of the true  convolution of the noisy data and took over 100 seconds to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, as we can see in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2,  the run times for all of our methods on the same grid took less than a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This indicates that the original  TT-SVD algorithm is practically unsuitable for removing data noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  The max TT-rank of the discretization of f(x) in the QTT format, F, is 17, yet we were able to achieve  our approximation using a max rank of Rmax = 10 for the max rank TT-SVD and max rank TT-RSVD  algorithms and ˆRmax = 15 for the QTT-FFT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, even if we do not know the exact TT-rank,  we can still compute a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2 shows run times for different grid sizes for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We can see that computing the  convolution with FFT is faster than our methods for these values of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' However, the convolution with our  QTT methods gets closer to the FFT run time as K increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This is shown in the last column of Table  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2 where we see the ratio of the max rank TT-SVD convolution method to the FFT convolution method  is getting smaller as K grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This helps verify our theoretical result that for some constant max rank  Rmax (and ˆRmax), the max rank TT-SVD convolution method is asymptotically faster than computing the  convolution with FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The amount of data needed for our method to outperform the FFT method may be  impractical for most real-world applications in 1-2 spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3 shows the number of elements to represent the data fξ fully versus how many elements are  required to store the data in the QTT-format with a prescribed max rank of Rmax = 10, Fξ, in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  As we can see, storing all the elements takes a lot of data and grows exponentially in K, while storing the  elements in the QTT format takes a lot less data and only grows linearly in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' These values for the QTT-data  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2: Top Left: True convolution of data without noise, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Top Right: Function data with noise, fξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Middle Left: True convolution of data with noise, Iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Middle Right: Convolution using the max rank  TT-SVD algorithm, IQTT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Bottom Left: Absolute error of Iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Bottom Right: Absolute error of IQTT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  storage can be found by looking at the size of the core tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For the tensor Fξ in the QTT format and  Iref  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  10  5  0  5  10  Xf(α) +s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  10  5  0  5  10  XI:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  10  5  0  5  10  XIQTT  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  10  5  0  5  10  X[I-Irefl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='01  0  10  5  0  5  10  XIQTT -Irefl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='01  0  10  5  0  5  10  XDenoising Convolution Algorithms  15  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2: Run time (seconds): Example 1 convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  K  Iξ  IQTT0  IQTTr  Iδ  IRTT  IQTT0/Iξ  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='325  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='369  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='485δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='414  65  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='067  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='653  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='694  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='479δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='609  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='7463  24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='41  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='86  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='10δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='80  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='6446  26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='77  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='75  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='68  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='93δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='97  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0763  28  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='6  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='7  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='0δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4339  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3: Data storage for Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  K  fξ  Fξ  16  65,536  2088  20  1,048,576  2888  24  16,777,216  3688  26  67,108,864  4088  28  268,435,456  4488  with a max TT-rank of Rmax = 10, we have the TT-cores  F(1)  ξ , F(K)  ξ  ∈ R1×2×2,  F(2)  ξ , F(K−1)  ξ  ∈ R2×2×4,  F(3)  ξ , F(K−2)  ξ  ∈ R4×2×8,  F(4)  ξ , F(K−3)  ξ  ∈ R8×2×10,  F(k)  ξ  ∈ R10×2×10,  k = 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Thus, the number of elements, Nel, that make up this QTT tensors is:  Nel = 2(1 × 2 × 2) + 2(2 × 2 × 4) + 2(4 × 2 × 8) + 2(8 × 2 × 10) + (K − 8)(10 × 2 × 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  The max rank TT-RSVD algorithm is not able to produce results as good as the max rank TT-SVD  (see Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1 for relative error comparison and Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2 for a run time comparison) but is still able to  produce a reasonably low error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' While the run time for the max rank TT-SVD is faster than the max rank  TT-RSVD for all of our methods, the max rank TT-RSVD can be faster for tensors with larger mode sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  This is due to the SVD in max rank TT-SVD algorithm with mode sizes, Mk, may be computed on a matrix  with mk = MkRmax rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In contrast, for the max rank TT-RSVD algorithm, the SVD is computed on a  matrix with mk = Rmax + p rows when Mk = 2 (such as for the QTT decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The difference in  the sizes of mk does not make up for the extra amount of work the RSVD algorithm does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Although this  paper focuses on the QTT-decomposition and thus Mk = 2, we believe this is important to note as the max  rank TT-RSVD algorithm can speed up the TT-decomposition for higher mode tensor data and still produce  accurate approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We verify this by computing the max rank TT-SVD algorithm and the max rank  TT-RSVD algorithm on a tensor with K = 8 modes with each mode of size Mk = 10, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Each  element of this tensor is taken from the uniform distribution U[0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The max rank TT-SVD algorithm took  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='57 seconds, and the max rank TT-RSVD algorithm only took 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='12 seconds, almost half the time of the  max rank TT-SVD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2  Example 2  If we were to choose a coarser resolution for the example of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', a wider sinc function), we could  reduce the noise using the standard convolution at the cost of smoothing out the solution’s peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Doing this  gives similar results for the true convolution and with our methods (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In this section, we show an  example where the ground reflectivity is very oscillatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Here, the resolution ∆x determined by the GAF  must be small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', the sinc function must be “skinny”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Otherwise, if the sinc window is close to or larger  than the characteristic scale of variation of the ground reflectivity, then the convolution can smooth out the  actual oscillations instead of just the noise, losing most of the information in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  We choose the ground reflectivity as  f(x) = e−(3x)2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='9 sin( 2xπ  5∆x) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4 cos( xπ  3∆x)),  x ∈ [−1, 1],  and  (fξ)j = f(xj) + ξj,  xj = −1 + ∆x  2 + j∆x,  j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , N − 1,  ∆x = 2  N ,  N = 2K−1 − 1,  with ξj ∼ N(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We use ∆x = 2∆x and the max TT-rank of the discretization of the smooth function  f(x) in the QTT-format is 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Again, we use the max ranks of Rmax = 10 and ˆRmax = 15 for the max rank TT-SVD and QTT-SVD  algorithms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' As the function is too oscillatory to see a lot of helpful information in the full graph  (see top left of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3), we show a zoomed-in plot of the graph of Iξ, IQTT0, and Iref (see top right of  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' While there is some error, the QTT-FFT convolution agrees with the true convolution Iref very  well, whereas Iξ has a more considerable noticeable difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This is verified by the graphs of the absolute  error given in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3, where the bottom left shows the error for Iξ, and the bottom right shows the error  for IQTT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3  Example 3  Let  f(x, y) = e−((2x)2+(2y)2)(sin(2πx) − cos(7πy) + cos(4πxy) − sin(3πxy),  (x, y) ∈ [−1, 1] × [−1, 1]  and  (fξ)j,k = f(xj, yk) + ξj,k,  xj = −1 + ∆x  2 + j∆x,  yk = −1 + ∆y  2 + k∆y,  j, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' , N − 1,  ∆x = ∆y = 2  N ,  N = 2K−1 − 1,  with ξj,k ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We have ∆x = ∆y = 2∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, the diameter of the main lobe of the sinc is 4∆x  on the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Here, we show a 2D example whose discretization of a smooth function f has a matrix  rank of 23 and a TT-rank of 26 when represented in the QTT format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We still use the ranks Rmax = 10 and  ˆRmax = 15 for our max rank TT-SVD (max rank TT-RSVD) and max rank QTT-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, our TT-ranks  are much smaller than the true TT-ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4 and Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1, notice that our method can still capture  the shape of the original function with an error that is an order of magnitude smaller than the error from the  Denoising Convolution Algorithms  17  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='3: Top Left: True convolution of data without noise, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Top Right: Zoomed in graph of Iξ, IQTT0,  and Iref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Bottom Left: Absolute error of Iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Bottom Right: Absolute error of IQTT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  true convolution using FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The plots on the bottom of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4 are a side view of the error graphs, as it  is easier to compare the errors in this view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The 2D examples are similar to the previous test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Thus, it  is reasonable to assume our method works about the same regardless of the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  In Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4, we compare the run times for the different methods of computing the convolution in two  spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We get similar results as the one-dimensional case, where the fastest run time is from  the convolution with FFT, but with the max rank TT-SVD and max rank TT-RSVD methods approaching  its run time asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' In 2D, when there is the same amount of data as in the 1D case (for example,  214×14 in 2D compared to 228 in 1D), the 2D examples do not run as fast as the 1D example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This is due to  the extra work in the 2D QTT-FFT algorithm from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Again we compare the amount of data stored in the full format versus in the QTT-format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Note that the  spatial dimension of the original function does not matter in how much storage it takes, just the dimensionality  of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For example, it takes just as much data to store a vector in R220 as it does to store a matrix in  R210×210 in the QTT format with a max rank of Rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  [I -Irefl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='005  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  XIQTT -Irefl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='005  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+page_content='5344  X18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4: Top Left: True convolution of data without noise, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Top Right: Function data with noise, fξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Middle Left: True convolution of data with noise, Iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Middle Right: Convolution using the max rank  TT-SVD algorithm, IQTT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Bottom Left: Absolute error of Iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Bottom Right: Absolute error of IQTT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Iref  2  1  0  1  1  0  0  1  1f(α) +$  2  1  0  7  1  0  0  1  12  1  0  1  1  0  0  1  12  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1  1  0  0  1  1[Is -Irefl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+page_content='5  1IIQTT。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' - Irefl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+page_content='5  1Denoising Convolution Algorithms  19  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4: Run times (seconds): Example 3 convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  K  Iξ  IQTT0  IQTTr  Iδ  IRTT  Ilr  IQTT0/Iξ  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+page_content='27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+page_content='121  14  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='67  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='8  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='05  1087δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='04  164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5  286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='2rank=2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='519  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='5: Data storage for Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  K  fξ  Fξ  8  65,536  2088  10  1,048,576  2888  12  16,777,216  3688  14  268,435,456  4488  8  Conclusions  In this paper, we have shown that the QTT decomposition, along with the QTT-FFT algorithm, can effectively  remove noise from signals with full TT-ranks when the true signal is of low rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' As we have seen in the  numerical examples, we could drastically remove the amount of noise from the signal compared to if we  took the convolution in the traditional way of using the FFT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This comes at the cost of run time,  but our methods still run at a reasonable speed which got closer to the FFT run time as the dimensionality  of the data increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This is demonstrated by three different examples, two in one spatial dimension and  one in two spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' We are even able to show that our method works on very oscillatory data  where it is required to have a sinc kernel with a narrow main lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Using approximate TT-ranks smaller than  the TT-ranks of the actual signal data, we are able to recover most of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' This indicates that as long  as the signal is reasonably smooth, the QTT decomposition can effectively be used for noise reduction for  high-dimensional data, even if the true TT-rank is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  From our three new approaches, the max rank TT-SVD convolution algorithm works much better than  the max rank TT-RSVD and the SV drop off TT-SVD convolution algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' As was stated before, the  max rank TT-RSVD can outperform the max rank TT-SVD algorithm when there are larger mode sizes  that are used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' For this reason, we present this algorithm, as we have not seen it in the literature  elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The SV drop off TT-SVD convolution algorithms do not produce as accurate of a method as the  max rank TT-SVD or the max rank TT-RSVD algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' however, in some cases, it does run faster, and  this method may give a higher degree of confidence that the truncated singular values are of little importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Unfortunately, this method can also lead to long run times, as is seen in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='4 when K = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  A  Randomized SVD  Here, we give a brief overview of the randomized SVD (RSVD) decomposition from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' To compute the  RSVD of the matrix A ∈ Rm×n, the first step is to find Q ∈ Rm×(k+p) such that  A ≈ QQ∗A  where Q has orthonormal columns and whose columns are approximations for the range of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Here, k is  the number of singular values that we want in our approximation to be close to the singular values of A, and  20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Leonard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tsunkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Utyuzhnikov  p is what is known as an oversampling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' To find Q, we use the following Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Algorithm 4: Solving the Fixed-Rank Problem  input : A, k, p  output : Q  Draw random matrix Ω ∈ Rn×(k+p) such that Ωi,j ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Let Y = AΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Compute QR factorization QR = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Once we have obtained Q, we can compute the low-rank RSVD using Algorithm 5 (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1  in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Algorithm 5: RSVD  input : A, Q, k  output : UΣV ∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let B = Q∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Compute SVD: ˜UΣV ∗ = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Let U = Q ˜U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  With these algorithms, we obtain an approximation ˜  A to A such that  ∥A − ˜  A∥ ≤ (1 + 11  �  k + p  �  min(m, n))σk+1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1)  with probability 1 − 6p−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' If we truncate the SVD to only the leading k singular values in Algorithm 5, then  the error on the left-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='1) only increases by at most σk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' The computational complexity for  each step of this algorithm is given as  O(mn(k + p))  O((k + p)2n)  O((k + p)2m),  Thus, for k + p < min(m, n), the overall algorithm requires O(mn(k + p)) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  Acknowledgments  The work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Chertock was supported in part by NSF grants DMS-1818684 and DMS-2208438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+page_content=' Leonard was supported in part by NSF grant DMS-1818684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+page_content=' Tensor-train decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' SIAM Journal on Scientific Computing, 33(5):2295–2317,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  [22] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Oseledets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Constructive representation of functions in low-rank tensor formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=',  37(1):1–18, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  [23] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Oseledets and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tyrtyshnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Breaking the curse of dimensionality, or how to use SVD in  many dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', 31(5):3744–3759, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  [24] Ivan Oseledets and Eugene Tyrtyshnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' TT-cross approximation for multidimensional arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Linear  Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', 432(1):70–88, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Rakhuba and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Oseledets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Fast multidimensional convolution in low-rank tensor formats via  cross approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=', 37(2):A565–A582, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  [26] Tianyi Shi, Maximilian Ruth, and Alex Townsend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Parallel algorithms for computing the tensor-train  decomposition, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='  [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Tucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Some mathematical notes on three-mode factor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
+page_content=' Psychometrika, 31:279–311,  1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FQT4oBgHgl3EQffTZu/content/2301.13339v1.pdf'}
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+SciPost Physics
+Submission
+Composite-boson formalism applied to strongly bound
+fermion pairs in a one-dimensional trap
+Mart´ın D. Jim´enez1, Eloisa Cuestas1, 2, Ana P. Majtey1 and Cecilia Cormick1⋆
+1 Instituto de F´ısica Enrique Gaviola, CONICET and Universidad Nacional de C´ordoba,
+Ciudad Universitaria, X5016LAE, C´ordoba, Argentina
+2 Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate
+University, Onna, Okinawa 904-0495, Japan
+⋆ cecilia.cormick@unc.edu.ar
+January 11, 2023
+Abstract
+We analyze a system of fermions in a one dimensional harmonic trap with at-
+tractive delta-interactions between different fermions species, as an approximate
+description of experiments involving atomic dimers. We solve the problem of two
+fermion pairs numerically using the so-called “coboson formalism” as an alterna-
+tive to techniques which are based on the single-particle basis. This allows us to
+explore the strongly bound regime, approaching the limit of infinite attraction
+in which the composite particles behave as hard-core bosons. Our procedure is
+computationally inexpensive and illustrates how the coboson toolbox is useful for
+ultracold atom systems even in absence of condensation.
+Contents
+1
+Introduction
+2
+2
+The procedure, step by step
+3
+2.1
+Single-pair solution
+3
+2.2
+Basis for two pairs
+5
+2.3
+Construction of the Hamiltonian
+6
+3
+Analytical considerations for infinite attraction
+7
+4
+Numerical study of the ground state for strong attraction
+8
+5
+Summary and conclusions
+12
+A Calculation of Hamiltonian and overlap matrix in position basis
+14
+B Spatial correlations for two fermions of equal kind
+15
+C Spatial correlations for fermions of different kinds
+17
+1
+arXiv:2301.03637v1  [cond-mat.quant-gas]  9 Jan 2023
+
+SciPost Physics
+Submission
+D Off-diagonal correlation parameter
+19
+E Correlations in momentum space
+19
+References
+21
+1
+Introduction
+The possibility to engineer atomic and molecular many-body systems by controlling and
+assembling simpler components has made enormous progress thanks to Feshbach resonances.
+In this way, molecular Bose-Einstein condensates have been formed starting from ultracold
+atomic gases [1, 2]. Similar setups have been used for the controlled observation of relevant
+phenomena in statistical physics such as Wigner crystals [3] and the BEC-BCS crossover [4,5].
+Within the field of ultracold Fermi gases, one-dimensional systems are known to exhibit very
+peculiar properties [6]. In particular, strongly bound fermion pairs reach a limit in which they
+behave as hard-core bosons, which in turn are related to non-interacting fermion models [7].
+We consider a one-dimensional scenario, with fermions of two different kinds in a harmonic
+trap and an attractive contact interaction leading to fermion pairing. The first steps towards
+the exact solution of the one-dimensional Fermi gas with contact interactions in a ring are due
+to Gaudin and Yang in 1967 [8,9]. For the trapped case most of the analytical work focuses
+on the strongly repulsive case, see [10] and references therein. Numerical approaches for this
+system include multiconfigurational time-dependent Hartree method [11], quantum diffusion
+Montecarlo [12], density matrix renormalization group [13] and a variety of quantum-chemical
+treatments such as coupled-cluster methods [14], among others. The vast body of literature
+in this field has been reviewed for instance in [6,15].
+Even though much effort has been devoted to this system, the usual numerical treatment
+takes as a basis the harmonic oscillator eigenstates, making computations very costly for strong
+attraction [14, 16–20]. Alternative procedures which are more efficient for strong attraction
+have been proposed in [21,22]. Here, as a different approach, we tackle the problem of two pairs
+with two fermions each in the context of coboson theory [23,24]. This theoretical framework,
+originally developed for excitons in semiconductors [23–25], has by now been applied to a
+variety of systems, including Bose-Einstein condensates [26], superconductors [27, 28] and
+Feshbach molecules [29].
+A very useful simplification often encountered in this treatment is the so-called coboson
+ansatz, which is analogous to a condensate formed by composite bosons and is the canonical-
+ensemble counterpart of the BCS ansatz [23,28]. Using tools from the coboson formalism, we
+show that the coboson ansatz does not provide a good approximation of the true ground state
+for the case of two pairs in the limit of strong interaction. This is to be expected in the light
+of previous results [30, 31] and also because the limit of infinitely bound pairs corresponds
+to hard-core bosons which are known to form only a quasi-condensate in 1D traps [32–34].
+However, the coboson formalism also provides tools to describe the state beyond the coboson
+ansatz [27,28]. We thus develop a representation of the problem in the coboson basis, i.e. in
+terms of the eigenstates of one pair of interacting fermions in the trap.
+2
+
+SciPost Physics
+Submission
+This basis is specially convenient and expected to work better for the regime of strong
+attraction, which is difficult to address numerically (see for instance Ref. [16]) and has been
+not studied exhaustively as the repulsive regime [6,15]. In this respect, our method is related
+with the perturbative approach in [35]. The case of two pairs is of particular relevance within
+the coboson formalism, however, the method we propose can be extended to larger systems.
+The motivation of our work can then be stated as i) to show that even if the coboson ansatz
+fails the correct ground state for this system can be recovered using the complete toolbox of
+the coboson formalism ii) to show that the two-body coboson basis is useful in the strongly
+attractive limit where the single-particle basis is not convenient.
+Besides the numerical convenience of using the coboson basis, studying this system within
+the coboson formalism leads to semi-analytical reliable results that can provide a safe ground
+to quantify the fractional statistics [36,37] of the one-dimensional Fermi gas [38–40]. This is
+a good starting point to analyze the relationship between anyonic statistics and the entangle-
+ment of the constituent particles of the composite boson, which has been pointed out to be
+the key to understand composite effects and ideal bosonic behavior [41–43].
+The basic steps of our procedure to tackle the problem of two trapped fermion pairs are
+the same as in [31] and are as follows:
+1. We solve the problem of a pair of interacting fermions in the trap. The operators B†
+n
+that create each single-pair eigenstate, and the corresponding energies En, will be the
+starting point of the treatment. We truncate the basis considering the states with the
+lowest energies, up to some quantum number nmax.
+2. From the single-pair basis operators B†
+n we form the two-coboson basis generated by the
+action on the vacuum of operators of the kind B†
+nB†
+m.
+3. We calculate the form of the Hamiltonian in this truncated coboson basis.
+4. Solving the corresponding generalized eigenvalue problem, we estimate the ground state
+for two pairs and analyze its properties.
+This method allows us to interpolate from the interaction strengths for which the single-
+particle basis is suitable [17–20], all the way to very strongly bound pairs approaching the
+limit of hard-core bosons. Using coboson-theory tools combined with Taylor expansions, we
+calculate several quantities of interest, including the energy and two-particle correlators.
+The work is presented as follows: in Sec. 2 we review how to write the problem in the
+coboson framework. Section 3 is devoted to analytical considerations for infinite attraction.
+In Sec. 4 we discuss our numerical results. A summary and conclusions are given in Sec. 5.
+Finally, several appendices with detailed calculations are included.
+2
+The procedure, step by step
+2.1
+Single-pair solution
+For definiteness we will assume that both fermion kinds, which we call a and b, have the same
+mass, and that the creation and annihilation operators corresponding to different fermion
+species commute (this last choice does not affect the final results). We also assume that the
+trapping potential is the same for both species.
+3
+
+SciPost Physics
+Submission
+The first step requires the solution of the single-pair problem, with a Hamiltonian given
+by:
+H1 =
+�
+α=a,b
+� p2
+α
+2m + mω2x2
+α
+2
+�
+− γ δ(xa − xb)
+(1)
+with γ > 0. This problem can be solved by separation of the center-of-mass and relative
+variables.
+The center-of-mass solution is given by the harmonic oscillator eigenfunctions
+corresponding to mass 2m. The relative motion has been solved in the general case in Refs.
+[44,45] but for simplicity we focus only on strongly bound pairs, so that the relative motion
+has a wavefunction of the form of an exponential,
+ψr(xr) ≃
+√
+λ e−λ|xr|,
+(2)
+and the energy associated with the relative motion can be approximated by:
+Eγ = −ℏ2λ2
+m
+,
+λ ≃ mγ
+2ℏ2 .
+(3)
+In this regime, the single-pair eigenfunctions are then approximately of the form:
+ψn(xa, xb) ≃ ϕn
+�xa + xb
+2
+� √
+λ e−λ|xa−xb|,
+(4)
+where ϕn are the harmonic oscillator eigenfunctions for a particle of mass 2m. The corre-
+sponding energies are:
+En = ℏω
+�
+n + 1
+2
+�
++ Eγ.
+(5)
+From these solutions, we define the coboson creation operators B†
+n such that:
+|˜n⟩ = B†
+n|v⟩,
+(6)
+where |˜n⟩ is the n-th single-pair eigenstate, and |v⟩ is the vacuum. In particular, the coboson
+operators B†
+n can be written in terms of field operators as:
+B†
+n ≃
+�
+dxadxb ψn(xa, xb)Ψ†
+a(xa)Ψ†
+b(xb).
+(7)
+For consistency, neglecting states where the internal motion is excited implies also a trun-
+cation in the center-of-mass states, so that the basis includes all single-pair eigenstates up to
+a certain energy cutoff. In particular, we keep only states where the index n associated with
+the center-of-mass motion is such that the excited internal states are well above the energy
+scales considered, i.e.:
+n ≪ |Eγ|
+ℏω = (λxω)2.
+(8)
+For convenience here we have defined a spatial scale xω associated with the harmonic oscillator,
+xω =
+�
+ℏ
+mω .
+(9)
+The inequality in Eq. (8) stresses once more the fact that our restricted basis is only appro-
+priate for strong attraction, when the size of each bound pair is very small compared with the
+spatial scale of the trap and thus λxω is large. It is also important to note that since Eq. (2)
+and therefore Eq. (4) are valid for λ xω ⪆ 5 all of our results rely on this condition [46].
+4
+
+SciPost Physics
+Submission
+2.2
+Basis for two pairs
+From the set of states corresponding to the lowest energies of the single-pair Hamiltonian,
+one can form states of the form:
+|˜n ˜m⟩ = B†
+nB†
+m|v⟩,
+(10)
+with n ≤ m (we note that the coboson creation operators commute) and |v⟩ the vacuum.
+Because of the fermionic character of the constituent particles, states generated in this form
+are neither normalized nor orthogonal [23]. We truncate this two-pair basis with the condition
+n + m ≤ nmax, and then approximate the ground state in the form:
+|GS⟩ =
+�
+m≤n
+cm,n|˜n ˜m⟩ .
+(11)
+An often useful approximation for the ground state of dilute systems of N pairs with
+short-range interactions is given by what we call the “coboson ansatz” [23]. This corresponds
+to the state obtained from the repeated application on the vacuum of the operator B0 that
+creates a single pair in its ground state:
+|N⟩ = (B†
+0)N
+√N!χN
+|v⟩,
+(12)
+where χN is a normalization constant. However, this can only provide a good approximation
+of the true ground state in systems which are expected to exhibit condensation at zero tem-
+perature. This is not the case in the problem we analyze [30,31,33,34]. In order to quantify
+the quality of the approximation, we study the fidelity F between the true ground state for
+two pairs, |GS⟩, and the coboson ansatz:
+F = |⟨GS|(B†
+0)2|v⟩|2
+⟨v|B2
+0(B†
+0)2|v⟩
+,
+(13)
+where the true ground state |GS⟩ is approximated numerically using the coboson basis given
+in Eq. (10) for two-pairs (N = 2).
+Even if the coboson ansatz is not a good approximation, one can still compute the ground
+state by means of the coboson formalism. In order to do this, we will work with the space
+generated by the coboson operators as in Eq. (10). First, we compute all overlaps between
+the relevant states from the expression:
+Skl,mn = ⟨v|BkBlB†
+mB†
+n|v⟩ = δmlδkn+δnlδkm−
+�
+⟨˜k|⊗⟨˜l|Xa| ˜m⟩⊗|˜n⟩+⟨˜k|⊗⟨˜l|Xb| ˜m⟩⊗|˜n⟩
+�
+. (14)
+Here Xα with α = a, b is an operator that exchanges the states of the two fermions of kind α,
+and it acts on a fictitious space where fermions of equal kind are treated as distinguishable.
+Since our goal is to find the ground state, instead of building an orthonormal basis, we keep
+the overlap matrix S to solve the corresponding generalized eigenvalue problem.
+The matrix S can be calculated following different strategies. In the coboson literature
+[23], the overlaps are evaluated in terms of matrix elements of the change of basis between
+single-pair eigenstates and the separable single-fermion basis. However, this procedure can
+be numerically costly and lead to large errors when many coefficients are non-negligible and
+5
+
+SciPost Physics
+Submission
+no analytical expression exists for the sums required. Thus, we resort to a different form of
+evaluation. Plugging the explicit form of the operators B†
+n given by Eq. (7) in all formulas,
+and using (anti)commutators, we can obtain an expression for the elements of the overlap
+matrix as:
+Smn,jk ≃
+�
+δmjδnk − λ2
+�
+dx dy1 dy2 dy3 dy4 δ(y1 + y2 − y3 − y4)
+ϕm(x) ϕn
+�
+x + y3 − y1 + y2
+2
+�
+ϕj
+�
+x + y3 − y1
+2
+�
+ϕk
+�
+x + y3 − y2
+2
+�
+e
+−λ �
+l
+|yl|
+�
++ same with j ↔ k.
+(15)
+Since we are interested in the case of strong attraction, the factors of the form e−λ|yl| allow
+us to perform a Taylor expansion in 1/(xωλ) for the harmonic oscillator functions ϕn. This
+is possible given the truncation of our basis in Eq. (8), which implies that the spatial scale
+associated with the center of mass is much longer than the pair size λ−1. In this form one can
+find approximate expressions for S from a lengthy but straightforward evaluation of spatial
+integrals. This procedure is explained in detail in Appendix A.
+2.3
+Construction of the Hamiltonian
+We now need to compute the Hamiltonian in the coboson basis. The Hamiltonian can be split
+in two parts, corresponding to the non-interacting terms and the interactions. The interaction
+part is quartic and can be written in terms of field operators as:
+Hint = −γ
+�
+dx Ψ†
+a(x)Ψ†
+b(x)Ψa(x)Ψb(x).
+(16)
+The Hamiltonian matrix elements in the coboson basis can be obtained from the expres-
+sion:
+⟨v|BkBlHB†
+mB†
+n|v⟩ = (En + Em)Skl,mn + ⟨v|BkBl
+�
+[Hint, B†
+m], B†
+n
+�
+|v⟩,
+(17)
+which is just a rewriting of the formulas in [23]. Notice that when using the coboson formal-
+ism the one-body term which contains the kinetic energy and trap potential is absorbed by
+quantities that were calculated when solving the single-pair case (first term on the right-hand-
+side in the above equation). In a similar spirit as for the calculation of the overlap matrix
+S, instead of following the standard expressions in [23] we estimate the Hamitonian elements
+using a Taylor expansion of spatial integrals.
+In particular, the last line of Eq. (17) can be written as:
+⟨v|BmBn
+�
+[Hint, B†
+j], B†
+k
+�
+|v⟩ ≃
+γλ2
+� �
+dxdydy′e−λ(|y|+|y′|+|y−y′|)ϕm(x)ϕn
+�
+x + y′ − y
+2
+�
+ϕj
+�
+x + y′ − y
+2
+�
+ϕk
+�
+x + y′
+2
+�
+−
+�
+dxdx′dydy′e−2λ(|y|+|y′|)ϕm(x)ϕn(x′)ϕj(x)ϕk(x′)δ
+�
+x − x′ + y + y′
+2
+� �
++ same with n ↔ m + same with j ↔ k + same with {j, k} ↔ {m, n}.
+(18)
+The details of the procedure involving the Taylor expansion of the Hamiltonian elements are
+also provided in Appendix A.
+6
+
+SciPost Physics
+Submission
+3
+Analytical considerations for infinite attraction
+Before presenting the results of our numerical approach, we note that the case of infinite
+attraction can be solved exactly. In this limit, fermions of different species are so strongly
+bound that they behave as point-like hard-core bosons of mass 2m, and the problem can be
+solved by means of fermionization [7]. According to this procedure, one must first consider
+the ground state of two identical non-interacting fermions of mass 2m in the trap. This state
+is given by:
+ψ2f(x1, x2) = 2mω
+ℏ√π e−mω(x2
+1+x2
+2)/ℏ(x1 − x2)
+(19)
+and corresponds to the antisymmetric combination of having one fermion in the trap ground
+state and another in the first excited state. Then, one obtains the wavefunction of the hard-
+core bosons as the symmetrized form of the previous expression, i.e.:
+ψhc(x1, x2) = 2mω
+ℏ√π e−mω(x2
+1+x2
+2)/ℏ|x1 − x2|,
+(20)
+where the subindex “hc” stands for “hard-core”.
+From these expressions we can calculate all properties of the ground state for λ → ∞. For
+instance, the asymptotic ground-state energy, excluding the binding energy Eγ of each pair,
+is found to be given by the sum of the two lowest energies of the harmonic oscillator. Thus,
+the total ground-state energy for very large λ is approximately 2Eγ + 2ℏω. We can define
+an effective interaction energy between pairs as ∆E = E2 − 2E1, where EN is the ground-
+state energy of N = 1, 2 pairs. Considering that a single pair has a ground-state energy of
+Eγ + ℏω/2, we then obtain an effective interaction energy which for very large attraction
+approaches ∆E = ℏω.
+Using the ground-state wavefunction as expressed above, one can also analytically calculate
+the fidelity between the true ground state and the coboson ansatz for infinite attraction. We
+find an asymptotic fidelity of F∞ = 2/π ≃ 0.64, which is lower than the one obtained in the
+same regime for two fermion pairs in translationally invariant models [30,31].
+Following the same lines, one can find the joint density of composite particles at positions
+x and x′ for the limit of infinite attraction. This is of the form:
+Dhc(x, x′) = 8
+π2
+(x − x′)2
+x4ω
+e−2(x′2+x2)/x2
+ω.
+(21)
+One can also write down the conditional probability P(x′|x) of finding a composite point-like
+particle at position x′ provided that another one was found at position x:
+Phc(x′|x) = 1
+xω
+�
+2
+π
+(x − x′)2
+x2 + x2ω/4 e−2(x′/xω)2.
+(22)
+Furthermore, one can calculate the asymptotic values of the coefficients in the expansion
+of the ground state in the coboson basis, Eqs. (10-11), obtaining for λ → ∞:
+c(∞)
+mn = −(2 − δmn) (−1)(m−n)/2
+√
+m!n!
+�
+1
+π
+(m + n)!
+(m/2 + n/2)!
+1
+2m+n
+1
+m + n − 1.
+(23)
+7
+
+SciPost Physics
+Submission
+This expression is valid for even and nonzero n+m, and here δmn is the Kronecker delta. For
+symmetry reasons the coefficients cmn vanish for odd n + m, and for n = m = 0 we find:
+c(∞)
+00
+=
+�
+1
+π.
+(24)
+Since the coboson ansatz corresponds to the repeated application of the coboson operator B†
+0,
+and for λ → ∞ the wavefunctions associated with the different B†
+m become orthogonal, the
+asymptotic value of c00 determines the asymptotic fidelity between the correct ground state
+and the coboson ansatz. The additional factor
+√
+2 in the fidelity comes from the definition of
+the coboson basis in Eq. (10), which does not include a prefactor 1/
+√
+2 for m = n.
+Before tackling the numerical treatment of the problem for strong but finite attraction,
+we note that also the limit of infinitesimal attraction can be treated analytically. For γ = 0,
+the ground state of the system is separable, with the two lowest oscillator levels occupied
+for both kinds of fermions. Then, the energy ∆E approaches 2ℏω. It is very important to
+notice that in this separable limit, the coboson normalization factor χ2 in Eq.(12) vanishes,
+and thus the coboson ansatz is not defined for γ = 0. Nevertheless, using perturbation theory
+together with analytical results for the Schmidt coefficients [46] one can calculate the limit
+value of the fidelity between the true ground state and the coboson ansatz, and find that as
+the attractive interaction strength approaches zero, F approaches a value of approximately
+0.37. Indeed, for γ ∼ 0 we obtain χ2 ∼ 0.342 θ2 and F ∼ θ2/8χ2 with θ ∼ γ/
+√
+2πℏωxω. We
+note, however, that the weakly bound case is not within the scope of our present study, and
+it has been extensively analyzed before [17–20].
+4
+Numerical study of the ground state for strong attraction
+In the following we perform a numerical study of the ground state according to the procedure
+outlined in Sec. 2. A delicate point in the calculation is the choice of the number of basis
+states. A very small number leads to a poor description of the system, whereas for a very
+large number it becomes unjustified to leave out the excited states of the relative motion, and
+it can also lead to numerical problems if the overlap matrix becomes worse conditioned. As
+a compromise, we choose the maximum center-of-mass energy included in our description to
+grow linearly with λ.
+In Fig. 1 (a) we show our results for the interaction energy ∆E = E2 −2E1 using a Taylor
+expansion for the calculation of both the overlap and the Hamiltonian matrices. We also plot
+in Fig. 1 (b) the fidelity F between the ansatz and the true ground state as a function of λ
+when choosing the energy in the truncated basis to be given by nmax = λxω. We note that our
+results show reasonable agreement with the known behaviour for infinite attraction. Notice
+that the difference between the numerical ∆E obtained for λxω ≃ 200 and the asymptotic
+value ℏω presented in Fig. 1 (a) is of about 4%, whereas the binding energy for this case is
+so large that ∆E is five orders of magnitude smaller than the total energy.
+As can be seen in the comparison provided in Fig. 2, for λxω = 30 the coefficients cm,n
+of the ground state in the form of Eq. (11) are already very close to the ones obtained from
+the hard-core boson limit given in Eqs. (23-24). This also hints at a procedure to perform
+approximate computations more efficiently: instead of taking the full basis as in Fig. 2, one
+can use a truncation inspired by the asymptotic values of the coefficients in Eqs. (23-24).
+8
+
+SciPost Physics
+Submission
+(a)
+(b)
+Figure 1: a) Energy for two pairs, excluding the trivial contribution equal to twice
+the single-pair energy, as a function of λ. b) Fidelity between the coboson ansatz
+and the numerically found ground state as a function of λ. The results are obtained
+from the lowest non-trivial order of the Taylor expansion (green stars) and the next
+non-zero higher-order corrections (black circles) as reported in Appendix A. The
+horizontal dashed red lines indicate the asymptotic values for λ → ∞.
+One can also directly approximate the state by taking the coboson basis in Eq. (10) to be a
+function of λ but the coefficients in this basis to be given by the asymptotic values, which
+gives a fast and compact approximation for the ground state. Indeed, the ground state found
+numerically for λxω = 30 has a fidelity of 0.993 with the state obtained taking the asymptotic
+values of the coefficients and truncating the basis in the same form.
+From the numerical solution of the problem one can characterize the ground state through
+several key properties. In particular, in Fig. 3 (a) we illustrate the spatial correlations between
+fermions of equal kind through the joint density distribution
+Daa(x, x′) = ⟨ψ|Ψ†
+a(x′)Ψ†
+a(x)Ψa(x)Ψa(x′)|ψ⟩,
+(25)
+evaluated for the case λxω = 30. The details of the calculation are provided in Appendix
+B. This plot displays clear signatures of Pauli exclusion as a sharp diagonal feature. Two
+identical fermions are most likely found apart from each other at a distance which is set by
+the spatial scale of the harmonic trap.
+For comparison, Fig. 3 (b) displays the joint density for fermions of different kinds:
+Dab(x, x′) = ⟨ψ|Ψ†
+b(x′)Ψ†
+a(x)Ψa(x)Ψb(x′)|ψ⟩.
+(26)
+This plot exhibits a strong diagonal correlation corresponding to particles that form a bound
+pair, with additional much broader peaks corresponding to particles belonging to different
+pairs. The calculation of Dab is explained in Appendix C.
+Another quantity that reflects the spatial correlations present in the ground state is the
+conditional probability Paa(x|0) to find one fermion of kind a at position x given that another
+identical fermion was found at the origin. This function is plotted in Fig. 4 (a), for the
+numerical solution with λxω = 30. For comparison we also show the conditional probability
+9
+
+SciPost Physics
+Submission
+Figure 2: Coefficients in the coboson decomposition from the numerical resolution of
+the problem based on a Taylor expansion for λxω = 30, in black circles. The index
+k here refers to a particular ordering of the m, n coefficients using a single label. For
+comparison we show the values according to the asymptotic expression in Eqs. (23-
+24) as red four-pointed stars, which overlap with the numerical resuls within the size
+of the symbols. The basis was truncated with nmax = λxω. The coefficients in the
+plot are normalized taking ctSc = 1. The vertical dashed light-gray lines delimitate
+sections of the basis containing states B†
+mB†
+n|v⟩ with a fixed value of m + n.
+Paa(x|0) obtained from the hard-core limit of λ → ∞ and from the coboson ansatz of Eq. (12)
+evaluated for λxω = 30. The corresponding formulas are given in Appendix B. The plots show
+qualitative agreement between the numerical results and the point-like hard-core boson limit,
+in sharp contrast with the coboson ansatz in its standard form.
+Indeed, the form of the
+conditional probability Paa(x|0) is similar to the probability distribution corresponding to
+the first excited state of the harmonic oscillator, the maxima of which are indicated with
+dotted vertical lines in the figure.
+In a similar manner one can compare the predictions for the spatial correlations of fermions
+of different kinds. To this aim, we consider the behaviour of the conditional particle density
+Dab(x|x′) indicating the density of fermions of kind a at position x conditioned on having
+found a fermion of kind b at position x′. We plot this quantity with x′ = 0 for the numerical
+solution corresponding to λxω = 30 in Fig. 4 (b), where we also plot the predictions of the
+point-like hard-core boson limit and the coboson ansatz for λxω = 30. The derivation of the
+corresponding formulas is shown in Appendix C. All three curves have a narrow peak around
+the origin, associated with the probability to find a fermion paired with the first one detected
+(in the limit λ → ∞ this peak is a delta function). The curves however differ strongly in the
+behaviour related with the probability to find the remaining particle of kind a. This second
+contribution to the conditional density has the same shape as Paa(x|0), and closely resembles
+the probability distribution for the first excited state of the harmonic oscillator of mass 2m,
+a behaviour which is not properly described by the standard coboson ansatz.
+Figures 3 and 4 were concerned with density distributions in space, associated with diago-
+nal terms of the system’s density matrix in space representation. Figure 5 a) shows in contrast
+10
+
+SciPost Physics
+Submission
+(a)
+(b)
+Figure 3: a) Joint density distribution Daa(x, x′) in units of x−2
+ω , for two fermions of
+kind a at positions x and x′ simultaneously. b) Joint density distribution Dab(x, x′),
+in units of x−2
+ω , for finding a fermion of kind a at position x and one of kind b at
+position x′ simultaneously. Both densities were obtained from the numerical solution
+for λxω = 30. Details of the calculations are given in Appendices B and C.
+an off-diagonal feature, namely the off-diagonal correlation function [17]:
+g2(x) =
+ρab(0, 0; x, x)
+�
+ρab(0, 0; 0, 0)ρab(x, x; x, x)
+,
+(27)
+where ρab is the reduced density matrix for two fermions of different kind. The quantity g2
+is an indicator of spatial two-particle coherence, and the coboson ansatz predicts a constant
+value g2(x) = 1 in the limit of infinite attraction. The numerical results (in black) show that
+this coherence decays within the typical scale set by the harmonic oscillator, but it stays high
+for all values with non-negligible particle densities. Nevertheless, the off-diagonal correlation
+we find is always smaller than the one corresponding to the hard-core limit, depicted in red
+for comparison. This is not due to a variation in the decay of the spatial coherence, as can be
+seen in Fig. 5 b). Rather, the difference between our numerical results and the limit λ → ∞
+is given by a different density profile, since the particle density at the origin is lower for finite
+λ than in the limit of infinite attraction.
+For the same numerically found ground state one can also characterize the properties in
+momentum space using similar techniques. In Fig. 6 we show the joint probability distribution
+for fermions of different kinds in momentum space. This plot displays a strong anti-diagonal
+peak which is the counterpart of the diagonal peak found for the joint probability distribution
+in position space, shown in Fig. 3 (b). The remaining features of the plot do not ressemble the
+state of two identical trapped fermions of mass 2m; this difference in the behaviour of position
+and momentum is typical of hard-core bosons [7,32,47]. The calculation of the joint density
+11
+
+2
+1
+0
+-2
+-2
+-1
+0
+1
+2
+0
+0.2
+0.4
+0.6
+0.82
+1
+m
+0
+1
+-2
+-2
+-1
+0
+1
+2
+0
+0.2
+0.4
+0.6
+0.8SciPost Physics
+Submission
+(a)
+(b)
+Figure 4: a) Conditional probability Paa(x|0) to find a fermion of kind a at position
+x when another fermion was already found at the origin. b) Conditional density
+Dab(x|0) indicating the density of fermions of kind a at position x conditioned on
+having found a fermion of kind b at the origin. In both plots the solid black curve is
+the numerical result with λxω = 30 and nmax = λxω. The dashed red curve is the
+analytical result for the probability obtained for the point-like hard-core boson limit,
+and the blue dash-dotted line is the probability predicted by the coboson ansatz in
+Eq. (12) for N = 2 and λxω = 30. Details of the calculations are given in Appendices
+B and C. The vertical light-gray lines indicate the positions ± xω/
+√
+2, which are the
+locations of the maxima of the conditional probability for λ → ∞.
+in momentum space is similar to the one of Dab(x, x′), but involves a Fourier transform of the
+coboson basis. The details are explained in Appendix E.
+5
+Summary and conclusions
+We have tackled the problem of two identical composite particles, each made of two distin-
+guishable fermions, inside a harmonic trap and with contact attractive interactions between
+fermions of different species. We explored the strongly bound regime using the coboson for-
+malism to build a compact basis of states, greatly reducing the computational requirements
+associated with the usual description in terms of single-particle eigenstates.
+We have studied the approach of the interaction energy to the limit of infinite attraction,
+corresponding to point-like hard-core bosons, and we have confirmed that the coboson ansatz
+in its standard form does not provide an accurate description of the ground state for any of the
+interaction strengths within our analysis. Since the energy of the coboson ansatz for N pairs
+can be approximated from the energy for one and two pairs [23] the coboson ansatz cannot
+provide a good estimation for the energy of a system made of N pairs. We have also shown
+that the point-like hard-core boson limit provides a good approximation of the coefficients
+when writing the ground state in the coboson basis. Furthermore, we have used the numerical
+results to characterize spatial correlations present in the ground state, both between fermions
+12
+
+SciPost Physics
+Submission
+(a)
+(b)
+(c)
+Figure 5: a) Off-diagonal correlation function g2(x), b) off-diagonal matrix elements
+and c) diagonal matrix elements of the reduced density matrix ρab. Black solid lines
+correspond to numerical results for λxω = 30, red dashed ones to point-like hard-core
+bosons and blue dash-dotted lines correspond to the prediction of the coboson ansatz
+for N = 2 and λxω = 30. Details are provided in the main text and in Appendix D.
+Notice that the vertical axis of subplot (a) does not begin at zero.
+of different and equal kinds, complementing previous work [17].
+The composite-boson procedure presented can be generalized to higher numbers of parti-
+cles and different forms of the trapping potential. Most importantly, we expect this approach
+to provide an additional tool to the ones usually applied for the description of experiments
+involving bosonic Feshbach molecules made of fermionic constituents in quasi one-dimensional
+settings.
+13
+
+SciPost Physics
+Submission
+Figure 6: Joint density distribution �Dab(k, k′), in units of x2
+ω, for finding a fermion
+of kind a with momentum k and one of kind b with momentum k′ simultaneously,
+obtained from the numerical solution for λxω = 30. Details are provided in the main
+text and in Appendix E.
+Acknowledgements
+We thank Thomas Busch for his careful reading of the manuscript and valuable comments.
+E. C. is grateful to Tran Duong Anh-Tai for his suggestions.
+Author contributions
+M. D. J. performed the numerical calculations with support by
+E. C. and A. P. M. All authors contributed to the derivation of the analytical formulas. C. C.
+coordinated the project and the writing of the draft.
+Funding information
+The authors acknowledge funding from grant PICT 2017-2583 from
+ANPCyT (Argentina).
+A
+Calculation of Hamiltonian and overlap matrix in position
+basis
+In the limit of very strong interaction, it makes sense to use that the wavefunctions for the
+center of mass vary over a scale which is much larger than the one for the relative motion.
+Thus, we start from Eq. (15) for the elements of the overlap matrix, use that all yj are of
+the order of λ−1, and perform a Taylor expansion in these small displacements. The lowest
+14
+
+40
+20
+0
+M
+-20
+-40
+-40
+-20
+0
+20
+40
+kaw
+5 10-4 10-3
+5 10-3 10-2SciPost Physics
+Submission
+orders give:
+Smn,jk ≃ δmjδnk + δnjδmk − 5
+λImn,jk +
+7
+8λ3
+�
+dx(2ϕmϕnϕ′
+jϕ′
+k + ϕmϕ′
+nϕjϕ′
+k + ϕ′
+mϕnϕjϕ′
+k
++ ϕmϕ′
+nϕ′
+jϕk + ϕ′
+mϕnϕ′
+jϕk + 2ϕ′
+mϕ′
+nϕjϕk) .
+(28)
+Here, all functions are evaluated at position x, the primes mean that a first derivative must
+be taken, and Imn,jk is an integral of a product of four single-particle harmonic-oscillator
+eigenstates:
+Ijk,lm =
+�
+dx ϕj(x)ϕk(x)ϕl(x)ϕm(x) .
+(29)
+These integrals are evaluated using known properties of the Hermite polynomials. In turn,
+the integrals with derivatives of the eigenfunctions can be written in terms of the elements
+Imn,jk using the relation:
+ϕ′
+n =
+�mω
+ℏ (√n ϕn−1 −
+√
+n + 1 ϕn+1),
+(30)
+keeping in mind that the ϕ are defined as the eigenfunctions of the harmonic oscillator with
+mass 2m.
+In the same way one can write an expression for the part of the Hamiltonian involving the
+commutator, Eq. (18). The dominant contributions give, after some manipulations:
+⟨v|BmBn[V †
+j , B†
+k]|v⟩ ≃ 2γImn,jk
+− γ
+8λ2
+�
+dx [ϕmϕnϕ′
+jϕ′
+k+3(ϕmϕ′
+nϕjϕ′
+k+ϕmϕ′
+nϕ′
+jϕk+ϕ′
+mϕnϕjϕ′
+k+ϕ′
+mϕnϕ′
+jϕk)+11ϕ′
+mϕ′
+nϕjϕk]
++
+γ
+128λ4
+�
+dx{21(ϕmϕ′
+nϕ′
+jϕ′′
+k + ϕmϕ′
+nϕ′′
+j ϕ′
+k + ϕ′
+mϕnϕ′
+jϕ′′
+k + ϕ′
+mϕnϕ′′
+j ϕ′
+k)
++ 4(ϕmϕ′′
+nϕjϕ′′
+k + ϕmϕ′′
+nϕ′′
+j ϕk + ϕ′′
+mϕnϕjϕ′′
+k + ϕ′′
+mϕnϕ′′
+j ϕk) + 27(ϕmϕ′′
+nϕ′
+jϕ′
+k + ϕ′′
+mϕnϕ′
+jϕ′
+k)
+− 22(ϕ′
+mϕ′
+nϕjϕ′′
+k + ϕ′
+mϕ′
+nϕ′′
+j ϕk) + 57ϕ′′
+mϕ′′
+nϕjϕk + ϕmϕnϕ′′
+j ϕ′′
+k},
+(31)
+where again all functions are evaluated at position x and the double primes mean that a
+second derivative must be taken. This formula can be calculated using similar steps as before.
+Putting this together with the part from (Ej + Ek)Smn,jk we can find a consistent expansion
+for the Hamiltonian up to this order.
+For our numerical calculations, we include the orders reported for S and H. One could
+improve this evaluation by considering higher orders of the Taylor expansion. However, we
+checked that for the parameter regimes studied the results obtained with these formulas are
+not significantly altered by excluding the higher order, as can be seen in Fig. 1.
+B
+Spatial correlations for two fermions of equal kind
+We first consider the joint density distribution for fermions of equal kind:
+Daa(x, x′) = ⟨ψ|Ψ†
+a(x′)Ψ†
+a(x)Ψa(x)Ψa(x′)|ψ⟩,
+(32)
+15
+
+SciPost Physics
+Submission
+of course, taking two fermions of kind b leads in our model to the same result. We note that
+this definition means that:
+��
+dx dx′ Daa(x, x′) = 2 .
+(33)
+We now show how we calculate this joint density for the numerically found ground state.
+Starting from the expansion of the state in the coboson basis, Eq. (11), we find:
+Daa(x, x′) =
+�
+m≤n
+�
+j≤l
+cmn cjl [J(jm)
+1
+(x)J(ln)
+1
+(x′) − J(jn)
+2
+(x, x′)J(ml)
+2
+(x, x′)]
++ same with n ↔ m, j ↔ l, and {n, l} ↔ {m, j}.
+(34)
+The result for the standard coboson ansatz corresponds to setting all cjl to zero except for
+c00. In the formula above we have introduced auxiliary integrals given by:
+J(jm)
+1
+(x) =
+�
+dx′ψj(x, x′)ψm(x, x′),
+(35)
+and
+J(jm)
+2
+(x, x′) =
+�
+dx′′ψj(x, x′′)ψm(x′, x′′) .
+(36)
+The integrals J2 account for fermion-exchange terms and are negligible unless x and x′ are
+close together within a distance of order 1/λ.
+From these formulas one can recover the vanishing of the conditional probability for x = x′
+for arbitrary states. For the limit λ → ∞, the joint density Daa tends to the expression given
+in Eq. (21) which was calculated from the ground state of two point-like hard-core bosons.
+In the limit of very large but finite attraction, one can resort to a Taylor expansion for the
+calculation of the integrals, in the same spirit as the calculations in Appendix A. For J1 we
+obtain:
+J(jm)
+1
+(x) ≃ ϕjϕm +
+1
+16λ2 [2ϕ′
+jϕ′
+m + ϕ′′
+j ϕm + ϕjϕ′′
+m]
++
+1
+256λ4 [6ϕ′′
+j ϕ′′
+m + 4ϕ(3)
+j ϕ′
+m + 4ϕ′
+jϕ(3)
+m + ϕ(4)
+j ϕn + ϕjϕ(4)
+m ] .
+(37)
+Here, all functions are evaluated at position x and we are using primes (double primes) over
+the functions to denote derivatives (second derivatives), whereas derivatives of higher order are
+indicated with superindices between parenthesis. We remind the reader that the ϕn indicate
+the oscillator eigenfunctions for mass 2m.
+The integral for J2 can be expanded as:
+J(jm)
+2
+(x, x′) ≃ e−λ|x−x′|�
+ϕj(x)ϕm(x′)(1 + λ|x − x′|)
+− 1
+4λ
+�
+ϕ′
+j(x)ϕm(x′) − ϕj(x)ϕ′
+m(x′)]λ(x − x′)(1 + λ|x − x′|)
+�
++
+1
+6λ2
+�1
+4ϕ′
+j(x)ϕ′
+m(x′)
+�
+3 + 3λ|x − x′| − λ3|x − x′|3�
++ 1
+8
+�
+ϕ′′
+j (x)ϕm(x′) + ϕj(x)ϕ′′
+m(x′)
+��
+3 + 3λ|x − x′| + 3λ2(x − x′)2 + 2λ3|x − x′|3���
+.
+(38)
+16
+
+SciPost Physics
+Submission
+From the joint density Daa(x, x′) one can also calculate the conditional probability Paa(x|x′)
+of finding a particle of kind a at position x when another of the same kind was found at position
+x′. This can be computed from:
+Paa(x|x′) =
+Daa(x, x′)
+⟨ψ|Ψ†
+a(x′)Ψa(x′)|ψ⟩
+,
+(39)
+so that:
+�
+dx Paa(x|x′) = 1
+∀ x′.
+(40)
+For the limit λ → ∞, the conditional probability Paa tends to the expression given in Eq. (22)
+calculated from the ground state of two point-like hard-core bosons.
+On the other hand, the standard coboson ansatz predicts for λ → ∞ a behaviour of the
+form:
+Daa(x, x′) =
+�
+2ϕ0(x)2ϕ0(x′)2
+if x ̸= x′
+0
+if x = x′,
+(41)
+so that
+Paa(x|x′) =
+�
+ϕ0(x)2
+if x ̸= x′
+0
+if x = x′.
+(42)
+C
+Spatial correlations for fermions of different kinds
+We now calculate spatial correlations between fermions of different kinds. In particular, we
+are interested in the joint particle density
+Dab(x, x′) = 4ρab(xa, xb; xa, xb) .
+(43)
+Here, ρab is the reduced density matrix of two different fermions in position basis and is given
+by [17]:
+ρab(xa, xb; x′
+a, x′
+b) = 1
+4⟨ψ|Ψ†
+a(xa)Ψ†
+b(xb)Ψb(x′
+b)Ψa(x′
+a)|ψ⟩ .
+(44)
+In the following we proceed to the calculation of the joint density for the general numerical
+solution. Replacing the expansion of the state in the coboson basis leads to:
+Dab(x, x′) =
+�
+m≤n
+�
+j≤l
+cmn cjl
+��
+δnlψm(x, x′)ψj(x, x′) + J(ln)
+1
+(x) J(jm)
+1
+(x′)
+− ψm(x, x′)J(jl|n)
+3
+(x, x′) − ψj(x, x′)J(mn|l)
+3
+(x, x′)
+�
++ same with n ↔ m, j ↔ l, and {n, l} ↔ {m, j}
+�
+.
+(45)
+Here, the J1 are given in Eq. (35), and the J3 contain interference terms given by:
+J(jl|n)
+3
+(x, x′) =
+�
+dydy′ψj(x, x − y) ψl(x′ + y′, x′)ψn(x − y, x′ + y′) .
+(46)
+Again, the result for the coboson ansatz is found setting all coefficients cjl to zero except for
+c00.
+17
+
+SciPost Physics
+Submission
+Resorting to the Taylor expansion J(jl|n)
+3
+(x, x′) can be approximated by
+J(jl|n)
+3
+(x, x′) ≃ e−λ|x−x′|
+�
+1
+2
+√
+λ
+ϕj(x)ϕl(x′)ϕn(x)
+�
+λ2(x − x′)2 + 3λ|x − x′| + 3
+�
++
+1
+12
+√
+λ
+�
+−ϕ′
+j(x)ϕl(x′)ϕn(x) + ϕj(x)ϕ′
+l(x′)ϕn(x) − 3ϕj(x)ϕl(x′)ϕ′
+n(x)
+�
+× (x − x′)
+�
+λ2(x − x′)2 + 3λ|x − x′| + 3
+�
++
+1
+24λ5/2
+�
+− 1
+4ϕ′
+j(x)ϕ′
+l(x′)ϕn(x) − 3
+4ϕj(x)ϕ′
+l(x′)ϕ′
+n(x) + 1
+4ϕ′
+j(x)ϕl(x′)ϕ′
+n(x)
++ 1
+2ϕj(x)ϕl(x′)ϕ′′
+n(x)
+�
+×
+�
+λ4(x − x′)4 + 2λ3|x − x′|3 − 3λ2(x − x′)2 − 15λ|x − x′| − 15
+�
++
+1
+12λ5/2
+�1
+2ϕ′
+j(x)ϕl(x′)ϕ′
+n(x) + 1
+8ϕ′′
+j (x)ϕl(x′)ϕn(x) + 1
+8ϕj(x)ϕ′′
+l (x′)ϕn(x)
++ 5
+8ϕj(x)ϕl(x′)ϕ′′
+n(x)
+�
+×
+�
+λ4(x − x′)4 + 4λ3|x − x′|3 + 9λ2(x − x′)2 + 15λ|x − x′| + 15
+�
+�
+.
+(47)
+We note that just as in Paa, the terms with J1 are the only ones that are non-negligible when
+x and x′ are at a distance much larger than 1/λ. Thus, Paa and Dab behave in the same
+way for e−λ|x−x′| ≪ 1, corresponding to detection of particles in different bound pairs. In the
+opposite limit of x close to x′, Dab has a peak of width 1/λ corresponding to detection of the
+particle forming a pair with the fermion detected at x′.
+We now calculate a quantity analogue to Paa(x|x′) but applying to fermions of different
+kinds. In particular, we wish to calculate the conditional probability Pab(x|x′) of finding a
+fermion of kind a at position x conditioned on having found a fermion of kind b at position
+x′. This, however, is trickier because after the detection of one fermion of kind b there are
+two remaining identical fermions of kind a.
+Thus, we choose to work with a conditional particle density Dab(x|x′) indicating the density
+of fermions of kind a at position x conditioned on having found a fermion of kind b at position
+x′. Since two identical fermions can never be found at the same place, this quantity is related
+with the conditional probability Pab, but its interpretation is more straightforward and, in
+contrast with a probability, Dab must be normalized to 2. More precisely, the conditional
+particle density is given by:
+Dab(x|x′) =
+Dab(x, x′)
+⟨ψ|Ψ†
+b(x′)Ψb(x′)|ψ⟩
+.
+(48)
+such that
+�
+dx Dab(x|x′) = 2
+∀ x′.
+(49)
+For infinite attraction, it holds that Dab(x|x′) = Paa(x|x′) + δ(x − x′).
+For the coboson
+ansatz, in the limit λ → ∞ one has Dab(x, x′) = 2ϕ0(x)2[ϕ0(x′)2 + δ(x − x′)] and accordingly
+Dab(x|x′) = ϕ0(x′)2 + δ(x − x′).
+18
+
+SciPost Physics
+Submission
+D
+Off-diagonal correlation parameter
+Here we provide the expression for the off-diagonal correlation parameter g2(x) defined in
+Eq. (27). The diagonal matrix elements appearing in the denominator are particular instances
+of the calculation in the previous section, so that one can use Eq. (45) evaluated for x′ = x.
+For the off-diagonal part, we plug the decomposition of the state in the coboson basis and
+apply (anti)commutators as in the previous sections.
+ρab(0, 0; x, x) ≃ 1
+4
+�
+m≤n
+�
+j≤l
+cmn cjl
+��
+ψj(x, x)ψm(0, 0)δnl + J(ln)
+2
+(x, 0)J(jm)
+2
+(x, 0)
+− ψj(x, x)J(mn|l)
+4
+(0) − ψm(0, 0)J(jl|n)
+4
+(x)
+�
++ same with n ↔ m, j ↔ l, and {n, l} ↔ {m, j}
+�
+,
+(50)
+where we have introduced a new integral expression:
+J(mn|l)
+4
+(x) =
+�
+dydy′ ψm(y, x)ψn(x, y′)ψl(y, y′),
+(51)
+that can be Taylor-expanded as follows:
+J(mn|l)
+4
+≃
+3
+2
+√
+λ
+ϕmϕnϕl
++
+5
+4λ5/2 [ϕ′
+mϕ′
+nϕl + 3ϕ′
+mϕnϕ′
+l + 3ϕmϕ′
+nϕ′
+l + ϕ′′
+mϕnϕl + ϕmϕ′′
+nϕl + 3ϕmϕnϕ′′
+l ],
+(52)
+with all functions evaluated at the same position.
+In the limit of infinite attraction the form of g2 can be calculated using the point-like
+hard-core boson solution. This gives:
+g2−hc(x) =
+4x
+√
+2π + xωerfc(
+√
+2 x/xω)
+�
+4x2 + x2ω
+,
+(53)
+where “erfc” is the complementary error function. This is a quite flat behaviour for g2, but
+still clearly different from the totally flat profile, g2(x) = 1 ∀ x, that is obtained from the
+standard coboson ansatz for λ → ∞.
+E
+Correlations in momentum space
+One can easily extend the results from the previous appendices to momentum space. In order
+to do this, we resort to the expresion of the coboson wavefunctions in momentum space:
+�ψn(k1, k2) =
+�
+2
+πλ
+xωe−iπn/2
+1 +
+�
+k1−k2
+2λ
+�2 ϕn[x2
+ω(k1 + k2)],
+(54)
+which is just the Fourier transform of Eq. (4). These functions are of order
+�
+xω/λ and decay
+in a scale of order λ for the relative momentum (k1 − k2)/2 and of order √n/xω for the
+center-of-mass momentum k1 + k2.
+19
+
+SciPost Physics
+Submission
+From this expression one can derive formulas for the correlations in momentum space
+following similar steps as before. One should only keep in mind that, in contrast to position
+space, the wavefunctions in momentum space are complex. In particular, we find for the
+momentum correlations between fermions of different kinds an equation which is analogous
+to Eq. (45):
+�Dab(k, k′) =
+�
+m≤n
+�
+j≤l
+cmn cjl
+��
+δnl �ψ∗
+m(k, k′) �ψj(k, k′)+ �J(nl)
+1
+(k) �J(mj)
+1
+(k′)− �ψ∗
+m(k, k′) �J(jl|n)
+3
+(k, k′)
+− �ψj(k, k′)[ �J(mn|l)
+3
+(k, k′)]∗�
++ same with n ↔ m, j ↔ l, and {n, l} ↔ {m, j}
+�
+.
+(55)
+Here, the asterisk denotes a complex conjugation and we have defined:
+�J(nl)
+1
+(k) =
+�
+dk′ �ψ∗
+n(k, k′) �ψl(k, k′),
+(56)
+and
+�J(jl|n)
+3
+(k, k′) =
+�
+dqdq′ �ψj(k, q′) �ψl(q, k′) �ψ∗
+n(q, q′) .
+(57)
+Replacing the form of the wavefunctions in momentum space one finds the integral ex-
+pression:
+�J(nl)
+1
+(k) = 2x2
+ω
+πλ eiπ(n−l)/2
+�
+dk′ ϕn(x2
+ωk′)ϕl(x2
+ωk′)
+�
+1 + ( k′−2k
+2λ )2�2
+.
+(58)
+Taking into account the restriction on the values of n, l within our basis, one can perform
+a Taylor expansion in k′/λ in the expression above. We stress that the values of k cannot
+be assumed to be much smaller than λ, since λ is indeed the typical scale for the relative
+momentum. In this way we obtain:
+�J(nl)
+1
+(k) ≃ 2x2
+ω
+πλ eiπ(n−l)/2
+�
+dk′ϕn(x2
+ωk′)ϕl(x2
+ωk′)
+�
+��
+1
+�
+k2
+λ2 + 1
+�2 +
+2kk′
+λ2
+�
+k2
+λ2 + 1
+�3 −
+(1 − 5k2
+λ2 )k′2
+2λ2
+�
+k2
+λ2 + 1
+�4
+�
+�� ,
+(59)
+which can be evaluated using properties of the Hermite polynomials.
+The integrals for �J3 can be cast in the form:
+�J(jl|n)
+3
+(k, k′) =
+�2x2
+ω
+πλ
+�3/2
+e−iπ(j+l−n)/2
+�
+dqdq′
+ϕj(x2
+ωq′)ϕl(x2
+ωq)ϕn[x2
+ω(q + q′ − k − k′)]
+�
+1 + ( q′−2k
+2λ )2
+� �
+1 + ( q−2k′
+2λ )2
+� �
+1 + ( k−k′+q−q′
+2λ
+)2
+� .
+(60)
+Performing a Taylor expansion here is justified for the q, q′ divided by λ in the denominator,
+but not for the same variables inside the wavefunction ϕ. This makes this calculation much
+20
+
+SciPost Physics
+Submission
+more involved. The Taylor expansion of the denominator gives:
+�J(jl|n)
+3
+(k, k′) ≃
+�2x2
+ω
+πλ
+�3/2
+e−iπ(j+l−n)/2
+�
+dqdq′ϕj(x2
+ωq′)ϕl(x2
+ωq)ϕn[x2
+ω(q + q′ − k − k′)]
+�
+��
+1
+�
+k2
+λ2 + 1
+�2 �
+k′2
+λ2 + 1
+�2 �
+(k−k′)2
+4λ2
++ 1
+�2
++
+(k − 3k′)
+�
+k′(k−k′)
+2λ2
+− 1
+�
+λ2
+�
+k2
+λ2 + 1
+�2 �
+k′2
+λ2 + 1
+�3 �
+(k−k′)2
+4λ2
++ 1
+�3 q +
+(3k − k′)
+�
+k(k−k′)
+2λ2
++ 1
+�
+λ2
+�
+k2
+λ2 + 1
+�3 �
+k′2
+λ2 + 1
+�2 �
+(k−k′)2
+4λ2
++ 1
+�3 q′
+�
+�� ,
+(61)
+Here one is still left with a non-trivial integral in q, q′. This can be solved using the decom-
+position formula
+ϕn (x + y) =
+∞
+�
+i,j=0
+Aij|nϕi (x) ϕj (y) ,
+(62)
+where the coefficients
+Aij|n =
+� �
+ϕn (x + y) ϕi (x) ϕj (y) dxdy
+(63)
+can be evaluated using properties of the Hermite polynomials. For numerical evaluation this
+summation must be truncated. Performing this one up to i, j = 100 good approximations are
+obtained. Then we are left with terms similar to those found in the calculation for �J1.
+The first term in Eq. (55) contains contributions of order xω/λ which decay in a scale
+of order λ for the relative momentum (k − k′)/2 and of order 1/xω for the center-of-mass
+momentum k + k′. The second term, involving �J1, contains contributions of order 1/λ2 which
+decay on a scale of order λ for k and k′ separately. We note that this contribution is broad and
+has a Lorentzian decay, whereas the decay of the contributions in the first term is Gaussian
+for the center of mass. Thus, they may be of the same order depending on the point where
+they are evaluated. In any case, the dominant feature is the anti-diagonal resulting from the
+first term. The remaining terms, containing �J3, have a similar behaviour as the first (i.e.
+with a strong anti-diagonal) but are one order smaller in 1/(λxω), which justifies using an
+expansion for �J3 to a lower order than for �J1.
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+Rev. A 81, 031608 (2010), doi:10.1103/PhysRevA.81.031608.
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diff --git a/ZdE2T4oBgHgl3EQfEgbW/content/tmp_files/load_file.txt b/ZdE2T4oBgHgl3EQfEgbW/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf,len=945
+page_content='SciPost Physics  Submission  Composite-boson formalism applied to strongly bound  fermion pairs in a one-dimensional trap  Mart´ın D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Jim´enez1, Eloisa Cuestas1, 2, Ana P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Majtey1 and Cecilia Cormick1⋆  1 Instituto de F´ısica Enrique Gaviola, CONICET and Universidad Nacional de C´ordoba,  Ciudad Universitaria, X5016LAE, C´ordoba, Argentina  2 Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate  University, Onna, Okinawa 904-0495, Japan  ⋆ cecilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='cormick@unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='ar  January 11, 2023  Abstract  We analyze a system of fermions in a one dimensional harmonic trap with at-  tractive delta-interactions between different fermions species, as an approximate  description of experiments involving atomic dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We solve the problem of two  fermion pairs numerically using the so-called “coboson formalism” as an alterna-  tive to techniques which are based on the single-particle basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This allows us to  explore the strongly bound regime, approaching the limit of infinite attraction  in which the composite particles behave as hard-core bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Our procedure is  computationally inexpensive and illustrates how the coboson toolbox is useful for  ultracold atom systems even in absence of condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Contents  1  Introduction  2  2  The procedure, step by step  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='1  Single-pair solution  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='2  Basis for two pairs  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='3  Construction of the Hamiltonian  6  3  Analytical considerations for infinite attraction  7  4  Numerical study of the ground state for strong attraction  8  5  Summary and conclusions  12  A Calculation of Hamiltonian and overlap matrix in position basis  14  B Spatial correlations for two fermions of equal kind  15  C Spatial correlations for fermions of different kinds  17  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='03637v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='quant-gas]  9 Jan 2023  SciPost Physics  Submission  D Off-diagonal correlation parameter  19  E Correlations in momentum space  19  References  21  1  Introduction  The possibility to engineer atomic and molecular many-body systems by controlling and  assembling simpler components has made enormous progress thanks to Feshbach resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In this way, molecular Bose-Einstein condensates have been formed starting from ultracold  atomic gases [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Similar setups have been used for the controlled observation of relevant  phenomena in statistical physics such as Wigner crystals [3] and the BEC-BCS crossover [4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Within the field of ultracold Fermi gases, one-dimensional systems are known to exhibit very  peculiar properties [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In particular, strongly bound fermion pairs reach a limit in which they  behave as hard-core bosons, which in turn are related to non-interacting fermion models [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  We consider a one-dimensional scenario, with fermions of two different kinds in a harmonic  trap and an attractive contact interaction leading to fermion pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The first steps towards  the exact solution of the one-dimensional Fermi gas with contact interactions in a ring are due  to Gaudin and Yang in 1967 [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For the trapped case most of the analytical work focuses  on the strongly repulsive case, see [10] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Numerical approaches for this  system include multiconfigurational time-dependent Hartree method [11], quantum diffusion  Montecarlo [12], density matrix renormalization group [13] and a variety of quantum-chemical  treatments such as coupled-cluster methods [14], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The vast body of literature  in this field has been reviewed for instance in [6,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Even though much effort has been devoted to this system, the usual numerical treatment  takes as a basis the harmonic oscillator eigenstates, making computations very costly for strong  attraction [14, 16–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Alternative procedures which are more efficient for strong attraction  have been proposed in [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Here, as a different approach, we tackle the problem of two pairs  with two fermions each in the context of coboson theory [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This theoretical framework,  originally developed for excitons in semiconductors [23–25], has by now been applied to a  variety of systems, including Bose-Einstein condensates [26], superconductors [27, 28] and  Feshbach molecules [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  A very useful simplification often encountered in this treatment is the so-called coboson  ansatz, which is analogous to a condensate formed by composite bosons and is the canonical-  ensemble counterpart of the BCS ansatz [23,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Using tools from the coboson formalism, we  show that the coboson ansatz does not provide a good approximation of the true ground state  for the case of two pairs in the limit of strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This is to be expected in the light  of previous results [30, 31] and also because the limit of infinitely bound pairs corresponds  to hard-core bosons which are known to form only a quasi-condensate in 1D traps [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  However, the coboson formalism also provides tools to describe the state beyond the coboson  ansatz [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We thus develop a representation of the problem in the coboson basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' in  terms of the eigenstates of one pair of interacting fermions in the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  2  SciPost Physics  Submission  This basis is specially convenient and expected to work better for the regime of strong  attraction, which is difficult to address numerically (see for instance Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' [16]) and has been  not studied exhaustively as the repulsive regime [6,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In this respect, our method is related  with the perturbative approach in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The case of two pairs is of particular relevance within  the coboson formalism, however, the method we propose can be extended to larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The motivation of our work can then be stated as i) to show that even if the coboson ansatz  fails the correct ground state for this system can be recovered using the complete toolbox of  the coboson formalism ii) to show that the two-body coboson basis is useful in the strongly  attractive limit where the single-particle basis is not convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Besides the numerical convenience of using the coboson basis, studying this system within  the coboson formalism leads to semi-analytical reliable results that can provide a safe ground  to quantify the fractional statistics [36,37] of the one-dimensional Fermi gas [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This is  a good starting point to analyze the relationship between anyonic statistics and the entangle-  ment of the constituent particles of the composite boson, which has been pointed out to be  the key to understand composite effects and ideal bosonic behavior [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The basic steps of our procedure to tackle the problem of two trapped fermion pairs are  the same as in [31] and are as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We solve the problem of a pair of interacting fermions in the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The operators B†  n  that create each single-pair eigenstate, and the corresponding energies En, will be the  starting point of the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We truncate the basis considering the states with the  lowest energies, up to some quantum number nmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' From the single-pair basis operators B†  n we form the two-coboson basis generated by the  action on the vacuum of operators of the kind B†  nB†  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We calculate the form of the Hamiltonian in this truncated coboson basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Solving the corresponding generalized eigenvalue problem, we estimate the ground state  for two pairs and analyze its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  This method allows us to interpolate from the interaction strengths for which the single-  particle basis is suitable [17–20], all the way to very strongly bound pairs approaching the  limit of hard-core bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Using coboson-theory tools combined with Taylor expansions, we  calculate several quantities of interest, including the energy and two-particle correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The work is presented as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 2 we review how to write the problem in the  coboson framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Section 3 is devoted to analytical considerations for infinite attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 4 we discuss our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' A summary and conclusions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Finally, several appendices with detailed calculations are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  2  The procedure, step by step  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='1  Single-pair solution  For definiteness we will assume that both fermion kinds, which we call a and b, have the same  mass, and that the creation and annihilation operators corresponding to different fermion  species commute (this last choice does not affect the final results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We also assume that the  trapping potential is the same for both species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  3  SciPost Physics  Submission  The first step requires the solution of the single-pair problem, with a Hamiltonian given  by:  H1 =  �  α=a,b  � p2  α  2m + mω2x2  α  2  �  − γ δ(xa − xb)  (1)  with γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This problem can be solved by separation of the center-of-mass and relative  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The center-of-mass solution is given by the harmonic oscillator eigenfunctions  corresponding to mass 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The relative motion has been solved in the general case in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  [44,45] but for simplicity we focus only on strongly bound pairs, so that the relative motion  has a wavefunction of the form of an exponential,  ψr(xr) ≃  √  λ e−λ|xr|,  (2)  and the energy associated with the relative motion can be approximated by:  Eγ = −ℏ2λ2  m  ,  λ ≃ mγ  2ℏ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (3)  In this regime, the single-pair eigenfunctions are then approximately of the form:  ψn(xa, xb) ≃ ϕn  �xa + xb  2  � √  λ e−λ|xa−xb|,  (4)  where ϕn are the harmonic oscillator eigenfunctions for a particle of mass 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The corre-  sponding energies are:  En = ℏω  �  n + 1  2  �  + Eγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (5)  From these solutions, we define the coboson creation operators B†  n such that:  |˜n⟩ = B†  n|v⟩,  (6)  where |˜n⟩ is the n-th single-pair eigenstate, and |v⟩ is the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In particular, the coboson  operators B†  n can be written in terms of field operators as:  B†  n ≃  �  dxadxb ψn(xa, xb)Ψ†  a(xa)Ψ†  b(xb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (7)  For consistency, neglecting states where the internal motion is excited implies also a trun-  cation in the center-of-mass states, so that the basis includes all single-pair eigenstates up to  a certain energy cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In particular, we keep only states where the index n associated with  the center-of-mass motion is such that the excited internal states are well above the energy  scales considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' :  n ≪ |Eγ|  ℏω = (λxω)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (8)  For convenience here we have defined a spatial scale xω associated with the harmonic oscillator,  xω =  �  ℏ  mω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (9)  The inequality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (8) stresses once more the fact that our restricted basis is only appro-  priate for strong attraction, when the size of each bound pair is very small compared with the  spatial scale of the trap and thus λxω is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' It is also important to note that since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (2)  and therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (4) are valid for λ xω ⪆ 5 all of our results rely on this condition [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  4  SciPost Physics  Submission  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='2  Basis for two pairs  From the set of states corresponding to the lowest energies of the single-pair Hamiltonian,  one can form states of the form:  |˜n ˜m⟩ = B†  nB†  m|v⟩,  (10)  with n ≤ m (we note that the coboson creation operators commute) and |v⟩ the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Because of the fermionic character of the constituent particles, states generated in this form  are neither normalized nor orthogonal [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We truncate this two-pair basis with the condition  n + m ≤ nmax, and then approximate the ground state in the form:  |GS⟩ =  �  m≤n  cm,n|˜n ˜m⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (11)  An often useful approximation for the ground state of dilute systems of N pairs with  short-range interactions is given by what we call the “coboson ansatz” [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This corresponds  to the state obtained from the repeated application on the vacuum of the operator B0 that  creates a single pair in its ground state:  |N⟩ = (B†  0)N  √N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='χN  |v⟩,  (12)  where χN is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' However, this can only provide a good approximation  of the true ground state in systems which are expected to exhibit condensation at zero tem-  perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This is not the case in the problem we analyze [30,31,33,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In order to quantify  the quality of the approximation, we study the fidelity F between the true ground state for  two pairs, |GS⟩, and the coboson ansatz:  F = |⟨GS|(B†  0)2|v⟩|2  ⟨v|B2  0(B†  0)2|v⟩  ,  (13)  where the true ground state |GS⟩ is approximated numerically using the coboson basis given  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (10) for two-pairs (N = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Even if the coboson ansatz is not a good approximation, one can still compute the ground  state by means of the coboson formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In order to do this, we will work with the space  generated by the coboson operators as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' First, we compute all overlaps between  the relevant states from the expression:  Skl,mn = ⟨v|BkBlB†  mB†  n|v⟩ = δmlδkn+δnlδkm−  �  ⟨˜k|⊗⟨˜l|Xa| ˜m⟩⊗|˜n⟩+⟨˜k|⊗⟨˜l|Xb| ˜m⟩⊗|˜n⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (14)  Here Xα with α = a, b is an operator that exchanges the states of the two fermions of kind α,  and it acts on a fictitious space where fermions of equal kind are treated as distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Since our goal is to find the ground state, instead of building an orthonormal basis, we keep  the overlap matrix S to solve the corresponding generalized eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The matrix S can be calculated following different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In the coboson literature  [23], the overlaps are evaluated in terms of matrix elements of the change of basis between  single-pair eigenstates and the separable single-fermion basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' However, this procedure can  be numerically costly and lead to large errors when many coefficients are non-negligible and  5  SciPost Physics  Submission  no analytical expression exists for the sums required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Thus, we resort to a different form of  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Plugging the explicit form of the operators B†  n given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (7) in all formulas,  and using (anti)commutators, we can obtain an expression for the elements of the overlap  matrix as:  Smn,jk ≃  �  δmjδnk − λ2  �  dx dy1 dy2 dy3 dy4 δ(y1 + y2 − y3 − y4)  ϕm(x) ϕn  �  x + y3 − y1 + y2  2  �  ϕj  �  x + y3 − y1  2  �  ϕk  �  x + y3 − y2  2  �  e  −λ �  l  |yl|  �  + same with j ↔ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (15)  Since we are interested in the case of strong attraction, the factors of the form e−λ|yl| allow  us to perform a Taylor expansion in 1/(xωλ) for the harmonic oscillator functions ϕn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This  is possible given the truncation of our basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (8), which implies that the spatial scale  associated with the center of mass is much longer than the pair size λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In this form one can  find approximate expressions for S from a lengthy but straightforward evaluation of spatial  integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This procedure is explained in detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='3  Construction of the Hamiltonian  We now need to compute the Hamiltonian in the coboson basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The Hamiltonian can be split  in two parts, corresponding to the non-interacting terms and the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The interaction  part is quartic and can be written in terms of field operators as:  Hint = −γ  �  dx Ψ†  a(x)Ψ†  b(x)Ψa(x)Ψb(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (16)  The Hamiltonian matrix elements in the coboson basis can be obtained from the expres-  sion:  ⟨v|BkBlHB†  mB†  n|v⟩ = (En + Em)Skl,mn + ⟨v|BkBl  �  [Hint, B†  m], B†  n  �  |v⟩,  (17)  which is just a rewriting of the formulas in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Notice that when using the coboson formal-  ism the one-body term which contains the kinetic energy and trap potential is absorbed by  quantities that were calculated when solving the single-pair case (first term on the right-hand-  side in the above equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In a similar spirit as for the calculation of the overlap matrix  S, instead of following the standard expressions in [23] we estimate the Hamitonian elements  using a Taylor expansion of spatial integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In particular, the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (17) can be written as:  ⟨v|BmBn  �  [Hint, B†  j], B†  k  �  |v⟩ ≃  γλ2  � �  dxdydy′e−λ(|y|+|y′|+|y−y′|)ϕm(x)ϕn  �  x + y′ − y  2  �  ϕj  �  x + y′ − y  2  �  ϕk  �  x + y′  2  �  −  �  dxdx′dydy′e−2λ(|y|+|y′|)ϕm(x)ϕn(x′)ϕj(x)ϕk(x′)δ  �  x − x′ + y + y′  2  � �  + same with n ↔ m + same with j ↔ k + same with {j, k} ↔ {m, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (18)  The details of the procedure involving the Taylor expansion of the Hamiltonian elements are  also provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  6  SciPost Physics  Submission  3  Analytical considerations for infinite attraction  Before presenting the results of our numerical approach, we note that the case of infinite  attraction can be solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In this limit, fermions of different species are so strongly  bound that they behave as point-like hard-core bosons of mass 2m, and the problem can be  solved by means of fermionization [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' According to this procedure, one must first consider  the ground state of two identical non-interacting fermions of mass 2m in the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This state  is given by:  ψ2f(x1, x2) = 2mω  ℏ√π e−mω(x2  1+x2  2)/ℏ(x1 − x2)  (19)  and corresponds to the antisymmetric combination of having one fermion in the trap ground  state and another in the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Then, one obtains the wavefunction of the hard-  core bosons as the symmetrized form of the previous expression, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' :  ψhc(x1, x2) = 2mω  ℏ√π e−mω(x2  1+x2  2)/ℏ|x1 − x2|,  (20)  where the subindex “hc” stands for “hard-core”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  From these expressions we can calculate all properties of the ground state for λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For  instance, the asymptotic ground-state energy, excluding the binding energy Eγ of each pair,  is found to be given by the sum of the two lowest energies of the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Thus,  the total ground-state energy for very large λ is approximately 2Eγ + 2ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We can define  an effective interaction energy between pairs as ∆E = E2 − 2E1, where EN is the ground-  state energy of N = 1, 2 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Considering that a single pair has a ground-state energy of  Eγ + ℏω/2, we then obtain an effective interaction energy which for very large attraction  approaches ∆E = ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Using the ground-state wavefunction as expressed above, one can also analytically calculate  the fidelity between the true ground state and the coboson ansatz for infinite attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We  find an asymptotic fidelity of F∞ = 2/π ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='64, which is lower than the one obtained in the  same regime for two fermion pairs in translationally invariant models [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Following the same lines, one can find the joint density of composite particles at positions  x and x′ for the limit of infinite attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This is of the form:  Dhc(x, x′) = 8  π2  (x − x′)2  x4ω  e−2(x′2+x2)/x2  ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (21)  One can also write down the conditional probability P(x′|x) of finding a composite point-like  particle at position x′ provided that another one was found at position x:  Phc(x′|x) = 1  xω  �  2  π  (x − x′)2  x2 + x2ω/4 e−2(x′/xω)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (22)  Furthermore, one can calculate the asymptotic values of the coefficients in the expansion  of the ground state in the coboson basis, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (10-11), obtaining for λ → ∞:  c(∞)  mn = −(2 − δmn) (−1)(m−n)/2  √  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  �  1  π  (m + n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (m/2 + n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  1  2m+n  1  m + n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (23)  7  SciPost Physics  Submission  This expression is valid for even and nonzero n+m, and here δmn is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For  symmetry reasons the coefficients cmn vanish for odd n + m, and for n = m = 0 we find:  c(∞)  00  =  �  1  π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (24)  Since the coboson ansatz corresponds to the repeated application of the coboson operator B†  0,  and for λ → ∞ the wavefunctions associated with the different B†  m become orthogonal, the  asymptotic value of c00 determines the asymptotic fidelity between the correct ground state  and the coboson ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The additional factor  √  2 in the fidelity comes from the definition of  the coboson basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (10), which does not include a prefactor 1/  √  2 for m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Before tackling the numerical treatment of the problem for strong but finite attraction,  we note that also the limit of infinitesimal attraction can be treated analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For γ = 0,  the ground state of the system is separable, with the two lowest oscillator levels occupied  for both kinds of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Then, the energy ∆E approaches 2ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' It is very important to  notice that in this separable limit, the coboson normalization factor χ2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (12) vanishes,  and thus the coboson ansatz is not defined for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Nevertheless, using perturbation theory  together with analytical results for the Schmidt coefficients [46] one can calculate the limit  value of the fidelity between the true ground state and the coboson ansatz, and find that as  the attractive interaction strength approaches zero, F approaches a value of approximately  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Indeed, for γ ∼ 0 we obtain χ2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='342 θ2 and F ∼ θ2/8χ2 with θ ∼ γ/  √  2πℏωxω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We  note, however, that the weakly bound case is not within the scope of our present study, and  it has been extensively analyzed before [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  4  Numerical study of the ground state for strong attraction  In the following we perform a numerical study of the ground state according to the procedure  outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' A delicate point in the calculation is the choice of the number of basis  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' A very small number leads to a poor description of the system, whereas for a very  large number it becomes unjustified to leave out the excited states of the relative motion, and  it can also lead to numerical problems if the overlap matrix becomes worse conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' As  a compromise, we choose the maximum center-of-mass energy included in our description to  grow linearly with λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 1 (a) we show our results for the interaction energy ∆E = E2 −2E1 using a Taylor  expansion for the calculation of both the overlap and the Hamiltonian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We also plot  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 1 (b) the fidelity F between the ansatz and the true ground state as a function of λ  when choosing the energy in the truncated basis to be given by nmax = λxω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We note that our  results show reasonable agreement with the known behaviour for infinite attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Notice  that the difference between the numerical ∆E obtained for λxω ≃ 200 and the asymptotic  value ℏω presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 1 (a) is of about 4%, whereas the binding energy for this case is  so large that ∆E is five orders of magnitude smaller than the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  As can be seen in the comparison provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 2, for λxω = 30 the coefficients cm,n  of the ground state in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (11) are already very close to the ones obtained from  the hard-core boson limit given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (23-24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This also hints at a procedure to perform  approximate computations more efficiently: instead of taking the full basis as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 2, one  can use a truncation inspired by the asymptotic values of the coefficients in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (23-24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  8  SciPost Physics  Submission  (a)  (b)  Figure 1: a) Energy for two pairs, excluding the trivial contribution equal to twice  the single-pair energy, as a function of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' b) Fidelity between the coboson ansatz  and the numerically found ground state as a function of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The results are obtained  from the lowest non-trivial order of the Taylor expansion (green stars) and the next  non-zero higher-order corrections (black circles) as reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The  horizontal dashed red lines indicate the asymptotic values for λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  One can also directly approximate the state by taking the coboson basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (10) to be a  function of λ but the coefficients in this basis to be given by the asymptotic values, which  gives a fast and compact approximation for the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Indeed, the ground state found  numerically for λxω = 30 has a fidelity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='993 with the state obtained taking the asymptotic  values of the coefficients and truncating the basis in the same form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  From the numerical solution of the problem one can characterize the ground state through  several key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In particular, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 3 (a) we illustrate the spatial correlations between  fermions of equal kind through the joint density distribution  Daa(x, x′) = ⟨ψ|Ψ†  a(x′)Ψ†  a(x)Ψa(x)Ψa(x′)|ψ⟩,  (25)  evaluated for the case λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The details of the calculation are provided in Appendix  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This plot displays clear signatures of Pauli exclusion as a sharp diagonal feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Two  identical fermions are most likely found apart from each other at a distance which is set by  the spatial scale of the harmonic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  For comparison, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 3 (b) displays the joint density for fermions of different kinds:  Dab(x, x′) = ⟨ψ|Ψ†  b(x′)Ψ†  a(x)Ψa(x)Ψb(x′)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (26)  This plot exhibits a strong diagonal correlation corresponding to particles that form a bound  pair, with additional much broader peaks corresponding to particles belonging to different  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The calculation of Dab is explained in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Another quantity that reflects the spatial correlations present in the ground state is the  conditional probability Paa(x|0) to find one fermion of kind a at position x given that another  identical fermion was found at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This function is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 4 (a), for the  numerical solution with λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For comparison we also show the conditional probability  9  SciPost Physics  Submission  Figure 2: Coefficients in the coboson decomposition from the numerical resolution of  the problem based on a Taylor expansion for λxω = 30, in black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The index  k here refers to a particular ordering of the m, n coefficients using a single label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For  comparison we show the values according to the asymptotic expression in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (23-  24) as red four-pointed stars, which overlap with the numerical resuls within the size  of the symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The basis was truncated with nmax = λxω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The coefficients in the  plot are normalized taking ctSc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The vertical dashed light-gray lines delimitate  sections of the basis containing states B†  mB†  n|v⟩ with a fixed value of m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Paa(x|0) obtained from the hard-core limit of λ → ∞ and from the coboson ansatz of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (12)  evaluated for λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The corresponding formulas are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The plots show  qualitative agreement between the numerical results and the point-like hard-core boson limit,  in sharp contrast with the coboson ansatz in its standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Indeed, the form of the  conditional probability Paa(x|0) is similar to the probability distribution corresponding to  the first excited state of the harmonic oscillator, the maxima of which are indicated with  dotted vertical lines in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In a similar manner one can compare the predictions for the spatial correlations of fermions  of different kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' To this aim, we consider the behaviour of the conditional particle density  Dab(x|x′) indicating the density of fermions of kind a at position x conditioned on having  found a fermion of kind b at position x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We plot this quantity with x′ = 0 for the numerical  solution corresponding to λxω = 30 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 4 (b), where we also plot the predictions of the  point-like hard-core boson limit and the coboson ansatz for λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The derivation of the  corresponding formulas is shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' All three curves have a narrow peak around  the origin, associated with the probability to find a fermion paired with the first one detected  (in the limit λ → ∞ this peak is a delta function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The curves however differ strongly in the  behaviour related with the probability to find the remaining particle of kind a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This second  contribution to the conditional density has the same shape as Paa(x|0), and closely resembles  the probability distribution for the first excited state of the harmonic oscillator of mass 2m,  a behaviour which is not properly described by the standard coboson ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Figures 3 and 4 were concerned with density distributions in space, associated with diago-  nal terms of the system’s density matrix in space representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Figure 5 a) shows in contrast  10  SciPost Physics  Submission  (a)  (b)  Figure 3: a) Joint density distribution Daa(x, x′) in units of x−2  ω , for two fermions of  kind a at positions x and x′ simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' b) Joint density distribution Dab(x, x′),  in units of x−2  ω , for finding a fermion of kind a at position x and one of kind b at  position x′ simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Both densities were obtained from the numerical solution  for λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Details of the calculations are given in Appendices B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  an off-diagonal feature, namely the off-diagonal correlation function [17]:  g2(x) =  ρab(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x, x)  �  ρab(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 0, 0)ρab(x, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x, x)  ,  (27)  where ρab is the reduced density matrix for two fermions of different kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The quantity g2  is an indicator of spatial two-particle coherence, and the coboson ansatz predicts a constant  value g2(x) = 1 in the limit of infinite attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The numerical results (in black) show that  this coherence decays within the typical scale set by the harmonic oscillator, but it stays high  for all values with non-negligible particle densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Nevertheless, the off-diagonal correlation  we find is always smaller than the one corresponding to the hard-core limit, depicted in red  for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This is not due to a variation in the decay of the spatial coherence, as can be  seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 5 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Rather, the difference between our numerical results and the limit λ → ∞  is given by a different density profile, since the particle density at the origin is lower for finite  λ than in the limit of infinite attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  For the same numerically found ground state one can also characterize the properties in  momentum space using similar techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 6 we show the joint probability distribution  for fermions of different kinds in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This plot displays a strong anti-diagonal  peak which is the counterpart of the diagonal peak found for the joint probability distribution  in position space, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The remaining features of the plot do not ressemble the  state of two identical trapped fermions of mass 2m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' this difference in the behaviour of position  and momentum is typical of hard-core bosons [7,32,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The calculation of the joint density  11  2  1  0  2  2  1  0  1  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='82  1  m  0  1  2  2  1  0  1  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='8SciPost Physics  Submission  (a)  (b)  Figure 4: a) Conditional probability Paa(x|0) to find a fermion of kind a at position  x when another fermion was already found at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' b) Conditional density  Dab(x|0) indicating the density of fermions of kind a at position x conditioned on  having found a fermion of kind b at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In both plots the solid black curve is  the numerical result with λxω = 30 and nmax = λxω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The dashed red curve is the  analytical result for the probability obtained for the point-like hard-core boson limit,  and the blue dash-dotted line is the probability predicted by the coboson ansatz in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (12) for N = 2 and λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Details of the calculations are given in Appendices  B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The vertical light-gray lines indicate the positions ± xω/  √  2, which are the  locations of the maxima of the conditional probability for λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  in momentum space is similar to the one of Dab(x, x′), but involves a Fourier transform of the  coboson basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The details are explained in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  5  Summary and conclusions  We have tackled the problem of two identical composite particles, each made of two distin-  guishable fermions, inside a harmonic trap and with contact attractive interactions between  fermions of different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We explored the strongly bound regime using the coboson for-  malism to build a compact basis of states, greatly reducing the computational requirements  associated with the usual description in terms of single-particle eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  We have studied the approach of the interaction energy to the limit of infinite attraction,  corresponding to point-like hard-core bosons, and we have confirmed that the coboson ansatz  in its standard form does not provide an accurate description of the ground state for any of the  interaction strengths within our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Since the energy of the coboson ansatz for N pairs  can be approximated from the energy for one and two pairs [23] the coboson ansatz cannot  provide a good estimation for the energy of a system made of N pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We have also shown  that the point-like hard-core boson limit provides a good approximation of the coefficients  when writing the ground state in the coboson basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Furthermore, we have used the numerical  results to characterize spatial correlations present in the ground state, both between fermions  12  SciPost Physics  Submission  (a)  (b)  (c)  Figure 5: a) Off-diagonal correlation function g2(x), b) off-diagonal matrix elements  and c) diagonal matrix elements of the reduced density matrix ρab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Black solid lines  correspond to numerical results for λxω = 30, red dashed ones to point-like hard-core  bosons and blue dash-dotted lines correspond to the prediction of the coboson ansatz  for N = 2 and λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Details are provided in the main text and in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Notice that the vertical axis of subplot (a) does not begin at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  of different and equal kinds, complementing previous work [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The composite-boson procedure presented can be generalized to higher numbers of parti-  cles and different forms of the trapping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Most importantly, we expect this approach  to provide an additional tool to the ones usually applied for the description of experiments  involving bosonic Feshbach molecules made of fermionic constituents in quasi one-dimensional  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  13  SciPost Physics  Submission  Figure 6: Joint density distribution �Dab(k, k′), in units of x2  ω, for finding a fermion  of kind a with momentum k and one of kind b with momentum k′ simultaneously,  obtained from the numerical solution for λxω = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Details are provided in the main  text and in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Acknowledgements  We thank Thomas Busch for his careful reading of the manuscript and valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' is grateful to Tran Duong Anh-Tai for his suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Author contributions  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' performed the numerical calculations with support by  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' All authors contributed to the derivation of the analytical formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  coordinated the project and the writing of the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Funding information  The authors acknowledge funding from grant PICT 2017-2583 from  ANPCyT (Argentina).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  A  Calculation of Hamiltonian and overlap matrix in position  basis  In the limit of very strong interaction, it makes sense to use that the wavefunctions for the  center of mass vary over a scale which is much larger than the one for the relative motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Thus, we start from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (15) for the elements of the overlap matrix, use that all yj are of  the order of λ−1, and perform a Taylor expansion in these small displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The lowest  14  40  20  0  M  20  40  40  20  0  20  40  kaw  5 10-4 10-3  5 10-3 10-2SciPost Physics  Submission  orders give:  Smn,jk ≃ δmjδnk + δnjδmk − 5  λImn,jk +  7  8λ3  �  dx(2ϕmϕnϕ′  jϕ′  k + ϕmϕ′  nϕjϕ′  k + ϕ′  mϕnϕjϕ′  k  + ϕmϕ′  nϕ′  jϕk + ϕ′  mϕnϕ′  jϕk + 2ϕ′  mϕ′  nϕjϕk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (28)  Here, all functions are evaluated at position x, the primes mean that a first derivative must  be taken, and Imn,jk is an integral of a product of four single-particle harmonic-oscillator  eigenstates:  Ijk,lm =  �  dx ϕj(x)ϕk(x)ϕl(x)ϕm(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (29)  These integrals are evaluated using known properties of the Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In turn,  the integrals with derivatives of the eigenfunctions can be written in terms of the elements  Imn,jk using the relation:  ϕ′  n =  �mω  ℏ (√n ϕn−1 −  √  n + 1 ϕn+1),  (30)  keeping in mind that the ϕ are defined as the eigenfunctions of the harmonic oscillator with  mass 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In the same way one can write an expression for the part of the Hamiltonian involving the  commutator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The dominant contributions give,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' after some manipulations:  ⟨v|BmBn[V †  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' B†  k]|v⟩ ≃ 2γImn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='jk  − γ  8λ2  �  dx [ϕmϕnϕ′  jϕ′  k+3(ϕmϕ′  nϕjϕ′  k+ϕmϕ′  nϕ′  jϕk+ϕ′  mϕnϕjϕ′  k+ϕ′  mϕnϕ′  jϕk)+11ϕ′  mϕ′  nϕjϕk]  +  γ  128λ4  �  dx{21(ϕmϕ′  nϕ′  jϕ′′  k + ϕmϕ′  nϕ′′  j ϕ′  k + ϕ′  mϕnϕ′  jϕ′′  k + ϕ′  mϕnϕ′′  j ϕ′  k)  + 4(ϕmϕ′′  nϕjϕ′′  k + ϕmϕ′′  nϕ′′  j ϕk + ϕ′′  mϕnϕjϕ′′  k + ϕ′′  mϕnϕ′′  j ϕk) + 27(ϕmϕ′′  nϕ′  jϕ′  k + ϕ′′  mϕnϕ′  jϕ′  k)  − 22(ϕ′  mϕ′  nϕjϕ′′  k + ϕ′  mϕ′  nϕ′′  j ϕk) + 57ϕ′′  mϕ′′  nϕjϕk + ϕmϕnϕ′′  j ϕ′′  k},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (31)  where again all functions are evaluated at position x and the double primes mean that a  second derivative must be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This formula can be calculated using similar steps as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Putting this together with the part from (Ej + Ek)Smn,jk we can find a consistent expansion  for the Hamiltonian up to this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  For our numerical calculations, we include the orders reported for S and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' One could  improve this evaluation by considering higher orders of the Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' However, we  checked that for the parameter regimes studied the results obtained with these formulas are  not significantly altered by excluding the higher order, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  B  Spatial correlations for two fermions of equal kind  We first consider the joint density distribution for fermions of equal kind:  Daa(x, x′) = ⟨ψ|Ψ†  a(x′)Ψ†  a(x)Ψa(x)Ψa(x′)|ψ⟩,  (32)  15  SciPost Physics  Submission  of course, taking two fermions of kind b leads in our model to the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We note that  this definition means that:  ��  dx dx′ Daa(x, x′) = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (33)  We now show how we calculate this joint density for the numerically found ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Starting from the expansion of the state in the coboson basis, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (11), we find:  Daa(x, x′) =  �  m≤n  �  j≤l  cmn cjl [J(jm)  1  (x)J(ln)  1  (x′) − J(jn)  2  (x, x′)J(ml)  2  (x, x′)]  + same with n ↔ m, j ↔ l, and {n, l} ↔ {m, j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (34)  The result for the standard coboson ansatz corresponds to setting all cjl to zero except for  c00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In the formula above we have introduced auxiliary integrals given by:  J(jm)  1  (x) =  �  dx′ψj(x, x′)ψm(x, x′),  (35)  and  J(jm)  2  (x, x′) =  �  dx′′ψj(x, x′′)ψm(x′, x′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (36)  The integrals J2 account for fermion-exchange terms and are negligible unless x and x′ are  close together within a distance of order 1/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  From these formulas one can recover the vanishing of the conditional probability for x = x′  for arbitrary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For the limit λ → ∞, the joint density Daa tends to the expression given  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (21) which was calculated from the ground state of two point-like hard-core bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In the limit of very large but finite attraction, one can resort to a Taylor expansion for the  calculation of the integrals, in the same spirit as the calculations in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For J1 we  obtain:  J(jm)  1  (x) ≃ ϕjϕm +  1  16λ2 [2ϕ′  jϕ′  m + ϕ′′  j ϕm + ϕjϕ′′  m]  +  1  256λ4 [6ϕ′′  j ϕ′′  m + 4ϕ(3)  j ϕ′  m + 4ϕ′  jϕ(3)  m + ϕ(4)  j ϕn + ϕjϕ(4)  m ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (37)  Here, all functions are evaluated at position x and we are using primes (double primes) over  the functions to denote derivatives (second derivatives), whereas derivatives of higher order are  indicated with superindices between parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We remind the reader that the ϕn indicate  the oscillator eigenfunctions for mass 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The integral for J2 can be expanded as:  J(jm)  2  (x, x′) ≃ e−λ|x−x′|�  ϕj(x)ϕm(x′)(1 + λ|x − x′|)  − 1  4λ  �  ϕ′  j(x)ϕm(x′) − ϕj(x)ϕ′  m(x′)]λ(x − x′)(1 + λ|x − x′|)  �  +  1  6λ2  �1  4ϕ′  j(x)ϕ′  m(x′)  �  3 + 3λ|x − x′| − λ3|x − x′|3�  + 1  8  �  ϕ′′  j (x)ϕm(x′) + ϕj(x)ϕ′′  m(x′)  ��  3 + 3λ|x − x′| + 3λ2(x − x′)2 + 2λ3|x − x′|3���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (38)  16  SciPost Physics  Submission  From the joint density Daa(x, x′) one can also calculate the conditional probability Paa(x|x′)  of finding a particle of kind a at position x when another of the same kind was found at position  x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This can be computed from:  Paa(x|x′) =  Daa(x, x′)  ⟨ψ|Ψ†  a(x′)Ψa(x′)|ψ⟩  ,  (39)  so that:  �  dx Paa(x|x′) = 1  ∀ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (40)  For the limit λ → ∞, the conditional probability Paa tends to the expression given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (22)  calculated from the ground state of two point-like hard-core bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  On the other hand, the standard coboson ansatz predicts for λ → ∞ a behaviour of the  form:  Daa(x, x′) =  �  2ϕ0(x)2ϕ0(x′)2  if x ̸= x′  0  if x = x′,  (41)  so that  Paa(x|x′) =  �  ϕ0(x)2  if x ̸= x′  0  if x = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (42)  C  Spatial correlations for fermions of different kinds  We now calculate spatial correlations between fermions of different kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In particular, we  are interested in the joint particle density  Dab(x, x′) = 4ρab(xa, xb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' xa, xb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (43)  Here, ρab is the reduced density matrix of two different fermions in position basis and is given  by [17]:  ρab(xa, xb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x′  a, x′  b) = 1  4⟨ψ|Ψ†  a(xa)Ψ†  b(xb)Ψb(x′  b)Ψa(x′  a)|ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (44)  In the following we proceed to the calculation of the joint density for the general numerical  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Replacing the expansion of the state in the coboson basis leads to:  Dab(x, x′) =  �  m≤n  �  j≤l  cmn cjl  ��  δnlψm(x, x′)ψj(x, x′) + J(ln)  1  (x) J(jm)  1  (x′)  − ψm(x, x′)J(jl|n)  3  (x, x′) − ψj(x, x′)J(mn|l)  3  (x, x′)  �  + same with n ↔ m, j ↔ l, and {n, l} ↔ {m, j}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (45)  Here, the J1 are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (35), and the J3 contain interference terms given by:  J(jl|n)  3  (x, x′) =  �  dydy′ψj(x, x − y) ψl(x′ + y′, x′)ψn(x − y, x′ + y′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (46)  Again, the result for the coboson ansatz is found setting all coefficients cjl to zero except for  c00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  17  SciPost Physics  Submission  Resorting to the Taylor expansion J(jl|n)  3  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x′) can be approximated by  J(jl|n)  3  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x′) ≃ e−λ|x−x′|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='ϕj(x)ϕl(x′)ϕn(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='λ2(x − x′)2 + 3λ|x − x′| + 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='−ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='j(x)ϕl(x′)ϕn(x) + ϕj(x)ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='l(x′)ϕn(x) − 3ϕj(x)ϕl(x′)ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='n(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='× (x − x′)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='λ2(x − x′)2 + 3λ|x − x′| + 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='24λ5/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='4ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='j(x)ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='l(x′)ϕn(x) − 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='4ϕj(x)ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='l(x′)ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='n(x) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='4ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='j(x)ϕl(x′)ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='n(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='2ϕj(x)ϕl(x′)ϕ′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='n(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='λ4(x − x′)4 + 2λ3|x − x′|3 − 3λ2(x − x′)2 − 15λ|x − x′| − 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='12λ5/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='2ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='j(x)ϕl(x′)ϕ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='n(x) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='8ϕ′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='j (x)ϕl(x′)ϕn(x) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='8ϕj(x)ϕ′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='l (x′)ϕn(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='+ 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='8ϕj(x)ϕl(x′)ϕ′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='n(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='λ4(x − x′)4 + 4λ3|x − x′|3 + 9λ2(x − x′)2 + 15λ|x − x′| + 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (47)  We note that just as in Paa, the terms with J1 are the only ones that are non-negligible when  x and x′ are at a distance much larger than 1/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Thus, Paa and Dab behave in the same  way for e−λ|x−x′| ≪ 1, corresponding to detection of particles in different bound pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In the  opposite limit of x close to x′, Dab has a peak of width 1/λ corresponding to detection of the  particle forming a pair with the fermion detected at x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  We now calculate a quantity analogue to Paa(x|x′) but applying to fermions of different  kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In particular, we wish to calculate the conditional probability Pab(x|x′) of finding a  fermion of kind a at position x conditioned on having found a fermion of kind b at position  x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This, however, is trickier because after the detection of one fermion of kind b there are  two remaining identical fermions of kind a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  Thus, we choose to work with a conditional particle density Dab(x|x′) indicating the density  of fermions of kind a at position x conditioned on having found a fermion of kind b at position  x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Since two identical fermions can never be found at the same place, this quantity is related  with the conditional probability Pab, but its interpretation is more straightforward and, in  contrast with a probability, Dab must be normalized to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' More precisely, the conditional  particle density is given by:  Dab(x|x′) =  Dab(x, x′)  ⟨ψ|Ψ†  b(x′)Ψb(x′)|ψ⟩  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (48)  such that  �  dx Dab(x|x′) = 2  ∀ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (49)  For infinite attraction, it holds that Dab(x|x′) = Paa(x|x′) + δ(x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  For the coboson  ansatz, in the limit λ → ∞ one has Dab(x, x′) = 2ϕ0(x)2[ϕ0(x′)2 + δ(x − x′)] and accordingly  Dab(x|x′) = ϕ0(x′)2 + δ(x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  18  SciPost Physics  Submission  D  Off-diagonal correlation parameter  Here we provide the expression for the off-diagonal correlation parameter g2(x) defined in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The diagonal matrix elements appearing in the denominator are particular instances  of the calculation in the previous section, so that one can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (45) evaluated for x′ = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  For the off-diagonal part, we plug the decomposition of the state in the coboson basis and  apply (anti)commutators as in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  ρab(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x) ≃ 1  4  �  m≤n  �  j≤l  cmn cjl  ��  ψj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x)ψm(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 0)δnl + J(ln)  2  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 0)J(jm)  2  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 0)  − ψj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x)J(mn|l)  4  (0) − ψm(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' 0)J(jl|n)  4  (x)  �  + same with n ↔ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' j ↔ l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' and {n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' l} ↔ {m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' j}  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (50)  where we have introduced a new integral expression:  J(mn|l)  4  (x) =  �  dydy′ ψm(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' x)ψn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' y′)ψl(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' y′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (51)  that can be Taylor-expanded as follows:  J(mn|l)  4  ≃  3  2  √  λ  ϕmϕnϕl  +  5  4λ5/2 [ϕ′  mϕ′  nϕl + 3ϕ′  mϕnϕ′  l + 3ϕmϕ′  nϕ′  l + ϕ′′  mϕnϕl + ϕmϕ′′  nϕl + 3ϕmϕnϕ′′  l ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (52)  with all functions evaluated at the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  In the limit of infinite attraction the form of g2 can be calculated using the point-like  hard-core boson solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This gives:  g2−hc(x) =  4x  √  2π + xωerfc(  √  2 x/xω)  �  4x2 + x2ω  ,  (53)  where “erfc” is the complementary error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This is a quite flat behaviour for g2, but  still clearly different from the totally flat profile, g2(x) = 1 ∀ x, that is obtained from the  standard coboson ansatz for λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  E  Correlations in momentum space  One can easily extend the results from the previous appendices to momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In order  to do this, we resort to the expresion of the coboson wavefunctions in momentum space:  �ψn(k1, k2) =  �  2  πλ  xωe−iπn/2  1 +  �  k1−k2  2λ  �2 ϕn[x2  ω(k1 + k2)],  (54)  which is just the Fourier transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' These functions are of order  �  xω/λ and decay  in a scale of order λ for the relative momentum (k1 − k2)/2 and of order √n/xω for the  center-of-mass momentum k1 + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  19  SciPost Physics  Submission  From this expression one can derive formulas for the correlations in momentum space  following similar steps as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' One should only keep in mind that, in contrast to position  space, the wavefunctions in momentum space are complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In particular, we find for the  momentum correlations between fermions of different kinds an equation which is analogous  to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (45):  �Dab(k, k′) =  �  m≤n  �  j≤l  cmn cjl  ��  δnl �ψ∗  m(k, k′) �ψj(k, k′)+ �J(nl)  1  (k) �J(mj)  1  (k′)− �ψ∗  m(k, k′) �J(jl|n)  3  (k, k′)  − �ψj(k, k′)[ �J(mn|l)  3  (k, k′)]∗�  + same with n ↔ m, j ↔ l, and {n, l} ↔ {m, j}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (55)  Here, the asterisk denotes a complex conjugation and we have defined:  �J(nl)  1  (k) =  �  dk′ �ψ∗  n(k, k′) �ψl(k, k′),  (56)  and  �J(jl|n)  3  (k, k′) =  �  dqdq′ �ψj(k, q′) �ψl(q, k′) �ψ∗  n(q, q′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (57)  Replacing the form of the wavefunctions in momentum space one finds the integral ex-  pression:  �J(nl)  1  (k) = 2x2  ω  πλ eiπ(n−l)/2  �  dk′ ϕn(x2  ωk′)ϕl(x2  ωk′)  �  1 + ( k′−2k  2λ )2�2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (58)  Taking into account the restriction on the values of n, l within our basis, one can perform  a Taylor expansion in k′/λ in the expression above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We stress that the values of k cannot  be assumed to be much smaller than λ, since λ is indeed the typical scale for the relative  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In this way we obtain:  �J(nl)  1  (k) ≃ 2x2  ω  πλ eiπ(n−l)/2  �  dk′ϕn(x2  ωk′)ϕl(x2  ωk′)  �  ��  1  �  k2  λ2 + 1  �2 +  2kk′  λ2  �  k2  λ2 + 1  �3 −  (1 − 5k2  λ2 )k′2  2λ2  �  k2  λ2 + 1  �4  �  �� ,  (59)  which can be evaluated using properties of the Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The integrals for �J3 can be cast in the form:  �J(jl|n)  3  (k, k′) =  �2x2  ω  πλ  �3/2  e−iπ(j+l−n)/2  �  dqdq′  ϕj(x2  ωq′)ϕl(x2  ωq)ϕn[x2  ω(q + q′ − k − k′)]  �  1 + ( q′−2k  2λ )2  � �  1 + ( q−2k′  2λ )2  � �  1 + ( k−k′+q−q′  2λ  )2  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (60)  Performing a Taylor expansion here is justified for the q, q′ divided by λ in the denominator,  but not for the same variables inside the wavefunction ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This makes this calculation much  20  SciPost Physics  Submission  more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The Taylor expansion of the denominator gives:  �J(jl|n)  3  (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' k′) ≃  �2x2  ω  πλ  �3/2  e−iπ(j+l−n)/2  �  dqdq′ϕj(x2  ωq′)ϕl(x2  ωq)ϕn[x2  ω(q + q′ − k − k′)]  �  ��  1  �  k2  λ2 + 1  �2 �  k′2  λ2 + 1  �2 �  (k−k′)2  4λ2  + 1  �2  +  (k − 3k′)  �  k′(k−k′)  2λ2  − 1  �  λ2  �  k2  λ2 + 1  �2 �  k′2  λ2 + 1  �3 �  (k−k′)2  4λ2  + 1  �3 q +  (3k − k′)  �  k(k−k′)  2λ2  + 1  �  λ2  �  k2  λ2 + 1  �3 �  k′2  λ2 + 1  �2 �  (k−k′)2  4λ2  + 1  �3 q′  �  �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  (61)  Here one is still left with a non-trivial integral in q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' This can be solved using the decom-  position formula  ϕn (x + y) =  ∞  �  i,j=0  Aij|nϕi (x) ϕj (y) ,  (62)  where the coefficients  Aij|n =  � �  ϕn (x + y) ϕi (x) ϕj (y) dxdy  (63)  can be evaluated using properties of the Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' For numerical evaluation this  summation must be truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Performing this one up to i, j = 100 good approximations are  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Then we are left with terms similar to those found in the calculation for �J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' (55) contains contributions of order xω/λ which decay in a scale  of order λ for the relative momentum (k − k′)/2 and of order 1/xω for the center-of-mass  momentum k + k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The second term, involving �J1, contains contributions of order 1/λ2 which  decay on a scale of order λ for k and k′ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' We note that this contribution is broad and  has a Lorentzian decay, whereas the decay of the contributions in the first term is Gaussian  for the center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Thus, they may be of the same order depending on the point where  they are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' In any case, the dominant feature is the anti-diagonal resulting from the  first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' The remaining terms, containing �J3, have a similar behaviour as the first (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  with a strong anti-diagonal) but are one order smaller in 1/(λxω), which justifies using an  expansion for �J3 to a lower order than for �J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content='  References  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
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+page_content=' Bayha, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Subramanian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
+page_content=' Heintze, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfEgbW/content/2301.03637v1.pdf'}
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+rsta.royalsocietypublishing.org
+Research
+Article submitted to journal
+Subject Areas:
+Fluid Mechanics - Physics
+Keywords:
+Elasto-inertial instability, Experiments,
+Torque scaling, Friction dynamics
+Author for correspondence:
+S. Amir Bahrani: e-mail:
+amir.bahrani@imt-nord-europe.fr
+Friction dynamics of
+elasto-inertial turbulence in
+Taylor-Couette flow of
+viscoelastic fluids
+M. Moazzen1, T. Lacassagne1, V. Thomy2
+and S. A. Bahrani1
+1IMT Nord Europe, Institut Mines-Télécom, Univ. Lille,
+Centre for Energy and Environment, F-59000 Lille,
+France
+2Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique
+Hauts-de-France, UMR 8520 - IEMN – Institut
+d’Electronique de Microélectronique et de
+Nanotechnologie, F-59000 Lille, France
+Dynamic properties of elasto-inertial turbulence (EIT)
+are studied in a Taylor-Couette geometry. EIT is
+a chaotic flow state that develops upon both non-
+negligible inertia and viscoelasticity. A combination
+of direct flow visualisation and torque measurement
+allows to verify the earlier onset of EIT compared to
+purely inertial instabilities (and inertial turbulence).
+The scaling of the pseudo-Nusselt number with
+inertia
+and
+elasticity
+is
+discussed
+here
+for
+the
+first
+time.
+Variations
+in
+the
+friction
+coefficient,
+temporal frequency spectra, and spatial power density
+spectra highlight that EIT undergoes an intermediate
+behavior before transitioning to its fully developed
+chaotic state that requires both high inertia and
+elasticity. During this transition the contribution of
+secondary flows to the overall friction dynamics is
+limited. This is expected to be of great interest in the
+aim of achieving efficiency mixing at low drag and
+low but finite Reynolds number.
+©
+The Authors.
+Published by the Royal Society under the terms of the
+Creative Commons Attribution License http://creativecommons.org/licenses/
+by/4.0/, which permits unrestricted use, provided the original author and
+source are credited.
+arXiv:2301.02047v1  [physics.flu-dyn]  5 Jan 2023
+
+2
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+1. Introduction
+The Taylor-Couette (TC) geometry consists in two concentric cylinders, with either one or both
+cylinders rotating. Since the seminal work of Taylor [1], it has been extensively used by researchers
+[2] thanks to its simplicity of use which allows to easily study flow instabilities. In the most
+familiar case where only the inner cylinder is rotating and the outer cylinder is stationary, a
+non-dimensional control parameter is the Reynolds number, which is defined as R = ρΩriδ/µ,
+where µ, ρ, Ω are the dynamic viscosity, fluid density and the rotational speed of the inner
+cylinder, respectively, and δ = ro − ri is the gap width, with ri and ro the inner and outer
+radii. The geometry can be characterised by two non dimensional parameters: the aspect ratio
+Γ = h/δ and radius ratio η = ri/ro, where h is length of the cylinder. Alternatively, the curvature
+ratio κ = ri/δ can be used. The aforementioned parameters are known to have an influence on
+stability, not only in Newtonian but also in non-Newtonian fluids. Indeed, the TC geometry is
+also widely used in the study of complex fluids [3, 4], such as dilute polymer solutions [5, 6]
+or suspensions of particles [7] among others. In particular, polymeric liquids flow which exhibit
+viscoelastic behavior have been studied with great interest, due to the existence of sets of specific
+flow regimes, and motivated by the ubiquity of viscoelasticity in daily life, industrial and natural
+applications, such as biology, pharmaceutics, paints, among others [8].
+The mechanism of instability in these fluids are different from those that occur in Newtonian
+cases. In Newtonian fluids, the instability comes from the destabilizing effect of the centrifugal
+force gradient (which comes from variations of kinetic momentum), and overcoming of it on the
+stabilizing effect of viscous drag force. In such fluids, at low Reynolds number, a purely azimuthal
+uniform shear flow develops, which is called circular Couette flow (CCF). It eventually becomes
+unstable upon increasing R as explained above, and secondary flows appear as axisymmetric
+counter-rotating vortices called Taylor vortex flow (TVF). Further increase of Reynolds number
+creates non-axisymmetric sinusoidal axial oscillations called wavy Taylor vortex flow (WVF).
+Eventually, additional wavelentghs appear and the flow transitions to turbulence [9, 10, 11, 12, 13].
+In non-Newtonian, viscoelastic fluids, the mechanism of instability, and subsequently its flow
+transition, is different. Polymers solutions are common viscoelastic fluids. Polymers are high
+molecular weight molecules made of a large number of monomers connected with covalent
+bonds, resulting in long linear, branched or network chains [14]). The arrangement (conformation)
+of the polymer chain at the rest condition is in the way that have maximum conformational
+entropy [15]. When the polymer coil is stretched, e.g because of an applied stress or deformation,
+it tends to recover its lost maximum entropy energy and return to its equilibrium chain structure.
+Due to this entropic tendency of polymers, elastic stresses are created in the chain which as a
+result of the stress difference between the flow direction and the direction perpendicular to it
+(direction of shear), which doesn’t exist in Newtonian fluids. In rotational flow such as Taylor-
+Couette Flows (TCF), curved streamlines induce a hoop stress, balanced by an adverse pressure
+gradient in the radial direction. A flow perturbation may cause a fluid particle to move towards
+a region of enhanced stretching, enhancing the local hoop stress and destabilizing the flow [16].
+Moreover, a part of the chain’s elastic deformation energy can be released elsewhere in the flow,
+further promoting instability. The elastic behavior of polymer solutions thus highly depend on
+deformation rates, and relaxation time of polymer chains. The degree of elastic response of a
+fluid subjected to a shear rate ˙γ is quantified by the Weissenberg number Wi, defined in the
+case of TCF as Wi = λe ˙γ with ˙γ = Ωri/δ the nominal shear rate in the gap. The elastic number
+El is then defined by El = Wi
+R = λe
+λv = λeµ
+ρδ2 and represents the competition between inertial and
+elastic effects, with λv = ρδ2/µ the viscous characteristic time. The resulting El depends only on
+the geometrical parameters and the properties of the fluid (which may themselves be shear-rate
+dependent, see below). El allows to classify fluids into 3 groups: weak (El < 10−2), moderate
+(10−2 < El < 1) and strong elasticity (El > 1) [17, 18, 19]. Based on the elasticity level, various
+instability and transition scenarios are observed. In the range of very low elasticity (i.e, El ≪ 1)
+the elastic effects are very weak compared to inertia effects and observed flow transition are
+
+3
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+comparable to the Newtonian case (CCF→TVF→WVF) as R increases [17, 20] with slightly
+shifted critical conditions Rc because of presence of light amount of polymer.
+At high values of elasticity, in the case of vanishing R, a purely elastic CCF-TVF transition
+is observed [21]. Subsequently in the case of elevated Wi another transition will lead to a
+chaotic regime called elastic turbulence [22, 23, 24], which exhibits turbulent like-characteristics
+in absence of inertia. When neither R nor El can be neglected, we find ourselves in the domain
+of elasto-inertial transitions. In particular, primary and secondary elasto-inertial instabilities
+manifest themselves in non-axisymmetric flow states [18, 19, 25, 26, 27, 28]. An increase in inertia
+(R) or elasticity (El) leads these pre-chaotic behaviors to transition to strongly unsteady states:
+"disordered oscillations" (DO) [27], "defect mediated turbulence (DMT) [19], "spatio-temporal
+intermittency" (STI) [29] or "merge-split transitions" (MST) [28], and all contribute to a gradual
+transition to elasto-inertial turbulence (EIT) [6, 18, 28]. A summary of several flow transition
+observed experimentally, as a function of geometrical parameter and viscoelastic fluid properties,
+are listed in table 1.
+The possibility of triggering such chaotic behavior opens extremely interesting perspectives
+in terms of mixing and intensification of transfers at low R. While transition scenarios are now
+relatively well identified in the literature, several questions remain to be tackled: what are the
+characteristics of EIT in TCF? What is the dynamic behavior of these flows in terms of friction and
+energy dissipation ? This work aims at addressing this last point in particular, by reporting for the
+first time friction and spatio-temporal properties of TCF of constant viscosity polymer solutions
+with shear-dependent viscoelasticity.
+2. Materials and methods
+Experiments were performed using aqueous Boger solutions of high molecular weight polymer
+of partially hydrolysed polyacrylamide (HPAM, Mw = 15 − 20 × 106 g/mol). At first, a stock
+aqueous polymer solution of 1000 ppm was prepared. Samples from this solution were then
+dissolved in pure water and mixed in glycerol in order to obtain different concentrations ˜cp of
+25, 50, 100, 150, 200, 250, 300, 350 ppm with base solution similar to that of our previous study [7]:
+41.8 % glycerol and 58.2% water (in volume) and 12.7% of salt (in mass). After preparation, the
+aqueous solutions are left at rest for 24 h before performing any other manipulation.
+Base fluid
+Figure 1. (a) Shear viscosity as a function of shear rate for HPAM polymer solution at various concentrations. The
+observed jump in viscosity at a concentration of 350 ppm (after 600 s−1) is related to an elastic instability in the rheometer.
+(b) Measured first normal stress N1 of the solutions. A line with slope of 2 is plotted as a guide to the eye. (c) Relaxation
+time obtained using our experimental methodology. Colour lines are fits to the experimental data.
+Rheological behavior of all working fluids was characterized using a rotational rheometer
+(Anton-Paar MCR 302) equipped with a cone-plane geometry (50 mm/1◦) with truncation gap of
+
+4
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+Study
+η
+Γ
+δ (mm)
+κ
+solution
+µp/µ
+El
+Rmax
+dR/dt∗
+FTO
+Present study
+0.914
+10
+1.5
+0.093
+HPAM/W
+0.075
+0.05-0.1
+200
+<0.01
+CCF-SVF-RSW-MWVF
+/G/S
+0.086-0.221
+0.11-0.35
+CCF-EIT
+Lacassagne et.
+0.77
+21.56
+6.26
+0.289
+PAAm/W/G
+0.148
+0.2263
+200
+0.33
+CCF-TVF-RSW-EIT
+al., (2020) [28]
+Martinez-Arias &
+0.909
+30
+5
+0.1
+PEO/PEG
+0.083-0.325
+0.06-0.17
+160
+<0.6
+CCF-TVF-RSW-EIT
+Peixinho (2017)[34]
+/W/IPA
+0,78
+0.71-1.09
+84
+CCF-RSW–EIT
+Dutcher et.
+0.912
+60.7
+6.69
+0.096
+PEO/W/G
+2.82
+0.1-0.21
+200-250
+0.68
+CCF-TVF-RSW-EIT
+al., (2013) [18]
+Dutcher et.
+0.912
+60.7
+6.69
+0.096
+PEO/W/G
+0.3
+0.00047
+200-250
+<0.68
+CCF-TVF-WVF-...-TTV
+al., (2011) [17]
+0.93
+0.0017
+CCF-TVF-WVF-MWVF-WVF-MWF-CWV-WTV-MT
+0.92
+0.0054
+CCF-TVF-WVF-MWVF-WVF-WTV / CWV
+0.78
+0.023
+CCF-TVF-WVF-MWVF-WVF
+Crumeyrolle et.
+0.883
+47
+5.9
+0.132
+PEO/W
+0 - 3.45
+0.002-0.03
+200
+N/A
+CCF-TVF-WVF
+al., (2005) [35]
+5.32-12.4
+0.07-0.5
+CCF-RSW
+Groisman et.
+0.829
+74
+7
+0.2
+PAAm/W
+0.82
+0.025
+N/A
+N/A
+CCF-TVF-WVF
+al., (1998) [36]
+/saccharose
+0.03-0.08
+CCF-TVF-RSW-DO
+0.09-0.15
+CCF-DO
+0.2-27
+CCF-DO
+Groisman et.
+0.708
+54
+7.85
+0.413
+PAAm/W
+0.008-0.25
+0.1-0.15
+N/A
+N/A
+CCF-TVF-RSW-DO
+al., (1996) [27]
+/saccharose
+0.15-0.22
+CCF-TVF-DO
+0.22-0.34
+CCF-DO
+Groisman et.
+0.708
+54
+7.85
+0.413
+PAAm/W
+0.78
+0.023-0.033
+N/A
+N/A
+CCF-RSW-DO
+al., (1993) [37]
+/saccharose
+Table 1. Some experimentally observed flow transition patterns in viscoelastic Boger fluids (constant viscosity µ assumed) with different fluid properties and geometrical parameters (rotating inner
+cylinder and stationary outer cylinder). DO = Disordered Oscillations, FP = Flame Pattern, SVF = Spiral Vortex Flow, MWVF = Modulated Wavy Vortex Flow, MT = Modulated Turbulence, RSW =
+Rotating Spiral Waves, TTV = Turbulent Taylor Vortices, CWV = Chaotic Wavy Vortex Flow. µp = µ − µs is the polymer contribution to the total viscosity. κ = δ/ri is the curvature ratio. W = Water, G
+= Glycerol, PEO = Polyethylène Oxyde, PAAm = Polyacrylamide, PEG= Polyethylène Glycol, IPA = Isopropyl Alcohol. N/A = not available.
+
+5
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+θ = 0.104 mm at a constant temperature of 22◦C. In order to find the viscosity of samples, steady-
+shear viscosity measurements were performed on a shear-rate interval of 0.01/s < ˙γ < 800/s.
+Samples viscosity remained constant over this range of shear rates. The flow curves for all samples
+are reported in figure 1. In order to evaluate the dynamic, shear rate dependent, relaxation times
+λe, the protocol detailed by [30], based on normal force measurement, was used. It consisted of
+several incremental steps during which constant shear rate was imposed. Subtracting the means
+of two values of the first normal stress difference N1 for each step gives a way to resolve the
+instrumental drift of the normal force and correct N1( ˙γ) values. A second correction is performed
+to remove the contribution of fluid inertia to the normal force given by the rheometer N1,tot,
+such that N1 = N1,tot + 0.15ρΩ2R2 with Ω = tan (θ) ˙γ the angular velocity (the 0.15 prefactor
+corresponds to inertial and secondary flow corrections and was proposed by [31]). N1 can be
+expressed as a power-law function of the shear rate; N1 = Ψ ˙γϵ, where Ψ and ϵ are constants.
+ϵ = 2 implies that the behavior follows the Oldroyd-B model [32]: N1 = 2(µ − µs)λe ˙γ2, and
+the viscosity µ is dominated by the Newtonian solvent contribution, µs. However, unlike the
+Oldroyd-B model, the relaxation time here also follows a power law (shear-rate dependent)
+function as λe = a ˙γb, where a and b are constants. As polymer concentration increases, b become
+more negative which means that λe becomes more sensitive to shear rate. A summary of the
+aforementioned coefficients is shown in table 2. This advanced viscoelasticity characterisation
+protocol allows to account for the effective shear-dependency of λe, and thus El, in constant
+viscosity fluids and thus increase the accuracy on the critical El values detection. As expected
+from [33] the method performs better for higher polymer concentrations, with less noise on the
+N1 and λe data. It here results in a poor fitting of λe data for the 25 ppm case only.
+Coef.
+˜cp (ppm)
+25
+50
+100
+150
+200
+250
+300
+350
+λe
+a
+0.057
+0.205
+0.265
+0.367
+0.42
+0.48
+0.52
+0.59
+b
+-0.1
+-0.28
+-0.3
+-0.34
+-0.35
+-0.36
+-0.37
+-0.375
+N1
+Ψ
+6.5e−5
+1.8e−4
+3.8e−4
+8e−4
+1.3e−3
+1.5e−4
+3.4e−3
+3.9e−3
+ϵ
+1.9
+1.8
+1.75
+1.7
+1.65
+1.65
+1.55
+1.55
+Table 2. Rheological parameter of HPAM polymer solution derived by fitting an Oldroyd-B model in order to find relaxation
+time, λe. The solutions follow the relation of λe = a ˙γb and N1 = Ψ ˙γϵ, where b and ϵ govern the viscoelastic behavior.
+The TCF experiments were performed in a Taylor-Couette cell mounted on the same rheometer
+as illustrated in figure 2. The geometrical parameters were: δ = 1.5 mm, η = 0.914 and Γ = 10.
+In the present study, only ramp-up experiments were performed (slow acceleration of the inner
+cylinder), combining torque measurements and visualisation using Iriodin particles (∼ 0.1% in
+mass) and a light source, following a protocol detailed in our previous work [7]. The inner
+cylinder acceleration rate was 0.0082 < dR
+dt∗ = ρ2riδ3
+µ2
+dΩ
+dt < 0.01, and the temperature was 22 (±
+0.4) °C.
+3. Results and discussion
+(a) Torque measurements
+Let T be the raw measured torque on the rheometer shaft and G = T /Tv = T /2πh
+�
+µ2/ρ
+�
+the
+dimensionless torque. G is thus simply a non-dimensional version T , scaled by torque-scale Tv.
+Figure 3 shows plots for G as a function of R (left) and Wi (right) for all polymer solutions. The
+discontinuity in G values indicates the onset of a secondary flow, as will be discussed in section
+(b). It appears that G increases with both increasing R and increasing Wi, since the shear-rate
+
+6
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+Spotlight
+EIT regime
+ snap shot
+h
+ro
+ri
+Outer cylinder
+Inner cylinder
+Air circulation
+HPAM solution
+Shaft
+Figure 2. Schematic of the Taylor-Couette apparatus mounted on the Anton-Paar MCR-102 rheometer with camera and
+light position for visualisation. As shown in the figure, the upper end and lower end of the gap are a free surface and a
+stationary wall, respectively.
+Figure 3. Plots of G as a function of R (left) and Wi (right) for all polymer concentrations ˜cp.
+increases. However, the increase rate of G after the discontinuity is reduced as ˜cp increases, which
+is a key feature of the drag dynamics of the unsteady flow state, as will be detailed in sections (c)
+and (d).
+(b) Transitions, flow states, and critical R
+Flow maps diagram (space-R diagrams), for ˜cp= 25, 100, 200, 350 ppm, coupled with plots of
+Wi and of the effective (pseudo) Nusselt number, N as a function R is presented in Figure 4.
+Wi increases with yet a decreasing slope (all the more decreasing that ˜cp increases), due to the
+shear-rate dependency of λe. N is defined as N = T /Tlam = G/Glam, with Glam = 2ηR/(1 +
+
+7
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+(a) 25 ppm
+(c) 200 ppm
+(b) 100 ppm
+(d) 350 ppm
+CCF
+CCF
+CCF
+CCF
+EIT
+EIT
+EIT
+Figure 4. Space-R diagram (top), Nusselt-R (bottom plots, left axis) and Wi-R (bottom plots, right axis) plots. The plots show the transitions from CCF to RSW (Rotating Spiral Waves) and TVF for
+a concentration of (a) 25 ppm and from CCF to EIT for (b) 100 ppm (c) 200 ppm (d) 350 ppm.
+
+8
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+η)(1 − η)2 the dimensionless torque for the laminar flow between infinitely long cylinders. N
+allows to go further into the interpretation, compared to G, by scaling the (non-dimensional)
+torque by its laminar value, which accounts for the geometry. It represents the dissipation rate
+of kinetic energy [38] and the ability of the flow to convey momentum radially. Critical values
+for all numbers (subscript .c) are found combining flow visualisation and torque measurement
+(jump in N values [34]). Here, a CCF-EIT transition is observed for ˜cp ≥ 50 ppm (corresponding
+El50
+c = 0.13), as can be seen clearly in both flow map (changing alignment of Iriodin flakes from
+purely azimuthal to random) and N − R diagram. This direct transition was previously observed
+by [27]. N related to CCF (R < Rc) is almost constant and is around one. This shows that the CCF
+torque depends linearly on viscosity (and viscosity does not depend on the shear rate variation).
+The abrupt change in N values clearly indicates the beginning of the EIT regime.
+Figure 5 shows the Rc at which the CCF-EIT transition (or the primary transition, for the 25
+ppm case) occurs as a function of ˜cp. The side color bar indicates the corresponding Elc. Rc for the
+25 ppm case is 134 or R25
+c = 0.88R0c, demonstrating that the CCF flow is destabilized compared
+to the Newtonian case (base solvent of this polymer for which the transition occurs at R0c = 150.8
+in this setup [7]). Rc decreases as the polymer concentration or the elastic number increases. Up
+to 150 ppm or El = 0.18, this reduction is very strong while for ˜cp > 150 ppm or El > 0.18, it
+becomes milder, which suggests that the presence effect of polymer after this concentration is less
+effective. The lower value of Elc for direct CCF-EIT (El50
+c = 0.12) is here slightly lower than the
+value of 0.22 reported by [27]. This minor discrepancy can be ascribed to the different relaxation
+time estimation protocol used (ours accounting for shear-rate dependency) and to the variations
+in geometrical parameters [39, 40].
+Figure 5. Rc for the onset of EIT (or for the primary instability, in the 25 ppm case) as a function of the ˜cp with marker
+colours indicating Elc values at the onset of EIT
+(c) Torque scaling in EIT
+Figure 6 shows all curves for N − Ta at all polymer concentrations, essentially revisiting the raw
+data from figure 3 but this time normalizing by laminar flow behaviour as allowed by the use of
+N. Ta = (1+η)6
+(64η4) R2 is proportional to R2 by a geometric constant. Both numbers can be equally
+used to quantify flow inertia when the geometry is kept constant, but Ta is more frequently
+
+9
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+encountered in the literature for Taylor-Couette Nusselt scalings. The color map indicates the R-
+dependent El value. At low Reynolds number, in the CCF state, all N are approximately constant,
+as expected. The slight variation in Nusselt number values (around 1) is due to wall effects that
+introduce an additional torque [41]. Increasing R, up to the instability limit, there is a slight
+increase in the slope of N, as noted previously by [34] or [42]. Passing through the critical point,
+an abrupt change in the value of N occurs. This jump intensifies with increasing ˜cp. Increasing
+the ˜cp, the overall value of the Nusselt number increases.
+Interestingly, after the onset of EIT and as ˜cp increases, the global slope of the N-Ta curve in
+the EIT regime gradually decreases, evolving to a Ta-independent Nusselt number region. This
+evolution occurs faster as the concentration of polymer increases: the rate of change and slopes
+are concentration dependent, but the asymptotic behaviour appears not to be.
+This can be interpreted as follows. After the onset of EIT, secondary chaotic flows arise and
+generate friction at the walls leading to a global increase in N. Increasing Ta or R, kinetic energy
+is injected in the flow. It is either dissipated by wall friction, which translates into an increase
+in N, or by elastic dissipation by the polymer chains, which is expected not to depend on R.
+Increasing ˜cp comes to promoting the second mechanism over the first, reduce the share of kinetic
+energy dissipated by viscosity, and thus the increase in N. This last point can be examined from
+another angle: by qualitatively observing the elastic threshold below which the transition to the
+asymptotic behaviour is gradual (and not sharp), it can be infered that even after the onset of
+EIT, the flow still requires a given amount of inertia and/or elasticity, i.e. a given Wi increase, for
+elastic energy transfers to balance inertial ones.
+Figure 6. N − Ta curve for all concentrations with El color bar that illustrates the evolution of the elastic number during
+the change of Ta.
+(d) Elasto-inertial drag coefficient
+In order to compare the data with Newtonian (laminar or turbulent) experiments and references
+[43], curves from figure 6 can be re-scaled to display the friction coefficient CMz defined as CMz ∼
+N/R [7], shown in figure 7.
+In the Newtonian case, the onset of TVF is known to stop the CMz ∼ R−1 decrease (see
+dashed line in figure 7), and the onset of WVF to make CMz decrease again with R. In TVF, the
+
+10
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+drag increases much faster than what is expected from laminar assumption, in WVF, it increases
+faster (but not so much): TVF is very efficient in conveying momentum radially, WVF not that
+much, since part of the energy is involved in axial motion. A drag measurement experiment was
+conducted for a Newtonian fluid (water) and is reported figure 7, for a turbulent state (R ≫ 1800).
+The slope of CMz is compared with Von Kármán gap theory [44]. The results show that our CMz
+follows the 1/�CMz = 3 ln(R
+�
+CMz) − 2.7 corresponding to ∼ R−1/3 against the work of [43].
+Back to the viscoelastic case, for the higher El fluids, the onset of EIT leads to a sharp
+increase in CMz and N. Yet, after this sharp onset, N tends to a constant value (see figure 6),
+and the friction coefficient asymptotically moves back to its "laminar" evolution, approximately
+(∼ R−1), as shown in figure 7. This suggests that the additional dissipation induced by the
+polymer chains and additional radial momentum transport is not Reynolds-dependent. The R-
+dependency on friction can be scaled by that of the laminar case, with no significant R-dependent
+contribution of the viscoelastic secondary flows. This supports the very recent DNS (Direct
+Numerical Simulation) of [45] suggesting that elasto-inertial TCF structures are not efficient in
+radial momentum convection, which is why they tend to merge and split [28] or create defects
+[19], thus helping transition to chaos. It is yet worth noting that the jump, i.e the additional
+dissipative contribution of the polymer, is itself El-dependent: the jump is higher when the elastic
+number increases. Between the jump and the asymptotic high El region, there is a transitional
+behaviour which can be understood as the establishment period of EIT. This establishment period
+displays a different slope, about R−2/3 for the 50 ppm case as illustrated on figure 7. For 25 ppm,
+the constant CMz on a narrow range shows the existence of intermediate regimes visually closer
+to TVF (RSW for Rotating Spiral Waves, see [28]), and apparently also in terms of torque dynamics
+(see [7]). Results also show that EIT is different from inertial turbulence modified by polymer for
+which drag is reduced. It would be interesting to see what happens when R tends to values for
+which inertial turbulence is expected, as done in [46] and confront with drag reduction theory.
+0
+6
+Turb.
+Figure 7. Friction coefficient CMz as a function of R for all polymer concentrations (curve labels). The color scale
+represents El values and the dashed line the -1 slope characteristic of a laminar flow. Slopes for R−2/3 and R−1 are
+illustrated by dotted lines and triangle, orange and red, respectively. In inset, an experiment for pure water in turbulent
+flow state (full black line) is reported along with its ∼ R−1/3 asymptotic trend and compared with the ∼ R−1/4 trend
+of [43]. The gray part of the curves correspond to points below the rheometer’s minimum torque capacity and should be
+discarded.
+
+0.2
+0.15
+0.1
+0.05
+20
+50
+100
+200
+R
+0.05
+0.1
+0.15
+0.2 
+0.25
+0.3
+0.350.2
+0.15
+0.1
+0.05
+20
+50
+100
+200
+R
+0.05
+0.1
+0.15
+0.2 
+0.25
+0.3
+0.350.2
+0.15
+0.1
+0.05
+20
+50
+100
+200
+R
+0.05
+0.1
+0.15
+0.2 
+0.25
+0.3
+0.3511
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+(e) Spatio-temporal analysis: frequency maps and spatial PSD
+Following the protocol detailed in [7], frequency maps for ˜cp = 25, 100, 200, 350 are shown in
+figure 8. Those plots shows the temporal FFT spectra of the reflected light intensity signal, for
+all R stacked vertically, the colorbar representing the intensity of the spectra (arbitrary units).
+Frequency maps display no particular spectral signature in the CCF domain, before Rc is reached,
+other than that of the inner cylinder rotation frequency fcylinder and its harmonics [28, 47]. Such
+ridges are still slightly visible for R > Rc, but the overall spectral signature changes, with a
+broadband distribution of energy from large to small temporal scales, a signature of the chaotic
+behaviour characteristic of EIT [18, 27, 28].
+CCF
+CCF
+CCF
+CCF
+EIT
+EIT
+EIT
+Figure 8. Frequency maps computed as [7] or [28], from the reflected light intensity signal, for all vertically stacked R. The
+color scale represents the FFT intensity from weak (yellow/light) to strong (purple/dark) energy content. Instantaneous
+change in intensity indicates the transition from CCF to EIT and the oblique dashed lines indicate the rotation frequency
+of the inner cylinder. For 25 ppm the first instability is RSW which results in the appearance of discrete peaks that are no
+longer seen in EIT.
+Further analysis of the spectral behaviour of EIT can be made by computing spatial power
+spectral density (PSD) of the intensity signal at constant Rc and El. This is achieved by extracting
+sets of n vertical lines In(z) at constant R on flow maps (e.g from figure 4), subtracting the average
+intensity profile ⟨I⟩ (z) = 1
+n
+�n
+j=1 Ij(z) in order to define the intensity fluctuation (i′)n(z) =
+In(z) − ⟨I⟩ (z), computing PSD((i′)n(z)) and averaging PSDs on n. Values are finally scaled by
+the spatial (over z) average of ⟨I⟩ (z) called I0. One typically uses n = 50, a number for which
+the convergence of spectra was deemed sufficient for the analysis that follows. PSD spectra for
+all polymer concentrations at R/Rc ≃ 1.2 are reported in figure 9 a), and spectra for the 350 ppm
+case at various R values in figure 9 b). An additional experiment was performed with water,
+in order to reach R ≃ 5100 and produce a flow map and PSD in the inertial turbulence regime
+(inset of figure 9 a). For that curve, one would expect to capture a -5/3 slope if i) turbulence is
+sufficiently developed and ii) the intensity fluctuation signal is in some way representative of the
+radial velocity fluctuations, as suggested by [48]. It appears from the inset of 9 a) that the inertial
+turbulence curve roughly follows the -5/3 trend at least at intermediate scales, which seems to
+validate both arguments. When considering spectra for EIT, the -5/3 slope is not expected in low
+R cases where inertial turbulence would not have been present. Different scaling exponents of
+EIT in various flow configurations (channel flows, TC flows...) can be found in the literature, in a
+range from -14/3 to -3, always steeper than the inertial case. The -3 slope has recently been put
+forward by [49] in their study of viscoelastic polymer jets as a universal spectral behaviour of EIT,
+after having been theoretically predicted by [50]. Recent numerical simulations by [45] have been
+
+12
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+able to retrieve this scaling in elasto-inertial TCF. This reference slope is also represented on figure
+9 b).
+Figure 9. Spatio-temporal dynamics: a) Spatial PSD spectra for all polymer concentrations at R/Rc = 1.2, with an inset
+showing typical spectrum for inertial turbulence in water. b) Spatial PSD spectra for ˜cp = 350 ppm at various R dotted
+and dashed black lines in b) denote k−5/3 and k−3 slopes [49, 50], respectively. The dotted blue line in a) and b) is a
+-7/3 slope fitting the experimental data.
+It here appears that all curves for EIT fall short of -3 trends, but still display a somehow
+universal slope with respect to polymer concentration (figure 9 a). An increase in Reynolds
+number seems to consolidate this slope by increasing the k span over which it applies (figure 9 b).
+The slope value is here around -7/3 (figure 9 a and b). It is worth mentioning that the visualisation
+method probes the flow from the outside and not in the bulk, and may be subject to boundary
+layer effects on the outer cylinder. So the value of the slope itself must be interpreted with care.
+Figure 9 yet confirms key findings [49] namely that of an apparent universal spectral slope of
+EIT, steeper than inertial turbulence. Additionally, our results suggest that EIT is intrinsically a
+combination of elasticity and inertia, as R helps develop the slope when increased.
+4. Conclusion
+In this work, new characterisations of the dynamics of EIT in Taylor-Couette flow of polymer
+solutions were presented. Combining flow visualisation and torque measurements allowed to
+detect CCF-EIT transition and to describe key dynamic features of EIT in TCF. In particular the
+scaling of N with Ta and its dependency on fluid elasticity has been discussed. Two sub-domains
+of EIT were reported: a transitional one for which energy dissipation is still dominated by inertia
+and a fully developed one for which elastic energy transfer become dominant. Spectral analysis
+support the idea of a chaotic developed EIT state for which PSD would exhibit a universal slope.
+Developed EIT displays an asymptotic "laminar-like" scaling for the friction coefficient: the wall
+friction is directly correlated to the base azimuthal flow and secondary flow structures do not
+play a role in wall friction as their energy is dissipated elastically. This requires a sufficient level
+of both elasticity and inertia, and is expected to be of great interest in the aim of achieving efficient
+mixing at low drag and low R.
+Authors’ Contributions. MM performed, analysed, interpreted the experimental work reported in this
+paper, and drafted the initial manuscript. TL and AB equally contributed to the design of the experiments
+(together with MM), conceptualisation of the project, guidance, and supervision of the work (together with
+
+13
+rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+VT). AB and VT secured funding. All authors contributed to the writing, reading, editing and approved the
+manuscript.
+Competing Interests. The author(s) declare that they have no competing interests.
+Funding. Financial support from Region Hauts de France and Institut Mines Télécom is gratefully
+acknowledged.
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+
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Bahrani1  1IMT Nord Europe, Institut Mines-Télécom, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Lille,  Centre for Energy and Environment, F-59000 Lille,  France  2Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Lille, CNRS, Centrale Lille, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Polytechnique  Hauts-de-France, UMR 8520 - IEMN – Institut  d’Electronique de Microélectronique et de  Nanotechnologie, F-59000 Lille, France  Dynamic properties of elasto-inertial turbulence (EIT)  are studied in a Taylor-Couette geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' EIT is  a chaotic flow state that develops upon both non-  negligible inertia and viscoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A combination  of direct flow visualisation and torque measurement  allows to verify the earlier onset of EIT compared to  purely inertial instabilities (and inertial turbulence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  The scaling of the pseudo-Nusselt number with  inertia  and  elasticity  is  discussed  here  for  the  first  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Variations  in  the  friction  coefficient,  temporal frequency spectra, and spatial power density  spectra highlight that EIT undergoes an intermediate  behavior before transitioning to its fully developed  chaotic state that requires both high inertia and  elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' During this transition the contribution of  secondary flows to the overall friction dynamics is  limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This is expected to be of great interest in the  aim of achieving efficiency mixing at low drag and  low but finite Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Introduction  The Taylor-Couette (TC) geometry consists in two concentric cylinders, with either one or both  cylinders rotating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Since the seminal work of Taylor [1], it has been extensively used by researchers  [2] thanks to its simplicity of use which allows to easily study flow instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In the most  familiar case where only the inner cylinder is rotating and the outer cylinder is stationary, a  non-dimensional control parameter is the Reynolds number, which is defined as R = ρΩriδ/µ,  where µ, ρ, Ω are the dynamic viscosity, fluid density and the rotational speed of the inner  cylinder, respectively, and δ = ro − ri is the gap width, with ri and ro the inner and outer  radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The geometry can be characterised by two non dimensional parameters: the aspect ratio  Γ = h/δ and radius ratio η = ri/ro, where h is length of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Alternatively, the curvature  ratio κ = ri/δ can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The aforementioned parameters are known to have an influence on  stability, not only in Newtonian but also in non-Newtonian fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Indeed, the TC geometry is  also widely used in the study of complex fluids [3, 4], such as dilute polymer solutions [5, 6]  or suspensions of particles [7] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In particular, polymeric liquids flow which exhibit  viscoelastic behavior have been studied with great interest, due to the existence of sets of specific  flow regimes, and motivated by the ubiquity of viscoelasticity in daily life, industrial and natural  applications, such as biology, pharmaceutics, paints, among others [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  The mechanism of instability in these fluids are different from those that occur in Newtonian  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In Newtonian fluids, the instability comes from the destabilizing effect of the centrifugal  force gradient (which comes from variations of kinetic momentum), and overcoming of it on the  stabilizing effect of viscous drag force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In such fluids, at low Reynolds number, a purely azimuthal  uniform shear flow develops, which is called circular Couette flow (CCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It eventually becomes  unstable upon increasing R as explained above, and secondary flows appear as axisymmetric  counter-rotating vortices called Taylor vortex flow (TVF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Further increase of Reynolds number  creates non-axisymmetric sinusoidal axial oscillations called wavy Taylor vortex flow (WVF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Eventually, additional wavelentghs appear and the flow transitions to turbulence [9, 10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  In non-Newtonian, viscoelastic fluids, the mechanism of instability, and subsequently its flow  transition, is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Polymers solutions are common viscoelastic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Polymers are high  molecular weight molecules made of a large number of monomers connected with covalent  bonds, resulting in long linear, branched or network chains [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The arrangement (conformation)  of the polymer chain at the rest condition is in the way that have maximum conformational  entropy [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' When the polymer coil is stretched, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='g because of an applied stress or deformation,  it tends to recover its lost maximum entropy energy and return to its equilibrium chain structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Due to this entropic tendency of polymers, elastic stresses are created in the chain which as a  result of the stress difference between the flow direction and the direction perpendicular to it  (direction of shear), which doesn’t exist in Newtonian fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In rotational flow such as Taylor-  Couette Flows (TCF), curved streamlines induce a hoop stress, balanced by an adverse pressure  gradient in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A flow perturbation may cause a fluid particle to move towards  a region of enhanced stretching, enhancing the local hoop stress and destabilizing the flow [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Moreover, a part of the chain’s elastic deformation energy can be released elsewhere in the flow,  further promoting instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The elastic behavior of polymer solutions thus highly depend on  deformation rates, and relaxation time of polymer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The degree of elastic response of a  fluid subjected to a shear rate ˙γ is quantified by the Weissenberg number Wi, defined in the  case of TCF as Wi = λe ˙γ with ˙γ = Ωri/δ the nominal shear rate in the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The elastic number  El is then defined by El = Wi  R = λe  λv = λeµ  ρδ2 and represents the competition between inertial and  elastic effects, with λv = ρδ2/µ the viscous characteristic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The resulting El depends only on  the geometrical parameters and the properties of the fluid (which may themselves be shear-rate  dependent, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' El allows to classify fluids into 3 groups: weak (El < 10−2), moderate  (10−2 < El < 1) and strong elasticity (El > 1) [17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Based on the elasticity level, various  instability and transition scenarios are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In the range of very low elasticity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='e, El ≪ 1)  the elastic effects are very weak compared to inertia effects and observed flow transition are  3  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='royalsocietypublishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='org Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  comparable to the Newtonian case (CCF→TVF→WVF) as R increases [17, 20] with slightly  shifted critical conditions Rc because of presence of light amount of polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  At high values of elasticity, in the case of vanishing R, a purely elastic CCF-TVF transition  is observed [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Subsequently in the case of elevated Wi another transition will lead to a  chaotic regime called elastic turbulence [22, 23, 24], which exhibits turbulent like-characteristics  in absence of inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' When neither R nor El can be neglected, we find ourselves in the domain  of elasto-inertial transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In particular, primary and secondary elasto-inertial instabilities  manifest themselves in non-axisymmetric flow states [18, 19, 25, 26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' An increase in inertia  (R) or elasticity (El) leads these pre-chaotic behaviors to transition to strongly unsteady states:  "disordered oscillations" (DO) [27], "defect mediated turbulence (DMT) [19], "spatio-temporal  intermittency" (STI) [29] or "merge-split transitions" (MST) [28], and all contribute to a gradual  transition to elasto-inertial turbulence (EIT) [6, 18, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A summary of several flow transition  observed experimentally, as a function of geometrical parameter and viscoelastic fluid properties,  are listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  The possibility of triggering such chaotic behavior opens extremely interesting perspectives  in terms of mixing and intensification of transfers at low R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' While transition scenarios are now  relatively well identified in the literature, several questions remain to be tackled: what are the  characteristics of EIT in TCF?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' What is the dynamic behavior of these flows in terms of friction and  energy dissipation ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This work aims at addressing this last point in particular, by reporting for the  first time friction and spatio-temporal properties of TCF of constant viscosity polymer solutions  with shear-dependent viscoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Materials and methods  Experiments were performed using aqueous Boger solutions of high molecular weight polymer  of partially hydrolysed polyacrylamide (HPAM, Mw = 15 − 20 × 106 g/mol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' At first, a stock  aqueous polymer solution of 1000 ppm was prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Samples from this solution were then  dissolved in pure water and mixed in glycerol in order to obtain different concentrations ˜cp of  25, 50, 100, 150, 200, 250, 300, 350 ppm with base solution similar to that of our previous study [7]:  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='8 % glycerol and 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='2% water (in volume) and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='7% of salt (in mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' After preparation, the  aqueous solutions are left at rest for 24 h before performing any other manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Base fluid  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' (a) Shear viscosity as a function of shear rate for HPAM polymer solution at various concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The  observed jump in viscosity at a concentration of 350 ppm (after 600 s−1) is related to an elastic instability in the rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  (b) Measured first normal stress N1 of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A line with slope of 2 is plotted as a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' (c) Relaxation  time obtained using our experimental methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Colour lines are fits to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Rheological behavior of all working fluids was characterized using a rotational rheometer  (Anton-Paar MCR 302) equipped with a cone-plane geometry (50 mm/1◦) with truncation gap of  4  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='royalsocietypublishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=', (1996) [27]  /saccharose  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='22  CCF-TVF-DO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='34  CCF-DO  Groisman et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='708  54  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='413  PAAm/W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='033  N/A  N/A  CCF-RSW-DO  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=', (1993) [37]  /saccharose  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Some experimentally observed flow transition patterns in viscoelastic Boger fluids (constant viscosity µ assumed) with different fluid properties and geometrical parameters (rotating inner  cylinder and stationary outer cylinder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' DO = Disordered Oscillations, FP = Flame Pattern, SVF = Spiral Vortex Flow, MWVF = Modulated Wavy Vortex Flow, MT = Modulated Turbulence, RSW =  Rotating Spiral Waves, TTV = Turbulent Taylor Vortices, CWV = Chaotic Wavy Vortex Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' µp = µ − µs is the polymer contribution to the total viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' κ = δ/ri is the curvature ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' W = Water, G  = Glycerol, PEO = Polyethylène Oxyde, PAAm = Polyacrylamide, PEG= Polyethylène Glycol, IPA = Isopropyl Alcohol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' N/A = not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  5  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='org Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A 0000000  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='104 mm at a constant temperature of 22◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In order to find the viscosity of samples, steady-  shear viscosity measurements were performed on a shear-rate interval of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='01/s < ˙γ < 800/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Samples viscosity remained constant over this range of shear rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The flow curves for all samples  are reported in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In order to evaluate the dynamic, shear rate dependent, relaxation times  λe, the protocol detailed by [30], based on normal force measurement, was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It consisted of  several incremental steps during which constant shear rate was imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Subtracting the means  of two values of the first normal stress difference N1 for each step gives a way to resolve the  instrumental drift of the normal force and correct N1( ˙γ) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A second correction is performed  to remove the contribution of fluid inertia to the normal force given by the rheometer N1,tot,  such that N1 = N1,tot + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='15ρΩ2R2 with Ω = tan (θ) ˙γ the angular velocity (the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='15 prefactor  corresponds to inertial and secondary flow corrections and was proposed by [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' N1 can be  expressed as a power-law function of the shear rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' N1 = Ψ ˙γϵ, where Ψ and ϵ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  ϵ = 2 implies that the behavior follows the Oldroyd-B model [32]: N1 = 2(µ − µs)λe ˙γ2, and  the viscosity µ is dominated by the Newtonian solvent contribution, µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' However, unlike the  Oldroyd-B model, the relaxation time here also follows a power law (shear-rate dependent)  function as λe = a ˙γb, where a and b are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' As polymer concentration increases, b become  more negative which means that λe becomes more sensitive to shear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A summary of the  aforementioned coefficients is shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This advanced viscoelasticity characterisation  protocol allows to account for the effective shear-dependency of λe, and thus El, in constant  viscosity fluids and thus increase the accuracy on the critical El values detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' As expected  from [33] the method performs better for higher polymer concentrations, with less noise on the  N1 and λe data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It here results in a poor fitting of λe data for the 25 ppm case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  ˜cp (ppm)  25  50  100  150  200  250  300  350  λe  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='5e−5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='8e−4  8e−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='3e−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='55  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Rheological parameter of HPAM polymer solution derived by fitting an Oldroyd-B model in order to find relaxation  time, λe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The solutions follow the relation of λe = a ˙γb and N1 = Ψ ˙γϵ, where b and ϵ govern the viscoelastic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  The TCF experiments were performed in a Taylor-Couette cell mounted on the same rheometer  as illustrated in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The geometrical parameters were: δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='5 mm, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='914 and Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  In the present study, only ramp-up experiments were performed (slow acceleration of the inner  cylinder), combining torque measurements and visualisation using Iriodin particles (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='1% in  mass) and a light source, following a protocol detailed in our previous work [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The inner  cylinder acceleration rate was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='0082 < dR  dt∗ = ρ2riδ3  µ2  dΩ  dt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='01, and the temperature was 22 (±  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='4) °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Results and discussion  (a) Torque measurements  Let T be the raw measured torque on the rheometer shaft and G = T /Tv = T /2πh  �  µ2/ρ  �  the  dimensionless torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' G is thus simply a non-dimensional version T , scaled by torque-scale Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 3 shows plots for G as a function of R (left) and Wi (right) for all polymer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The  discontinuity in G values indicates the onset of a secondary flow, as will be discussed in section  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It appears that G increases with both increasing R and increasing Wi, since the shear-rate  6  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  Spotlight  EIT regime  snap shot  h  ro  ri  Outer cylinder  Inner cylinder  Air circulation  HPAM solution  Shaft  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Schematic of the Taylor-Couette apparatus mounted on the Anton-Paar MCR-102 rheometer with camera and  light position for visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' As shown in the figure, the upper end and lower end of the gap are a free surface and a  stationary wall, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Plots of G as a function of R (left) and Wi (right) for all polymer concentrations ˜cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' However, the increase rate of G after the discontinuity is reduced as ˜cp increases, which  is a key feature of the drag dynamics of the unsteady flow state, as will be detailed in sections (c)  and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  (b) Transitions, flow states, and critical R  Flow maps diagram (space-R diagrams), for ˜cp= 25, 100, 200, 350 ppm, coupled with plots of  Wi and of the effective (pseudo) Nusselt number, N as a function R is presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Wi increases with yet a decreasing slope (all the more decreasing that ˜cp increases), due to the  shear-rate dependency of λe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' N is defined as N = T /Tlam = G/Glam, with Glam = 2ηR/(1 +  7  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='org Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A 0000000  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  (a) 25 ppm  (c) 200 ppm  (b) 100 ppm  (d) 350 ppm  CCF  CCF  CCF  CCF  EIT  EIT  EIT  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Space-R diagram (top), Nusselt-R (bottom plots, left axis) and Wi-R (bottom plots, right axis) plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The plots show the transitions from CCF to RSW (Rotating Spiral Waves) and TVF for  a concentration of (a) 25 ppm and from CCF to EIT for (b) 100 ppm (c) 200 ppm (d) 350 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  8  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  η)(1 − η)2 the dimensionless torque for the laminar flow between infinitely long cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' N  allows to go further into the interpretation, compared to G, by scaling the (non-dimensional)  torque by its laminar value, which accounts for the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It represents the dissipation rate  of kinetic energy [38] and the ability of the flow to convey momentum radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Critical values  for all numbers (subscript .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='c) are found combining flow visualisation and torque measurement  (jump in N values [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Here, a CCF-EIT transition is observed for ˜cp ≥ 50 ppm (corresponding  El50  c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='13), as can be seen clearly in both flow map (changing alignment of Iriodin flakes from  purely azimuthal to random) and N − R diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This direct transition was previously observed  by [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' N related to CCF (R < Rc) is almost constant and is around one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This shows that the CCF  torque depends linearly on viscosity (and viscosity does not depend on the shear rate variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  The abrupt change in N values clearly indicates the beginning of the EIT regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 5 shows the Rc at which the CCF-EIT transition (or the primary transition, for the 25  ppm case) occurs as a function of ˜cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The side color bar indicates the corresponding Elc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Rc for the  25 ppm case is 134 or R25  c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='88R0c, demonstrating that the CCF flow is destabilized compared  to the Newtonian case (base solvent of this polymer for which the transition occurs at R0c = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='8  in this setup [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Rc decreases as the polymer concentration or the elastic number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Up  to 150 ppm or El = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='18, this reduction is very strong while for ˜cp > 150 ppm or El > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='18, it  becomes milder, which suggests that the presence effect of polymer after this concentration is less  effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The lower value of Elc for direct CCF-EIT (El50  c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='12) is here slightly lower than the  value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='22 reported by [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This minor discrepancy can be ascribed to the different relaxation  time estimation protocol used (ours accounting for shear-rate dependency) and to the variations  in geometrical parameters [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Rc for the onset of EIT (or for the primary instability, in the 25 ppm case) as a function of the ˜cp with marker  colours indicating Elc values at the onset of EIT  (c) Torque scaling in EIT  Figure 6 shows all curves for N − Ta at all polymer concentrations, essentially revisiting the raw  data from figure 3 but this time normalizing by laminar flow behaviour as allowed by the use of  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Ta = (1+η)6  (64η4) R2 is proportional to R2 by a geometric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Both numbers can be equally  used to quantify flow inertia when the geometry is kept constant, but Ta is more frequently  9  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='royalsocietypublishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  encountered in the literature for Taylor-Couette Nusselt scalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The color map indicates the R-  dependent El value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' At low Reynolds number, in the CCF state, all N are approximately constant,  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The slight variation in Nusselt number values (around 1) is due to wall effects that  introduce an additional torque [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Increasing R, up to the instability limit, there is a slight  increase in the slope of N, as noted previously by [34] or [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Passing through the critical point,  an abrupt change in the value of N occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This jump intensifies with increasing ˜cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Increasing  the ˜cp, the overall value of the Nusselt number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Interestingly, after the onset of EIT and as ˜cp increases, the global slope of the N-Ta curve in  the EIT regime gradually decreases, evolving to a Ta-independent Nusselt number region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This  evolution occurs faster as the concentration of polymer increases: the rate of change and slopes  are concentration dependent, but the asymptotic behaviour appears not to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  This can be interpreted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' After the onset of EIT, secondary chaotic flows arise and  generate friction at the walls leading to a global increase in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Increasing Ta or R, kinetic energy  is injected in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It is either dissipated by wall friction, which translates into an increase  in N, or by elastic dissipation by the polymer chains, which is expected not to depend on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Increasing ˜cp comes to promoting the second mechanism over the first, reduce the share of kinetic  energy dissipated by viscosity, and thus the increase in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This last point can be examined from  another angle: by qualitatively observing the elastic threshold below which the transition to the  asymptotic behaviour is gradual (and not sharp), it can be infered that even after the onset of  EIT, the flow still requires a given amount of inertia and/or elasticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' a given Wi increase, for  elastic energy transfers to balance inertial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' N − Ta curve for all concentrations with El color bar that illustrates the evolution of the elastic number during  the change of Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  (d) Elasto-inertial drag coefficient  In order to compare the data with Newtonian (laminar or turbulent) experiments and references  [43], curves from figure 6 can be re-scaled to display the friction coefficient CMz defined as CMz ∼  N/R [7], shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  In the Newtonian case, the onset of TVF is known to stop the CMz ∼ R−1 decrease (see  dashed line in figure 7), and the onset of WVF to make CMz decrease again with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In TVF, the  10  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='royalsocietypublishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='org Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  drag increases much faster than what is expected from laminar assumption, in WVF, it increases  faster (but not so much): TVF is very efficient in conveying momentum radially, WVF not that  much, since part of the energy is involved in axial motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A drag measurement experiment was  conducted for a Newtonian fluid (water) and is reported figure 7, for a turbulent state (R ≫ 1800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  The slope of CMz is compared with Von Kármán gap theory [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The results show that our CMz  follows the 1/�CMz = 3 ln(R  �  CMz) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='7 corresponding to ∼ R−1/3 against the work of [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Back to the viscoelastic case, for the higher El fluids, the onset of EIT leads to a sharp  increase in CMz and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Yet, after this sharp onset, N tends to a constant value (see figure 6),  and the friction coefficient asymptotically moves back to its "laminar" evolution, approximately  (∼ R−1), as shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This suggests that the additional dissipation induced by the  polymer chains and additional radial momentum transport is not Reynolds-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The R-  dependency on friction can be scaled by that of the laminar case, with no significant R-dependent  contribution of the viscoelastic secondary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This supports the very recent DNS (Direct  Numerical Simulation) of [45] suggesting that elasto-inertial TCF structures are not efficient in  radial momentum convection, which is why they tend to merge and split [28] or create defects  [19], thus helping transition to chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It is yet worth noting that the jump, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='e the additional  dissipative contribution of the polymer, is itself El-dependent: the jump is higher when the elastic  number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Between the jump and the asymptotic high El region, there is a transitional  behaviour which can be understood as the establishment period of EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This establishment period  displays a different slope, about R−2/3 for the 50 ppm case as illustrated on figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' For 25 ppm,  the constant CMz on a narrow range shows the existence of intermediate regimes visually closer  to TVF (RSW for Rotating Spiral Waves, see [28]), and apparently also in terms of torque dynamics  (see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Results also show that EIT is different from inertial turbulence modified by polymer for  which drag is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It would be interesting to see what happens when R tends to values for  which inertial turbulence is expected, as done in [46] and confront with drag reduction theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  0  6  Turb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Friction coefficient CMz as a function of R for all polymer concentrations (curve labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The color scale  represents El values and the dashed line the -1 slope characteristic of a laminar flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Slopes for R−2/3 and R−1 are  illustrated by dotted lines and triangle, orange and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In inset, an experiment for pure water in turbulent  flow state (full black line) is reported along with its ∼ R−1/3 asymptotic trend and compared with the ∼ R−1/4 trend  of [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The gray part of the curves correspond to points below the rheometer’s minimum torque capacity and should be  discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  (e) Spatio-temporal analysis: frequency maps and spatial PSD  Following the protocol detailed in [7], frequency maps for ˜cp = 25, 100, 200, 350 are shown in  figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Those plots shows the temporal FFT spectra of the reflected light intensity signal, for  all R stacked vertically, the colorbar representing the intensity of the spectra (arbitrary units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Frequency maps display no particular spectral signature in the CCF domain, before Rc is reached,  other than that of the inner cylinder rotation frequency fcylinder and its harmonics [28, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Such  ridges are still slightly visible for R > Rc, but the overall spectral signature changes, with a  broadband distribution of energy from large to small temporal scales, a signature of the chaotic  behaviour characteristic of EIT [18, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  CCF  CCF  CCF  CCF  EIT  EIT  EIT  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Frequency maps computed as [7] or [28], from the reflected light intensity signal, for all vertically stacked R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The  color scale represents the FFT intensity from weak (yellow/light) to strong (purple/dark) energy content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Instantaneous  change in intensity indicates the transition from CCF to EIT and the oblique dashed lines indicate the rotation frequency  of the inner cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' For 25 ppm the first instability is RSW which results in the appearance of discrete peaks that are no  longer seen in EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Further analysis of the spectral behaviour of EIT can be made by computing spatial power  spectral density (PSD) of the intensity signal at constant Rc and El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This is achieved by extracting  sets of n vertical lines In(z) at constant R on flow maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='g from figure 4), subtracting the average  intensity profile ⟨I⟩ (z) = 1  n  �n  j=1 Ij(z) in order to define the intensity fluctuation (i′)n(z) =  In(z) − ⟨I⟩ (z), computing PSD((i′)n(z)) and averaging PSDs on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Values are finally scaled by  the spatial (over z) average of ⟨I⟩ (z) called I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' One typically uses n = 50, a number for which  the convergence of spectra was deemed sufficient for the analysis that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' PSD spectra for  all polymer concentrations at R/Rc ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='2 are reported in figure 9 a), and spectra for the 350 ppm  case at various R values in figure 9 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' An additional experiment was performed with water,  in order to reach R ≃ 5100 and produce a flow map and PSD in the inertial turbulence regime  (inset of figure 9 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' For that curve, one would expect to capture a -5/3 slope if i) turbulence is  sufficiently developed and ii) the intensity fluctuation signal is in some way representative of the  radial velocity fluctuations, as suggested by [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It appears from the inset of 9 a) that the inertial  turbulence curve roughly follows the -5/3 trend at least at intermediate scales, which seems to  validate both arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' When considering spectra for EIT, the -5/3 slope is not expected in low  R cases where inertial turbulence would not have been present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Different scaling exponents of  EIT in various flow configurations (channel flows, TC flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=') can be found in the literature, in a  range from -14/3 to -3, always steeper than the inertial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The -3 slope has recently been put  forward by [49] in their study of viscoelastic polymer jets as a universal spectral behaviour of EIT,  after having been theoretically predicted by [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Recent numerical simulations by [45] have been  12  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='royalsocietypublishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='org Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  able to retrieve this scaling in elasto-inertial TCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This reference slope is also represented on figure  9 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Spatio-temporal dynamics: a) Spatial PSD spectra for all polymer concentrations at R/Rc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='2, with an inset  showing typical spectrum for inertial turbulence in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' b) Spatial PSD spectra for ˜cp = 350 ppm at various R dotted  and dashed black lines in b) denote k−5/3 and k−3 slopes [49, 50], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The dotted blue line in a) and b) is a  7/3 slope fitting the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  It here appears that all curves for EIT fall short of -3 trends, but still display a somehow  universal slope with respect to polymer concentration (figure 9 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' An increase in Reynolds  number seems to consolidate this slope by increasing the k span over which it applies (figure 9 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  The slope value is here around -7/3 (figure 9 a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' It is worth mentioning that the visualisation  method probes the flow from the outside and not in the bulk, and may be subject to boundary  layer effects on the outer cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' So the value of the slope itself must be interpreted with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Figure 9 yet confirms key findings [49] namely that of an apparent universal spectral slope of  EIT, steeper than inertial turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Additionally, our results suggest that EIT is intrinsically a  combination of elasticity and inertia, as R helps develop the slope when increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Conclusion  In this work, new characterisations of the dynamics of EIT in Taylor-Couette flow of polymer  solutions were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Combining flow visualisation and torque measurements allowed to  detect CCF-EIT transition and to describe key dynamic features of EIT in TCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' In particular the  scaling of N with Ta and its dependency on fluid elasticity has been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Two sub-domains  of EIT were reported: a transitional one for which energy dissipation is still dominated by inertia  and a fully developed one for which elastic energy transfer become dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Spectral analysis  support the idea of a chaotic developed EIT state for which PSD would exhibit a universal slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Developed EIT displays an asymptotic "laminar-like" scaling for the friction coefficient: the wall  friction is directly correlated to the base azimuthal flow and secondary flow structures do not  play a role in wall friction as their energy is dissipated elastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' This requires a sufficient level  of both elasticity and inertia, and is expected to be of great interest in the aim of achieving efficient  mixing at low drag and low R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Authors’ Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' MM performed, analysed, interpreted the experimental work reported in this  paper, and drafted the initial manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' TL and AB equally contributed to the design of the experiments  (together with MM), conceptualisation of the project, guidance, and supervision of the work (together with  13  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='royalsocietypublishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='org Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  VT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' AB and VT secured funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' All authors contributed to the writing, reading, editing and approved the  manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Competing Interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' The author(s) declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Financial support from Region Hauts de France and Institut Mines Télécom is gratefully  acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  References  1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Taylor, “VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Stability of a viscous liquid contained between two rotating cylinders,”  Philosophical Transactions of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Series A, Containing Papers of a  Mathematical or Physical Character, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' 223, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Andereck and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Hayot, “Ordered and turbulent patterns in Taylor-Couette flow,” tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Topayev, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Nouar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Bernardin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Neveu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Bahrani, “Taylor-vortex flow in shear-  thinning fluids,” Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content='  4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Lopez, “Vortex merging and splitting events in viscoelastic Taylor–Couette flow,” Journal  of Fluid Mechanics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' Samanta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Dubief, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Holzner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
+page_content=' Morozov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content='  47 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdA0T4oBgHgl3EQfGf9T/content/2301.02047v1.pdf'}
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+Giving life to robotic skins 
+ 
+Ahmad Rafsanjani1a, Fergal B. Coulter2b, André R. Studart2c 
+ 
+1 Center for Soft Robotics, SDU Biorobotics, University of Southern Denmark, Odense, Denmark 
+2 Complex Materials, Department of Materials, ETH Zurich, Zurich, Switzerland 
+ 
+Correspondence:  
+a ahra@mmmi.sdu.dk 
+b fergal.coulter@mat.ethz.ch 
+c andre.studart@mat.ethz.ch 
+ 
+Abstract 
+The skin of humanoid robots often lacks human tactility and the inherent self-repair capability of 
+biological tissues. Recently, researchers have grown a living, self-healing skin on a robot finger by 
+subsequent culturing of human dermal and epidermal cells. Here, we highlight the significance of this 
+study alongside challenges toward developing biohybrid robots equipped with sensate and adaptive 
+living robotic skins. 
+ 
+ 
+ 
+ 
+ 
+Figure 1. Living skin equivalent grown on a robotic finger featuring dermal and epidermal layers with 
+water repellency, low adhesion, and self-healing characteristics. 
+ 
+ 
+
+Skinequivalent
+Epidermal layer
+Waterrepellency
+Lowadhesion
+Self-healingInteractions between humanoid robots with humans and the environment has been focused almost 
+exclusively on the face and voice, overlooking the importance of the skin – the largest organ in our 
+body. Conversely, touch communicates distinct emotions among humans, such as anger, fear, disgust, 
+love, gratitude, and sympathy [1]. Our skin is an active sensory organ, a socially expressive means, a 
+permeable regulatory filter, and a self-healing protective layer [2]. In contrast, the skin of existing 
+humanoid robots is a passive layer, the sole function of which is shielding the robot’s internal structure 
+from the external world. Robotics has taken tremendous leaps in generating extremely complex human 
+gaits, as seen in the latest Atlas robot (Boston Dynamics) that can parkour with jumps and vaults like a 
+real pro. However, the rigid insensate skin of existing humanoid robots is remarkably limited when it 
+comes to interactions with humans or adapting to dynamic environments. In recent years, the 
+underexplored world of robotic skins has attracted researchers from many disciplines to augment the 
+interaction affordance of robots.  
+Research on robotic skins – covering fingertips to whole-body of robots – has been primarily focused 
+on integrating electronic skins, known as E-skins, onto the external surface of robots to give them a 
+sense of touch [3]. As an illustrative example, researchers developed a multilayered electronic dermis 
+on a robot’s fingertip made up of conductive and piezoresistive textiles encased in silicone rubber. Such 
+an artificial skin can transform tactile information from object grasping into a neuromorphic signal and 
+pass it to the peripheral nerves of an amputee to elicit sensory perceptions of touch and pain [4]. Unlike 
+E-skins where electrons are responsible for sensing and transmission of tactile information, human skin 
+sensors rely on ionic current. Ionic skins bring robotic skin equivalents closer to their biological 
+counterparts and, in the simplest form, can be constructed by sandwiching a stretchable dielectric layer 
+between two stretchable ionically conductive layers usually made of salt-containing hydrogels [5]. The 
+assembly of charged poly(acrylic acid) and neutral polyacrylamide hydrogels resulted in a piezoionic 
+sensor skin with a built-in potential difference that operates by pressure-driven ion flux and can find 
+application in neuro-prosthetic robots [6]. Although essential in several applications, upscaling these 
+and other tactile sensing technologies to the whole-body area is even more challenging, since it involves 
+the organization and calibration of several spatially distributed discrete sensors and big data processing. 
+Roboticists demonstrated the feasibility of covering almost the entire surface of a human-sized 
+humanoid robot with more than a thousand self-calibrated multimodal sensing tiles, enabling the spatial 
+perception of temperature, pressure, acceleration, and proximity [7].  
+Besides the sense of touch, self-healing and damage resilience have also been features of major interest 
+in the development of soft materials and artificial skins for robotic applications [8]. The idea has been 
+to imbue electronic skins with the ability to restore sensing functionalities by using materials that can 
+re-establish broken interconnects or fill up cracks after damage. Such self-healing capabilities have been 
+achieved in electronic skins using, for example, liquid metals as fluid conductive elements and dynamic 
+supramolecular polymers and hydrogels as structural materials. In spite of these remarkable feats, self-
+healing artificial materials still suffer from excessive creep, dependence on external stimulus for healing 
+and tendency to deteriorate with the number of healing events. 
+While research on robotic skins previously concentrated primarily on reproducing perceptual functions, 
+giving robots a lifelike appearance, and generating a humanlike tactile feeling with self-repair 
+capabilities is imperative for their broader dissemination and acceptance by humans in the medical care, 
+nursing centers, and service industry. Recently, Takeuchi and coworkers developed a living human skin 
+equivalent grown on a robot finger [9]. This skin exhibits a self-healing ability and can bear the 
+deformation induced by the joint motions of the finger. A novelty of this research lies in the fabrication 
+of an in vitro cell culture that uniformly covers a three-dimensional three-joint finger, demonstrating 
+the implementation of a human-like skin on a fundamental building block of a humanoid robot hand. 
+Living fibroblasts from connective tissue of human skin are used to build a dermal layer, which is 
+covered with an epidermal layer containing living keratinocytes responsible for the barrier function of 
+the human skin. Emphasis is placed on the fixation of the newly grown bilayer skin to the substrate, 
+where anchor points were created to prevent slippage or incorrect movement of the coating. To 
+manufacture the skin, the bare robotic finger was first inserted into a cylindrical rubber mold, with a 
+small gap between the two. A collagen solution containing living fibroblasts was then poured into the 
+mold and incubated. After three days, a mature dermis formed on the irregular three-dimensional 
+
+surface of the finger that featured an anchor fixation at its root to minimize tissue shrinkage. Finally, 
+the epidermal layer was grown atop the dermis tissue by seeding and culturing living keratinocytes from 
+different sides of the mold for another two weeks to complete the construction of the skin equivalent. 
+Histological analysis on frozen sections of stained skin equivalent proved a seamless adhesion of 
+epidermis and dermis tissues and the formation of uniform layers. 
+The skin equivalent consisting of dermal and epidermal layers showed a significantly lower capacitance 
+and a higher electrical resistance compared to the control sample containing only the dermis layer. This 
+confirmed the effectiveness of the epidermis as an electrical barrier for the robot skin. Water retention 
+tests demonstrated that the presence of the epidermis tissue makes the skin equivalent almost 
+impermeable compared to the pure dermis equivalent. Moreover, the epidermis layer is sufficiently 
+water repellent and non-adherent to form water droplets on its surface and to efficiently handle 
+electrostatically charged objects (e.g., polystyrene foam beads). In addition to the epidermis, the 
+performance of the dermal layer as the load-bearing component of the fabricated skin was also 
+evaluated. Mechanical characterization of the dermis equivalent revealed a strain-hardening regime akin 
+to human skin. However, the measured tensile strength of the artificial living skin is orders of magnitude 
+lower than the values reported for the biological counterpart. While this low mechanical strength 
+remains an open challenge, the skin showed cell-mediated self-repair capabilities and was sufficiently 
+robust to withstand the stresses developed during motion of the finger. The self-healing capability of 
+the living robotic skin is demonstrated by applying an acellular collagen sheet to an intentional wound 
+on the dorsal dermis of the robotic finger. The grafted collagen sheet successfully sealed the wound and 
+seamlessly adhered to the dermis equivalent after one week. 
+In the past, we have seen many examples of humanoid robots with skin equivalents that are fabricated 
+using artificial polymers such as silicone rubber or foamed latex. While humanoids can be exquisitely 
+painted, sculpted, and accented with make-up to improve their human-likeness, very often these robots 
+fall into the category of not looking quite human enough to be convincing, or worse – they fall into the 
+so-called uncanny valley - whereby people are left unsettled by the perceivable but subtle differences 
+in the visible versus expected movement. Human-like resemblance can be improved by modifying the 
+extensibility and compliance of the polymers themselves. Techniques used in theatrical prosthetics such 
+as chain extenders or deadeners can modify the movement of the elastomer skin, improving on how 
+wrinkling or folding occurs due to movement. Nonetheless, this is no substitute for the real thing, and 
+these new conformal cell culturing techniques developed by Takeuchi and coworkers should facilitate 
+an improved experience for future interactions with humanoid robots.  
+Some caveats regarding the concept of creating and sustaining a true living skin over a robot include a 
+need for blood vessels and capillaries to carry artificial blood containing oxygen and nutrients required 
+to maintain the cells. Alongside this, a distribution network should exist for stem cells to replace and 
+remove apoptotic skin cells, a particularly challenging task given the delicate nature of undifferentiated 
+cells. Integrating a sensor network into the dermis equivalent, potentially based on ionic currents, will 
+also be required to add perception to the repertoire of living robotic skin functions. Finally, upscaling 
+from a robot finger to a human-sized humanoid robot cannot be achieved with current traditional casting 
+methods and requires recruiting digital manufacturing techniques such as multiaxial, multi-material, 
+and multifunctional bioprinting of complex, soft materials [10]. Despite all these shortcomings, 
+Takeuchi and coworkers showcased that giving life to robotic skins can enable the realization of 
+expressive, protective, and self-healing biohybrid robots. 
+ 
+Acknowledgment  
+A.R was supported by the Villum Young Investigator grant 37499 and the Danish Council for 
+Independent Research (DFF) through the DFF Sapere Aude grant 1051-00075B. A.R.S. and F.B.C. 
+thank the financial support from the Swiss National Science Foundation (grant 200020_204614) and 
+from the Strategic Focus Area Advanced Manufacturing (SFA‐AM) of the Swiss ETH domain, as part 
+of the Manufhaptics project. 
+ 
+ 
+
+References: 
+[1] Hertenstein, M.J., Keltner, D., App, B., Bulleit, B.A. and Jaskolka, A.R., 2006. Touch communicates 
+distinct emotions. Emotion, 6(3), p.528. 
+[2] Hu, Y. and Hoffman, G., 2022. What Can a Robot’s Skin Be? Designing Texture-Changing Skin 
+for Human-Robot Social Interaction. ACM Transactions on Human-Robot Interaction. 
+[3] Shih, B., Shah, D., Li, J., Thuruthel, T.G., Park, Y.L., Iida, F., Bao, Z., Kramer-Bottiglio, R. and 
+Tolley, M.T., 2020. Electronic skins and machine learning for intelligent soft robots. Science 
+Robotics, 5(41), p.eaaz9239. 
+[4] Osborn, L.E., Dragomir, A., Betthauser, J.L., Hunt, C.L., Nguyen, H.H., Kaliki, R.R. and Thakor, 
+N.V., 2018. Prosthesis with neuromorphic multilayered e-dermis perceives touch and pain. Science 
+Robotics, 3(19), p.eaat3818. 
+[5] Sun, J.Y., Keplinger, C., Whitesides, G.M. and Suo, Z., 2014. Ionic skin. Advanced Materials, 
+26(45), pp.7608-7614. 
+[6] Dobashi, Y., Yao, D., Petel, Y., Nguyen, T.N., Sarwar, M.S., Thabet, Y., Ng, C.L., Scabeni Glitz, 
+E., Nguyen, G.T.M., Plesse, C. and Vidal, F., 2022. Piezoionic mechanoreceptors: Force-induced 
+current generation in hydrogels. Science, 376(6592), pp.502-507. 
+[7] Cheng, G., Dean-Leon, E., Bergner, F., Olvera, J.R.G., Leboutet, Q. and Mittendorfer, P., 2019. A 
+comprehensive realization of robot skin: Sensors, sensing, control, and applications. Proceedings of the 
+IEEE, 107(10), pp.2034-2051.  
+[8] Bilodeau, R.A. and Kramer, R.K., 2017. Self-healing and damage resilience for soft robotics: A 
+review. Frontiers in Robotics and AI, 4, p.48. 
+[9] Kawai, M., Nie, M., Oda, H., Morimoto, Y., Takeuchi, S., 2022. Living skin on a robot. Matter, 
+5(7), pp.2190-2208. 
+[10] Schaffner, M., Rühs, P.A., Coulter, F., Kilcher, S. and Studart, A.R., 2017. 3D printing of bacteria 
+into functional complex materials. Science Advances, 3(12), p.eaao6804. 
+ 
+
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+page_content=' Coulter2b, André R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Studart2c  1 Center for Soft Robotics, SDU Biorobotics, University of Southern Denmark, Odense, Denmark 2 Complex Materials, Department of Materials, ETH Zurich, Zurich, Switzerland  Correspondence:  a ahra@mmmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
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+page_content='ch  Abstract The skin of humanoid robots often lacks human tactility and the inherent self-repair capability of biological tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Recently, researchers have grown a living, self-healing skin on a robot finger by subsequent culturing of human dermal and epidermal cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Here, we highlight the significance of this study alongside challenges toward developing biohybrid robots equipped with sensate and adaptive living robotic skins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Living skin equivalent grown on a robotic finger featuring dermal and epidermal layers with water repellency, low adhesion, and self-healing characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  Skinequivalent Epidermal layer Waterrepellency Lowadhesion Self-healingInteractions between humanoid robots with humans and the environment has been focused almost exclusively on the face and voice, overlooking the importance of the skin – the largest organ in our body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Conversely, touch communicates distinct emotions among humans, such as anger, fear, disgust, love, gratitude, and sympathy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Our skin is an active sensory organ, a socially expressive means, a permeable regulatory filter, and a self-healing protective layer [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' In contrast, the skin of existing humanoid robots is a passive layer, the sole function of which is shielding the robot’s internal structure from the external world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Robotics has taken tremendous leaps in generating extremely complex human gaits, as seen in the latest Atlas robot (Boston Dynamics) that can parkour with jumps and vaults like a real pro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' However, the rigid insensate skin of existing humanoid robots is remarkably limited when it comes to interactions with humans or adapting to dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' In recent years, the underexplored world of robotic skins has attracted researchers from many disciplines to augment the interaction affordance of robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Research on robotic skins – covering fingertips to whole-body of robots – has been primarily focused on integrating electronic skins, known as E-skins, onto the external surface of robots to give them a sense of touch [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  As an illustrative example, researchers developed a multilayered electronic dermis on a robot’s fingertip made up of conductive and piezoresistive textiles encased in silicone rubber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Such an artificial skin can transform tactile information from object grasping into a neuromorphic signal and pass it to the peripheral nerves of an amputee to elicit sensory perceptions of touch and pain [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Unlike E-skins where electrons are responsible for sensing and transmission of tactile information, human skin sensors rely on ionic current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Ionic skins bring robotic skin equivalents closer to their biological counterparts and, in the simplest form, can be constructed by sandwiching a stretchable dielectric layer between two stretchable ionically conductive layers usually made of salt-containing hydrogels [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' The assembly of charged poly(acrylic acid) and neutral polyacrylamide hydrogels resulted in a piezoionic sensor skin with a built-in potential difference that operates by pressure-driven ion flux and can find application in neuro-prosthetic robots [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Although essential in several applications, upscaling these and other tactile sensing technologies to the whole-body area is even more challenging, since it involves the organization and calibration of several spatially distributed discrete sensors and big data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  Roboticists demonstrated the feasibility of covering almost the entire surface of a human-sized humanoid robot with more than a thousand self-calibrated multimodal sensing tiles, enabling the spatial perception of temperature, pressure, acceleration, and proximity [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Besides the sense of touch, self-healing and damage resilience have also been features of major interest in the development of soft materials and artificial skins for robotic applications [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' The idea has been to imbue electronic skins with the ability to restore sensing functionalities by using materials that can re-establish broken interconnects or fill up cracks after damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Such self-healing capabilities have been achieved in electronic skins using, for example, liquid metals as fluid conductive elements and dynamic supramolecular polymers and hydrogels as structural materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' In spite of these remarkable feats, self- healing artificial materials still suffer from excessive creep, dependence on external stimulus for healing and tendency to deteriorate with the number of healing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' While research on robotic skins previously concentrated primarily on reproducing perceptual functions, giving robots a lifelike appearance, and generating a humanlike tactile feeling with self-repair capabilities is imperative for their broader dissemination and acceptance by humans in the medical care, nursing centers, and service industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  Recently, Takeuchi and coworkers developed a living human skin equivalent grown on a robot finger [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' This skin exhibits a self-healing ability and can bear the deformation induced by the joint motions of the finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' A novelty of this research lies in the fabrication of an in vitro cell culture that uniformly covers a three-dimensional three-joint finger, demonstrating the implementation of a human-like skin on a fundamental building block of a humanoid robot hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Living fibroblasts from connective tissue of human skin are used to build a dermal layer, which is covered with an epidermal layer containing living keratinocytes responsible for the barrier function of the human skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Emphasis is placed on the fixation of the newly grown bilayer skin to the substrate, where anchor points were created to prevent slippage or incorrect movement of the coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' To manufacture the skin, the bare robotic finger was first inserted into a cylindrical rubber mold, with a small gap between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' A collagen solution containing living fibroblasts was then poured into the mold and incubated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' After three days, a mature dermis formed on the irregular three-dimensional  surface of the finger that featured an anchor fixation at its root to minimize tissue shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Finally, the epidermal layer was grown atop the dermis tissue by seeding and culturing living keratinocytes from different sides of the mold for another two weeks to complete the construction of the skin equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Histological analysis on frozen sections of stained skin equivalent proved a seamless adhesion of epidermis and dermis tissues and the formation of uniform layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' The skin equivalent consisting of dermal and epidermal layers showed a significantly lower capacitance and a higher electrical resistance compared to the control sample containing only the dermis layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' This confirmed the effectiveness of the epidermis as an electrical barrier for the robot skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Water retention tests demonstrated that the presence of the epidermis tissue makes the skin equivalent almost impermeable compared to the pure dermis equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Moreover, the epidermis layer is sufficiently water repellent and non-adherent to form water droplets on its surface and to efficiently handle electrostatically charged objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=', polystyrene foam beads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' In addition to the epidermis, the performance of the dermal layer as the load-bearing component of the fabricated skin was also evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Mechanical characterization of the dermis equivalent revealed a strain-hardening regime akin to human skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  However, the measured tensile strength of the artificial living skin is orders of magnitude lower than the values reported for the biological counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' While this low mechanical strength remains an open challenge, the skin showed cell-mediated self-repair capabilities and was sufficiently robust to withstand the stresses developed during motion of the finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' The self-healing capability of the living robotic skin is demonstrated by applying an acellular collagen sheet to an intentional wound on the dorsal dermis of the robotic finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' The grafted collagen sheet successfully sealed the wound and seamlessly adhered to the dermis equivalent after one week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' In the past, we have seen many examples of humanoid robots with skin equivalents that are fabricated using artificial polymers such as silicone rubber or foamed latex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' While humanoids can be exquisitely painted, sculpted, and accented with make-up to improve their human-likeness, very often these robots fall into the category of not looking quite human enough to be convincing, or worse – they fall into the so-called uncanny valley - whereby people are left unsettled by the perceivable but subtle differences in the visible versus expected movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Human-like resemblance can be improved by modifying the extensibility and compliance of the polymers themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  Techniques used in theatrical prosthetics such as chain extenders or deadeners can modify the movement of the elastomer skin, improving on how wrinkling or folding occurs due to movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Nonetheless, this is no substitute for the real thing, and these new conformal cell culturing techniques developed by Takeuchi and coworkers should facilitate an improved experience for future interactions with humanoid robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Some caveats regarding the concept of creating and sustaining a true living skin over a robot include a need for blood vessels and capillaries to carry artificial blood containing oxygen and nutrients required to maintain the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Alongside this, a distribution network should exist for stem cells to replace and remove apoptotic skin cells, a particularly challenging task given the delicate nature of undifferentiated cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Integrating a sensor network into the dermis equivalent, potentially based on ionic currents, will also be required to add perception to the repertoire of living robotic skin functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' Finally, upscaling from a robot finger to a human-sized humanoid robot cannot be achieved with current traditional casting methods and requires recruiting digital manufacturing techniques such as multiaxial, multi-material, and multifunctional bioprinting of complex, soft materials [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  Despite all these shortcomings, Takeuchi and coworkers showcased that giving life to robotic skins can enable the realization of expressive, protective, and self-healing biohybrid robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='  Acknowledgment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='R was supported by the Villum Young Investigator grant 37499 and the Danish Council for Independent Research (DFF) through the DFF Sapere Aude grant 1051-00075B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
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+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content=' thank the financial support from the Swiss National Science Foundation (grant 200020_204614) and from the Strategic Focus Area Advanced Manufacturing (SFA‐AM) of the Swiss ETH domain, as part of the Manufhaptics project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
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+page_content=' Science Advances, 3(12), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
+page_content='eaao6804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQfXgDe/content/2301.02295v1.pdf'}
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+arXiv:2301.08757v1  [hep-th]  20 Jan 2023
+HU-EP-23/03-RTG
+Quadratic twist-noncommutative gauge theory
+Tim Meier∗ and Stijn J. van Tongeren†
+Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin,
+IRIS Geb¨aude, Zum Grossen Windkanal 2, 12489 Berlin
+Studies of noncommutative gauge theory have mainly focused on noncommutative spacetimes
+with constant noncommutative structure, with little known about actions for noncommutative 4D
+Yang-Mills theory beyond this case. We construct an action for Yang-Mills theory on a quadratically-
+noncommutative spacetime, i.e. of quantum-plane type, obtained from a Drinfeld twist, with star-
+gauge symmetry. Applied to supersymmetric Yang-Mills theory, this gives a candidate AdS/CFT
+dual of string theory on a related deformation of AdS5×S5, which is expected to be integrable in
+the planar limit.
+Noncommutativity between space-time coordinates is
+a likely feature of quantum gravity [1], actively studied
+from numerous angles [2–4]. In string theory, noncom-
+mutative gauge theory appears in the low energy dynam-
+ics of open strings [5], and thereby the AdS/CFT cor-
+respondence [6–8]. Despite the formal and phenomeno-
+logical relevance of noncommutative gauge theory, it is
+not clear how to write actions to all orders in the non-
+commutativity, when going beyond the case of constant
+noncommutativity.
+In this letter we consider a non-
+commutative Drinfeld-twist deformation of Minkowski
+space, with quadratically-coordinate-dependent noncom-
+mutativity, and construct an all-order action for Yang-
+Mills theory with star-gauge symmetry. Beyond provid-
+ing a first example of a noncommutative Yang-Mills the-
+ory action with quadratic noncommutativity, our choice
+of deformation is motivated by the AdS/CFT correspon-
+dence and integrability. Applied to maximally supersym-
+metric Yang-Mills theory, our noncommutative deforma-
+tion provides a concrete candidate gauge theory dual of
+a particular Yang-Baxter deformation [9–11] of the fa-
+mously integrable AdS5×S5 superstring [12, 13], as con-
+jectured in [14].
+This opens the door to investigating
+integrability for a range of novel planar noncommutative
+gauge theories.
+We consider noncommutative field theory in the usual
+spirit of Weyl quantization, trading noncommuting field
+operators for a noncommutative product – the star prod-
+uct – between commutative fields [15, 16].
+Our star
+product is obtained from a Drinfeld twist, whereby it
+automatically comes with clear algebraic properties and
+a natural differential calculus [2], and twists rather than
+plainly breaks Poincar´e symmetry [17, 18]. The original
+Groenewold-Moyal noncommutative deformation can be
+viewed as a twist, and it is well known how to construct
+a Yang-Mills action in this case [16]. For general twists,
+however, it is not clear how to define a suitable dual field
+strength tensor, and construct an action for noncommu-
+tative Yang-Mills theory. For example, the approaches
+of [19, 20] for κ-Minkowski space, were necessarily per-
+turbative, and only solved to leading order in the non-
+commutativity. To our knowledge, the only non-constant
+case known to all orders, is the U(1) Yang-Mills theory
+studied in [21] for a particular twist with linear noncom-
+mutativity, where standard Hodge duality suffices.
+We show how a twisted version of Hodge duality allows
+us to define a noncommutative Yang-Mills action for a
+twist based on two commuting Lorentz generators, with a
+noncommutative structure with quadratic coordinate de-
+pendence. Our construction moreover provides a broader
+framework that covers all known Poincar´e-based twist-
+deformations of Minkowski space, including non-abelian
+ones, whose r matrices are unimodular [22]. Given our
+motivations in AdS/CFT, we also discuss how to cou-
+ple our theory to (adjoint) matter, define our deformed
+maximally-supersymmetric Yang-Mills theory, and dis-
+cuss its AdS/CFT interpretation.
+LORENTZ-DEFORMED MINKOWSKI SPACE
+In deformation quantization, a noncommutative space-
+time is described via a regular spacetime whose function
+algebra is equipped with a noncommutative (star) prod-
+uct. The noncommutative product we consider for func-
+tions on R1,3 is based on the Drinfeld twist
+F = e
+iλ
+2 (M01⊗M23−M23⊗M01),
+(1)
+built from two commuting Lorentz generators, M01 and
+M23, with Mµν = i(xµ∂ν − xν∂µ). It defines our non-
+commutative star product via
+f(x) ⋆ g(x) ≡ µ(F−1(f(x), g(x))),
+(2)
+where µ(h(x), z(x)) = h(x)z(x) is the ordinary pointwise
+product of functions, and λ is our deformation parame-
+ter. We will refer to this as the Lorentz twist, or defor-
+mation. In contrast to the familiar Groenewold-Moyal
+star product associated to
+FGM = e− iθµν
+4
+(∂µ⊗∂ν−∂ν⊗∂µ),
+where θ is a constant antisymmetric matrix, the bidif-
+ferential operator appearing in the exponent of our twist
+has nontrivial coordinate dependence [23].
+
+2
+Our star product is associative because M01 and M23
+commute [24]. Its noncommutative structure is
+xµ ⋆ xν = Rσµρνxρ ⋆ xσ,
+(3)
+with 24 nonzero components for R,
+R0
+0
+0
+0 = 1,
+R0
+0
+2
+2 = cosh λ,
+R0
+1
+2
+3 = −i sinh λ, (4)
+and others obtained by index permutation symmetries of
+the twist (1), namely 2 ↔ 3 or (0, 1) ↔ (2, 3) combined
+with a sign change of λ, and 0 ↔ 1.
+Under (conventional) conjugation we have f ⋆ g = g⋆f.
+As our goal is gauge theory, we need differential calcu-
+lus, now suitably twisted [2]. Using standard differential
+calculus on Minkowski space we define
+dxµ ⋆ f = µ(F−1(dxµ, f)),
+(5)
+where the vector fields M01 and M23 act via Lie deriva-
+tives, summarized as
+dxµ ⋆ f = dxν ¯F ν
+µ(f),
+(6)
+with
+¯Fµν =
+
+
+
+cosh λM23
+2
+− sinh λM23
+2
+0
+0
+− sinh λM23
+2
+cosh λM23
+2
+0
+0
+0
+0
+cos λM01
+2
+− sin λM01
+2
+0
+0
+sin λM01
+2
+cos λM01
+2
+
+
+
+ν
+µ
+.
+Commuting functions through forms gives rise to
+dxµ ⋆ f = Rνµ(f) ⋆ dxν,
+(7)
+with the R matrix Rν µ = ¯Fν ρ ¯Fρν. Both R and ¯F are
+vector-field-valued elements of the Lorentz group, in the
+sense that, raising and lowering indices with the usual
+Minkowski metric,
+RµνRρν = ¯Fµν ¯F ρν = δρ
+µ.
+(8)
+We also define a star-wedge product
+dxµ ∧⋆ dxν = ˆµ(F−1(dxµ, dxν)),
+(9)
+where ˆµ(a, b) = a ∧ b is the regular wedge product. Con-
+cretely
+dxµ ∧⋆ dxν = ¯Fσµρνdxσ ∧ dxρ,
+(10)
+with ¯Fσµρν = Rσµρν|λ→λ/2. We will mostly work with
+star forms
+ω = ω⋆
+µν...ρ ⋆ dxµ ∧⋆ dxν ∧⋆ . . . ∧⋆ dxρ,
+(11)
+but occasionally will also express them as regular forms
+ω = ωµν...ρdxµ ∧ dxν ∧ . . . ∧ dxρ.
+(12)
+Our star forms are totally R-antisymmetric, e.g.
+dxµ ∧⋆ dxν = −Rρµσνdxσ ∧⋆ dxρ.
+We will use the ordinary exterior derivative, which has
+the desired product rule
+d(ω ∧⋆ χ) = dω ∧⋆ χ + (−1)pω ∧⋆ dχ.
+(13)
+for p and q forms ω and χ, respectively, as it com-
+mutes with Lie derivatives. Under conjugation we have
+ω ∧⋆ χ = (−1)pqχ ∧⋆ ω.
+Our star product is graded-cyclic under an integral,
+�
+ω ∧⋆ χ = (−1)pq
+�
+χ ∧⋆ ω
+(14)
+when χ∧⋆ ω is a top form, upon integration by parts [25].
+HODGE DUALITY
+To define our twisted Hodge duality we take a natural
+generalization of the Levi-Civita symbol,
+dxµ ∧⋆ dxν ∧⋆ dxρ ∧⋆ dxσ = ǫµνρσd4x,
+(15)
+where the volume form d4x = dx0 ∧⋆ dx1 ∧⋆ dx2 ∧⋆ dx3 =
+dx0 ∧ dx1 ∧ dx2 ∧ dx3 is not deformed. By definition ǫ is
+R antisymmetric, and conjugates as
+ǫµνρσ = ǫσρνµ.
+It is also graded cyclic. Its 32 nonzero components are
+ǫ0123 = −ǫ0132 = ǫ0231 = −ǫ0321 = 1,
+ǫ1212 = −ǫ0202 = ǫ1313 = −ǫ0303 = i sinh λ,
+ǫ0312 = −ǫ0213 = cosh λ,
+(16)
+plus others related by graded cyclicity.
+In regular Hodge duality we can freely permute indices
+on the Levi-Civita symbol for signs, giving many equiv-
+alent definitions of a dual form. The appropriate choice
+in our twisted setting is
+∗ dxµ1 ∧⋆ . . . ∧⋆ dxµk = (−1)σ(k)
+(4−k)! ǫµk+1...µ4
+µ1...µkdxµ4 ∧⋆ . . . ∧⋆ dxµk+1,
+(17)
+
+3
+where σ(p) denotes the signature of the reversal of p ob-
+jects, i.e.
+σ(1) = σ(4) = 0, σ(2) = σ(3) = 1.
+The
+reversed index contraction in the dual form is essential.
+Restricted to basis star forms, this twisted Hodge du-
+ality commutes with Lie derivatives along vector fields
+in the Poincar´e algebra, and hence with our star prod-
+uct. This allows us to consistently extend it star-linearly
+to arbitrary forms, where it continues to commute with
+Poincar´e Lie derivatives and our star product.
+Our Hodge duality has all other usual properties, ap-
+propriately twisted, to be discussed in detail in [22]. It
+preserves R antisymmetry and reality of star forms, and
+for a p form ω we have
+∗ ∗ω = −(−1)pω.
+(18)
+For equal-degree p forms ω and χ we also have
+ω ∧⋆ ∗χ = (−1)σ(p)+1p!ω⋆
+µ...ν ⋆ Rκ
+µ . . . Rρ
+νχ⋆ρ...κ d4x,
+∗ω ∧⋆ χ = (−1)p+σ(p)+1p!Rµ
+κ . . . Rν
+ρω⋆
+µ...ν ⋆ χ⋆ρ...κ d4x,
+so that
+�
+ω ∧⋆ ∗χ =
+�
+χ ∧⋆ ∗ω.
+(19)
+upon integration by parts.
+Related to Hodge duality
+commuting with star products, we have
+[dxµ ∧⋆ dxν ∧⋆ dxρ ∧⋆ dxσ ⋆, f] = 0,
+(20)
+for
+any
+f,
+which
+concretely
+follows
+from
+star-
+commutativity of ǫ and the volume form, being constant
+and Lorentz invariant respectively. This implies that ǫ is
+an invariant of the R matrix, namely
+ǫτκζφR µ
+τ R ν
+κ R ρ
+ζ R σ
+φ
+= ǫµνρσ.
+(21)
+A similar form of Hodge duality was discussed for q-
+Minkowski space in [26], see also [27].
+YANG-MILLS THEORY
+We would like to consider Yang-Mills theory on
+Lorentz-deformed Minkowski space. Because its gauge
+transformations are functions, they are affected by the
+star product, and it is natural to consider star-gauge
+transformations [16]. A fundamental field Φ then trans-
+forms as
+δεΦ(x) = iε(x) ⋆ Φ(x),
+under a gauge transformation by ε ∈ h, where h is the
+Lie algebra of the gauge group H[28].
+Working in terms of forms, we have
+d (δεΦ) = d (iε ⋆ Φ)
+= idε ⋆ Φ + iε ⋆ dΦ,
+and we can define the covariant derivative
+DΦ = dΦ + iA ⋆ Φ,
+(22)
+with
+δεA = dε + i [ε ⋆, A] ,
+δε (DΦ) = iε ⋆ DΦ.
+Next we define the field strength tensor
+G = dA − iA ∧⋆ A,
+(23)
+which transforms star-covariantly
+δεG = i [ε ⋆, G] .
+We now consider a natural deformation of the commuta-
+tive Yang-Mills action,
+SNC-YM =
+�
+Tr G ∧⋆ ∗G.
+(24)
+Since our Hodge dual commutes with star products, ∗G
+transforms star covariantly.
+Since our star product is
+cyclic under integration, this action is gauge invariant.
+To illustrate this nontrivial point, let us derive the
+transformation of ∗G in components. Starting from
+G = G⋆
+µν ⋆ dxµ ∧⋆ dxν,
+using eqs. (7), we find
+δεG⋆
+µν = iε ⋆ G⋆
+µν − iG⋆
+ρσ ⋆ R ρ
+µ R σ
+ν ε.
+(25)
+The transformation of ∗G is then
+δε (∗G) = iε ⋆ (∗G)
+− iG⋆
+µνǫξκτλ ⋆ dxρ ∧⋆ dxσ ⋆ Rξ
+σRκ
+ρR µ
+τ R ν
+λ ε
+= i [ε ⋆, ∗G] ,
+(26)
+where we used eqs. (8), and (21).
+In star components our action reads
+SNC-YM =
+�
+Tr G⋆
+µν ⋆ RρµRσνG⋆σρd4x.
+(27)
+Expressed in unstarred components, repeated integration
+by parts gives
+SNC-YM =
+�
+Tr GµνGνµ d4x,
+(28)
+where
+Gµν = ∂[µAν] − i ¯Fρκστ ¯F[ν|
+σ(Aκ) ⋆ ¯F ρ
+|µ](Aτ),
+(29)
+showing that the kinetic term for the gauge field is un-
+deformed, while the interaction terms are deformed.
+Our action has twisted Poincar´e symmetry in the spirit
+of [17, 18], meaning the following. In the commutative
+
+4
+setting, the Poincar´e algebra acts on individual fields via
+Lie derivatives, which by the product rule combine to Lie
+derivatives of the Lagrangian [29]. For Poincar´e genera-
+tors these are total derivatives, leaving the action invari-
+ant. Introducing a coproduct ∆(ξ) = ξ ⊗ 1 + 1 ⊗ ξ for
+generators ξ, the product rule takes the form
+ξ(µ(f, g)) = µ(∆(ξ)(f, g)),
+with multiple coproducts extending this to products in-
+volving more fields.
+Our twisted product is similarly
+compatible with a twisted coproduct
+ξ(f ⋆ g) = ξ(µ(F−1(f, g))) = µ(F−1∆F(ξ)(f, g)),
+where ∆F = F∆F−1. Since every product in our ac-
+tion is a star product, if we let the Poincar´e algebra act
+(nonlocally) on products of fields by this twisted coprod-
+uct, the result is still a total derivative, and an invari-
+ant action. The specific twisted Poincar´e algebra for our
+Lorentz twist is discussed in [30].
+MATTER FIELDS AND SUPERSYMMETRIC
+YANG-MILLS THEORY
+We can readily couple our theory to matter. For ad-
+joint scalars for instance, we can write
+SNC-φ =
+�
+Tr Dφ† ∧⋆ ∗Dφ +
+�
+Tr(φ† ⋆ φ)⋆n d4x. (30)
+where
+Dφ = dφ − i [A ⋆, φ] .
+Gauge invariance follows as for star-Yang-Mills theory.
+Working with forms allows us to straightforwardly de-
+fine actions, while guessing e.g.
+the component forms
+of eqs.
+(27-29) would be difficult.
+To tackle fermions
+in similar spirit, we combine left and right-handed Weyl
+spinors ψα and ¯ψ ˙α with Grassmann-valued basis spinors
+sα and ¯s ˙α to form the Grassmann-even ψ = ψαsα and
+¯ψ = ¯ψ ˙αs ˙α. We then take our twist to act via the left
+and right-handed Weyl representation of the Poincar´e al-
+gebra, on sα and ¯s ˙α respectively. These spinors play an
+analogous role to forms, in components resulting in spinor
+analogues of eqs. (5-10).
+We now assemble our γ matrices into a convenient
+object, taking the Pauli matrices σi, i = 1, 2, 3, and
+σ0 = 12×2 to form
+σ = σµα ˙αsα¯s ˙αdxµ = σ⋆
+µα ˙αsα ⋆ ¯s ˙α ⋆ dxµ.
+Coupled by the Pauli matrices, the transformation prop-
+erties of the spinors and one form cancel, making σ
+Lorentz invariant, hence star commutative. For adjoint
+fermions we then define the kinetic action
+SNC-ψ =
+� � �
+d2sd2¯s Tr ¯ψ ⋆ σ ∧⋆ ∗Dψ,
+(31)
+where
+Dψ = dψ − i [A ⋆, ψ] ,
+and the Grassmann integrals over the basis spinors ex-
+tract the appropriate components. Gauge invariance of
+this action follows as before, since σ is star commuta-
+tive. Combined with an adjoint scalar φ, we can form
+the gauge-invariant Yukawa-like interactions
+� �
+d2¯s Tr ¯ψ ⋆ φ ⋆ ¯ψ d4x,
+and
+� �
+d2s Tr ψ ⋆ φ ⋆ ψ d4x.
+We use these ingredients to define the action for
+maximally-supersymmetric Yang-Mills theory (SYM) on
+Lorentz-deformed R1,3 as
+SNC-SYM = 1
+4g2 Tr
+�
+G ∧⋆ ∗G + Tr
+�
+DφIJ ∧⋆ ∗DφIJ − g2
+16Tr
+�
+d4x
+�
+φIJ, φKL�
+⋆ ⋆ [φIJ, φKL]⋆
+(32)
++ Tr
+�
+d2sd2¯s
+�
+¯ψI ⋆ σ ∧⋆ ∗DψI + ig
+2 Tr
+�
+d2s
+�
+d4x ψI ⋆
+�
+φIJ, ψJ
+�
+⋆ − ig
+2 Tr
+�
+d2¯s
+�
+d4x ¯ψI ⋆
+�
+φIJ, ¯ψJ
+�
+⋆ .
+where ψI, I = 1, 2, 3, 4, are the four fermions of SYM,
+and the φIJ = −φJI contain the six real scalars. This de-
+formation of SYM classically has twisted superconformal
+symmetry [31].
+As the dilatation generator commutes
+with our twist, it is conventionally scale invariant.
+With gauge algebra u(n) this action provides a can-
+didate AdS/CFT dual to the Yang-Baxter deformation
+of the AdS5×S5 superstring [9–11] for the r matrix
+r = M01 ⊗M23 −M23 ⊗M01, as conjectured in [14] based
+on a shared twisted symmetry structure, and conceptu-
+ally in line with the discussion in [32]. The corresponding
+
+5
+AdS5 background is deformed to
+ds2 = −ρ2dα2 + r2dθ2
+z2 − ˜λ2ρ2r2/z2 + dθ2 + dr2 + dz2
+z2
+,
+(33)
+B = ˜λ
+ρ2r2
+z4 − ˜λ2ρ2r2 dα ∧ dθ,
+e−2(φ−φ0) = 1 − λ2 ρ2r2
+z4 ,
+in Rindler coordinates (ρ, α) in the (x0, x1) plane, and
+polar coordinates (r, θ) in the (x2, x3) plane, of AdS5 in
+the Poincar´e patch. It is further supported by nontrivial
+Ramond-Ramond forms.
+The deformation parameters
+are related as λ =
+�
+g2
+YMNc˜λ/2π.
+OUTLOOK
+We have constructed an action for noncommutative
+Yang-Mills theory with star-gauge symmetry, for the
+Lorentz twist with quadratically-coordinate-dependent
+noncommutativity. Our construction relies on properties
+of the twist and R matrix, combined with our nontrivial
+twisted Hodge duality, and, for SYM, on our fermionic
+extension of twisted differential calculus.
+There are various open questions surrounding our de-
+formation at the quantum level, for instance regarding
+the form of UV/IR mixing and its presumable absence in
+SYM, and the fate of twisted symmetry. At the classical
+level, noncommutative gauge theories admit an under-
+lying L∞ algebraic structure [33, 34], and it would be
+interesting to investigate this for our deformation, and
+contrast it with the braided noncommutative gauge the-
+ories of [34, 35].
+Applied to SYM, the Lorentz deformation gives the
+natural AdS/CFT dual of a related Yang-Baxter defor-
+mation of the AdS5 string. Our construction in fact ex-
+tends to all noncommutative spacetimes described by the
+known Drinfeld twists of the Poincar´e algebra, with uni-
+modular r matrix, providing candidate gauge theory du-
+als for a large class of Yang-Baxter deformations of the
+AdS5 string [22]. However, while matching planar sym-
+metry structures between gauge and string theory is cer-
+tainly promising, the actual decoupling limit underlying
+these dualities can be subtle. For constant noncommu-
+tativity, while the space-like and light-like cases are fine
+[36], time-like noncommutativity results in a noncommu-
+tative open string rather than gauge theory [37, 38]. Our
+Lorentz deformation mixes these cases, appearing space-
+like inside the light-cone in the (x0, x1) plane, but time-
+like outside it, and in general this decoupling limit needs
+careful analysis. However, even in cases with a subtle
+decoupling limit, remnants of a duality to our type of
+noncommutative gauge theory are likely to survive at the
+planar level.
+We expect planar Lorentz-deformed SYM to be inte-
+grable, based on the integrability of its proposed string
+dual.
+At the classical level this should take the form
+of Yangian invariance [39, 40], now twisted similarly to
+[41].
+At the quantum level, we should find a spectral
+problem described by an integrable spin chain, similar
+to the famous dilatation operator of undeformed SYM
+[42].
+The Lorentz deformation is particularly natural
+in this regard, as it preserves dilatation symmetry. We
+have defined a suitable related spectral problem in pla-
+nar Lorentz-deformed SYM, and are in the process of
+extracting its integrable structure [43] – which we expect
+to relate to the twisted spin chain of [32] – building on a
+planar equivalence theorem [22] in the spirit of Filk [44].
+We hope this will pave the way to integrable AdS/CFT
+for general (homogeneous) Yang-Baxter deformations of
+the AdS5 string, and its lower dimensional cousins.
+ACKNOWLEDGEMENTS.
+We would like to thank Riccardo Borsato, Ben Hoare,
+and Anna Pacho�l for discussions, and Gleb Arutyunov,
+Riccardo Borsato, Jerzy Lukierski, Anna Pacho�l, and
+Richard Szabo for valuable comments on the draft. TM’s
+research is funded by the Deutsche Forschungsgemein-
+schaft (DFG, German Research Foundation) - Projek-
+tnummer 417533893/GRK2575 “Rethinking Quantum
+Field Theory”. The work of ST is supported by the Ger-
+man Research Foundation via the Emmy Noether pro-
+gram “Exact Results in Extended Holography”. ST is
+supported by LT.
+∗ tmeier@physik.hu-berlin.de
+† svantongeren@physik.hu-berlin.de
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+
diff --git a/h9FAT4oBgHgl3EQf9h7F/content/tmp_files/load_file.txt b/h9FAT4oBgHgl3EQf9h7F/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..d16b2cc2bb675dcf8200ecd6f375904fa0ec3566
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@@ -0,0 +1,412 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf,len=411
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='08757v1  [hep-th]  20 Jan 2023  HU-EP-23/03-RTG  Quadratic twist-noncommutative gauge theory  Tim Meier∗ and Stijn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' van Tongeren†  Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin,  IRIS Geb¨aude, Zum Grossen Windkanal 2, 12489 Berlin  Studies of noncommutative gauge theory have mainly focused on noncommutative spacetimes  with constant noncommutative structure, with little known about actions for noncommutative 4D  Yang-Mills theory beyond this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' We construct an action for Yang-Mills theory on a quadratically-  noncommutative spacetime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' of quantum-plane type, obtained from a Drinfeld twist, with star-  gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Applied to supersymmetric Yang-Mills theory, this gives a candidate AdS/CFT  dual of string theory on a related deformation of AdS5×S5, which is expected to be integrable in  the planar limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Noncommutativity between space-time coordinates is  a likely feature of quantum gravity [1], actively studied  from numerous angles [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' In string theory, noncom-  mutative gauge theory appears in the low energy dynam-  ics of open strings [5], and thereby the AdS/CFT cor-  respondence [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Despite the formal and phenomeno-  logical relevance of noncommutative gauge theory, it is  not clear how to write actions to all orders in the non-  commutativity, when going beyond the case of constant  noncommutativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  In this letter we consider a non-  commutative Drinfeld-twist deformation of Minkowski  space, with quadratically-coordinate-dependent noncom-  mutativity, and construct an all-order action for Yang-  Mills theory with star-gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Beyond provid-  ing a first example of a noncommutative Yang-Mills the-  ory action with quadratic noncommutativity, our choice  of deformation is motivated by the AdS/CFT correspon-  dence and integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Applied to maximally supersym-  metric Yang-Mills theory, our noncommutative deforma-  tion provides a concrete candidate gauge theory dual of  a particular Yang-Baxter deformation [9–11] of the fa-  mously integrable AdS5×S5 superstring [12, 13], as con-  jectured in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  This opens the door to investigating  integrability for a range of novel planar noncommutative  gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We consider noncommutative field theory in the usual  spirit of Weyl quantization, trading noncommuting field  operators for a noncommutative product – the star prod-  uct – between commutative fields [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Our star  product is obtained from a Drinfeld twist, whereby it  automatically comes with clear algebraic properties and  a natural differential calculus [2], and twists rather than  plainly breaks Poincar´e symmetry [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' The original  Groenewold-Moyal noncommutative deformation can be  viewed as a twist, and it is well known how to construct  a Yang-Mills action in this case [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' For general twists,  however, it is not clear how to define a suitable dual field  strength tensor, and construct an action for noncommu-  tative Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' For example, the approaches  of [19, 20] for κ-Minkowski space, were necessarily per-  turbative, and only solved to leading order in the non-  commutativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' To our knowledge, the only non-constant  case known to all orders, is the U(1) Yang-Mills theory  studied in [21] for a particular twist with linear noncom-  mutativity, where standard Hodge duality suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We show how a twisted version of Hodge duality allows  us to define a noncommutative Yang-Mills action for a  twist based on two commuting Lorentz generators, with a  noncommutative structure with quadratic coordinate de-  pendence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Our construction moreover provides a broader  framework that covers all known Poincar´e-based twist-  deformations of Minkowski space, including non-abelian  ones, whose r matrices are unimodular [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Given our  motivations in AdS/CFT, we also discuss how to cou-  ple our theory to (adjoint) matter, define our deformed  maximally-supersymmetric Yang-Mills theory, and dis-  cuss its AdS/CFT interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  LORENTZ-DEFORMED MINKOWSKI SPACE  In deformation quantization, a noncommutative space-  time is described via a regular spacetime whose function  algebra is equipped with a noncommutative (star) prod-  uct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' The noncommutative product we consider for func-  tions on R1,3 is based on the Drinfeld twist  F = e  iλ  2 (M01⊗M23−M23⊗M01),  (1)  built from two commuting Lorentz generators, M01 and  M23, with Mµν = i(xµ∂ν − xν∂µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' It defines our non-  commutative star product via  f(x) ⋆ g(x) ≡ µ(F−1(f(x), g(x))),  (2)  where µ(h(x), z(x)) = h(x)z(x) is the ordinary pointwise  product of functions, and λ is our deformation parame-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' We will refer to this as the Lorentz twist, or defor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' In contrast to the familiar Groenewold-Moyal  star product associated to  FGM = e− iθµν  4  (∂µ⊗∂ν−∂ν⊗∂µ),  where θ is a constant antisymmetric matrix, the bidif-  ferential operator appearing in the exponent of our twist  has nontrivial coordinate dependence [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  2  Our star product is associative because M01 and M23  commute [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Its noncommutative structure is  xµ ⋆ xν = Rσµρνxρ ⋆ xσ,  (3)  with 24 nonzero components for R,  R0  0  0  0 = 1,  R0  0  2  2 = cosh λ,  R0  1  2  3 = −i sinh λ, (4)  and others obtained by index permutation symmetries of  the twist (1), namely 2 ↔ 3 or (0, 1) ↔ (2, 3) combined  with a sign change of λ, and 0 ↔ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Under (conventional) conjugation we have f ⋆ g = g⋆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  As our goal is gauge theory, we need differential calcu-  lus, now suitably twisted [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Using standard differential  calculus on Minkowski space we define  dxµ ⋆ f = µ(F−1(dxµ, f)),  (5)  where the vector fields M01 and M23 act via Lie deriva-  tives, summarized as  dxµ ⋆ f = dxν ¯F ν  µ(f),  (6)  with  ¯Fµν =  \uf8eb  \uf8ec  \uf8ed  cosh λM23  2  − sinh λM23  2  0  0  − sinh λM23  2  cosh λM23  2  0  0  0  0  cos λM01  2  − sin λM01  2  0  0  sin λM01  2  cos λM01  2  \uf8f6  \uf8f7  \uf8f8  ν  µ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Commuting functions through forms gives rise to  dxµ ⋆ f = Rνµ(f) ⋆ dxν,  (7)  with the R matrix Rν µ = ¯Fν ρ ¯Fρν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Both R and ¯F are  vector-field-valued elements of the Lorentz group, in the  sense that, raising and lowering indices with the usual  Minkowski metric,  RµνRρν = ¯Fµν ¯F ρν = δρ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (8)  We also define a star-wedge product  dxµ ∧⋆ dxν = ˆµ(F−1(dxµ, dxν)),  (9)  where ˆµ(a, b) = a ∧ b is the regular wedge product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Con-  cretely  dxµ ∧⋆ dxν = ¯Fσµρνdxσ ∧ dxρ,  (10)  with ¯Fσµρν = Rσµρν|λ→λ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' We will mostly work with  star forms  ω = ω⋆  µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='ρ ⋆ dxµ ∧⋆ dxν ∧⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' ∧⋆ dxρ,  (11)  but occasionally will also express them as regular forms  ω = ωµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='ρdxµ ∧ dxν ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' ∧ dxρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (12)  Our star forms are totally R-antisymmetric, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  dxµ ∧⋆ dxν = −Rρµσνdxσ ∧⋆ dxρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We will use the ordinary exterior derivative, which has  the desired product rule  d(ω ∧⋆ χ) = dω ∧⋆ χ + (−1)pω ∧⋆ dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (13)  for p and q forms ω and χ, respectively, as it com-  mutes with Lie derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Under conjugation we have  ω ∧⋆ χ = (−1)pqχ ∧⋆ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Our star product is graded-cyclic under an integral,  �  ω ∧⋆ χ = (−1)pq  �  χ ∧⋆ ω  (14)  when χ∧⋆ ω is a top form, upon integration by parts [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  HODGE DUALITY  To define our twisted Hodge duality we take a natural  generalization of the Levi-Civita symbol,  dxµ ∧⋆ dxν ∧⋆ dxρ ∧⋆ dxσ = ǫµνρσd4x,  (15)  where the volume form d4x = dx0 ∧⋆ dx1 ∧⋆ dx2 ∧⋆ dx3 =  dx0 ∧ dx1 ∧ dx2 ∧ dx3 is not deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' By definition ǫ is  R antisymmetric, and conjugates as  ǫµνρσ = ǫσρνµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  It is also graded cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Its 32 nonzero components are  ǫ0123 = −ǫ0132 = ǫ0231 = −ǫ0321 = 1,  ǫ1212 = −ǫ0202 = ǫ1313 = −ǫ0303 = i sinh λ,  ǫ0312 = −ǫ0213 = cosh λ,  (16)  plus others related by graded cyclicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  In regular Hodge duality we can freely permute indices  on the Levi-Civita symbol for signs, giving many equiv-  alent definitions of a dual form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' The appropriate choice  in our twisted setting is  ∗ dxµ1 ∧⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' ∧⋆ dxµk = (−1)σ(k)  (4−k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' ǫµk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='µ4  µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='µkdxµ4 ∧⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' ∧⋆ dxµk+1,  (17)  3  where σ(p) denotes the signature of the reversal of p ob-  jects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  σ(1) = σ(4) = 0, σ(2) = σ(3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  The  reversed index contraction in the dual form is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Restricted to basis star forms, this twisted Hodge du-  ality commutes with Lie derivatives along vector fields  in the Poincar´e algebra, and hence with our star prod-  uct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' This allows us to consistently extend it star-linearly  to arbitrary forms, where it continues to commute with  Poincar´e Lie derivatives and our star product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Our Hodge duality has all other usual properties, ap-  propriately twisted, to be discussed in detail in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' It  preserves R antisymmetry and reality of star forms, and  for a p form ω we have  ∗ ∗ω = −(−1)pω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (18)  For equal-degree p forms ω and χ we also have  ω ∧⋆ ∗χ = (−1)σ(p)+1p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='ω⋆  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='ν ⋆ Rκ  µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Rρ  νχ⋆ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='κ d4x,  ∗ω ∧⋆ χ = (−1)p+σ(p)+1p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='Rµ  κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Rν  ρω⋆  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='ν ⋆ χ⋆ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='κ d4x,  so that  �  ω ∧⋆ ∗χ =  �  χ ∧⋆ ∗ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (19)  upon integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Related to Hodge duality  commuting with star products, we have  [dxµ ∧⋆ dxν ∧⋆ dxρ ∧⋆ dxσ ⋆, f] = 0,  (20)  for  any  f,  which  concretely  follows  from  star-  commutativity of ǫ and the volume form, being constant  and Lorentz invariant respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' This implies that ǫ is  an invariant of the R matrix, namely  ǫτκζφR µ  τ R ν  κ R ρ  ζ R σ  φ  = ǫµνρσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (21)  A similar form of Hodge duality was discussed for q-  Minkowski space in [26], see also [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  YANG-MILLS THEORY  We would like to consider Yang-Mills theory on  Lorentz-deformed Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Because its gauge  transformations are functions, they are affected by the  star product, and it is natural to consider star-gauge  transformations [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' A fundamental field Φ then trans-  forms as  δεΦ(x) = iε(x) ⋆ Φ(x),  under a gauge transformation by ε ∈ h, where h is the  Lie algebra of the gauge group H[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Working in terms of forms, we have  d (δεΦ) = d (iε ⋆ Φ)  = idε ⋆ Φ + iε ⋆ dΦ,  and we can define the covariant derivative  DΦ = dΦ + iA ⋆ Φ,  (22)  with  δεA = dε + i [ε ⋆, A] ,  δε (DΦ) = iε ⋆ DΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Next we define the field strength tensor  G = dA − iA ∧⋆ A,  (23)  which transforms star-covariantly  δεG = i [ε ⋆, G] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We now consider a natural deformation of the commuta-  tive Yang-Mills action,  SNC-YM =  �  Tr G ∧⋆ ∗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (24)  Since our Hodge dual commutes with star products, ∗G  transforms star covariantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Since our star product is  cyclic under integration, this action is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  To illustrate this nontrivial point, let us derive the  transformation of ∗G in components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Starting from  G = G⋆  µν ⋆ dxµ ∧⋆ dxν,  using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' (7), we find  δεG⋆  µν = iε ⋆ G⋆  µν − iG⋆  ρσ ⋆ R ρ  µ R σ  ν ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (25)  The transformation of ∗G is then  δε (∗G) = iε ⋆ (∗G)  − iG⋆  µνǫξκτλ ⋆ dxρ ∧⋆ dxσ ⋆ Rξ  σRκ  ρR µ  τ R ν  λ ε  = i [ε ⋆, ∗G] ,  (26)  where we used eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' (8), and (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  In star components our action reads  SNC-YM =  �  Tr G⋆  µν ⋆ RρµRσνG⋆σρd4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (27)  Expressed in unstarred components, repeated integration  by parts gives  SNC-YM =  �  Tr GµνGνµ d4x,  (28)  where  Gµν = ∂[µAν] − i ¯Fρκστ ¯F[ν|  σ(Aκ) ⋆ ¯F ρ  |µ](Aτ),  (29)  showing that the kinetic term for the gauge field is un-  deformed, while the interaction terms are deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Our action has twisted Poincar´e symmetry in the spirit  of [17, 18], meaning the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' In the commutative  4  setting, the Poincar´e algebra acts on individual fields via  Lie derivatives, which by the product rule combine to Lie  derivatives of the Lagrangian [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' For Poincar´e genera-  tors these are total derivatives, leaving the action invari-  ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Introducing a coproduct ∆(ξ) = ξ ⊗ 1 + 1 ⊗ ξ for  generators ξ, the product rule takes the form  ξ(µ(f, g)) = µ(∆(ξ)(f, g)),  with multiple coproducts extending this to products in-  volving more fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Our twisted product is similarly  compatible with a twisted coproduct  ξ(f ⋆ g) = ξ(µ(F−1(f, g))) = µ(F−1∆F(ξ)(f, g)),  where ∆F = F∆F−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Since every product in our ac-  tion is a star product, if we let the Poincar´e algebra act  (nonlocally) on products of fields by this twisted coprod-  uct, the result is still a total derivative, and an invari-  ant action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' The specific twisted Poincar´e algebra for our  Lorentz twist is discussed in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  MATTER FIELDS AND SUPERSYMMETRIC  YANG-MILLS THEORY  We can readily couple our theory to matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' For ad-  joint scalars for instance, we can write  SNC-φ =  �  Tr Dφ† ∧⋆ ∗Dφ +  �  Tr(φ† ⋆ φ)⋆n d4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' (30)  where  Dφ = dφ − i [A ⋆, φ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Gauge invariance follows as for star-Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Working with forms allows us to straightforwardly de-  fine actions, while guessing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  the component forms  of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  (27-29) would be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  To tackle fermions  in similar spirit, we combine left and right-handed Weyl  spinors ψα and ¯ψ ˙α with Grassmann-valued basis spinors  sα and ¯s ˙α to form the Grassmann-even ψ = ψαsα and  ¯ψ = ¯ψ ˙αs ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' We then take our twist to act via the left  and right-handed Weyl representation of the Poincar´e al-  gebra, on sα and ¯s ˙α respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' These spinors play an  analogous role to forms, in components resulting in spinor  analogues of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' (5-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We now assemble our γ matrices into a convenient  object, taking the Pauli matrices σi, i = 1, 2, 3, and  σ0 = 12×2 to form  σ = σµα ˙αsα¯s ˙αdxµ = σ⋆  µα ˙αsα ⋆ ¯s ˙α ⋆ dxµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Coupled by the Pauli matrices, the transformation prop-  erties of the spinors and one form cancel, making σ  Lorentz invariant, hence star commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' For adjoint  fermions we then define the kinetic action  SNC-ψ =  � � �  d2sd2¯s Tr ¯ψ ⋆ σ ∧⋆ ∗Dψ,  (31)  where  Dψ = dψ − i [A ⋆, ψ] ,  and the Grassmann integrals over the basis spinors ex-  tract the appropriate components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Gauge invariance of  this action follows as before, since σ is star commuta-  tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Combined with an adjoint scalar φ, we can form  the gauge-invariant Yukawa-like interactions  � �  d2¯s Tr ¯ψ ⋆ φ ⋆ ¯ψ d4x,  and  � �  d2s Tr ψ ⋆ φ ⋆ ψ d4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We use these ingredients to define the action for  maximally-supersymmetric Yang-Mills theory (SYM) on  Lorentz-deformed R1,3 as  SNC-SYM = 1  4g2 Tr  �  G ∧⋆ ∗G + Tr  �  DφIJ ∧⋆ ∗DφIJ − g2  16Tr  �  d4x  �  φIJ, φKL�  ⋆ ⋆ [φIJ, φKL]⋆  (32)  + Tr  �  d2sd2¯s  �  ¯ψI ⋆ σ ∧⋆ ∗DψI + ig  2 Tr  �  d2s  �  d4x ψI ⋆  �  φIJ, ψJ  �  ⋆ − ig  2 Tr  �  d2¯s  �  d4x ¯ψI ⋆  �  φIJ, ¯ψJ  �  ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  where ψI, I = 1, 2, 3, 4, are the four fermions of SYM,  and the φIJ = −φJI contain the six real scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' This de-  formation of SYM classically has twisted superconformal  symmetry [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  As the dilatation generator commutes  with our twist, it is conventionally scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  With gauge algebra u(n) this action provides a can-  didate AdS/CFT dual to the Yang-Baxter deformation  of the AdS5×S5 superstring [9–11] for the r matrix  r = M01 ⊗M23 −M23 ⊗M01, as conjectured in [14] based  on a shared twisted symmetry structure, and conceptu-  ally in line with the discussion in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' The corresponding  5  AdS5 background is deformed to  ds2 = −ρ2dα2 + r2dθ2  z2 − ˜λ2ρ2r2/z2 + dθ2 + dr2 + dz2  z2  ,  (33)  B = ˜λ  ρ2r2  z4 − ˜λ2ρ2r2 dα ∧ dθ,  e−2(φ−φ0) = 1 − λ2 ρ2r2  z4 ,  in Rindler coordinates (ρ, α) in the (x0, x1) plane, and  polar coordinates (r, θ) in the (x2, x3) plane, of AdS5 in  the Poincar´e patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' It is further supported by nontrivial  Ramond-Ramond forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  The deformation parameters  are related as λ =  �  g2  YMNc˜λ/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  OUTLOOK  We have constructed an action for noncommutative  Yang-Mills theory with star-gauge symmetry, for the  Lorentz twist with quadratically-coordinate-dependent  noncommutativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Our construction relies on properties  of the twist and R matrix, combined with our nontrivial  twisted Hodge duality, and, for SYM, on our fermionic  extension of twisted differential calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  There are various open questions surrounding our de-  formation at the quantum level, for instance regarding  the form of UV/IR mixing and its presumable absence in  SYM, and the fate of twisted symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' At the classical  level, noncommutative gauge theories admit an under-  lying L∞ algebraic structure [33, 34], and it would be  interesting to investigate this for our deformation, and  contrast it with the braided noncommutative gauge the-  ories of [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Applied to SYM, the Lorentz deformation gives the  natural AdS/CFT dual of a related Yang-Baxter defor-  mation of the AdS5 string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Our construction in fact ex-  tends to all noncommutative spacetimes described by the  known Drinfeld twists of the Poincar´e algebra, with uni-  modular r matrix, providing candidate gauge theory du-  als for a large class of Yang-Baxter deformations of the  AdS5 string [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' However, while matching planar sym-  metry structures between gauge and string theory is cer-  tainly promising, the actual decoupling limit underlying  these dualities can be subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' For constant noncommu-  tativity, while the space-like and light-like cases are fine  [36], time-like noncommutativity results in a noncommu-  tative open string rather than gauge theory [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Our  Lorentz deformation mixes these cases, appearing space-  like inside the light-cone in the (x0, x1) plane, but time-  like outside it, and in general this decoupling limit needs  careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' However, even in cases with a subtle  decoupling limit, remnants of a duality to our type of  noncommutative gauge theory are likely to survive at the  planar level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We expect planar Lorentz-deformed SYM to be inte-  grable, based on the integrability of its proposed string  dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  At the classical level this should take the form  of Yangian invariance [39, 40], now twisted similarly to  [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  At the quantum level, we should find a spectral  problem described by an integrable spin chain, similar  to the famous dilatation operator of undeformed SYM  [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  The Lorentz deformation is particularly natural  in this regard, as it preserves dilatation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' We  have defined a suitable related spectral problem in pla-  nar Lorentz-deformed SYM, and are in the process of  extracting its integrable structure [43] – which we expect  to relate to the twisted spin chain of [32] – building on a  planar equivalence theorem [22] in the spirit of Filk [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We hope this will pave the way to integrable AdS/CFT  for general (homogeneous) Yang-Baxter deformations of  the AdS5 string, and its lower dimensional cousins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  We would like to thank Riccardo Borsato, Ben Hoare,  and Anna Pacho�l for discussions, and Gleb Arutyunov,  Riccardo Borsato, Jerzy Lukierski, Anna Pacho�l, and  Richard Szabo for valuable comments on the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' TM’s  research is funded by the Deutsche Forschungsgemein-  schaft (DFG, German Research Foundation) - Projek-  tnummer 417533893/GRK2575 “Rethinking Quantum  Field Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' The work of ST is supported by the Ger-  man Research Foundation via the Emmy Noether pro-  gram “Exact Results in Extended Holography”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' ST is  supported by LT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  ∗ tmeier@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='de  † svantongeren@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='de  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Doplicher,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Fredenhagen,  and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  Roberts,  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
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+page_content='04066 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' van Tongeren, to appear ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  [23] Changing coordinates does not fundamentally change  this, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' in polar coordinates in the (x2, x3) plane,  while M23 becomes ∂θ, we should consider eiθ rather than  θ as an operator before the Weyl map, and M23(eiθ) is  not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Moreover, the natural Rindler type coor-  dinates for M01, inconveniently do not cover R1,3 in one  go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  [24] Formally, all twist-star products are associative thanks to  the cocycle condition on the twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' We are purposefully  introducing only minimal formal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
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+page_content='  [28] Star-gauge transformations only close for transforma-  tions in the fundamental representation of u(n) (or  gl(n, C), undesirable for other reasons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Other gauge alge-  bras can be considered in a universal enveloping algebra  approach [45], with the apparently infinitely many as-  sociated degrees of freedom reduced to finitely many via  the Seiberg-Witten map [5], at least perturbatively in the  deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' For our AdS/CFT applications,  u(n) suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  [29] In our covariant notation, this relies on the fact that  Poincar´e Lie derivatives commute with Hodge duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content='  The same is required, and holds, in our twisted setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
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+page_content=' Schupp, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Wess,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
+page_content=' C 21, 383 (2001), arXiv:hep-th/0104153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FAT4oBgHgl3EQf9h7F/content/2301.08757v1.pdf'}
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+arXiv:2301.03901v1  [gr-qc]  10 Jan 2023
+Constraining study of circular orbits and accretion disk around nonlinear
+electrodynamics black hole
+A. Ditta,1, ∗ G. Mustafa,2, † G. Abbas,1, ‡ Farruh Atamurotov,3, 4, 5, 6, § and Kimet Jusufi7, ¶
+1Department of mathematics, The Islamia University of Bahawalpur,
+Bahawalpur-63100, Pakistan
+2Department of Physics, Zhejiang Normal University, Jinhua 321004, China
+3Inha University in Tashkent, Ziyolilar 9, Tashkent 100170, Uzbekistan
+4Akfa University, Milliy Bog’ Street 264, Tashkent 111221, Uzbekistan
+5National University of Uzbekistan, Tashkent 100174, Uzbekistan
+6Tashkent State Technical University, Tashkent 100095, Uzbekistan
+7Physics Department, State University of Tetovo,
+Ilinden Street nn, 1200, Tetovo, North Macedonia
+The very latest observation of M87 supermassive black hole (BH) by the Event Horizon Telescope
+(EHT) provides the accretion onto BHs is an interesting study in the theory of gravity. We study
+the geodesics structure and accretion near a nonlinear electrodynamics BH in strong and weak field
+approximations. These approximations provide the disc-like structure under the geodesic motion
+and accretion around the BH. Near the equatorial plane, we provide some new reasons to make
+circular orbits and accretion of test particles around the BH. Then we investigate perturbations,
+the critical speed of the fluid and the mass accretion rate of particles around the central object.
+The physical validity of this study shows that the parameter β and Q play an important role in the
+circular orbits and the mass accretion rate in strong and weak field approximations.
+Keywords: Circular orbits; Accretion; Nonlinear Electrodynamics black hole.
+I.
+INTRODUCTION
+Black holes are quite fascinating objects revealed by the theory of general relativity (GR). In the universe, they
+are known as robust sources of the gravitational field and are also likely to have a high spin and magnetic intensity.
+Owing to these features, black holes are the supreme research laboratory for studying both gravity and matter in
+astrophysical experiments. Currently, some observational evidence has confirmed the presence of BHs. The first was
+the innovation of gravitational waves brought out from a binary BH merger by LIGO and Virgo collaboration [1].
+Another amazing revolutionary is the first image of M87∗ BH shadow [2, 3] and the very recently released image
+of Sgr A∗ [4] by the EHT through a very long baseline interferometry. Furthermore, the perceived electromagnetic
+spectrum of accretion disks can be responsible for the existence of BHs [5–7]. These latest achievements provide an
+informative response to the appreciation of the theory of GR and the nature of accretion disks near supermassive BHs
+in the strong gravity regime around the event horizon and may be assumed as a way to analyze modified theories of
+gravity.
+Accretion disks are formed due to the rotation of gaseous materials that travel in bounded orbits around the central
+mass because of the gravitational force, such as neutron stars, supermassive BHs, and Young Stellar Objects in Active
+Galactic Nuclei. In such systems, particle orbits are stable, but when the orbits of these materials become unstable,
+an accretion will occur. It is generally believed that massive central objects such as BHs capture particles from fluid in
+their vicinity and increase their mass through the accretion process. So, the study of the geodesic motion of particles
+in the vicinity of BH and particularly the analysis of some characteristic radii such as innermost stable circular orbits
+(risco) and marginally bound orbits (rmb) are remarkable issues for a suspicious study of the subject matter. These
+radii are essential in the study of BH accretion disks, where the inner edge of the disk coincides with the innermost
+stable circular orbit (ISCO) and the efficiency of the energy released [8].
+∗Electronic address: adsmeerkhan@gmail.com
+†Electronic address: gmustafa3828@gmail.com
+‡Electronic address: ghulamabbas@iub.edu.pk
+§Electronic address: atamurotov@yahoo.com
+¶Electronic address: kimet.jusufi@unite.edu.mk
+
+2
+The locality of stable or unstable circular orbits is consistent with the minimum or maximum of the effective
+potential correspondingly. In Newtonian theory, the effective potential has a minimum for any value of the angular
+momentum, and then it has no minimum radius of a stable circular orbit, (ISCO) [9]. But this position is altered when
+the effective potential has a difficult form liable to the particle angular momentum and other parameters. Therefore,
+in GR and for the particles moving near the Schwarzschild BH, the effective potential has two extrema for any value
+of angular momentum. But, only for a particular value of angular momentum do the two points happen together.
+This point presents ISCO where is placed at r = 3rg [9–14] where rg is the Schwarzschild radius.
+According to the metrics, the effects of spacetime affect the positions of these radii and some parameters such as
+specific energy, angular momentum, and angular velocity. A lot of research problems are dedicated to studying these
+radii and their physical structure. The effects of ISCO near the Kerr BH were studied by Ruffini et al. [15] and
+Bardeen et al. [16]. Even Hobson et al. [17] defined these properties in their textbook on GR. The efficiency of
+accretion disks, η, for Schwarzschild and Kerr BHs was calculated by Novikov and Thorne [18]. The Kerr-like metric
+was raised by Johannsen and Psaltis [19] and then Johannsen [20] formulated the accretion disks near such BHs. The
+geodesic structure and the circular orbits of charged particles near weakly magnetized rotating BHs are obtained by
+Tursunov et al. [21]. Since in an accretion disk the particles move in stable orbits but when perturbations (due to
+restoring forces) act on the particles, oscillations near the circular orbit are produced in radial and vertical directions
+with epicyclic frequencies. Therefore, the study of orbital and epicyclic frequencies plays an important role in the
+physics of accretion disks near the BHs. Isper [22, 23], Wagoner [24], Kato [11] and Ortega-Rodriges et al. [12] have
+studied in this field.
+The exact and analytic solutions obtained by Bondi [25], for characterizing different astrophysical scenarios have
+been involved in the continuous evolution of the accretion theory. Furthermore, analytic solutions are fundamental
+equipment as benchmark tests for numerical codes [26]. This model was prolonged by Michel [27] to a relativistic
+regime by assuming a Schwarzschild BH. Then again, analytic solutions to the known wind accretion scene have been
+developed by Bondi and Hoyle [28], further Hoyle and Littleton [29] in the Newtonian framework and also by Tejeda
+and AguayoOrtiz [30] in the relativistic framework of a Schwarzschild BH. Numerous analytic and numerical analyses
+have further prolonged the work of spherical accretion, e.g. [31–33], and also of wind accretion, e.g. [34–36]. Further,
+Abbas and Ditta obtained the useful properties of accretion for motivation of GR onto a class of BHs [37–45].
+So, there is a blank space in the literature because no study has been carried out on accretion disks for nonlinear
+electrodynamic BH with weak and strong field approximations. However, this study is the motivation of GR and
+we can honestly fill this blank in the present paper. Our main aspect of this paper is to study the non-rotating
+BH solutions of an accretion disk in strong and weak fields. For this, we assume an equatorial plane with a polar
+coordinates system. Firstly, we see the variations in horizons near the geometry of BHs. Then we analyze the circular
+orbits and effective potential as well.
+In order to study the locations of circular orbits such as innermost stable
+circular orbits risco marginally bound circular orbits rmb and photon sphere rph. The maximum radiation efficiency
+and binding energy with epicyclic frequencies are also investigated. Finally, the critical accretion with some dynamical
+parameters of isothermal fluid is obtained.
+Recently, Tretyakova [46], analyzed geodesics in the Horndeski BH observational properties. Further, Salahshoor
+and Nozari [47] used a similar method to analyze the circular orbits and accretion disks in detail in a class of
+Horndeski/Galileon BHs. In the present study, we also introduce a general class of nonlinear electrodynamic BH and
+used a similar procedure to find out the new accretion results for the motivation of GR. The arrangement of this paper
+is as follows: In section 2, we introduce the nonlinear electrodynamic BH spacetime. The wide-range calculations
+for a test particle motion are formulated in section 3. Further, the circular mechanism and oscillations are reviewed
+in the subsequent sections respectively. In section 4, we found the critical speed of the flow, mass accretion, and its
+time variation for the perfect fluid. In section 5, we analyzed the physical significance of all of these results for strong
+and weak fields. We summarized the results of the paper in section 6 by saying that the parameters Q and β have
+greatly suppressed strong and weak field approximations. Throughout the calculations, we consider the geometric
+units G = c = 1, and the spacetime signature (−, +, +, +).
+II.
+NONLINEAR ELECTRODYNAMIC BLACK HOLE
+The non-rotating nonlinear electrodynamics BH solutions are obtained by the general gravitational action [48, 49]
+which is given as
+S =
+� √−gd4x
+� R
+2k2 −
+F
+2βF + 1
+�
+,
+(1)
+
+3
+where g = det(gµν), R is a Ricci scalar, k−1 is a reduced Plank mass, F =
+1
+4FµνF µν and β is the dimensionless
+parameter of nonlinear electrodynamic BH. For a solution, we consider the spherically symmetric spacetime
+ds2 = −f(r)dt2 +
+1
+f(r)dr2 + r2(dθ2 + sin2 θdφ2),
+(2)
+with
+f(r) = 1 − 2M
+r
++ Q2
+r2 − C2k2
+2r2
++ C2k2
+30r2 (5ξ3 − 22ξ2 + 32ξ),
+(3)
+where
+ξ = 12
+√
+3√βr2 − βCλ3/4
+12βCλ1/4
+,
+(4)
+whereas
+λ = 6
+3√
+6r2(
+3√
+2β2/3 3√
+Cγ2/3 − 8
+3√
+3βC)
+β4/3C5/3 3√γ
+,
+(5)
+γ =
+√
+3
+�
+256βC2 + 27r4 + 9r2.
+(6)
+The strong and weak field solutions can be obtained from the metric function f(r) by applying r → 0 and r → ∞
+respectively, such that
+f(r)s = 1 − 2M
+r
++ Q2
+r2 − C2k2
+2r2 + 16C3/2k2
+15β1/4r ,
+(7)
+f(r)w = 1 − 2M
+r
++ Q2
+r2 − βC4k2
+10r6 .
+(8)
+It is noted that in the strong field limit f(r)s, the effects of nonlinear electrodynamics are highly effective and cannot
+be detached unless. However, in weak field limit, the last term goes more rapidly as r → ∞ such that f(r)w and the
+metric function f(r) are asymptotically Reissner-Nordstrom and hence the effects of nonlinear electrodynamics can
+simply be removed by applying β → 0. Fig. (1) has the following key points:
+• In the upper left plot, the curve passes through the circular disk (green and blue) and has an event and two
+horizons (inner and outer) in a strong field for different values of M.
+• In the upper right plot, the black curve has an event horizon while the red curve has two horizons for different
+values of Q in a strong field.
+• An event and two horizons are also present in the left bottom plot for the weak field.
+• The right bottom plot has an event horizon for β = 0 in a weak field and has no two horizons.
+It is noted that in a strong field, for smaller values of M, the curves are shifted outward to an event horizon. In the
+case of a weak field, for larger values of M, the curves are shifted inward to an event horizon. Both the left plots have
+Cauchy horizons in strong and weak fields, while the plots have not.
+III.
+GENERAL FORMULISM OF GEODESIC MOTION
+In this section, we formulate the general results for the geodesic motion of test particles by the underlying static and
+symmetric spacetime. For these results, we consider two Killing vectors ζt = ∂t and ζφ = ∂φ parallel two constants of
+motion E and L (conserved energy and angular momentum) by the trajectory as follows.
+E = −gµνζµ
+t uν ≡ −ut,
+L = gµνζµ
+φuν ≡ uφ,
+(9)
+where uµ = dxµ
+dτ = (ut, ur, uθ, uφ) is the 4-velocity of the moving particles and the particles obey the normalization
+condition uµuµ = −1, we obtain
+grr(ur)2 + gθθ(uθ)2 + gtt(ut)2 + gφφ(uφ)2 = −1.
+(10)
+
+4
+0
+2
+4
+6
+8
+10
+-0.5
+0.0
+0.5
+1.0
+1.5
+r
+f(r)
+0
+2
+4
+6
+8
+10
+-0.5
+0.0
+0.5
+1.0
+1.5
+r
+f(r)
+2
+4
+6
+8
+10
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+r
+f(r)
+0
+2
+4
+6
+8
+10
+-0.5
+0.0
+0.5
+1.0
+1.5
+r
+f(r)
+FIG. 1: Variation of horizon of strong and weak fields for M, Q and β. Upper plots for the strong field while lower plots for
+the weak field. This behavior is for the equations (7, 8) and the parameters are M = 1, k = 1, C = 1 and different values of Q
+and β.
+Taking (θ = π/2), for the equatorial plane then from Eqs. (9) and (10), we obtain
+ut =
+E
+f(r),
+(11)
+uθ = 0,
+uφ =
+L
+r2 ,
+ur =
+�
+f(r)
+�
+−1 + E2
+f(r) − L2
+r2
+�� 1
+2
+.
+Now, the conserved energy equation with an effective potential Veff for the motion of the test particle can be written
+as
+(ur)2 + Veff = E2.
+(12)
+Veff = f(r)
+�
+1 + L2
+r2
+�
+.
+(13)
+Clearly, the effective potential of particles depends on strong and weak fields parameter f(r) and the angular momen-
+tum. The analysis of effective potential is very important in geodesic motion. Since the location of the circular orbits
+can be found by the local extrema of the effective potential. In Fig. (2), the effective potential has the following key
+points:
+• There is no extremum for L = 0 (black curve) in strong field.
+
+5
+2
+4
+6
+8
+10 r
+1
+2
+3
+4
+5
+Veff
+2
+4
+6
+8
+10 r
+0.5
+1.0
+1.5
+2.0
+2.5
+3 �
+�
+Veff
+2
+4
+6
+8
+10 r
+0.5
+1.0
+1.5
+2.0
+Veff
+2
+4
+6
+8
+10 r
+0.5
+1.0
+1.5
+Veff
+FIG. 2: Variation of Veff of strong and weak fields for L, Q and β. Left plots for the strong field while right plots for the weak
+field. This behavior is for the equations (64) and (65).
+risco
+rmb
+rph
+rsing
+0.2
+0.4
+0.6
+0.8
+1.0 β
+0.5
+1.0
+1.5
+2.0
+2.5
+� ��
+r
+0.2
+0.4
+0.6
+0.8
+1.0 β
+0
+1
+2
+3
+4
+5
+r
+FIG. 3: Variation of the characteristic radii of strong and weak fields w.r.t. β for nonlinear electrodynamics BH.
+• The first extremum is observed at Veff = 1.95 for L = 2.2
+√
+3 (red curve).
+• Veff increases for increasing values of L and the maximum potential is seen at L = 7.
+• The curve shifted outward for lager values of L and inward for lager values of β and Q.
+It is noted that the location of the innermost stable circular orbits is represented by the point of extremum in a
+strong field at the distance r > 2. The potential increases for increasing the angular momentum but, it decreases
+for increasing the parameter β and Q. The maximum potential of the strong field is larger than the weak field. The
+characteristic radii curves versus r and β for strong and weak field are plotted in Fig. (3), therefore these radii have
+the following structure:
+
+6
+• The black solution curves (rsing) shifted inward to the singularity for M = 1, k = 1, and C = 0.9.
+• The radius of the photon sphere (rph) shifted outward to the singularity for M = 1, C = 0.9 and Q = 0.5.
+• The blue solution curves (rmb) shifted outward to the singularity for M = 1, k = 1, C = 0.9 and Q = 0.5.
+• The red solution curves (risco) shifted outward to the singularity for M = 1, k = 1, C = 0.9 and Q = 0.5.
+We have noted that for a small value of β = 0.1, all the radius falls onto the singularity very quickly. While for large
+values of β, the radii shifted outward to the singularity. So, the radius in a black circular disk is near the singularity
+while the radius in a red circular disk is away from the BH.
+A.
+Circular motions
+Here, we study the circular motion of test particles in an equatorial plane. The radial component for the circular
+motion r must be constant such that ur = ˙ur = 0. So, from Eq. (12), we have Veff = E2 and d/drVeff = 0. Using
+these relations, we find specific energy E, angular momentum L, angular velocity Ωφ, and angular momentum l given
+as.
+E2 =
+2rf 2(r)
+2rf(r) − r2f ′(r),
+(14)
+L2 =
+r3f ′(r)
+2rf(r) − r2f
+′(r),
+(15)
+Ωφ = dφ
+dt ≡ uφ
+ut ⇒ Ω2
+φ = 1
+2rf ′(r),
+(16)
+l2 = L2
+E2 =
+r3
+2rf 2(r)f ′(r).
+(17)
+For being real specific energy and angular momentum, the following condition must be held
+2rf(r) − r2f ′(r) > 0.
+(18)
+Hence, this condition is for the real existence of circular orbits and its limited area can be well-founded by solving
+the above inequality. The relations E2 < 1, E2 = 1 are for the bound and marginally bound orbits hold respectively.
+From Eq. (14), we have the useful equation
+r2f ′(r) + 2rf(r)[f(r) − 1] = 0.
+(19)
+Generally solving the above equation, one can be easily found the marginally bound orbits. Then from Eqs. (14) and
+(15), the energy and momentum will diverge at the radius r. So the diverge relation is given by
+2rf(r) − r2f ′(r) = 0.
+(20)
+This relation is very important for the characteristic of the photon sphere. Since the photon moves in a photon sphere
+on circular orbits. Therefore, the comparison of strong and weak fields plays an important role in the significance of
+circular orbits. The specific energy variation is represented in Fig. (4)and has the following key points:
+• Increases the specific energy and decreases the bound orbit radius for smaller values of β and larger values of Q
+in a strong field.
+• decreases the specific energy and very small change in the bound orbit radius for smaller values of β and larger
+values of Q in a weak field.
+• We plotted the top plots for taking fix value of β and variation in Q.
+• We plotted the bottom plots for taking fix value of Q and variation in β.
+The angular momentum variation is represented in Fig. (5) and has the following key points:
+• The angular momentum decreases for the increasing value of Q and increases for the increasing value of β in a
+strong field.
+
+7
+2
+4
+6
+8
+10 r
+0.88
+0 �
+� �
+� �
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1 
+ 
+E2
+2
+4
+6
+8
+10 r
+  
+  ! "
+# $% &
+E2
+2
+4
+6
+8
+10 r
+' (
+)
+* +
+,
+- .
+/
+2 4
+5
+6 7
+8
+E2
+2.00
+2.02
+2.04
+2.06
+2.08
+2.10
+0.658
+0.660
+0.662
+0.664
+0.666
+0.668
+9 : ;< =
+2
+4
+>
+?
+10 r
+@ A
+B
+C D
+E
+F G
+H
+I J
+K
+L M
+N
+E2
+FIG. 4: Variation of energy of strong and weak fields for Q and β. Left plots for the strong field while right plots for the weak
+field. This behavior is for the equation (71), (72).
+• The angular momentum increases for the increasing value of Q and also increases for the increasing value of β
+in the weak field.
+• We plotted the top plots for taking fix value of β and variation in Q.
+• We plotted the bottom plots for taking fix value of Q and variation in β.
+B.
+Stable circular orbits and radiation energy flux
+Stable circular orbits depend on the local minima of the effective potential that is found by the relations d2
+dr2 Veff > 0
+and for marginally stable circular orbits
+d2
+dr2 Veff = 0. Then from Eq. (13), we get
+d2
+dr2 Veff = f ′′(r)(1 + L2
+r2 ) − 4f ′(r)L2
+r3 + 6f(r) L2
+(r)4 .
+(21)
+The accretion possibly corresponds to the characteristic r < risco. It is noted that the falling materials from rest at
+infinity accrete onto the BH, the released gravitational energy of the falling materials may change into radiation and
+this energy is the cause of the most energetic astrophysical phenomena. The radiation energy flux over the accretion
+disk is due to the radiant energy corresponding to the specific energy E, angular momentum L and angular velocity
+Ωφ studied by Kato et al. [10]. Then we have
+K = −
+˙MΩφ,r
+4π√−g(E − LΩφ)2
+�
+(E − LΩφ)L,rdr,
+(22)
+where K is a radiation flux, Ωφ,r = dΩφ
+dr ,
+˙M is an accretion rate and g is the parameter which is given by
+g = det(gµν) = −r4 sin2 θ.
+(23)
+
+8
+2
+4
+6
+8
+10 r
+2
+3
+4
+5
+6
+O
+8
+L2
+2
+4
+6
+8
+10 r
+2
+4
+6
+8
+L2
+2
+4
+6
+8
+10 r
+2
+4
+6
+8
+10
+12
+14
+16
+L2
+2
+4
+6
+8
+10 r
+2
+4
+6
+8
+10
+12
+14
+16
+L2
+FIG. 5: Variation of angular momentum of strong and weak fields for Q and β. Left plots for the strong field while right plots
+for the weak field. This behavior is for the equations (73), (74).
+As our work restrictions are in the equatorial plane, so sin θ = 1. From the Eqs (13)-(16), we have
+K(r) =
+−
+˙M
+4πr4
+r
+�
+2f ′(r)
+(24)
+×[2f(r) − rf ′(r)][rf ′′(r) − f ′(r)]
+[2f(r) + rf ′(r)]2
+� r
+mb
+Z(r)dr,
+where
+Z(r) =
+�
+r
+2f ′(r)
+[2f(r) + rf ′(r)][−f ′′(r)rf(r) + 2rf ′2(r) − 3f ′(r)f(r)]
+[2f(r) − rf ′(r)]2
+.
+(25)
+As the steady-state accretion disk is considered in thermodynamical equilibrium, then the radiation emitted from
+the disk in the form of black body radiation. So, we supposed the relation K(r) = σT 4(r) among the radiation and
+temperature. The detailed study of this relation was explained by [50] and further debated by [51]. The radiation
+flux is represented in Fig. (6) and has the following key points:
+• The radiation flux has a maximum for Q = 0 and it is decreased for some different values of charge in the left
+plot of a strong field.
+• In the right plot of the strong field, the flux is increased for some different values of β.
+• The radiation flux has a minimum for Q = 0 and it is increased for some different values of charge in the left
+plot of the weak field.
+• The same picture is seen in the right plot.
+The temperature behavior can be seen by using the relation K = σT 4. The temperature profile is represented in Fig.
+(7) and has the following key points:
+
+9
+No Charge
+Q=0
+0
+1
+2
+3
+4
+5
+6
+7
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+r
+×105
+0
+1
+2
+3
+4
+5
+6
+7
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+r
+×105
+0
+1
+2
+3
+4
+5
+6
+7
+0.0
+0.1
+0.2
+0.3
+0.4
+r
+P×10
+Q
+0
+1
+2
+3
+4
+5
+6
+7
+0.0
+0.1
+0.2
+0.3
+0.4
+r
+R×10
+S
+FIG. 6: Variation of emission rate depend on radius r and non linear electrodynamic parameters Q and β in strong and weak
+fields. The black curves show the picture for no charge that is Q = 0.
+• The temperature attains a maximum position for Q = 0 and it is decreased for some different values of Q and
+β in a strong field.
+• The three curves (green, blue, and Yellow) are close to the bound radius between (1, 5).
+• The same behavior is seen in the weak field but only the difference is that these three curves (green, blue and
+Yellow) are close to the bound radius between (2, 4).
+The efficiency of the accreting fluid is another important analysis of the accretion disk. The maximum accreting
+efficiency of transforming energy into the radiative flux of particles between ISCO and infinity is defined as the ratio
+of the binding energy of the ISCO to the rest of mass energy. Therefore, the relations of efficiency and maximum
+accreting efficiency are η = 1 − E and η∗ = 1 − EISCO respectively. The efficiency behavior is represented in Fig. (8)
+and have the following key points:
+• The efficiency decreases for increasing values of Q and β and vice versa in the unstable orbits and the stable
+orbits are inward to the singularity in strong and weak fields.
+• Radial frequency denotes by yellow curves while the other curves are for the vertical frequency.
+• We plotted the top plots for taking fix value of β and variation in Q.
+• We plotted the bottom plots for taking fix value of Q and variation in β.
+C.
+Oscillations
+In the accretion process, various types of Oscillatory motions due to restoring forces are expected.
+Restoring
+forces perform the horizontal and vertical oscillations of perturbations in accretion disks. These two oscillations are
+produced by the rotation of the accretion disk. In this rotation, the equilibrium position is seen when a particle is
+
+10
+0
+1
+2
+3
+4
+5
+6
+7
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+r
+×
+10
+T
+σ
+0
+1
+2
+3
+4
+5
+6
+7
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+r
+×
+10
+U
+σ
+0
+1
+2
+3
+4
+5
+6
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+V W XY
+r
+Z×
+10
+[
+σ
+0
+1
+2
+3
+4
+5
+6
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+\ ] ^_
+r
+`×
+10
+a
+σ
+FIG. 7: Variation of temperature depend on radius r and non linear electrodynamic parameters Q and β in strong and weak
+fields. The black curves show the picture for no charge that is Q = 0.
+transformed to the radial direction. Further, the accretion disk is also composed of the centrifugal force as a result of
+the gravitational field of the central mass. When the former take over the latter then the fluid element will be pushed
+outward or inward to come back to its original position because of the epicyclic frequency Ωr. The fluid element
+produced the harmonic oscillation with vertical frequency Ωθ by the restoring force in the equatorial plane (the plane
+where the element of the fluid is perturbed in a vertical direction) [10]. Therefore, three frequencies are noted in a
+general discussion for the motion of the fluid element in the accretion disk. Harmonic vertical motion with vertical
+frequency {Ωθ, the harmonic radial motion with radial frequency Ωr and the circular motion with orbital frequency
+Ωφ}. In this section, we formulate the radial and vertical motions around the circular equatorial plane.
+For the radial and vertical motions, we take the relations 1
+2
+� dr
+dt
+�2 = V (r)
+eff and 1
+2
+� dθ
+dt
+�2 = V (θ)
+eff respectively. From
+Eq. (9), we put uθ = 0 for the radial motion, and for vertical motion, we put ur = 0. By setting ur = dr
+dτ = dr
+st ut and
+uθ = dθ
+dτ = dθ
+st ut, we have
+1
+2
+�dr
+dt
+�2
+= −1
+2
+f 3(r)
+E2
+�
+1 + E2
+f(r) +
+L2
+r2 sin2 θ
+�
+= V (r)
+eff.
+(26)
+1
+2
+�dθ
+dt
+�2
+= −1
+2
+f 2(r)
+r2E2
+�
+1 + E2
+f(r) +
+L2
+r2 sin2 θ
+�
+= V (θ)
+eff.
+For more explanation of radial and vertical epicyclic frequencies around the circular orbit, some perturbations δr and
+δθ are required. So, taking the time derivative of radial Eq. (26), we get
+d2r
+dt2 =
+dV (r)
+eff
+dr
+.
+(27)
+The deviation δr = r−r0 is for the perturbed particle from its original radius r = r0 and it has the following perturbed
+equation of motion
+d2δr
+dt2 =
+dV (r)
+eff
+dr
+δr ⇒ δ¨r + Ω2
+rδr = 0,
+(28)
+
+11
+10
+20
+30
+40
+50 r
+-0.02
+0.00
+0.02
+0.04
+0.06
+η
+10
+20
+30
+40
+50 r
+-0.02
+-0.01
+0.00
+0.01
+0.02
+b c de
+f g hi
+j k lm
+η
+10
+20
+30
+40
+50 r
+-0.02
+0.00
+0.02
+0.04
+0.06
+η
+5.00 5.02 5.04 5.06 5.08 5.10
+0.0754
+0.0756
+n o pq r s
+t u vw x y
+z { |} ~ 
+10
+20
+30
+40
+50 r
+-0.02
+0.00
+0.02
+0.04
+0.06
+0.08
+η
+FIG. 8: Variation of efficiency of strong and weak fields for Q and β. Left plots for the strong field while right plots for the
+weak field. The circular disks denote the maximum efficiency. This behavior is for the equation 1 − E, where E is taken from
+(71) and (72).
+the double dots represent the differential with time t and Ω2(r) = − d2
+dθ2 V (r)
+eff whereas applying the same method for
+vertical direction with deviation δθ = θ − θ0, we obtain
+d2δθ
+dt2 =
+dV (θ)
+eff
+dr
+δθ ⇒ δ¨θ + Ω2
+θδθ = 0,
+(29)
+where Ω2(θ) = − d2
+dθ2 V θ
+eff. Then from Eq. (26), we have
+Ω2
+θ = f 2(r)L2
+r4E2
+.
+(30)
+and
+Ω2
+r =
+1
+2r4E2 [(L2 + r2)r2f 2(r) − f(r)r4E2]f ′′(r)
+(31)
++2[(L2 + r2)f(r) − r2E2]r2f(r)f ′′(r) + 2r2f(r)f ′2(r)(L2 + r2)
+−2r
+�
+[(L2 + r2)2f(r) + r2E2]rf ′(r) + 4L2f 2(r)
+�
+f ′(r)
+−4L2f 2(r)(−3
+2f(r) + rf ′(r))].
+IV.
+BASIC DYNAMICAL EQUATIONS
+In this section, we find some basic calculations of accretion which were found by Babichev et al. [51, 52]. For these
+calculations, we take the Energy-momentum tensor of a perfect fluid
+T µν = (ρ + p)uµuν + pgµν,
+(32)
+
+12
+where ρ and p are energy density and pressure of accreting fluid whereas uµ is the four-velocity of the fluid. In an
+equatorial plane θ = π/2, the general form of the four-velocity is given by
+uµ = dxµ
+dτ = (ut, ur, 0, 0),
+(33)
+where τ denotes the proper time of the geodesic motion of the particles. The steady-state and spherically symmetric
+flow obey the normalization condition uµuµ = −1, we obtain
+ut =
+�
+f(r) + (ur)2
+f(r)
+,
+(34)
+due to square root, ut and ur may be positive or negative which specifies the forward or backward time conditions.
+As for ur < 0 (inward flow), the accretion occurs and the velocity of the fluid is negative whereas for ur > 0 (outward
+flow) and velocity of the fluid is positive. Therefore, the conservation laws of energy and momentum are necessary
+for the accretion analysis. Energy conservation is given by
+T µν
+;µ = 0 ⇒ T µν
+;µ =
+1
+√−g (√−gT µν),µ + Γν
+αµT αµ = 0,
+(35)
+where ; shows the covariant derivative whereas √−g = r2 sin2 θ and Γ is the second kind of Christoffel symbol. By
+simplifications, we obtain
+r2ur(ρ + p)
+�
+f(r) + (ur)2 = N0,
+(36)
+where N0 represents the integration constant. The relation between conservation law and four-velocity via uµT µν
+;ν = 0,
+we calculate
+(ρ + p);νuµuµuν + (ρ + p)uµ
+;νuµuν + (ρ + p)uµuµuν
+;ν + p,νgµνuµ + puµgµν
+;ν = 0.
+(37)
+By the conditions uµuµ = −1 and gµν
+;ν = 0, the above equation reduces to
+(ρ + p)uν
+;ν + uνρν = 0.
+(38)
+Taking the non-zero components, we obtain
+ρ′
+ρ + p + u′
+u + 2r
+r2 = 0.
+(39)
+By integration
+r2ur exp
+�
+dρ
+ρ + p = −N1,
+(40)
+where N1 represents an integration constant. Using ur < 0, the minus sign is taken on the right-hand side, so we get
+(ρ + p)
+�
+[(ur)2 + f(r)] exp
+�
+−
+�
+dρ
+ρ + p
+�
+= N2,
+(41)
+where N2 represents an integration constant. By using the above setup, the mass flux is given by
+(ρuµ);µ ≡
+1
+√−g (√−gρuµ),µ = 0,
+(42)
+and also it can be written as
+1
+√−g (√−gρuµ),r +
+1
+√−g (√−gρuµ),θ = 0.
+(43)
+Therefore the conservation mass equation is given by
+r2ρur = N3,
+(44)
+where N3 is constant of integration.
+
+13
+A.
+Dynamical parameters
+To continue further, we take isothermal fluid with the equation of state p = kρ whereas k is the state parameter.
+The flow must be flowing at a constant temperature in these fluids. Throughout the accretion, the sound speed
+remains constant for such fluids p ∝ ρ. Then, from Eqs. (40), (41) and (44), we have
+�ρ + p
+ρ
+� �
+[(ur)2 + f(r)] exp
+�
+−
+�
+dρ
+ρ + p
+�
+= N4,
+(45)
+where N4 is the integration constant. Using p = kρ in above equation, we get
+u(r) =
+�
+1
+k + 1
+� �
+(N4)2
+f(r) − (k + 1)2.
+(46)
+Therefore, the radial velocity of strong and weak fields is given by
+u(r)s =
+�
+1
+k + 1
+� �
+�
+�
+�
+(N4)2
+�
+1 − 2M
+r + Q2
+r2 − C2k2
+2r2 + 16C3/2k2
+15β1/4r
+� − (k + 1)2.
+(47)
+u(r)w =
+�
+1
+k + 1
+� �
+�
+�
+�
+(N4)2
+�
+1 − 2M
+r + Q2
+r2 − βC4k2
+10r6
+� − (k + 1)2.
+(48)
+From Eq. (44), we obtain the density of the fluid, given by
+ρ(r) = N3
+r2
+(k + 1)
+�
+(N4)2
+f(r) − (k + 1)2
+.
+(49)
+Therefore, the energy density of strong and weak fields is given by
+ρ(r)s = N3
+r2
+(k + 1)
+�
+(N4)2
+�
+1− 2M
+r + Q2
+r2 − C2k2
+2r2 + 16C3/2k2
+15β1/4r
+� − (k + 1)2
+.
+(50)
+ρ(r)w = N3
+r2
+(k + 1)
+�
+(N4)2
+�
+1− 2M
+r + Q2
+r2 − βC4k2
+10r6
+� − (k + 1)2
+.
+(51)
+The velocity variation is represented in Fig. (9) and has the following key points:
+• The fluid velocity increases by decreasing the charge values (Q = 1, 0.9, 0.8), and decreases the bound radius in
+a strong field.
+• The same behavior is seen for the variation of (β = 1, 2, 3) in the strong field.
+• The Schwarzschild behavior is seen for taking β = 0 and Q = 0 in weak field.
+• The blue and red plots show the nonlinear effects of fluid velocity for taking different values of Q and β in a
+weak field.
+Fig. (10) represents the density variation and has the following key points:
+• The fluid density increases by increasing the charge values (Q = 0, 0.5, 1.0), and curves shifted outward to the
+bound radius in a strong field.
+• The density decreases by increasing the values of (β = 1, 2, 3) in strong field.
+• The same behavior is seen in a weak field that is the density increases for increasing Q and decreases for
+increasing β.
+
+14
+2
+4
+6
+8
+10 r
+1.3
+1.4
+1.5
+1.6
+1.7
+€
+2
+4
+6
+8
+10 r
+1.30
+1.35
+1.40
+1.45
+1.50
+1.55
+
+β = 0, Q = 0
+Schwarzschild behavior at
+2
+4
+6
+8
+10 r
+1.6
+1.8
+2.0
+‚
+Schwarzschild behavior at
+β = 0, Q = 0
+2
+4
+6
+8
+10 r
+1.6
+1.8
+2.0
+ƒ
+FIG. 9: Variation of velocity of strong and weak fields for Q and β, also the equation of sate parameter k = 0.5.
+2
+4
+6
+8
+10 r
+0.2
+0.4
+0.6
+0.8
+ρ
+2
+4
+6
+8
+10 r
+0.2
+0.4
+0.6
+0.8
+ρ
+2
+4
+6
+8
+10 r
+0.10
+0.15
+0.20
+0.25
+„ …
+† ‡
+ρ
+2.00
+2.02
+2.04
+2.06
+2.08
+2.10
+0.290
+0.295
+0.300
+0.305
+0.310
+2
+4
+6
+8
+10 r
+0.10
+0.15
+0.20
+0.25
+0.30
+ρ
+FIG. 10: Variation of density of fluid of strong and weak fields for Q and β, also the equation of sate parameter k = 0.5.
+
+15
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+ˆ ‰ Š ‹Œ
+ Ž  ‘
+’ “ ” •–
+— ˜ ™ š›
+œ  ž Ÿ 
+¡ ¢ £ ¤¥
+¦ § ¨ ©ª
+r
+M
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+« ¬ ­ ®¯
+° ± ² ³´
+µ ¶ · ¸¹
+º » ¼ ½¾
+r
+M
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+0.02
+¿ÀÁÂ
+ÃÄÅÆ
+ÇÈÉÊ
+ËÌÍÎ
+0.07
+r
+M
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+0.020
+0.025
+ÏÐÑÒÓ
+ÔÕÖ×Ø
+ÙÚÛÜÝ
+Þßàáâ
+r
+M
+FIG. 11: Variation of the mass accretion rate of strong and weak fields w.r.t. Q and β for nonlinear electrodynamic BH.
+The mass accretion rate for the strong and weak fields are plotted in Fig. (11), therefore the plots have the following
+structure:
+• The maximum accretion rate of strong field occurs for no charge (Q = 0) between the distance 1.0, 2.05 (black
+curve).
+• The minimum accretion rate of strong field occurs for charge (Q = 0.75) between the distance 1.0, 1.4 (yellow
+curve).
+• The maximum accretion rate of weak field occurs for no charge (Q = 0) between the distance 1.0, 2.2 (black
+curve).
+• The minimum accretion rate of weak field occurs for charge (Q = 0.75) between the distance 1.4, 1.75 (yellow
+curve).
+In the strong and weak fields, we have noted that for large values of Q, the mass accretion rate decreases and the
+curves are inward to the smaller radii. While for small values of Q, the mass accretion rate increases, and the curves
+are outward to the larger radii. The small variation of mass accretion rate occurs all the solution curves pass through
+the circular disk for the different values of β.
+B.
+Mass expansion
+The mass of the BH is non-static for quintessence in astrophysical cases. Due to some processes such as accretion
+onto the BH and Hawking radiation, the mass will be changed slowly. The rate of change of accretion mass can be
+achieved by integrating the flux of fluid over the locality of BH and it is denoted by
+˙M. Therefore, it is given by
+˙M = −4πr2ur(ρ + p)
+�
+f(r) + (ur)2 ≡ −4πN0,
+(52)
+
+16
+where N0 = −N1N2 and N2 = (p∞ + ρ∞)
+�
+f(r∞) gives
+˙M = 4πN1(ρ∞ + p∞)
+�
+f(r∞)M 2.
+(53)
+Also, we take the time evolution of the mass of the BH, for this, the above equation can be written in the following
+form
+dM
+M 2 = Fdt,
+(54)
+where F = 4πN1(ρ∞ + p∞)
+�
+A(r∞), by integrating, we obtain
+Mt =
+Mi
+1 − FtMi
+≡
+Mi
+1 −
+t
+tcr
+,
+(55)
+where tcr =
+�
+4πN1(ρ∞ + p∞)
+�
+f(r∞)Mi
+�−1
+is the critical accretion of time evolution.
+Therefore, the required
+expression of BH mass accretion rate is given by
+˙M = 4πN1(ρ + p)M 2.
+(56)
+C.
+Critical accretion
+Since the fluid element is at rest far from the BH whereas it moves inwards, then it must be passed through the
+critical point where the velocity of the moving fluid is equal to the sound speed. The maximum accretion occurs if
+the moving fluid towards the critical point. Taking h = h(ρ) constant enthalpy then the fluid come to be barotropic.
+For this, the equation of state is given by [53].
+dh
+h = V 2 dρ
+ρ ,
+(57)
+where V is the local speed of sound. Then this equation gives ln h = V 2 ln n. So, from Eqs. (44), (45) and (57), we
+obtain
+�� u
+ut
+�2
+− V 2
+�
+(ln u),r =
+1
+r2(ut)
+�
+2rV 2(ut)2 − 1
+2r2f ′(r)
+�
+,
+(58)
+where critical points are denoted by the subscripted letter c and one can be found the solution of the local speed of
+sound at these points.
+V 2
+c =
+� uc
+utc
+�2
+.
+(59)
+At the sonic points, we have
+2rV 2
+c (utc)2 − 1
+2r2
+cf ′
+rc = 0.
+(60)
+We obtain the radial velocity at the critical point by putting (59) into (60), given by
+(uc)2 = 1
+4rr2
+cf ′
+rc.
+(61)
+By using Eqs. (34), (60) and (61), we obtain
+r2
+cf ′
+rc = 4rV 2
+c [f(rc) + (uc)2],
+(62)
+finally, it produces the local speed of sound, given by
+V 2
+c =
+r2
+cf ′
+rc
+r2cf ′rc + 4rf(rc).
+(63)
+
+17
+V.
+CIRCULAR EQUATORIAL GEODESICS
+The explicit form of the effective potential is important for the proceeding of circular geodesics. Hence, it is directed
+by Eq. (13), give as
+V s
+eff =
+�
+1 − 2M
+r
++ Q2
+r2 − C2k2
+2r2
++ 16C3/2k2
+15β1/4r
+� �
+1 + L2
+r2
+�
+,
+(64)
+V w
+eff =
+�
+1 − 2M
+r
++ Q2
+r2 − βC4k2
+10r6
+� �
+1 + L2
+r2
+�
+,
+(65)
+where
+d2
+dr2 Veff > 0 is the condition for the presence of the ISCO and also the Eq.
+(20) locate the ISCO at
+r ≥ 3
+�
+M − 8C3/2k2
+15β1/4 +
+�
+9
+�
+M − 8C3/2k2
+15β1/4
+�2
+− 8Q2 + 4C2k2
+�
+, as
+risco = 3
+
+M − 8C3/2k2
+15β1/4 +
+�
+9
+�
+M − 8C3/2k2
+15β1/4
+�2
+− 8Q2 + 4C2k2
+
+ ,
+(66)
+it is the required characteristic radius of the ISCO in the equatorial plane. It has been noted that the ISCO is
+an important study for the accretion process around the BH, Also, some other circular orbits are necessary for the
+completion of this process. Generally, the circular orbits proceed only when its radius is greater than the photon
+radius rph < r < rmb, as a result, if the particles fall onto the BH whereas for r > rmb then they must be move on
+the stable circular orbits. The results of circular orbits such as photon sphere rph, circular orbit rcirc and marginally
+bound orbit rmb along with singularity rsing at f(r) = 0 are given by
+rsing =
+1
+60
+�
+60M − 32C3/2k2
+β1/4
++
+�
+(32C3/2k2 − 60Mβ1/4)2 − 120(−15C2k2β1/4 + 30Q2β1/4)β1/4
+β1/4
+�
+,
+(67)
+rph = 3
+2
+
+M − 8C3/2k2
+15β1/4 + 1
+2
+�
+9
+�
+M − 8C3/2k2
+15β1/4
+�2
+− 8Q2 + 4C2k2
+
+ ,
+(68)
+rcirc > 3
+2
+
+M − 8C3/2k2
+15β1/4 + 1
+2
+�
+9
+�
+M − 8C3/2k2
+15β1/4
+�2
+− 8Q2 + 4C2k2
+
+ ,
+(69)
+rmb =
+1
+60
+�
+60M − 32C3/2k2
+β1/4
++
+�
+(−32C3/2k2 + 6015Mβ1/4)2 + 120(15C2k2β1/4 − 30Q2β1/4)β1/4
+β1/4
+�
+.
+(70)
+Now, we calculate the specific energy, specific angular momentum, angular velocity, and angular momentum of a
+moving particle in circular orbits for strong and weak fields. Therefore,
+E2
+s =
+2
+�
+1 − 2M
+r + Q2
+r2 − C2k2
+2r2 + 16C3/2k2
+15β1/4r
+�2
+2 +
+2
+�
+−5C2k2+10Q2−15Mr+ 8C3/2k2r
+β1/4
+�
+5r2
+.
+(71)
+E2
+w =
+2
+�
+1 − 2M
+r + Q2
+r2 − βC4k2
+10r6
+�2
+2 + 4Q2
+r2 − 6M
+r − 4C4k2β
+5r6
+.
+(72)
+L2
+s =
+C2k2 − 2Q2 + 2Mr − 16C3/2k2r
+15β1/4
+2 +
+2
+�
+−5C2k2+10Q2−15Mr+ 8C3/2k2r
+β1/4
+�
+5r2
+.
+(73)
+L2
+w =
+−2Q2 + 2Mr + 3C4k2β
+5r4
+2 + 4Q2
+r2 − 6M
+r − 4C4k2β
+5r6
+.
+(74)
+
+18
+Ω2
+φs =
+1
+4r2
+�C2k2
+r3
+− 2Q2
+r3
++ 2M
+r2 − 16C3/2k2
+15β1/4r2
+�2
+(75)
+Ω2
+φw =
+1
+4r2
+�−2Q2
+r3
++ 2M
+r2 + 3C4k2β
+5r7
+�2
+(76)
+l2
+s =
+C2k2 − 2Q2 + 2Mr − 16C3/2k2r
+15β1/4
+2
+�
+1 − 2M
+r + Q2
+r2 − C2k2
+2r2 + 16C3/2k2
+15β1/4r
+�2 .
+(77)
+l2
+w =
+−2Q2 + 2Mr + 3C4k2β
+5r4
+2
+�
+1 − 2M
+r + Q2
+r2 − βC4k2
+10r6
+�2 .
+(78)
+A.
+Epicyclic frequencies
+If a particle is moving in a circular orbit, then it achieves small oscillations in the direction of radial and vertical
+frequencies. These oscillations are the effects of perturbation on a moving particle in a circular orbit. So, the required
+frequencies are given by
+Ω2
+θs = 1
+r
+�C2k2
+r3
+− 2Q2
+r3
++ 2M
+r2 − 16C3/2k2
+15β1/4r2
+�
+(79)
+Ω2
+θw = 1
+r
+�−2Q2
+r3
++ 2M
+r2 + 3C4k2β
+5r7
+�
+(80)
+The radial and vertical frequency variation is represented in Fig. (12) and has the following key points:
+• The radial frequency is larger than the vertical frequencies for increasing value of Q and the same behavior is
+seen for the parameter β, so we take only one picture of behavior in a strong field.
+• The same behavior can be seen in the weak field.
+• We plotted the ratios plots between radial and vertical frequencies.
+• Interestingly, the ratio plots show the same behavior as the frequency plot that is radial frequency is greater
+than the vertical frequencies in strong and weak fields.
+VI.
+CONCLUSION
+In this paper, we studied the geodesic motion and accretion process of a test particle’s near a nonlinear electrody-
+namic BH in strong and weak field approximation. In this framework, we considered the equatorial plane and analyzed
+the circular geodesics with their stabilities, oscillations for small perturbations, unstable orbits and accretion of the
+fluid flowing onto the BH in a general form. Further, the effective potential, specific energy, angular momentum,
+epicyclic frequencies, characteristic radii, emission rate, and the mass evolution of the BHs have been studied. Then
+some general solutions under the strong and weak field limits are obtained by considering the equation of state p = kρ
+in the isothermal fluid.
+• The metric parameters of strong and weak fields suggest that the nonlinear electrodynamic effects can not be
+removed in the horizon structure but we can remove these effects for a far-distant observer. The effects of
+parameters Q and β are considered for each case of strong and weak fields and some solutions are compared to
+the Schwarzschild solution. The weak field analysis has shown that these solutions have a deviation from the
+Schwarzschild solution (where recovered by β = 0 and Q = 0). In Fig. (1), Al the horizons occur between the
+distance r = (0, 2) near the singularity, while for the maximum radius the curves are away from the singularity.
+All the plots have two horizons except for the bottom right plot.
+• In Fig. (2), the nonlinear electrodynamic parameters affect the effective potential, it can be seen that it is
+maximum for increased values of the parameters in a strong field. In a weak field, it maximizes for decreased
+
+19
+Radial Frequency
+Vertical Frequency
+1
+2
+3
+4
+5 r
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30Ω
+1
+2
+3
+4
+5 r
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Ω
+0
+2
+4
+6
+8
+10 r
+0.65
+0.70
+0.75
+0.80
+0.85
+Ωθ
+Ωr
+2
+4
+6
+8
+10 r
+0.65
+0.70
+0.75
+0.80
+Ωθ
+Ωr
+FIG. 12: Variation of frequency of strong and weak fields for Q and β. Left plots for the strong field while right plots for the
+weak field. This behavior is for the equations (75) and (76).
+values of the parameters, and the stable circular orbits are located at large distances while unstable circular
+orbits are located at small radii. We have noted the location of the characteristic radii (rsing, rph, rmb, risco)
+in Fig. (3). These radii have considerable deviation around the strong and weak fields. The unstable circular
+orbits fall onto the central mass at the distance β = 0.05 whereas the stable circular orbits are away from the
+central mass at the maximum β in a strong field. The pictures of the radii clearly show that the radius rISCO
+is greater than the other radii whereas the radius rsing is smaller than the other radii in strong and weak fields.
+The energy raises for increasing Q and decreasing β in a strong field. It can be seen from the energy diagram
+a very small deviation occur in the radius of the bound orbit. The angular momentum increases for decreasing
+Q and it raises for decreasing β whereas the unstable orbits inward to ISCO and stable orbits are outward
+to ISCO in a strong field. In a weak field, the angular momentum raises for decreasing Q and β whereas the
+unstable orbits inward to ISCO and stable orbits outward to ISCO.
+• Increasing the parameters Q and β, the efficiency of accretion decreases in a very small range (0.04, 0.08) in all
+plots for strong and weak fields. One can be seen that the unstable orbits have the maximum distances whereas
+the stable orbits have the minimum distances from the singularity. The epicyclic frequencies diagram shows that
+the radial frequency is greater than the vertical frequencies for increasing values of Q and β in strong and weak
+fields. It has been seen that the necessity of these variables is close to the central mass but extreme from the
+BH it is weak. From the behavior, see that Ωr < Ωθ. The energy profile shows that it is increased for increasing
+the values of charge Q and vice versa in a strong field. Also, it is decreased for increasing the values of β and
+vice versa. In a weak field, the specific energy decreases, and a very small change in the bound orbit radius is
+noted for smaller values of β and larger values of Q.
+• The radiation flux decreases at the singularity and has a maximum position away from the singularity in the
+vicinity of the strong field. By increasing the charge Q, the flux decreases and the minimum of the flux turns
+to the singularity but the reverse happens for the parameter β. Both the parameters Q and β show the same
+pictures in the case of a weak field. Therefore, the dependence on these parameters is very important in the
+vicinity of the strong and weak fields. These behaviors are different for temperature. All the solution curves are
+
+20
+bounded around the radius. The starting and ending points of these curves are in the bound radius r. Therefore,
+the maximum temperature happens for the value Q = 0, and is decreased for other values.
+In addition, we have investigated the radial velocity, energy density, and mass accretion rate of strong and weak
+fields by considering isothermal fluid and equation of state parameter k = 1/2. It has been noted that the radial
+velocity raises at the smaller radii for both parameters in the vicinity of the strong field but far from the field, the
+fluids have no radial velocity. Accretion takes place if the speed of fluids equal to the speed of sound and the flow
+is subsonic before the critical point. This flow will be supersonic around the BH so the speed of flow increases and
+passes through the critical point in the locality of strong and weak fields. The picture of a weak field represents that
+the radial velocity of the Schwarzschild BH is greater than the nonlinear electrodynamics BH. Therefore, the velocity
+increases by increasing the parameters, the speed of flow equals the speed of sound nearby the BH. From the density
+picture, it has been seen that the density of the fluid increases at larger radii for both parameters in the vicinity
+of strong and weak fields. All the solution curves turn down to the singularity and maximum density happens near
+to the singularity. Finally, we have analyzed the mass accretion rate, decreasing the parameters, The accretion rate
+increases and it achieves the maximum position at Q = 0 in the locality of a strong field. All the solution curves turn
+down at larger radii and are bound in the right plot of a strong field. In the locality of a weak field, the solution
+curves pass through the circular disk away from the singularity, accretion rate increases for increasing the parameters.
+Hence, the accretion rate depends on the metric parameters and the nature of the fluid.
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diff --git a/i9E2T4oBgHgl3EQfdAdf/content/tmp_files/load_file.txt b/i9E2T4oBgHgl3EQfdAdf/content/tmp_files/load_file.txt
new file mode 100644
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+page_content=' Ditta,1, ∗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Mustafa,2, † G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' ‡ Farruh Atamurotov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' § and Kimet Jusufi7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' ¶  1Department of mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The Islamia University of Bahawalpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Bahawalpur-63100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Pakistan  2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Zhejiang Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Jinhua 321004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' China  3Inha University in Tashkent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ziyolilar 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Tashkent 100170,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Uzbekistan  4Akfa University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Milliy Bog’ Street 264,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Tashkent 111221,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Uzbekistan  5National University of Uzbekistan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Tashkent 100174,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Uzbekistan  6Tashkent State Technical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Tashkent 100095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Uzbekistan  7Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' State University of Tetovo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Ilinden Street nn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 1200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Tetovo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' North Macedonia  The very latest observation of M87 supermassive black hole (BH) by the Event Horizon Telescope  (EHT) provides the accretion onto BHs is an interesting study in the theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' We study  the geodesics structure and accretion near a nonlinear electrodynamics BH in strong and weak field  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' These approximations provide the disc-like structure under the geodesic motion  and accretion around the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Near the equatorial plane, we provide some new reasons to make  circular orbits and accretion of test particles around the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then we investigate perturbations,  the critical speed of the fluid and the mass accretion rate of particles around the central object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The physical validity of this study shows that the parameter β and Q play an important role in the  circular orbits and the mass accretion rate in strong and weak field approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Keywords: Circular orbits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Accretion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Nonlinear Electrodynamics black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  INTRODUCTION  Black holes are quite fascinating objects revealed by the theory of general relativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In the universe, they  are known as robust sources of the gravitational field and are also likely to have a high spin and magnetic intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Owing to these features, black holes are the supreme research laboratory for studying both gravity and matter in  astrophysical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Currently, some observational evidence has confirmed the presence of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The first was  the innovation of gravitational waves brought out from a binary BH merger by LIGO and Virgo collaboration [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Another amazing revolutionary is the first image of M87∗ BH shadow [2, 3] and the very recently released image  of Sgr A∗ [4] by the EHT through a very long baseline interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Furthermore, the perceived electromagnetic  spectrum of accretion disks can be responsible for the existence of BHs [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' These latest achievements provide an  informative response to the appreciation of the theory of GR and the nature of accretion disks near supermassive BHs  in the strong gravity regime around the event horizon and may be assumed as a way to analyze modified theories of  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Accretion disks are formed due to the rotation of gaseous materials that travel in bounded orbits around the central  mass because of the gravitational force, such as neutron stars, supermassive BHs, and Young Stellar Objects in Active  Galactic Nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In such systems, particle orbits are stable, but when the orbits of these materials become unstable,  an accretion will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' It is generally believed that massive central objects such as BHs capture particles from fluid in  their vicinity and increase their mass through the accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So, the study of the geodesic motion of particles  in the vicinity of BH and particularly the analysis of some characteristic radii such as innermost stable circular orbits  (risco) and marginally bound orbits (rmb) are remarkable issues for a suspicious study of the subject matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' These  radii are essential in the study of BH accretion disks, where the inner edge of the disk coincides with the innermost  stable circular orbit (ISCO) and the efficiency of the energy released [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  ∗Electronic address: adsmeerkhan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='com  †Electronic address: gmustafa3828@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='com  ‡Electronic address: ghulamabbas@iub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='pk  §Electronic address: atamurotov@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='com  ¶Electronic address: kimet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='jusufi@unite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='mk  2  The locality of stable or unstable circular orbits is consistent with the minimum or maximum of the effective  potential correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In Newtonian theory, the effective potential has a minimum for any value of the angular  momentum, and then it has no minimum radius of a stable circular orbit, (ISCO) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' But this position is altered when  the effective potential has a difficult form liable to the particle angular momentum and other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore,  in GR and for the particles moving near the Schwarzschild BH, the effective potential has two extrema for any value  of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' But, only for a particular value of angular momentum do the two points happen together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  This point presents ISCO where is placed at r = 3rg [9–14] where rg is the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  According to the metrics, the effects of spacetime affect the positions of these radii and some parameters such as  specific energy, angular momentum, and angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' A lot of research problems are dedicated to studying these  radii and their physical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The effects of ISCO near the Kerr BH were studied by Ruffini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [15] and  Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Even Hobson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [17] defined these properties in their textbook on GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The efficiency of  accretion disks, η, for Schwarzschild and Kerr BHs was calculated by Novikov and Thorne [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The Kerr-like metric  was raised by Johannsen and Psaltis [19] and then Johannsen [20] formulated the accretion disks near such BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The  geodesic structure and the circular orbits of charged particles near weakly magnetized rotating BHs are obtained by  Tursunov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Since in an accretion disk the particles move in stable orbits but when perturbations (due to  restoring forces) act on the particles, oscillations near the circular orbit are produced in radial and vertical directions  with epicyclic frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, the study of orbital and epicyclic frequencies plays an important role in the  physics of accretion disks near the BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Isper [22, 23], Wagoner [24], Kato [11] and Ortega-Rodriges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [12] have  studied in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The exact and analytic solutions obtained by Bondi [25], for characterizing different astrophysical scenarios have  been involved in the continuous evolution of the accretion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Furthermore, analytic solutions are fundamental  equipment as benchmark tests for numerical codes [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This model was prolonged by Michel [27] to a relativistic  regime by assuming a Schwarzschild BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then again, analytic solutions to the known wind accretion scene have been  developed by Bondi and Hoyle [28], further Hoyle and Littleton [29] in the Newtonian framework and also by Tejeda  and AguayoOrtiz [30] in the relativistic framework of a Schwarzschild BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Numerous analytic and numerical analyses  have further prolonged the work of spherical accretion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [31–33], and also of wind accretion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Further,  Abbas and Ditta obtained the useful properties of accretion for motivation of GR onto a class of BHs [37–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  So, there is a blank space in the literature because no study has been carried out on accretion disks for nonlinear  electrodynamic BH with weak and strong field approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' However, this study is the motivation of GR and  we can honestly fill this blank in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Our main aspect of this paper is to study the non-rotating  BH solutions of an accretion disk in strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' For this, we assume an equatorial plane with a polar  coordinates system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Firstly, we see the variations in horizons near the geometry of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then we analyze the circular  orbits and effective potential as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  In order to study the locations of circular orbits such as innermost stable  circular orbits risco marginally bound circular orbits rmb and photon sphere rph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The maximum radiation efficiency  and binding energy with epicyclic frequencies are also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Finally, the critical accretion with some dynamical  parameters of isothermal fluid is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Recently, Tretyakova [46], analyzed geodesics in the Horndeski BH observational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Further, Salahshoor  and Nozari [47] used a similar method to analyze the circular orbits and accretion disks in detail in a class of  Horndeski/Galileon BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In the present study, we also introduce a general class of nonlinear electrodynamic BH and  used a similar procedure to find out the new accretion results for the motivation of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The arrangement of this paper  is as follows: In section 2, we introduce the nonlinear electrodynamic BH spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The wide-range calculations  for a test particle motion are formulated in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Further, the circular mechanism and oscillations are reviewed  in the subsequent sections respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In section 4, we found the critical speed of the flow, mass accretion, and its  time variation for the perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In section 5, we analyzed the physical significance of all of these results for strong  and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' We summarized the results of the paper in section 6 by saying that the parameters Q and β have  greatly suppressed strong and weak field approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Throughout the calculations, we consider the geometric  units G = c = 1, and the spacetime signature (−, +, +, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  NONLINEAR ELECTRODYNAMIC BLACK HOLE  The non-rotating nonlinear electrodynamics BH solutions are obtained by the general gravitational action [48, 49]  which is given as  S =  � √−gd4x  � R  2k2 −  F  2βF + 1  �  ,  (1)  3  where g = det(gµν), R is a Ricci scalar, k−1 is a reduced Plank mass, F =  1  4FµνF µν and β is the dimensionless  parameter of nonlinear electrodynamic BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' For a solution, we consider the spherically symmetric spacetime  ds2 = −f(r)dt2 +  1  f(r)dr2 + r2(dθ2 + sin2 θdφ2),  (2)  with  f(r) = 1 − 2M  r  + Q2  r2 − C2k2  2r2  + C2k2  30r2 (5ξ3 − 22ξ2 + 32ξ),  (3)  where  ξ = 12  √  3√βr2 − βCλ3/4  12βCλ1/4  ,  (4)  whereas  λ = 6  3√  6r2(  3√  2β2/3 3√  Cγ2/3 − 8  3√  3βC)  β4/3C5/3 3√γ  ,  (5)  γ =  √  3  �  256βC2 + 27r4 + 9r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (6)  The strong and weak field solutions can be obtained from the metric function f(r) by applying r → 0 and r → ∞  respectively, such that  f(r)s = 1 − 2M  r  + Q2  r2 − C2k2  2r2 + 16C3/2k2  15β1/4r ,  (7)  f(r)w = 1 − 2M  r  + Q2  r2 − βC4k2  10r6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (8)  It is noted that in the strong field limit f(r)s, the effects of nonlinear electrodynamics are highly effective and cannot  be detached unless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' However, in weak field limit, the last term goes more rapidly as r → ∞ such that f(r)w and the  metric function f(r) are asymptotically Reissner-Nordstrom and hence the effects of nonlinear electrodynamics can  simply be removed by applying β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (1) has the following key points:  In the upper left plot, the curve passes through the circular disk (green and blue) and has an event and two  horizons (inner and outer) in a strong field for different values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  In the upper right plot, the black curve has an event horizon while the red curve has two horizons for different  values of Q in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  An event and two horizons are also present in the left bottom plot for the weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The right bottom plot has an event horizon for β = 0 in a weak field and has no two horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  It is noted that in a strong field, for smaller values of M, the curves are shifted outward to an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In the  case of a weak field, for larger values of M, the curves are shifted inward to an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Both the left plots have  Cauchy horizons in strong and weak fields, while the plots have not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  GENERAL FORMULISM OF GEODESIC MOTION  In this section, we formulate the general results for the geodesic motion of test particles by the underlying static and  symmetric spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' For these results, we consider two Killing vectors ζt = ∂t and ζφ = ∂φ parallel two constants of  motion E and L (conserved energy and angular momentum) by the trajectory as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  E = −gµνζµ  t uν ≡ −ut,  L = gµνζµ  φuν ≡ uφ,  (9)  where uµ = dxµ  dτ = (ut, ur, uθ, uφ) is the 4-velocity of the moving particles and the particles obey the normalization  condition uµuµ = −1, we obtain  grr(ur)2 + gθθ(uθ)2 + gtt(ut)2 + gφφ(uφ)2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (10)  4  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  r  f(r)  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  r  f(r)  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  r  f(r)  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  r  f(r)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 1: Variation of horizon of strong and weak fields for M, Q and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Upper plots for the strong field while lower plots for  the weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This behavior is for the equations (7, 8) and the parameters are M = 1, k = 1, C = 1 and different values of Q  and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Taking (θ = π/2), for the equatorial plane then from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (9) and (10), we obtain  ut =  E  f(r),  (11)  uθ = 0,  uφ =  L  r2 ,  ur =  �  f(r)  �  −1 + E2  f(r) − L2  r2  �� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Now, the conserved energy equation with an effective potential Veff for the motion of the test particle can be written  as  (ur)2 + Veff = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (12)  Veff = f(r)  �  1 + L2  r2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (13)  Clearly, the effective potential of particles depends on strong and weak fields parameter f(r) and the angular momen-  tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The analysis of effective potential is very important in geodesic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Since the location of the circular orbits  can be found by the local extrema of the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (2), the effective potential has the following key  points:  There is no extremum for L = 0 (black curve) in strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  5  2  4  6  8  10 r  1  2  3  4  5  Veff  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  3 �  �  Veff  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  Veff  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  Veff  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 2: Variation of Veff of strong and weak fields for L, Q and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Left plots for the strong field while right plots for the weak  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This behavior is for the equations (64) and (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  risco  rmb  rph  rsing  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0 β  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  � ��  r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0 β  0  1  2  3  4  5  r  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 3: Variation of the characteristic radii of strong and weak fields w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' β for nonlinear electrodynamics BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The first extremum is observed at Veff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='95 for L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  √  3 (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Veff increases for increasing values of L and the maximum potential is seen at L = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The curve shifted outward for lager values of L and inward for lager values of β and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  It is noted that the location of the innermost stable circular orbits is represented by the point of extremum in a  strong field at the distance r > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The potential increases for increasing the angular momentum but, it decreases  for increasing the parameter β and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The maximum potential of the strong field is larger than the weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The  characteristic radii curves versus r and β for strong and weak field are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (3), therefore these radii have  the following structure:  6  The black solution curves (rsing) shifted inward to the singularity for M = 1, k = 1, and C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The radius of the photon sphere (rph) shifted outward to the singularity for M = 1, C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='9 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The blue solution curves (rmb) shifted outward to the singularity for M = 1, k = 1, C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='9 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The red solution curves (risco) shifted outward to the singularity for M = 1, k = 1, C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='9 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We have noted that for a small value of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='1, all the radius falls onto the singularity very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' While for large  values of β, the radii shifted outward to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So, the radius in a black circular disk is near the singularity  while the radius in a red circular disk is away from the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Circular motions  Here, we study the circular motion of test particles in an equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The radial component for the circular  motion r must be constant such that ur = ˙ur = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (12), we have Veff = E2 and d/drVeff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Using  these relations, we find specific energy E, angular momentum L, angular velocity Ωφ, and angular momentum l given  as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  E2 =  2rf 2(r)  2rf(r) − r2f ′(r),  (14)  L2 =  r3f ′(r)  2rf(r) − r2f  ′(r),  (15)  Ωφ = dφ  dt ≡ uφ  ut ⇒ Ω2  φ = 1  2rf ′(r),  (16)  l2 = L2  E2 =  r3  2rf 2(r)f ′(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (17)  For being real specific energy and angular momentum, the following condition must be held  2rf(r) − r2f ′(r) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (18)  Hence, this condition is for the real existence of circular orbits and its limited area can be well-founded by solving  the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The relations E2 < 1, E2 = 1 are for the bound and marginally bound orbits hold respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (14), we have the useful equation  r2f ′(r) + 2rf(r)[f(r) − 1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (19)  Generally solving the above equation, one can be easily found the marginally bound orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (14) and  (15), the energy and momentum will diverge at the radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So the diverge relation is given by  2rf(r) − r2f ′(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (20)  This relation is very important for the characteristic of the photon sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Since the photon moves in a photon sphere  on circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, the comparison of strong and weak fields plays an important role in the significance of  circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The specific energy variation is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (4)and has the following key points:  Increases the specific energy and decreases the bound orbit radius for smaller values of β and larger values of Q  in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  decreases the specific energy and very small change in the bound orbit radius for smaller values of β and larger  values of Q in a weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We plotted the top plots for taking fix value of β and variation in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We plotted the bottom plots for taking fix value of Q and variation in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The angular momentum variation is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (5) and has the following key points:  The angular momentum decreases for the increasing value of Q and increases for the increasing value of β in a  strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  7  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='88  0 �  � �  � �  \x0e \x0f  \x10 \x11  \x12 \x13  \x14 \x15  \x16 \x17  1 \x18  \x19 \x1a  E2  2  4  6  8  10 r  \x1b  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' "  # $% &  E2  2  4  6  8  10 r  \' (  )  +  ,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  /  2 4  5  6 7  8  E2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='06  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='658  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='660  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='662  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='664  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='666  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='668  9 : ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='< =  2  4  >  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  10 r  @ A  B  C D  E  F G  H  I J  K  L M  N  E2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 4: Variation of energy of strong and weak fields for Q and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Left plots for the strong field while right plots for the weak  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This behavior is for the equation (71), (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The angular momentum increases for the increasing value of Q and also increases for the increasing value of β  in the weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We plotted the top plots for taking fix value of β and variation in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We plotted the bottom plots for taking fix value of Q and variation in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Stable circular orbits and radiation energy flux  Stable circular orbits depend on the local minima of the effective potential that is found by the relations d2  dr2 Veff > 0  and for marginally stable circular orbits  d2  dr2 Veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (13), we get  d2  dr2 Veff = f ′′(r)(1 + L2  r2 ) − 4f ′(r)L2  r3 + 6f(r) L2  (r)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (21)  The accretion possibly corresponds to the characteristic r < risco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' It is noted that the falling materials from rest at  infinity accrete onto the BH, the released gravitational energy of the falling materials may change into radiation and  this energy is the cause of the most energetic astrophysical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The radiation energy flux over the accretion  disk is due to the radiant energy corresponding to the specific energy E, angular momentum L and angular velocity  Ωφ studied by Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then we have  K = −  ˙MΩφ,r  4π√−g(E − LΩφ)2  �  (E − LΩφ)L,rdr,  (22)  where K is a radiation flux, Ωφ,r = dΩφ  dr ,  ˙M is an accretion rate and g is the parameter which is given by  g = det(gµν) = −r4 sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (23)  8  2  4  6  8  10 r  2  3  4  5  6  O  8  L2  2  4  6  8  10 r  2  4  6  8  L2  2  4  6  8  10 r  2  4  6  8  10  12  14  16  L2  2  4  6  8  10 r  2  4  6  8  10  12  14  16  L2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 5: Variation of angular momentum of strong and weak fields for Q and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Left plots for the strong field while right plots  for the weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This behavior is for the equations (73), (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  As our work restrictions are in the equatorial plane, so sin θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' From the Eqs (13)-(16), we have  K(r) =  −  ˙M  4πr4  r  �  2f ′(r)  (24)  ×[2f(r) − rf ′(r)][rf ′′(r) − f ′(r)]  [2f(r) + rf ′(r)]2  � r  mb  Z(r)dr,  where  Z(r) =  �  r  2f ′(r)  [2f(r) + rf ′(r)][−f ′′(r)rf(r) + 2rf ′2(r) − 3f ′(r)f(r)]  [2f(r) − rf ′(r)]2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (25)  As the steady-state accretion disk is considered in thermodynamical equilibrium, then the radiation emitted from  the disk in the form of black body radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So, we supposed the relation K(r) = σT 4(r) among the radiation and  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The detailed study of this relation was explained by [50] and further debated by [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The radiation  flux is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (6) and has the following key points:  The radiation flux has a maximum for Q = 0 and it is decreased for some different values of charge in the left  plot of a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  In the right plot of the strong field, the flux is increased for some different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The radiation flux has a minimum for Q = 0 and it is increased for some different values of charge in the left  plot of the weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The same picture is seen in the right plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The temperature behavior can be seen by using the relation K = σT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The temperature profile is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (7) and has the following key points:  9  No Charge  Q=0  0  1  2  3  4  5  6  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='30  r  ×105  0  1  2  3  4  5  6  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
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+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='30  r  ×105  0  1  2  3  4  5  6  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  r  P×10  Q  0  1  2  3  4  5  6  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  r  R×10  S  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 6: Variation of emission rate depend on radius r and non linear electrodynamic parameters Q and β in strong and weak  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The black curves show the picture for no charge that is Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The temperature attains a maximum position for Q = 0 and it is decreased for some different values of Q and  β in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The three curves (green, blue, and Yellow) are close to the bound radius between (1, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The same behavior is seen in the weak field but only the difference is that these three curves (green, blue and  Yellow) are close to the bound radius between (2, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The efficiency of the accreting fluid is another important analysis of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The maximum accreting  efficiency of transforming energy into the radiative flux of particles between ISCO and infinity is defined as the ratio  of the binding energy of the ISCO to the rest of mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, the relations of efficiency and maximum  accreting efficiency are η = 1 − E and η∗ = 1 − EISCO respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The efficiency behavior is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (8)  and have the following key points:  The efficiency decreases for increasing values of Q and β and vice versa in the unstable orbits and the stable  orbits are inward to the singularity in strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Radial frequency denotes by yellow curves while the other curves are for the vertical frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We plotted the top plots for taking fix value of β and variation in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We plotted the bottom plots for taking fix value of Q and variation in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Oscillations  In the accretion process, various types of Oscillatory motions due to restoring forces are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Restoring  forces perform the horizontal and vertical oscillations of perturbations in accretion disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' These two oscillations are  produced by the rotation of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In this rotation, the equilibrium position is seen when a particle is  10  0  1  2  3  4  5  6  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
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+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='30  r  ×  10  T  σ  0  1  2  3  4  5  6  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
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+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='30  r  ×  10  U  σ  0  1  2  3  4  5  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  V W XY  r  Z×  10  [  σ  0  1  2  3  4  5  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  \\ ] ^_  r  `×  10  a  σ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 7: Variation of temperature depend on radius r and non linear electrodynamic parameters Q and β in strong and weak  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The black curves show the picture for no charge that is Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  transformed to the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Further, the accretion disk is also composed of the centrifugal force as a result of  the gravitational field of the central mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' When the former take over the latter then the fluid element will be pushed  outward or inward to come back to its original position because of the epicyclic frequency Ωr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The fluid element  produced the harmonic oscillation with vertical frequency Ωθ by the restoring force in the equatorial plane (the plane  where the element of the fluid is perturbed in a vertical direction) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, three frequencies are noted in a  general discussion for the motion of the fluid element in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Harmonic vertical motion with vertical  frequency {Ωθ, the harmonic radial motion with radial frequency Ωr and the circular motion with orbital frequency  Ωφ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In this section, we formulate the radial and vertical motions around the circular equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  For the radial and vertical motions, we take the relations 1  2  � dr  dt  �2 = V (r)  eff and 1  2  � dθ  dt  �2 = V (θ)  eff respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' From  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (9), we put uθ = 0 for the radial motion, and for vertical motion, we put ur = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' By setting ur = dr  dτ = dr  st ut and  uθ = dθ  dτ = dθ  st ut, we have  1  2  �dr  dt  �2  = −1  2  f 3(r)  E2  �  1 + E2  f(r) +  L2  r2 sin2 θ  �  = V (r)  eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (26)  1  2  �dθ  dt  �2  = −1  2  f 2(r)  r2E2  �  1 + E2  f(r) +  L2  r2 sin2 θ  �  = V (θ)  eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  For more explanation of radial and vertical epicyclic frequencies around the circular orbit, some perturbations δr and  δθ are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So, taking the time derivative of radial Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (26), we get  d2r  dt2 =  dV (r)  eff  dr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (27)  The deviation δr = r−r0 is for the perturbed particle from its original radius r = r0 and it has the following perturbed  equation of motion  d2δr  dt2 =  dV (r)  eff  dr  δr ⇒ δ¨r + Ω2  rδr = 0,  (28)  11  10  20  30  40  50 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='06  η  10  20  30  40  50 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  b c de  f g hi  j k lm  η  10  20  30  40  50 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='06  η  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0754  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0756  n o pq r s  t u vw x y  z { |} ~ \x7f  10  20  30  40  50 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='08  η  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 8: Variation of efficiency of strong and weak fields for Q and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Left plots for the strong field while right plots for the  weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The circular disks denote the maximum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This behavior is for the equation 1 − E, where E is taken from  (71) and (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  the double dots represent the differential with time t and Ω2(r) = − d2  dθ2 V (r)  eff whereas applying the same method for  vertical direction with deviation δθ = θ − θ0, we obtain  d2δθ  dt2 =  dV (θ)  eff  dr  δθ ⇒ δ¨θ + Ω2  θδθ = 0,  (29)  where Ω2(θ) = − d2  dθ2 V θ  eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (26), we have  Ω2  θ = f 2(r)L2  r4E2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (30)  and  Ω2  r =  1  2r4E2 [(L2 + r2)r2f 2(r) − f(r)r4E2]f ′′(r)  (31)  +2[(L2 + r2)f(r) − r2E2]r2f(r)f ′′(r) + 2r2f(r)f ′2(r)(L2 + r2)  −2r  �  [(L2 + r2)2f(r) + r2E2]rf ′(r) + 4L2f 2(r)  �  f ′(r)  −4L2f 2(r)(−3  2f(r) + rf ′(r))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  BASIC DYNAMICAL EQUATIONS  In this section, we find some basic calculations of accretion which were found by Babichev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' For these  calculations, we take the Energy-momentum tensor of a perfect fluid  T µν = (ρ + p)uµuν + pgµν,  (32)  12  where ρ and p are energy density and pressure of accreting fluid whereas uµ is the four-velocity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In an  equatorial plane θ = π/2, the general form of the four-velocity is given by  uµ = dxµ  dτ = (ut, ur, 0, 0),  (33)  where τ denotes the proper time of the geodesic motion of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The steady-state and spherically symmetric  flow obey the normalization condition uµuµ = −1, we obtain  ut =  �  f(r) + (ur)2  f(r)  ,  (34)  due to square root, ut and ur may be positive or negative which specifies the forward or backward time conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  As for ur < 0 (inward flow), the accretion occurs and the velocity of the fluid is negative whereas for ur > 0 (outward  flow) and velocity of the fluid is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, the conservation laws of energy and momentum are necessary  for the accretion analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Energy conservation is given by  T µν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='µ = 0 ⇒ T µν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='µ =  1  √−g (√−gT µν),µ + Γν  αµT αµ = 0,  (35)  where ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' shows the covariant derivative whereas √−g = r2 sin2 θ and Γ is the second kind of Christoffel symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' By  simplifications, we obtain  r2ur(ρ + p)  �  f(r) + (ur)2 = N0,  (36)  where N0 represents the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The relation between conservation law and four-velocity via uµT µν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='ν = 0,  we calculate  (ρ + p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='νuµuµuν + (ρ + p)uµ  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='νuµuν + (ρ + p)uµuµuν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='ν + p,νgµνuµ + puµgµν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (37)  By the conditions uµuµ = −1 and gµν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='ν = 0, the above equation reduces to  (ρ + p)uν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='ν + uνρν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (38)  Taking the non-zero components, we obtain  ρ′  ρ + p + u′  u + 2r  r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (39)  By integration  r2ur exp  �  dρ  ρ + p = −N1,  (40)  where N1 represents an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Using ur < 0, the minus sign is taken on the right-hand side, so we get  (ρ + p)  �  [(ur)2 + f(r)] exp  �  −  �  dρ  ρ + p  �  = N2,  (41)  where N2 represents an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' By using the above setup, the mass flux is given by  (ρuµ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='µ ≡  1  √−g (√−gρuµ),µ = 0,  (42)  and also it can be written as  1  √−g (√−gρuµ),r +  1  √−g (√−gρuµ),θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (43)  Therefore the conservation mass equation is given by  r2ρur = N3,  (44)  where N3 is constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  13  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Dynamical parameters  To continue further, we take isothermal fluid with the equation of state p = kρ whereas k is the state parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The flow must be flowing at a constant temperature in these fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Throughout the accretion, the sound speed  remains constant for such fluids p ∝ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (40), (41) and (44), we have  �ρ + p  ρ  � �  [(ur)2 + f(r)] exp  �  −  �  dρ  ρ + p  �  = N4,  (45)  where N4 is the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Using p = kρ in above equation, we get  u(r) =  �  1  k + 1  � �  (N4)2  f(r) − (k + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (46)  Therefore, the radial velocity of strong and weak fields is given by  u(r)s =  �  1  k + 1  � �  �  �  �  (N4)2  �  1 − 2M  r + Q2  r2 − C2k2  2r2 + 16C3/2k2  15β1/4r  � − (k + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (47)  u(r)w =  �  1  k + 1  � �  �  �  �  (N4)2  �  1 − 2M  r + Q2  r2 − βC4k2  10r6  � − (k + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (48)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (44), we obtain the density of the fluid, given by  ρ(r) = N3  r2  (k + 1)  �  (N4)2  f(r) − (k + 1)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (49)  Therefore, the energy density of strong and weak fields is given by  ρ(r)s = N3  r2  (k + 1)  �  (N4)2  �  1− 2M  r + Q2  r2 − C2k2  2r2 + 16C3/2k2  15β1/4r  � − (k + 1)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (50)  ρ(r)w = N3  r2  (k + 1)  �  (N4)2  �  1− 2M  r + Q2  r2 − βC4k2  10r6  � − (k + 1)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (51)  The velocity variation is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (9) and has the following key points:  The fluid velocity increases by decreasing the charge values (Q = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8), and decreases the bound radius in  a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The same behavior is seen for the variation of (β = 1, 2, 3) in the strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The Schwarzschild behavior is seen for taking β = 0 and Q = 0 in weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The blue and red plots show the nonlinear effects of fluid velocity for taking different values of Q and β in a  weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (10) represents the density variation and has the following key points:  The fluid density increases by increasing the charge values (Q = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0), and curves shifted outward to the  bound radius in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The density decreases by increasing the values of (β = 1, 2, 3) in strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The same behavior is seen in a weak field that is the density increases for increasing Q and decreases for  increasing β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  14  2  4  6  8  10 r  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='7  \x80  2  4  6  8  10 r  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='55  \x81  β = 0, Q = 0  Schwarzschild behavior at  2  4  6  8  10 r  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  \x82  Schwarzschild behavior at  β = 0, Q = 0  2  4  6  8  10 r  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  \x83  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 9: Variation of velocity of strong and weak fields for Q and β, also the equation of sate parameter k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  ρ  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  ρ  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  \x84  \x86 \x87  ρ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='06  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='290  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='295  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='305  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='310  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='30  ρ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 10: Variation of density of fluid of strong and weak fields for Q and β, also the equation of sate parameter k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
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+page_content='2  \x88 \x89 \x8a \x8b\x8c  \x8d \x8e \x8f \x90\x91  \x92 \x93 \x94 \x95\x96  \x97 \x98 \x99 \x9a\x9b  \x9c \x9d \x9e \x9f  ¡ ¢ £ ¤¥  ¦ § ¨ ©ª  r  M\uf110  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  « ¬ \xad ®¯  ° ± ² ³´  µ ¶ · ¸¹  º » ¼ ½¾  r  M\uf110  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='02  ¿ÀÁÂ  ÃÄÅÆ  ÇÈÉÊ  ËÌÍÎ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='07  r  M\uf110  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='025  ÏÐÑÒÓ  ÔÕÖ×Ø  ÙÚÛÜÝ  Þßàáâ  r  M\uf110  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 11: Variation of the mass accretion rate of strong and weak fields w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Q and β for nonlinear electrodynamic BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The mass accretion rate for the strong and weak fields are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (11), therefore the plots have the following  structure:  The maximum accretion rate of strong field occurs for no charge (Q = 0) between the distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05 (black  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The minimum accretion rate of strong field occurs for charge (Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='75) between the distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4 (yellow  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The maximum accretion rate of weak field occurs for no charge (Q = 0) between the distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2 (black  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The minimum accretion rate of weak field occurs for charge (Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='75) between the distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='75 (yellow  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  In the strong and weak fields, we have noted that for large values of Q, the mass accretion rate decreases and the  curves are inward to the smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' While for small values of Q, the mass accretion rate increases, and the curves  are outward to the larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The small variation of mass accretion rate occurs all the solution curves pass through  the circular disk for the different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Mass expansion  The mass of the BH is non-static for quintessence in astrophysical cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Due to some processes such as accretion  onto the BH and Hawking radiation, the mass will be changed slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The rate of change of accretion mass can be  achieved by integrating the flux of fluid over the locality of BH and it is denoted by  ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, it is given by  ˙M = −4πr2ur(ρ + p)  �  f(r) + (ur)2 ≡ −4πN0,  (52)  16  where N0 = −N1N2 and N2 = (p∞ + ρ∞)  �  f(r∞) gives  ˙M = 4πN1(ρ∞ + p∞)  �  f(r∞)M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (53)  Also, we take the time evolution of the mass of the BH, for this, the above equation can be written in the following  form  dM  M 2 = Fdt,  (54)  where F = 4πN1(ρ∞ + p∞)  �  A(r∞), by integrating, we obtain  Mt =  Mi  1 − FtMi  ≡  Mi  1 −  t  tcr  ,  (55)  where tcr =  �  4πN1(ρ∞ + p∞)  �  f(r∞)Mi  �−1  is the critical accretion of time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Therefore, the required  expression of BH mass accretion rate is given by  ˙M = 4πN1(ρ + p)M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (56)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Critical accretion  Since the fluid element is at rest far from the BH whereas it moves inwards, then it must be passed through the  critical point where the velocity of the moving fluid is equal to the sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The maximum accretion occurs if  the moving fluid towards the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Taking h = h(ρ) constant enthalpy then the fluid come to be barotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  For this, the equation of state is given by [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  dh  h = V 2 dρ  ρ ,  (57)  where V is the local speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then this equation gives ln h = V 2 ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (44), (45) and (57), we  obtain  �� u  ut  �2  − V 2  �  (ln u),r =  1  r2(ut)  �  2rV 2(ut)2 − 1  2r2f ′(r)  �  ,  (58)  where critical points are denoted by the subscripted letter c and one can be found the solution of the local speed of  sound at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  V 2  c =  � uc  utc  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (59)  At the sonic points, we have  2rV 2  c (utc)2 − 1  2r2  cf ′  rc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (60)  We obtain the radial velocity at the critical point by putting (59) into (60), given by  (uc)2 = 1  4rr2  cf ′  rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (61)  By using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (34), (60) and (61), we obtain  r2  cf ′  rc = 4rV 2  c [f(rc) + (uc)2],  (62)  finally, it produces the local speed of sound, given by  V 2  c =  r2  cf ′  rc  r2cf ′rc + 4rf(rc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (63)  17  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  CIRCULAR EQUATORIAL GEODESICS  The explicit form of the effective potential is important for the proceeding of circular geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Hence, it is directed  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (13), give as  V s  eff =  �  1 − 2M  r  + Q2  r2 − C2k2  2r2  + 16C3/2k2  15β1/4r  � �  1 + L2  r2  �  ,  (64)  V w  eff =  �  1 − 2M  r  + Q2  r2 − βC4k2  10r6  � �  1 + L2  r2  �  ,  (65)  where  d2  dr2 Veff > 0 is the condition for the presence of the ISCO and also the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (20) locate the ISCO at  r ≥ 3  �  M − 8C3/2k2  15β1/4 +  �  9  �  M − 8C3/2k2  15β1/4  �2  − 8Q2 + 4C2k2  �  , as  risco = 3  \uf8eb  \uf8edM − 8C3/2k2  15β1/4 +  �  9  �  M − 8C3/2k2  15β1/4  �2  − 8Q2 + 4C2k2  \uf8f6  \uf8f8 ,  (66)  it is the required characteristic radius of the ISCO in the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' It has been noted that the ISCO is  an important study for the accretion process around the BH, Also, some other circular orbits are necessary for the  completion of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Generally, the circular orbits proceed only when its radius is greater than the photon  radius rph < r < rmb, as a result, if the particles fall onto the BH whereas for r > rmb then they must be move on  the stable circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The results of circular orbits such as photon sphere rph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' circular orbit rcirc and marginally  bound orbit rmb along with singularity rsing at f(r) = 0 are given by  rsing =  1  60  �  60M − 32C3/2k2  β1/4  +  �  (32C3/2k2 − 60Mβ1/4)2 − 120(−15C2k2β1/4 + 30Q2β1/4)β1/4  β1/4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (67)  rph = 3  2  \uf8eb  \uf8edM − 8C3/2k2  15β1/4 + 1  2  �  9  �  M − 8C3/2k2  15β1/4  �2  − 8Q2 + 4C2k2  \uf8f6  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (68)  rcirc > 3  2  \uf8eb  \uf8edM − 8C3/2k2  15β1/4 + 1  2  �  9  �  M − 8C3/2k2  15β1/4  �2  − 8Q2 + 4C2k2  \uf8f6  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (69)  rmb =  1  60  �  60M − 32C3/2k2  β1/4  +  �  (−32C3/2k2 + 6015Mβ1/4)2 + 120(15C2k2β1/4 − 30Q2β1/4)β1/4  β1/4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (70)  Now, we calculate the specific energy, specific angular momentum, angular velocity, and angular momentum of a  moving particle in circular orbits for strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore,  E2  s =  2  �  1 − 2M  r + Q2  r2 − C2k2  2r2 + 16C3/2k2  15β1/4r  �2  2 +  2  �  −5C2k2+10Q2−15Mr+ 8C3/2k2r  β1/4  �  5r2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (71)  E2  w =  2  �  1 − 2M  r + Q2  r2 − βC4k2  10r6  �2  2 + 4Q2  r2 − 6M  r − 4C4k2β  5r6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (72)  L2  s =  C2k2 − 2Q2 + 2Mr − 16C3/2k2r  15β1/4  2 +  2  �  −5C2k2+10Q2−15Mr+ 8C3/2k2r  β1/4  �  5r2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (73)  L2  w =  −2Q2 + 2Mr + 3C4k2β  5r4  2 + 4Q2  r2 − 6M  r − 4C4k2β  5r6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (74)  18  Ω2  φs =  1  4r2  �C2k2  r3  − 2Q2  r3  + 2M  r2 − 16C3/2k2  15β1/4r2  �2  (75)  Ω2  φw =  1  4r2  �−2Q2  r3  + 2M  r2 + 3C4k2β  5r7  �2  (76)  l2  s =  C2k2 − 2Q2 + 2Mr − 16C3/2k2r  15β1/4  2  �  1 − 2M  r + Q2  r2 − C2k2  2r2 + 16C3/2k2  15β1/4r  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (77)  l2  w =  −2Q2 + 2Mr + 3C4k2β  5r4  2  �  1 − 2M  r + Q2  r2 − βC4k2  10r6  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (78)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Epicyclic frequencies  If a particle is moving in a circular orbit, then it achieves small oscillations in the direction of radial and vertical  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' These oscillations are the effects of perturbation on a moving particle in a circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' So, the required  frequencies are given by  Ω2  θs = 1  r  �C2k2  r3  − 2Q2  r3  + 2M  r2 − 16C3/2k2  15β1/4r2  �  (79)  Ω2  θw = 1  r  �−2Q2  r3  + 2M  r2 + 3C4k2β  5r7  �  (80)  The radial and vertical frequency variation is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (12) and has the following key points:  The radial frequency is larger than the vertical frequencies for increasing value of Q and the same behavior is  seen for the parameter β, so we take only one picture of behavior in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The same behavior can be seen in the weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  We plotted the ratios plots between radial and vertical frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Interestingly, the ratio plots show the same behavior as the frequency plot that is radial frequency is greater  than the vertical frequencies in strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  CONCLUSION  In this paper, we studied the geodesic motion and accretion process of a test particle’s near a nonlinear electrody-  namic BH in strong and weak field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In this framework, we considered the equatorial plane and analyzed  the circular geodesics with their stabilities, oscillations for small perturbations, unstable orbits and accretion of the  fluid flowing onto the BH in a general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Further, the effective potential, specific energy, angular momentum,  epicyclic frequencies, characteristic radii, emission rate, and the mass evolution of the BHs have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Then  some general solutions under the strong and weak field limits are obtained by considering the equation of state p = kρ  in the isothermal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The metric parameters of strong and weak fields suggest that the nonlinear electrodynamic effects can not be  removed in the horizon structure but we can remove these effects for a far-distant observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The effects of  parameters Q and β are considered for each case of strong and weak fields and some solutions are compared to  the Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The weak field analysis has shown that these solutions have a deviation from the  Schwarzschild solution (where recovered by β = 0 and Q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (1), Al the horizons occur between the  distance r = (0, 2) near the singularity, while for the maximum radius the curves are away from the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  All the plots have two horizons except for the bottom right plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (2), the nonlinear electrodynamic parameters affect the effective potential, it can be seen that it is  maximum for increased values of the parameters in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In a weak field, it maximizes for decreased  19  Radial Frequency  Vertical Frequency  1  2  3  4  5 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='30Ω  1  2  3  4  5 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='5  Ω  0  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='85  Ωθ  Ωr  2  4  6  8  10 r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='80  Ωθ  Ωr  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 12: Variation of frequency of strong and weak fields for Q and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Left plots for the strong field while right plots for the  weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This behavior is for the equations (75) and (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  values of the parameters, and the stable circular orbits are located at large distances while unstable circular  orbits are located at small radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' We have noted the location of the characteristic radii (rsing, rph, rmb, risco)  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' These radii have considerable deviation around the strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The unstable circular  orbits fall onto the central mass at the distance β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='05 whereas the stable circular orbits are away from the  central mass at the maximum β in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The pictures of the radii clearly show that the radius rISCO  is greater than the other radii whereas the radius rsing is smaller than the other radii in strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The energy raises for increasing Q and decreasing β in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' It can be seen from the energy diagram  a very small deviation occur in the radius of the bound orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The angular momentum increases for decreasing  Q and it raises for decreasing β whereas the unstable orbits inward to ISCO and stable orbits are outward  to ISCO in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In a weak field, the angular momentum raises for decreasing Q and β whereas the  unstable orbits inward to ISCO and stable orbits outward to ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Increasing the parameters Q and β, the efficiency of accretion decreases in a very small range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='08) in all  plots for strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' One can be seen that the unstable orbits have the maximum distances whereas  the stable orbits have the minimum distances from the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The epicyclic frequencies diagram shows that  the radial frequency is greater than the vertical frequencies for increasing values of Q and β in strong and weak  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' It has been seen that the necessity of these variables is close to the central mass but extreme from the  BH it is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' From the behavior, see that Ωr < Ωθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The energy profile shows that it is increased for increasing  the values of charge Q and vice versa in a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Also, it is decreased for increasing the values of β and  vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In a weak field, the specific energy decreases, and a very small change in the bound orbit radius is  noted for smaller values of β and larger values of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  The radiation flux decreases at the singularity and has a maximum position away from the singularity in the  vicinity of the strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' By increasing the charge Q, the flux decreases and the minimum of the flux turns  to the singularity but the reverse happens for the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Both the parameters Q and β show the same  pictures in the case of a weak field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, the dependence on these parameters is very important in the  vicinity of the strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' These behaviors are different for temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' All the solution curves are  20  bounded around the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The starting and ending points of these curves are in the bound radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore,  the maximum temperature happens for the value Q = 0, and is decreased for other values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  In addition, we have investigated the radial velocity, energy density, and mass accretion rate of strong and weak  fields by considering isothermal fluid and equation of state parameter k = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' It has been noted that the radial  velocity raises at the smaller radii for both parameters in the vicinity of the strong field but far from the field, the  fluids have no radial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Accretion takes place if the speed of fluids equal to the speed of sound and the flow  is subsonic before the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' This flow will be supersonic around the BH so the speed of flow increases and  passes through the critical point in the locality of strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' The picture of a weak field represents that  the radial velocity of the Schwarzschild BH is greater than the nonlinear electrodynamics BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Therefore, the velocity  increases by increasing the parameters, the speed of flow equals the speed of sound nearby the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' From the density  picture, it has been seen that the density of the fluid increases at larger radii for both parameters in the vicinity  of strong and weak fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' All the solution curves turn down to the singularity and maximum density happens near  to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Finally, we have analyzed the mass accretion rate, decreasing the parameters, The accretion rate  increases and it achieves the maximum position at Q = 0 in the locality of a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' All the solution curves turn  down at larger radii and are bound in the right plot of a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' In the locality of a weak field, the solution  curves pass through the circular disk away from the singularity, accretion rate increases for increasing the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Hence, the accretion rate depends on the metric parameters and the nature of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbott and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' (LIGO Scientific Collaboration and Virgo Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 116, 061102 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Akiyama  and  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  (Event  Horizon  Telescope  Collaboration),  Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 875, L1 (2019),  arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='11238 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='GA] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  Akiyama  and  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
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+page_content=' Wagoner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 311, 259 (1999), arXiv:astro-ph/9805028 [astro-ph] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Bondi, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 112, 195 (1952).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Rezzolla and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Zanotti, Relativistic Hydrodynamics (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Michel, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 15, 153 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Bondi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Hoyle, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 104, 273 (1944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Hoyle and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Lyttleton, Proceedings of the Cambridge Philosophical Society 35, 405 (1939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [30] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Tejeda and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Aguayo-Ortiz, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 487, 3607 (2019), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='04923 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='HE] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Karkowski and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Malec, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' D 87, 044007 (2013), arXiv:1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='3618 [gr-qc] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Mach and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Malec, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' D 88, 084055 (2013), arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='1546 [gr-qc] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [33] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Chaverra and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Sarbach, Classical and Quantum Gravity 32, 155006 (2015), arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='01641 [gr-qc] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Hunt, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 154, 141 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ruffert, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 427, 342 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Lora-Clavijo and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Guzm´an, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 429, 3144 (2013), arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='2139 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='HE] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [37] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ditta, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' A 33, 1850070 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  21  [38] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ditta, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 51, 43 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [39] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ditta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Jawad, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Umair Shahzad, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 51, 136 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ditta, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' C 80, 1212 (2020), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='12035 [gr-qc] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ditta and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' 52, 77 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [42] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Azam, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ditta, Chinese Journal of Physics 69, 143 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Ditta and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas, New Astronomy 81, 101437 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content='  [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
+page_content=' Abbas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E2T4oBgHgl3EQfdAdf/content/2301.03901v1.pdf'}
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+arXiv:2301.05000v1  [math.GR]  12 Jan 2023
+A SAMPLE ITERATED SMALL CANCELLATION THEORY FOR
+GROUPS OF BURNSIDE TYPE
+IGOR LYSENOK
+Abstract. We develop yet another technique to present the free Burnside group B(m, n)
+of odd exponent n with m ≥ 2 generators as a group satisfying a certain iterated small
+cancellation condition. Using the approach, we provide a reasonably accessible proof that
+B(m, n) is infinite with a moderate bound n > 2000 on the odd exponent n.
+1. Introduction
+The free m-generated Burnside group B(m, n) of exponent n is, by definition, the relatively
+free group in the variety of groups satisfying the identity xn = 1, i.e. B(m, n) ≃ Fm/F n
+m where
+Fm is the free group of rank m and F n
+m is the subgroup of Fm generated by all n-th powers.
+Obtaining a structural information about groups B(m, n) is known to be a difficult problem.
+The primary question of this sort is whether B(m, n) is finite for given m, n ≥ 2.
+The
+question is known as the Burnside problem [1] and it is still not completely answered. The
+group is shown to be finite for exponents n = 2, 3 [1], n = 4 [14] and n = 6 [7]. A negative
+solution to the Burnside problem is given by the Novikov–Adian theorem [11, 8] stating that
+the Burnside group B(m, n) of odd exponent n ≥ 665 with m ≥ 2 generators is infinite. As
+for now, infiniteness of B(m, n) is established for exponents of the form n = 665r or n ≥ 8000
+and any number m ≥ 2 of generators. Note that B(m, r) is a homomorphic image of B(m, n)
+if n is a multiple of r, so in this case infiniteness of B(m, r) implies infiniteness of B(m, n).
+The case when the exponent n does not have a large odd divisor was treated in [4, 9].
+Although it is believable that free Burnside groups B(m, n) are infinite for considerably
+lower values of n (and there are several announcements of results of this sort) the lowest
+published and carefully checked bound is still 665, obtained by Adian [8] for the case of odd
+exponent n.
+A principal step in understanding the structure of the group B(m, n) in the infinite case
+was made in the fundamental work by Novikov and Adian [11] and its improved version [8].
+One of the ingredients of the proof was a tightly interweaved version of the small cancellation
+theory similar to one developed by Tartakovski˘ı [15]. It was also shown in [8] that for m ≥ 2
+and odd n ≥ 665 the group B(m, n) has several properties similar to key properties of small
+cancellation groups. A basic one is layered Dehn’s property: a freely reduced nonempty word
+representing the identity in the group contains a large part of a defining relator modulo
+relations of the previous layer. This easily implies that any such word should contain a
+subword of the form Xt for sufficiently large t which in turn implies that B(m, n) is infinite.
+Unfortunately, the approach due to Novikov–Adian, even in its polished and improved form
+in [8], is extremely technical and has a complicated logical structure. Several later works
+[12, 13, 3, 2] pursued the goal to find a more conceptually explicit and technically simpler
+This research was supported by the Russian Science Foundation (project No. 21-11-00318).
+1
+
+approach to infinite Burnside groups, and more generally, to “infinite quotient of bounded
+exponent” phenomena in wider classes of groups as in [5, 3, 2].
+As an underlying basic
+idea, all these approaches utilize a small cancellation theory in a more or less explicit form
+though based on different implementation techniques. It was eventually realized that iterated
+small cancellation theory is indeed a relevant framework to present Burnside groups of large
+exponents as well as many other examples of infinitely presented groups of a “monster”
+nature. In an explicit form, a relevant version of the theory was formulated by Gromov and
+Delzant [3] and Coulon [2]. However, both approaches need extremely large exponents to be
+applied to Burnside groups. (In fact, the both incorporate “non-constructive” tools so that
+the proof does not provide any explicit lower bound on the exponent n.)
+Two questions naturally arise. What is the lower bound on the exponent n for which the
+iterated small cancellation approach can be applied to Burnside groups B(m, n)? Do we
+need a sophisticated technical framework to use the approach for reasonably small values of
+the exponent; for example, for values which are about several hundreds or less?
+The main goal of the present paper is to develop a sample version of the iterated small
+cancellation theory specially designed for free Burnside groups B(m, n) with a “moderate”
+lower bound on the exponent n. More precisely, our technique works for odd exponents
+n > 2000.
+We consider our approach as a first approximation and an introduction to a considerably
+more technical result on infiniteness of Burnside groups with substantially smaller bounds
+on the exponent.
+2. The iterated small cancellation condition
+2.1.
+We fix a group G given by a graded presentation
+(2-1)
+�
+A
+�� R = 1 (R ∈
+�
+α≥1
+Xα)
+�
+.
+Here we assume that the set of defining relators is partitioned into the union of subsets Xα
+indexed by a positive integer α. We call cyclic shifts of words R ∈ X±1
+α
+relators of rank α.
+Thus, the set of all relators of rank α is symmetrized, i.e. closed under cyclic shifts and
+taking inverses.
+With the presentation of G, there are naturally associated level groups Gα defined by all
+relations of rank up to α, i.e.
+(2-2)
+Gα =
+�
+A
+�� R = 1 (R ∈
+�
+β≤α
+Xβ)
+�
+2.2.
+Our small cancellation condition depends on two positive real-valued parameters λ
+and Ω satisfying
+(2-3)
+λ ≤ 1
+24,
+λΩ ≥ 20.
+We introduce also two other parameters with fixed value:
+ρ = 1 − 9λ,
+ζ = 1
+20.
+The role of λ, Ω, ρ and ζ can be described as follows:
+• λ is an analog of the small cancellation parameter in the classical condition C′(λ);
+2
+
+• Ω is the lower bound on the size of a relator R of rank α in terms of the length
+function | · |α−1 associated with Gα−1 (defined below in 2.7); see condition (S1)
+in 2.8.
+• ρ is the reduction threshold used in the definition of a reduced in Gα word. In-
+formally, a reduced in Gα word cannot contain more that ρ-th part of a relator of
+rank α up to closeness in Gα−1.
+• ζ is the rank scaling factor; it determines how the function | · |α rescales when
+incrementing the rank.
+2.3.
+For any α ≥ 0, we introduce the set Hα of bridge words of rank α recursively by setting
+H0 = {the empty word},
+Hα = {uSv | u, v ∈ Hα−1, S is a subword of a relator of rank α}.
+The definition immediately implies that Hα−1 ⊆ Hα. Note also that all sets Hα are closed
+under taking inverses.
+2.4.
+We call two elements x, y ∈ Gα close if x = uyv for some u, v ∈ Hα. This relation will
+be often used in the case when x and y are represented by words in the generators A. In
+that case we say that words X and Y are close in rank α if they represent close elements
+of Gα, or, equivalently, X = uY v in Gα for some u, v ∈ Hα.
+2.5.
+For α ≥ 0, the set Rα of words reduced in Gα, the set of fragments of rank α and the
+length function | · |α are defined by joint recursion.
+A word X in the generators A is reduced in G0 if X is freely reduced. A word X is reduced
+in Gα for α ≥ 1 if it is reduced in Gα−1 and the following is true: if a subword S of a
+relator R of rank α is close in rank α − 1 to a subword of X then
+|S|α−1 ≤ ρ|R|α−1.
+A word X is cyclically reduced in Gα if any cyclic shift of X is reduced in Gα.
+2.6.
+A nonempty word F is a fragment of rank α ≥ 1 if F is reduced in Gα−1 and is close
+in rank α − 1 to a subword P of a word of the form Rk where R is a relator of rank α. (In
+almost all situations P will be a subword of a cyclic shift of R.) A fragment of rank 0 is a
+word of length 1, i.e. a single letter of the alphabet A±1.
+It is convenient to assume that each fragment F of rank α ≥ 1 is considered with fixed
+associated words P, u, v and a relator R of rank α such that F = uPv in Gα−1, u, v ∈ Hα−1
+and P is a subword of Rk for some k > 0, i.e. a fragment is formally a quintuple (F, P, u, v, R).
+2.7.
+A fragmentation of rank α of a (linear or cyclic) word X is a partition of X into
+nonempty subwords of fragments of ranks β ≤ α. If F is a fragmentation of rank α of X
+then by definition, the weight of F in rank α is defined by
+weightα(F) = mα + ζmα−1 + ζ2mα−2 + · · · + ζαm0
+where mβ is the number of subwords of fragments of rank β in F. Here we assume that each
+subword in F is assigned a unique rank β.
+We now define a semi-additive length function | · |α on words in the generators A:
+|X|α = min{weightα(F) | F is a fragmentation of rank α of X}.
+Note that |X|0 is the usual length |X| of X.
+3
+
+2.8.
+The iterated small cancellation condition consists of the following three conditions
+(S0)–(S3) where the quantifier ‘for all α ≥ 1’ is assumed.
+(S0) (“Relators are reduced”) Any relator of rank α is cyclically reduced in Gα−1.
+(S1) (“Relators are large”) Any relator R of rank α satisfies
+|R|α−1 ≥ Ω.
+(S2) (“Small overlapping”) For i = 1, 2, let Si be a starting segment of a relator Ri of
+rank α.
+Assume that S1 = uS2v in Gα−1 for some u, v ∈ Hα−1 and |S1|α−1 ≥
+λ|R1|α−1. Then R1 = uR2u−1 in Gα−1.
+2.9.
+It can be proved that a group G satisfying conditions (S0)–(S2) possesses core properties
+of small cancellation groups, in particular, a version of Dehn’s property. We will impose,
+however, an extra condition on the graded presentation of G which implies cyclicity of all
+finite subgroups of groups Gα and avoids difficulties caused by existence of non-cyclic finite
+subgroups in the case of Burnside groups B(m, n) of even exponent n.
+(S3) (“No inverse conjugate relators”) No relator of rank α is conjugate in Gα−1 to its
+inverse.
+As we see below, this condition is satisfied if each relator R of rank α has the form Rn
+0
+where the exponent n (which can vary for different R) is odd and R0 is a non-power in Gα−1.
+See Corollary 13.11.
+Starting from Section 8, we will use a mild extra assumption on the graded presentation
+(2-1) by requiring it to be normalized in the following sense. The assumption is not essential
+and just makes arguments simpler (mainly due to Lemma 8.1) slightly improving bounds on
+the constants.
+2.10. Definition. We call a graded presentation (2-1) normalized if the following assertions
+hold:
+(i) Every relator R ∈ Xα has the form R ≖ Rt
+0 where R0 represents a non-power element
+of Gα−1 (i.e. R0 does not represent in Gα−1 an element of the form gk for k ≥ 2); we
+call R0 the root of a relator R.
+(ii) If R, S ∈ Xα and R ̸= S then R and S are not conjugate in Gα−1.
+Note that the condition to be normalized is not restrictive: every graded presentation can
+be replaced with a normalized one (although formally speaking, this replacement could affect
+the iterated small cancellation condition; however, in real applications this would hardly be
+the case).
+Remark. Checking conditions (S0)–(S3) requires knowledge about groups Gα−1. Thus pre-
+senting a group by relations satisfying the iterated small cancellation condition already re-
+quires a proof of properties of groups Gα by induction on the rank.
+3. Main results
+As in the case of classical small cancellation, the iterated small cancellation condition
+has strong consequences on the presented group G. A basic one is an analog of the Dehn
+property: every non-empty freely reduced word representing the trivial element of the group
+“contains a large part” of a relator.
+4
+
+In what follows, we assume that a group G is given by a normalized graded presentation
+satisfying conditions (S0)–(S3) above and for any α ≥ 0, Gα denotes the group defined by
+all relations of ranks up to α. We say that a word X is reduced in G if it is reduced in Gα
+for all α ≥ 0. The following theorem is an immediate consequence of Proposition 7.6.
+Theorem 1. Let X be a non-empty word in the generators A. If X reduced in Gα then
+X ̸= 1 in Gα. If X is reduced in G then X ̸= 1 in G.
+By expanding the definition of a reduced word in G we get an equivalent formulation
+which is more in the spirit of the small cancellation theory.
+Corollary. Let X be a freely reduced non-empty word. If X = 1 in G then for some α ≥ 1, X
+has a subword close in Gα−1 to a subword P of a relator R of rank α with |P|α−1 ≥ ρ|R|α−1.
+In the classical small cancellation theory, existence of a Dehn reduced representatives for
+group elements is a simple consequence of the fact that a word containing more than a half of
+a relator can be shortened by applying the corresponding relation. This approach does not
+work in our version of the iterated small cancellation and existence of reduced representatives
+is a nontrivial fact proved below and formulated in Proposition 11.1 and Corollary 14.8.
+Theorem 2. Every element of Gα can be represented by a word reduced in Gα.
+Every
+element of G can be represented by a word reduced in G.
+Many other properties of groups Gα and G are established in Sections 5–14. Our principal
+result shows that our version of the iterated small cancellation theory can be applied to
+free Burnside groups of odd exponent n with a moderate lower bound on n. The following
+theorem is a consequence of Propositions 16.8 and Corollary 16.10 (see also Remark 15.4).
+Theorem 3. For odd n > 2000 and m ≥ 2, the free Burnside group B(n, m) has a normalized
+graded presentation
+�
+A
+�� Cn = 1 (C ∈
+�
+α≥1
+Eα)
+�
+satisfying conditions (S0)–(S3) with λ = 80
+n , Ω = 0.25n.
+The following theorem is a well known property of Burnside groups of sufficiently large
+odd exponent. It is direct consequence of Propositions 9.14 and 16.6 (the definition of ω is
+given in 4.19).
+Theorem 4. Let n > 2000 be odd. Let X be a non-empty freely reduced word that is equal 1
+in B(m, n). Then X has a subword of the form C480 where C ∈ �
+α≥1 Eα.
+Note that, with existence of infinite aperiodic words in the 2-letter alphabet (see for
+example [8, §I.3]) this implies infiniteness of B(n, m) for odd n > 2000 and m ≥ 2.
+Some remarks. The present approach has much in common with paper [9]. However, the
+approach in [9] was based on the assumption that defining relations of the group under
+consideration are of the form xn = 1 for sufficiently large n. Although the general scheme of
+a large portion of our proofs is the same as in [9], our arguments are in different technical
+environment.
+We tried to make the iterated small cancellation condition as simple possible. In particular,
+we use a simple version of closeness in groups Gα (see 2.3 and 2.4). However, when presenting
+5
+
+the free Burnside group as an iterated small cancellation group, this version is not optimal
+for the bound on the exponent. A more refined version would significantly lower the bound.
+Nevertheless, we consider the bound n > 2000 on the exponent as a reasonable balance
+between its optimality and the complexity of definitions and proofs.
+The whole approach relies essentially on the simultaneous induction on the rank α. Since
+the proof of required statements about groups Gα needs a comprehensive analysis of certain
+types of relations in groups of previous ranks, the number of inductive hypotheses in quite
+large (several tens). We think that a large number of inductive hypotheses is an unavoidable
+feature of any “small cancellation” approach to infinite Burnside groups with a reasonably
+small lower bound on the exponent. Note that in the “basic” small cancellation theory in
+Sections 5–7 we use Proposition 7.8 (with its immediate consequence Proposition 7.9) as the
+only inductive hypothesis.
+We briefly mention essential ingredients of our approach.
+Sections 5–7 are devoted to analysis of van Kampen diagrams over the presentation (2-2)
+of the group Gα. In 5.1 we introduce diagrams with a special marking of the boundary so
+that the boundary loops of a diagram are divided into sides and bridges. The label of a side
+is a word reduced in Gα and bridges are “small” sections between sides labeled by bridge
+words of rank α. According to the marking, there are diagrams of bigon, trigon, etc. type.
+We then analyze a global structure of a diagram with marked boundary using the notion of
+contiguity subdiagram (see 6.5). For the quantitative analysis, we use a version of discrete
+connection in the spirit of [10] and the corresponding discrete analog of the Gauss-Bonnet
+formula (Proposition 7.3). The main outcomes are the bound on the total size of sides of
+a diagram with no bonds (Propositions 7.9 and 7.12) and the “single layered” structure of
+diagrams of small complexity (Propositions 7.11 and 7.13).
+The results of Sections 5–7 serve as a background for further analysis of relations in Gα.
+The most important type of relations under consideration are “closeness” relations in Gα of
+the form X = uY v where X, Y ∈ Rα and u, v ∈ Hα. The structural description of diagrams
+over the presentation of Gα transfers naturally to the language of the Cayley graph Γα of Gα,
+see 9.4. In Γα, words in the generators of the group are represented by paths and relations
+in Gα are represented by loops.
+The relation X = uY v becomes a loop X−1uYv in Γα
+which can be viewed as a coarse bigon; we say also that paths X and Y are close. The single
+layered structure of the filling diagram implies one-to-one correspondence between fragments
+of rank α in X and in Y that come from the 2-cells of the diagram, called active fragments
+of rank α with respect to the coarse bigon X−1uYv. To express the correspondence, we use
+the compatibility relation, defined in 8.6, on the set of fragments of rank α in Γα (i.e. paths
+in Γα labeled by fragments of rank α): if K and M are the corresponding active fragments
+of rank α in X and Y, respectively, then K and M−1 are compatible (Proposition 9.7).
+In Section 9 we perform this passage from diagrams over the presentation of Gα to the
+Cayley graph Γα. We establish several properties of coarse bigons, trigons and more generally,
+coarse polygons in Γα. We consider also conjugacy relations in Gα which are represented by
+parallel infinite lines in Γα (see 4.3).
+A fundamental property of close paths X and Y in Γα with label(X), label(Y) ∈ Rα is that
+the correspondence between fragments of rank α in X and Y extends to non-active ones. If
+K is a fragment in X of sufficiently large size then there exists a fragment of M of rank α
+in Y such that K is compatible with either M or M−1, with possible exceptions of extreme
+6
+
+positions of K in X (Proposition 10.6). Speaking informally, fragments of rank α play the
+role of letters when coincidence of words is replaced by closeness in Gα. This property of
+close paths X and Y in Γα and its analogs for coarse trigons in Gα (Proposition 10.7) and for
+conjugacy relations in Gα (Propositions 10.10 and 10.12) provide a technical base to analyze
+further properties of groups Gα and G. In particular, the correspondence between fragments
+of rank α in coarse bigons, under an appropriate adaptation, is crucial when we consider in
+Section 13 close in Gα periodic words.
+In Section 11 we prove that any element of Gα can be represented by a reduced word
+(Proposition 11.1) and is conjugate to an element represented by a cyclically reduced word
+and, moreover, by a strongly cyclically reduced word if it has infinite order (definition 4.15,
+Proposition 11.5).
+Sections 12 and 13 are preparatory for analysis of periodic relations over Gα. In Section 12
+we introduce the set of coarsely periodic words over Gα which are close (in a stronger sense
+then defined in 2.4) to periodic words with a strongly reduced in Gα period (Definition 12.4).
+The main result of Section 13, Proposition 13.4, is an analog of a well known property of
+periodic words stating that if two periodic words have a sufficiently large overlapping (for
+example, if they overlap for at least two of each of the periods) then they have a common
+period.
+In the last two Sections 15 and 16 we define a set of defining relations of the form Cn = 1
+(C ∈ �
+α≥1 Eα) for the Burnside group B(m, n) and prove that this set satisfies the iterated
+small cancellation condition (S0)–(S3). More precisely, in Definitions 15.1–15.3 we describe
+the recursive step to define Eα+1 given Eβ for β ≤ α, i.e. given the presentation of Gα. The
+principal idea to build sets Eα can be roughly described as “classification of periodic words
+by depth of periodicity” and is similar to one used in [11, 8]. Note that other approaches
+[12, 13, 4, 5, 3, 2] to groups of “Burnside type” use construction of periodic relations Cn = 1
+where for the next rank, C are chosen to be “short in size” with respect to the current group.
+We believe that the “depth of periodicity” approach, allthough more technical in several
+aspects, gives a more optimal lower bound on the exponent n.
+4. Preliminaries
+Starting from Section 5 we assume fixed a value of rank α ≥ 0 and a presentation (2-2) of a
+group Gα with relators R ∈ Xβ defined for all ranks β ≤ α. We assume that the presentation
+of Gα is normalized and satisfies conditions (S0)–(S3) and inequalities (2-3) for all ranks up
+to the fixed value α. In the proofs we will use forward references to statements for smaller
+values of rank, as already established. We will use references like “Proposition 2.3α−1” or
+“Lemma 3.4<α” etc. which mean “statement of Proposition 2.3 for rank α − 1” or “statement
+of Lemma 3.4 for all ranks β < α” respectively. With a few exceptions, statements whose
+formulation includes the case α = 0, are trivial or follow directly from definitions in that
+case.
+4.1. Words. We fix a set A of generators for a group G. By a word we always mean a group
+word over the alphabet A±1 = A ∪ {a−1 | a ∈ A}. We use notation X ≖ Y for identical
+equality of words X and Y . By X◦ we denote the cyclic word represented by a plain word X.
+A subword Y of a word X is always considered with an associated occurrence of Y in X
+that is clear from the context. To make it formal, we associate with a subword Y of X a pair
+of words (U, V ) such that UY V ≖ X. If Y is a subword of X with an associated pair (U, V )
+7
+
+then writing Y ≖ WZ we mean that W and Z are viewed as subwords of X with associated
+pairs (U, ZV ) and (UW, V ) respectively. Note that ‘subword Y of X1’ and ‘subword Y of
+X2’ are formally two distinct objects if X1 ̸= X2. It will be always clear from the context
+which ambient word is assumed for Y .
+A periodic word with period A, or an A-periodic word for short, is any subword of At
+for t > 0. According to the convention about subwords, an A-periodic word P is always
+considered with an associated occurrence of P in a word At.
+A partition of a word X is a representation of X as concatenation X = X1 · X2 · . . . · Xk of
+some subwords Xi. A word X is covered by a collection of words (Yi)i if X admits a partition
+X = X1 · X2 · . . . · Xk such that Xi is a subword of some Yti and ti ̸= tj for i ̸= j.
+4.2. Graphs. We use the term ‘graph’ as a synonym for ‘combinatorial 1-complex’. Edges of
+a graph are considered as having one of the two possible directions, so formally all our graphs
+are directed. By ι(e) and τ(e) we denote the starting and the ending vertices of an edge e,
+respectively, and e−1 denotes the inverse edge. An A-labeling on a graph Γ is a function from
+the set of edges of Γ with values in A±1 ∪ {1} such that label(e−1) = label(e)−1 for any e;
+here 1 denotes the empty word. An A-labeling naturally transfers to paths in Γ, so the label
+of a path P is a word in A±1. If P is a path in Γ then ι(P) and τ(P) denote the starting and
+the ending vertices of P, respectively. For any vertex a of Γ, there is the unique empty path
+at a. We identify this empty path with vertex a itself, so ι(a) = τ(a) = a and label(a) = 1.
+A path is simple if it visits no vertex twice. Two paths are disjoint if they have no common
+and no mutually inverse edges. A line in Γ is a bi-infinite path (we do not assume that lines
+have no loops).
+If X and Y are subpaths of a simple path Z then we write X ≪ Y if Z = Z1XZ2YZ3 for
+some Zi and X < Y if Z = Z1XuZ2 = Z1vYZ2 for some Zi and non-empty u and v. Although
+both relations depend on Z, it will be always clear from the context which Z is assumed.
+Clearly, if neither X and Y is contained in the other then either X < Y or Y < X. The union
+X ∪ Y of subpaths X and Y of Z is the shortest subpath of Z containing both X and Y.
+The Cayley graph Γ(G, A) of a group G with a generating set A is naturally viewed as
+an A-labeled graph. We identify vertices of Γ(G, A) with elements of G, so if ι(P) = a and
+τ(P) = b then label(P) is a word representing a−1b.
+The group G acts on Γ(G, A) by left multiplication.
+A path P in Γ(G, A) labeled by an A-periodic word is an A-periodic segment. An A-
+periodic line is a bi-infinite path labeled by A∞. Since an A-periodic word is assumed to
+have an associated occurrence in some At, an A-periodic segment P can be uniquely extended
+to an A-periodic line called the infinite periodic extension of P. If P and Q are A-periodic
+segments, P is a subpath of Q and the both have the same infinite periodic extension then
+Q is a periodic extension of P.
+We define also the translation element sA,P ∈ G that shifts the infinite periodic extension L
+of P forward by a period A. By definition, sA,P can be computed as follows. Take any vertex a
+on L such that the label of L at a starts with A. Then sA,P = aAa−1.
+If L1 and L2 are two periodic lines with periods A1 and A2 respectively then L1 and L2 are
+parallel if sA1,L1 = sA2,L2.
+4.3. Mapping relations in the Cayley graph. It follows from the definition of the Cayley
+graph that a word X in the generators A represents the identity of G if and only if some
+8
+
+(and therefore, any) path X in Γ(G, A) with label(X) ≖ X is a loop. Thus relations in G
+are represented by loops in Γ(G, A). This representation will be our basic tool to analyze
+relations in a group using geometric properties of its Cayley graph.
+We will often use the following notational convention. If X1X2 . . . Xn = 1 is a relation in
+a group G then we represent it by a loop X1X2 . . . Xn in the Cayley graph of G typed with
+the same letters in sans serif where, by default, label(Xi) ≖ Xi for all i.
+We represent also conjugacy relations in G by parallel periodic lines in Γ(G, A) as follows.
+Let X = Z−1Y Z in G. Consider a loop X−1Z−1YZ′ in Γ(G, A) with label(X) ≖ X, label(Y) ≖
+Y and label(Z) ≖ label(Z′) ≖ Z. We extend X to an X-periodic line L1 = . . . X−1X0X1 . . .
+with label(Xi) ≖ X and X0 = X and, in a similar way, extend Y to a Y -periodic line
+L2 = . . . Y−1Y0Y1 . . . with label(Yi) ≖ Y and Y0 = Y. Then we get a pair of parallel lines L1
+and L2 that represents conjugacy of X and Y in G.
+We will be freely switch between the language of paths in Cayley graphs and word relations.
+4.4. Van Kampen diagrams. Let G be a group with a presentation P = ⟨A | R⟩. A diagram ∆
+over P is a finite 2-complex ∆ embedded in R2 with a given A-labeling of the 1-skeleton ∆(1)
+such that the label of the boundary loop of every 2-cell of ∆ is either empty, has the form
+a±1a∓1 for a ∈ A or is a relator in R±1. Note that here we use an extended version of the
+widely used definition by allowing boundary loops of 2-cells labeled with empty word or freely
+cancellable pair of letters. This allows us to avoid technical issues related to singularities
+(see [13, §11.5] or [9, §4]).
+By default, all diagrams are assumed to be connected.
+We refer to 2-cells of a diagram ∆ simply as to cells; 1-cells and 0-cells are edges and
+vertices as usual. By δD we denote the boundary loop of a cell D and by δ∆ we denote
+the unique boundary loop of ∆ in case when ∆ is simply connected. We fix an orientation
+of R2 and assume that boundary loops of cells of ∆ and boundary loops of ∆ are positively
+oriented with respect to the cell or to the diagram, respectively. This means, for example,
+that (δD)−1 is a boundary loop of the diagram ∆−D obtained by removal of a cell D from ∆.
+Note that boundary loops of ∆ and of its cells are defined up to cyclic shift.
+According to van Kampen lemma ([6, Theorem V.1.1] and [13, Theorem 11.1]) a word X
+in the generators A represents the identity in G if and only if there exists a simply connected
+diagram ∆ over P with label(δ∆) ≖ X. Words X and Y represent conjugate elements of G if
+and only if there exists an annular (i.e. homotopy equivalent to an annulus) diagram over P
+with boundary loops X and Z such that label(X) ≖ X and label(Z) ≖ Y −1 ([6, Lemma V.5.2]
+and [13, Theorem 11.2]).
+If Σ is a subdiagram of ∆ then ∆ − Σ denotes the subdiagram of ∆ obtained as the
+topological closure of the complement ∆ \ Σ.
+Let ∆ and ∆′ be diagrams over P such that ∆′ is obtained from ∆ by either
+• contracting an edge e with label(e) ≖ 1 to a vertex,
+• contracting a cell D with label(δD) ≖ 1 to a vertex, or
+• contracting a cell D with label(δD) ≖ a±1a∓1, a ∈ A, to an edge labeled a±1.
+We call the inverse transition from ∆′ to ∆ an elementary refinement. A sequence of ele-
+mentary refinements is a refinement.
+There are several common use cases for refinement:
+9
+
+• Any diagram can be made by refinement non-singular, i.e. homeomorphic to a punc-
+tured disk. In particular, any simply connected diagram can be refined to a non-
+singular disk.
+• If C is a boundary loop of ∆ represented as a product C = X1 . . . Xk of paths Xi then,
+after refinement, the corresponding boundary loop of a new diagram ∆′ becomes
+X′
+1 . . . X′
+k where each Xi refines to a nonempty path X′
+i (see the definition in 4.5).
+4.5. Combinatorially continuous maps of graphs. We consider the class of maps between
+A-labeled graphs which are label preserving and can be realized as continuous maps of
+topological spaces. More precisely, a map φ : Λ → Λ′ between A-labeled graphs Λ and Λ′ is
+combinatorially continuous if
+• φ sends vertices to vertices and edges to edges or vertices; for any edge e of Λ, φ(e) is
+a vertex only if e has the empty label; if φ(e) is an edge then label(φ(e)) = label(e).
+• if φ(e) is an edge then φ preserves the starting and the ending vertices of e; if φ(e)
+is a vertex then φ(e) = φ(ι(e)) = φ(τ(e)).
+A combinatorially continuous map φ : Λ → Λ′ extends in a natural way to the map
+denoted also by φ, from the set of paths in Λ to the set of paths in Λ′. Clearly, φ preserves
+path labels.
+If a diagram ∆′ is obtained from a diagram ∆ by refinement then we have a combinatorially
+continuous map φ : ∆′(1) → ∆(1) induced by the sequence of contractions ∆′ → ∆. If P is a
+path in ∆ and P′ = φ(P) then P refines to P′.
+4.6. Mapping diagrams in Cayley graphs. It is well known that simply connected diagrams
+can be viewed as combinatorial surfaces in the Cayley complex of a group. Since we do not
+make use of two-dimensional structure, we adapt this view to the case of Cayley graphs.
+If ∆ is a simply connected diagram over P then there exists a combinatorially continuous
+map φ : ∆(1) → Γ(G, A). Any two such maps φ1, φ2 : ∆(1) → Γ(G, A) differ by translation
+by some element g ∈ G, i.e. φ1 = tgφ2 where tg : x �→ gx is the translation.
+In particular, if X is a loop in Γ(G, A) and for the boundary loop ¯X of ∆ we have label(¯X) =
+label(X) then there is a map φ : ∆(1) → Γ(G, A) such that φ(¯X) ≖ X. In this case we say
+that ∆ fills X via φ.
+If ∆ is not simply connected then we can consider a combinatorially continuous map
+φ : ˜∆(1) → Γ(G, A) where ˜∆ is the universal cover of ∆.
+Again, any two such maps
+φ1, φ2 : ˜∆(1) → Γ(G, A) differ by translation by an element of G. The set {Li}i of boundary
+loops of ∆ lifts to a (possibly infinite) set of bi-infinite boundary lines {˜Lj
+i}i,j of ˜∆ and thus
+produces a set of lines {φ(˜Lj
+i)}i,j in Γ(G, A). Each φ(˜Lj
+i) can be viewed as an Pi-periodic line
+with period Pi = label(Li). We will be interested mainly in the case when ∆ is an annular
+diagram, i.e. homotopy equivalent to a circle. In this case, boundary loops L1 and L2 of ∆
+produce two Pi-periodic lines φ(˜Li) (i = 1, 2) in Γ(G, A) such that φ(˜L1) and φ(˜L2)−1 are
+parallel.
+4.7. Definition. Let ∆ and ∆′ be diagrams of the same homotopy type over a presentation
+of a group G. We assume that a label preserving bijection Li �→ L′
+i is given between boundary
+loops of ∆ and ∆′ (which is usually clear from the context). We say that ∆ and ∆′ have
+the same frame type if there exist combinatorially continuous maps φ : ˜∆(1) → Γ(G, A) and
+10
+
+ψ : ˜∆′(1) → Γ(G, A) such that for each i we have the same sets of lines (or loops if ∆ and ∆′
+are simply connected) {φ(˜Lj
+i)}j = {ψ(˜L′j
+i )}j.
+The following two observations follow easily from the definition.
+4.8. Lemma. Two simply connected diagrams ∆ and ∆′ have the same frame type if and
+only if the labels of their boundary loops are equal words.
+Let ∆ and ∆′ be annular diagrams with boundary loops {L1, L2} and {L′
+1, L′
+2}. Then ∆
+and ∆′ have the same frame type if and only if the following is true. Take any vertices ai
+on Li (i = 1, 2) and let p be a path from a1 to a2 in ∆. Then there exist vertices a′
+i on L′
+i
+(i = 1, 2) and a path p′ from a′
+1 to a′
+2 in ∆′ such that the label of Li read at ai and the label
+of L′
+i read at a′
+i are equal words and label(p) = label(p′) in G.
+4.9. Lemma. Diagrams ∆ and ∆′ have the same frame type in the following two cases:
+• ∆′ is obtained from ∆ by refinement;
+• ∆′ is obtained from ∆ by cutting off a simply connected subdiagram and replacing it
+with another simply connected subdiagram.
+4.10. Groups Gα. Throughout the paper we will study a fixed family of groups Gα given by
+a presentation (2-2). Consequently, most of the related terminology will involve rank α as
+a parameter (though in some cases, it is not mentioned explicitly; for example, the already
+introduced measure µf(F) of fragments of rank α formally depends on α).
+Diagrams over the presentation of Gα are referred simply as diagrams over Gα.
+For
+1 ≤ β ≤ α, a cell of a diagram D over Gα with label(δD) ∈ Xβ is a cell of rank β. Cells with
+trivial boundary labels (i.e. empty or of the form aa−1) are cells of rank 0.
+The Cayley graph of Gα is denoted Γα. Note that if β > α then we have a natural covering
+map Γβ → Γα of labeled graphs. A loop L in Γα lifts to Γβ as a loop if and only if label(L) = 1
+in Gβ.
+4.11. Pieces. By a piece of rank α we call any (including empty) subword of a relator of
+rank α. If S is a subword of a cyclic shift of a relator R then we say also that S is a piece
+of R. We admit that a piece of rank α be the empty word. Note that our definition differs
+from the traditional view on a piece in the small cancellation theory as a common starting
+segment of two distinct relators.
+We assume that a piece S of rank α always has an associated relator R of rank α such
+that S is a start of R; so formally a piece of rank α should be viewed as a pair of the form
+(S, R). Associated relators are naturally inherited under taking subwords and inversion: if S
+is a piece of rank α with associated relator R = ST and S = S1S2 then S1 and S2 are viewed
+as pieces of rank α with associated relators R and S2TS1 respectively and S−1 is viewed as
+a piece of rank α with associated relator S−1T −1.
+For pieces of rank α we use a “measure” µ(S) ∈ [0, 1] defined by µ(S) =
+|S|α−1
+|R◦|α−1 as in (8-1)
+where R is the associated relator. (Recall that R◦ denotes the cyclic word represented by R.)
+If for some β, S is a path in Γβ or in a diagram over the presentation of Gβ and S is labeled
+by a piece of a relator of rank α (or by an R-periodic word where R is a relator of rank α)
+then we abbreviate µ(label(S)) simply as µ(S).
+4.12. Reformulation of conditions (S2) and (S3) in terms of Cayley graph. The following
+conditions on the presentation (2-1) are equivalent to (S2) and (S3), respectively.
+11
+
+(S2-Cayley) Let Li (i = 1, 2) be an Ri-periodic line in Γα−1 where Ri is a relator of rank α.
+If L1 and L2 have close subpaths P1 and P2 with |Pi| ≤ |Ri| and µ(P) ≥ γ then L1 and L2 are
+parallel.
+(S3-Cayley) There are no parallel R-periodic and R−1-periodic lines in Γα−1 where R is a
+relator of rank α.
+4.13. Bridge partition. We define also a bridge partition of rank α of a word w ∈ Hα as
+follows. A bridge partition of rank 0 is empty. A bridge partition of rank α ≥ 1 either
+• has the form w1 ·S ·w2 where wi ∈ Hα−1 and S is a piece of rank α called the central
+piece of w; or
+• is a single factor w itself in the case w ∈ Hα−1.
+If w is a bridge word of rank α endowed with a bridge partition u · S · v and ST is the
+relator of rank α associated with S then w′ = uT −1v is a bridge word of rank α equal to w
+in Gα. We say that w′ is obtained from w by switching. In this case we assume also that
+w′ is endowed with the bridge partition u · T −1 · v. Thus, applying the switching operation
+twice results in the initial word w.
+We will be considering paths in Cayley graphs Γβ labeled by bridge words of rank α. We
+call them bridges of rank α (with a slight abuse of terminology, we will also use this term
+in Section 5 for boundary paths with appropriate label in diagrams over the presentation
+of Gα). If w is bridge of rank α in Γβ then a bridge partition of rank α of w is either a
+factorization w = u · S · v where u and v are bridges of rank α − 1 and label(S) is a piece
+of rank α or a trivial factorization with the single factor w if w is bridge of rank α − 1. In
+the former case, if also β ≥ α, we define the switching operation on w in a similar way as
+in the case of words; namely, we take the word w′ obtained from w ≖ label(w) by switching
+and consider the path w′ with label(w′) ≖ w′ starting at the same vertex as w. Since w = w′
+in Γβ, bridges w and w′ have the same endpoints.
+4.14.
+The following properties of the function | · |α follow from the definition:
+(i) |X|α + |Y |α − 1 ≤ |XY |α ≤ |X|α + |Y |α; in particular, if Y is a subword of X then
+|Y |α ≤ |X|α.
+(ii) More generally, if a collection of words (Xi)i covers a (plain or cyclic) word X then
+|X|α ≤
+�
+i
+|Xi|α.
+If (Xi)1≤i≤k is a collection k of disjoint subwords of X then
+�
+i
+|Xi|α ≤ |X|α + k.
+(iii) |X|α ≤ ζ|X|α−1.
+(iv) |X◦|α = min{|Y |α | Y is a cyclic shift of X}.
+If X is a path in Γβ or in a diagram over the presentation of Gβ then we use abbreviation
+|X|α = |label(X)|α.
+4.15. Reduced words. The set of words reduced in Gα is denoted Rα. The definition imme-
+diately implies that Rα is closed under taking subwords.
+A word X is strongly cyclically reduced in Gα if any power Xt is reduced in Gα.
+12
+
+4.16. Coarse polygon relations. A relation in Gα of the form X1u1 . . . Xmum = 1 where
+words Xi are reduced in Gα and ui are bridge words of rank α, is called a coarse m-gon
+relation in Gα. We can write coarse polygon relations in different forms. For example, a
+coarse bigon relation can be written as X = uY v where X and Y are reduced in Gα and
+u, v ∈ Hα. In this form, the relation represents closeness of words X and Y in Gα.
+4.17.
+We transfer some terminology from words to paths in Γα.
+We call paths in Γα with label reduced in Gα simply reduced. Note that, according to
+Proposition 7.6, a reduced path X in Γα is simple. This implies that we can correctly treat
+the ordering of subpaths of X, intersections of subpaths, unions etc.
+Two vertices of Γα are close if they can be joined by a bridge of rank α (see 4.13). Two
+paths X and Y in Γα are close if their starting vertices and their ending vertices are close.
+We say that a loop P = X1u1X2u2, . . . , Xrur in Γα is a coarse r-gon if each Xi is reduced
+and each ui is a bridge of rank α. Paths Xi are sides of P.
+Note that paths X and Y in Γα are close if and only if X−1uYv is a coarse bigon for some u
+and v.
+4.18. Symmetry. All concepts (i.e. relations, functions etc.) and statements involving paths
+in the Cayley graphs Γα are invariant under the action of Gα in a natural way. For example,
+if paths X and Y in Γα are close then paths gX and gY are also close for any g ∈ Gα. We
+adopt a convention (which is essential for the invariance) that the action of Gα is extended
+onto extra data associated with paths in Γα: for example, if F is a fragment of rank β with
+base P then then gF is considered as a fragment of rank β with base gP and so on. This
+implies, for example, that µf(F) = µf(gF) for any g ∈ Gα.
+We will implicitly use symmetry with respect to inversion. For example, if F is a fragment
+of rank β with base P then F−1 is a fragment of rank β with base P−1 and µf(F−1) = µf(F).
+If a statement admits two symmetric forms then only one of them is formulated (as in case
+of Lemma 10.15, for instance).
+4.19. Numerical parameters. In many cases, it will be notationally more convenient to use
+instead of Ω its inverse:
+ω = 1
+Ω.
+Note that by (2-3),
+(4-1)
+ω ≤
+1
+480
+and
+λ ≥ 20ω.
+We will extensively use ω as a unit to measure pieces and fragments of rank α.
+Condition (S1) in 2.8 will be often used in the following form: if P is a piece of a relator R
+of rank α then
+(4-2)
+µ(P) ≤ ω|P|α−1.
+For reader’s convenience, we list our other global numerical parameters indicating the
+places where they first appeared.
+ν =
+ζ
+1 − 2ζ = 1
+18,
+θ = 1
+6(5 − 22ν) = 17
+27
+(Proposition 7.4),
+13
+
+η = 1 + 2ν
+θ
+= 30
+17
+(Proposition 7.9),
+ξ0 = 7λ − 1.5ω
+(Proposition 9.7),
+ξ1 = ξ0 − 2.6ω
+(Definition 12.2),
+ξ2 = ξ1 − 2λ − 3.4ω
+(Definition 12.4).
+5. Diagrams with marked boundary
+5.1. Boundary marking of rank α. We start with introducing a class of diagrams over the
+presentation (2-2) of Gα with extra data which, in particular, represent coarse polygon
+relations in Gα.
+Let ∆ be a non-singular diagram over the presentation (2-2).
+We say that ∆ has a
+boundary marking of rank α if for each boundary loop L of ∆, there is fixed a representation
+as a product L = X1u1 . . . Xmum of nonempty paths Xi and ui where labels of Xi are reduced
+in Gα and the label of each ui belongs to Hα. Paths Xi are called sides and paths ui are
+called bridges of ∆. We allow also that the whole boundary loop L of ∆ is viewed a side
+called a cyclic side. In this case we require that the label of L is cyclically reduced in Gα.
+If X1u1 . . . Xmum = 1 is a coarse polygon relation in Gα then there exists a disk diagram
+with boundary label X1u1 . . . Xmum such that label(Xi) ≖ Xi and label(ui) ≖ ui for all i.
+Refining ∆ if necessary (see 4.4) we can assume that ∆ is non-singular and all paths Xi
+and ui are nonempty, i.e. ∆ satisfies the definition above. In a similar way, we can associate
+with a conjugacy relation in Gα an annular diagram over the presentation of Gα with an
+appropriate boundary marking.
+Unless otherwise stated, “a diagram of rank α” will always mean “a non-singular diagram
+over the presentation (2-2) with a fixed boundary marking of rank α”. We use terms “di-
+agrams of monogon, bigon, trigon type etc.”
+to name disk diagrams of rank α with the
+appropriate number of sides.
+5.2. Complexity. If ∆ is a diagram of rank α then by b(∆) we denote the number of bridges
+of ∆. We define the complexity c(∆) of ∆ by
+c(∆) = b(∆) − 2χ(∆).
+5.3. Decrementing the rank. Let ∆ be a diagram of rank α ≥ 1. By ∆α−1 we denote the
+diagram over the presentation of Gα−1 obtained by removal from ∆ of all cells of rank α. Up
+to refinement of ∆, we assume that ∆α−1 is non-singular.
+We assume that every bridge w of ∆ is given a bridge partition of rank α as defined in 4.13,
+i.e. for some bridges w a factorization w = u · S · v is fixed where label(u), label(v) ∈ Hα−1
+and label(S) is a piece of rank α, and for all other w we have label(w) ∈ Hα−1. In the case
+when w has a nontrivial bridge partition u · S · v we say that w has native rank α and call S
+the central arc of u.
+We will be always assuming that all factors u, v and S are nonempty paths (this can be
+achieved by refinement).
+We then define a naturally induced boundary marking of rank α−1 of ∆α−1 (see Figure 1):
+• Sides of ∆ become sides of ∆α−1; we have also extra sides of ∆α−1 defined as follows.
+• If D is a cell of rank α of ∆ then boundary loop (δD)−1 of ∆α−1 becomes a cyclic
+side of ∆α−1.
+14
+
+• For each bridge w of rank α of ∆ we do the following. If the bridge partition of w
+is of the form u = v · S · w then we take v and w as bridges of ∆α−1 and the central
+arc S as a side of ∆α−1. Otherwise we have label(w) ∈ Hα−1 and we take w as a
+bridge of ∆α−1.
+cells of rank α
+∆
+∆α−1
+Figure 1. Producing ∆α−1 from ∆. Sides of ∆ and ∆α−1 are drawn by thicker lines
+5.4. Cell cancellation. We introduce two types of elementary reductions of a diagram ∆ of
+rank α ≥ 1. In both cases, we reduce the number of cells of rank α. As in 5.3, we assume
+that a bridge partition is fixed for each bridge ∆.
+Let C and D be two cells of rank α of ∆. We say that C and D form a cell-cell cancellable
+pair if there exists a simple path p joining two vertices a and b in the boundaries of C and D
+respectively, so that the label of the path QpRp−1 is equal 1 in Gα−1 where Q and R are
+boundary loops of C and D starting at a and b respectively see Figure 2a). In this case,
+a
+b
+p
+C
+D
+Q
+R
+Θ
+D
+C
+v
+S
+w
+∆
+∆
+∆
+a
+b
+c
+T
+Figure 2.
+we can perform the procedure of cell-cell cancellation as follows. We remove cells C and D
+from ∆, cut the remaining diagram along p and fill in the resulting region by a diagram Θ
+over the presentation of Gα−1 (see Figure 2b). The boundary marking of the new diagram
+naturally inherits the boundary marking of ∆ and the labels of sides and bridges are not
+changed.
+Now let u be a bridge of native rank α of ∆ with bridge partition u = v ·S·w. The label S
+of S has an associated relator R of rank α such that R ≖ ST for some T (according to the
+convention in 4.11). We attach a cell C of rank α to ∆ along S so that (ST)−1 becomes
+the label of the boundary loop (ST)−1 of C (see Figure 2c). For the new diagram ∆ ∪ C we
+15
+
+define the boundary marking of rank α with a new bridge vT−1w instead of u. We call this
+operation switching of u.
+If C and another cell D of rank α of ∆ form a cell-cell cancellation pair in ∆ ∪ C then we
+say that u and D form a bridge-cell cancellable pair. In this case, after performing a cell-cell
+cancellation in ∆ ∪ C we obtain a diagram ∆′ having one cell of rank α less than ∆. We will
+refer to this reduction step as bridge-cell cancellation.
+5.5. Definition (Reduced diagram). Let ∆ be a diagram of rank α ≥ 1 with fixed bridge
+partitions for all bridges of ∆. We say that ∆ is reduced if it has no cancellable pairs after
+any refinement.
+5.6. Remark. In what follows, we will be assuming that a diagram ∆ of rank α ≥ 1 has fixed
+bridge partitions of all bridges of ∆ if it is required by context. In particular, this applies
+when we consider the subdiagram ∆α−1 and the property of ∆ to be reduced.
+5.7. Reduction process. If a diagram ∆ of rank α is not reduced then, after possible re-
+finement, we obtain a cancellable pair which can be removed by performing the reduction
+procedure described above. Thus, any diagram of rank α ≥ 1 can be transformed to a re-
+duced one. Note that we use a sequence of transformations of the following two types in the
+reduction process:
+• transformations preserving the frame type (see Lemma 4.9);
+• bridge switching.
+Thus, after reduction the new diagram ¯∆ has the same frame type as ∆ up to bridge
+switching.
+The following observation follows from definitions 5.4 and 5.5 and will be used without
+explicit reference.
+5.8. Proposition. Let Σ be a subdiagram of a reduced diagram ∆ of rank α ≥ 1 such that
+the central arc of any bridge of Σ is either a subpath of the central arc of a bridge of ∆ or a
+subpath of (δD)−1 where D is a cell of rank α of ∆. Then Σ is reduced as well.
+6. Reduction to the previous rank
+6.1. Definition. Let ∆ be a diagram of rank α. A bond in ∆ is a simple path u satisfying
+the following conditions:
+(i) u joins two vertices on sides of ∆ and intersects the boundary of ∆ only at the
+endpoints of u;
+(ii) label(u) is equal in Gα to a word in Hα.
+(iii) u is not homotopic in ∆ (rel endpoints) to a subpath of a side of ∆;
+(iv) u does not cut off from ∆ a simply connected subdiagram with boundary loop u±1pvq
+where p is an end of a side of ∆, v is a bridge of ∆, q is a start of a side of ∆ and
+labels of p and q are empty words. See Figure 3.
+6.2.
+In most cases, we will assume that the label of a bond u already belongs to Hα. Note
+that this condition can always be achieved by cutting ∆ along u and attaching a subdiagram
+with boundary loop u±1v where label(v) ∈ Hα and its mirror copy, see Figure 4.
+6.3. Definition. A diagram of rank α is small if it has no bonds after any refinement.
+16
+
+u
+u
+v
+p
+q
+Figure 3. Excluded cases in (iii) and (iv)
+u
+u
+v
+u′
+Figure 4.
+The following observation is straightforward.
+6.4. Proposition.
+(i) The property of a diagram ∆ of rank α to be small depends only on the frame type
+of ∆.
+(ii) The property of a diagram of rank α to be small is preserved under switching of
+bridges.
+(iii) If ∆ is a small diagram of rank 0 with c(∆) > 0 then labels of all sides of ∆ are
+empty words.
+6.5. Definition. Let ∆ be a diagram of rank α ≥ 1. A disk subdiagram Π of ∆α−1 is a
+contiguity subdiagram of ∆ if the boundary loop of Π has the form Pu1Qu2 where P−1 and Q−1
+are nonempty subpaths of sides of ∆α−1 and each of the two paths ui is either a bond in ∆α−1
+with label(ui) ∈ Hα−1 or a bridge of ∆α−1. Note that here we use Definition 6.1 with rank
+α − 1 instead of α.
+The paths P±1 and Q±1 are contiguity arcs of Π. If P−1 and Q−1 occur, respectively, in
+sides S and T of ∆α−1 then we say that Π is a contiguity subdiagram of S to T (or between S
+and T).
+According to definition 2.4, if P and Q are contiguity arcs of a contiguity subdiagram with
+boundary loop Pu1Qu2 then labels of P−1 and Q are close in Gα−1.
+6.6. Lemma (small cancellation in reduced diagrams). Let ∆ be a reduced diagram of rank α.
+Let Π be a contiguity subdiagram of ∆ with boundary loop δΠ = PuQv where P and Q are
+the contiguity arcs of Π. Assume that P−1 occurs in the boundary loop of a cell D of rank α
+and Q−1 occurs in a side S of ∆α−1. Then:
+(i) If S is a side of ∆ then µ(P) < ρ;
+17
+
+(ii) If S is the boundary loop of a cell D′ distinct from D then µ(P) < λ;
+(iii) If S is the central arc of a bridge of ∆ then µ(P) < λ;
+Proof. If S is a side of ∆ then the label of S is reduced in Gα (or cyclically reduced in Gα
+if S is a cyclic side), as defined in 5.1. Then µ(P) < ρ by the definition of a reduced word
+in 2.5.
+Assume that µ(P) ≥ γ and S = δD′ where D′ is a cell distinct from D. Let R and R′ be
+boundary loops of D and D′ starting at the initial and terminal vertices of u, respectively.
+By the small cancellation condition (S2) we have label(R) = label(uR′u−1) in Gα−1, hence D
+and D′ form a cell-cell cancellable pair contrary to the hypothesis that ∆ is reduced.
+If µ(label(P)) ≥ λ and S is the central arc of a bridge of ∆ then in a similar way we see
+that D and S form a cell-bridge cancellable pair.
+□
+Note that the lemma leaves uncovered a possibility when S = δD, i.e. when Π is a contiguity
+subdiagram of D to itself. This case needs a special consideration.
+6.7. Definition. A cell D of rank α in a diagram ∆ of rank α ≥ 1 is folded if there exists a
+simple path u joining two vertices a and b in the boundary of D so that label(PQuQPu−1) = 1
+in Gα−1 where P and Q are subpaths of δD from a to b and from b to a respectively (Figure 5).
+Q
+P
+u
+a
+b
+Figure 5.
+6.8. Lemma (no folded cells). Assume that no relator of rank α is conjugate in Gα−1 to its
+inverse. Then folded cells do not exist. Consequently, if Π is a contiguity subdiagram of a
+cell of rank α to itself then for a contiguity arc P of Π we have µ(label(P)) < λ.
+Proof. The first statement is an immediate consequence of Definition 6.7. If Π is a contiguity
+subdiagram of a cell D of rank α to itself and P is a contiguity arc of Π with µ(label(P)) ≥ λ
+then, as in the proof of Lemma 6.6, we conclude that D is a folded cell.
+□
+6.9.
+We will be considering finite sets of disjoint contiguity subdiagrams of a diagram ∆ of
+rank α ≥ 1. Our goal is to produce a maximal, in an appropriate sense, such a set.
+Let {Πi} be a finite set of pairwise disjoint contiguity subdiagrams of ∆. Each connected
+component Θ of the complement ∆α−1 − � Πi is a diagram of rank α − 1 with a naturally
+induced boundary marking of rank α − 1 defined as follows:
+18
+
+• Bridges of ∆α−1 occurring in the boundary of Θ become bridges of Θ;
+• If u is a bond of ∆α−1 occurring in the boundary of some contiguity subdiagram Πi
+and u−1 occurs in the boundary of Θ then u−1 becomes a bridge of Θ;
+• The rest of the boundary of Θ consists of subpaths of sides of ∆α−1, or possibly
+cyclic sides of ∆α−1, which are viewed as sides of Θ.
+The following observation follows easily by induction on the number of contiguity subdi-
+agrams in a set {Πi}.
+6.10. Lemma. Let {Πi} be a set of r pairwise disjoint contiguity subdiagrams of a diagram ∆
+of rank α ≥ 1. Let {Θj} be the set of all connected components of the complement ∆α−1 −
+�
+i Πi. Then
+�
+j
+c(Θj) = c(∆α−1),
+�
+j
+χ(Θj) = χ(∆α−1) + r.
+6.11. Proposition. Let ∆ be a diagram of rank α ≥ 1. Then there exists another diagram ∆′
+of rank α and a finite set {Πi} of pairwise disjoint contiguity subdiagrams of ∆′ such that:
+(i) ∆′ is obtained from ∆ by replacing its subdiagram ∆α−1 with another subdiagram
+over the presentation of Gα−1 of the same frame type; in particular, ∆ and ∆′ have
+the same boundary marking and the same frame type.
+(ii) any connected component Θ of ∆′
+α−1 − �
+i Πi is a small diagram of rank α − 1.
+(iii) if c(∆α−1) > 0 then c(Θ) > 0 for each connected component Θ of ∆′
+α−1 − �
+i Πi.
+Proof. Let ∆ be a diagram of rank α and let {Πi} be a finite set of pairwise disjoint contiguity
+subdiagrams of ∆. Assume that a connected component Θ of ∆α−1 −�
+i Πi has a bond, pos-
+sibly after refinement. We describe how to obtain from {Πi} a new set of disjoint contiguity
+subdiagrams by either increasing the set or increasing the part of ∆ covered by {Πi}. We
+track on two inductive parameters: the number N of connected components of ∆α−1 −�
+i Πi
+and the total length L of sides of these components.
+Refining Θ inside ∆ we may assume that Θ has a bond u. An easy analysis shows that any
+bond in Θ is also a bond in ∆α−1. Performing surgery as described in 6.2 we may assume
+that the label of u belongs to Hα−1.
+Observe that u cuts Θ into a subdiagram Θ1 or two subdiagrams Θ1 and Θ2 which inherit
+the boundary marking of rank α −1. From the definition of complexity c(∗) we immediately
+see that c(Θ) = �
+i c(Θi) in either of the two cases. Since u is not homotopic to a subpath
+of a side of Θ we have c(Θi) ≥ 0 for each Θi. We change the set {Πi} depending on the
+following two cases:
+Case 1: u cuts Θ into two subdiagrams Θ1 and Θ2 and at least one of them, say Θ1,
+satisfies c(Θ1) = 0. Then Θ1 is a simply connected subdiagram with two bridges, and hence
+a contiguity subdiagram of ∆. Note that if for both Θ1 and Θ2 we have c(Θ1) = c(Θ2) = 0
+then ∆ has no cells of rank α and is itself a contiguity subdiagram.
+We then can take
+{Πi} = {∆}. We assume that this is not the case.
+Let v be the other bridge of Θ1. If u is a bridge of ∆α−1 then we simply add Θ1 to the
+set {Πi}. Otherwise v−1 is a bond of ∆α−1 occurring in the boundary loop of some Πi;
+then we attach Θ1 to Πi (see Figure 6. Note that the label of at least one side of Θ1 is
+19
+
+nonempty (by condition (iv) of Definition 6.1 applied to Θ and u). Hence after performing
+this operation, L is strictly decreased and N is not changed.
+Θ2
+Θ1
+Π
+v
+u
+Figure 6.
+Case 2: Case 1 does not hold. We refine ∆ so that u “bifurcates” into two paths u′ and u′′
+(Figure 7) and obtain a “degenerate” contiguity subdiagram Π of ∆ between u′ and u′′. We
+then add Π to the set {Πi}. The operation strictly increases N not changing L.
+∆α−1
+u
+u′
+u′′
+Π
+Figure 7.
+Starting from the empty set of contiguity subdiagrams Πi, we perform recursively the
+procedure described above. Each step we either decrease L not changing N or increase N
+not changing L. Furthermore, each time there is at most one connected component Θ of
+∆α−1 − �
+i Πi with c(Θ) ≤ 0 and it exists only if c(∆α−1) ≤ 0 for the initial diagram ∆. By
+Lemma 6.10, N is bounded from above, so the procedure terminates after finitely many steps.
+Upon termination, all connected components of ∆α−1 −�
+i Πi become small by construction.
+□
+6.12. Definition. We say that a set {Πi} satisfying the conclusion of Proposition 6.11 is a
+tight set of contiguity subdiagrams of ∆′.
+7. Global bounds on diagrams
+7.1.
+Let ∆ be a diagram of rank α ≥ 1 and {Πj} a set of disjoint contiguity subdiagrams
+of ∆. We have a tiling of ∆ by subdiagrams of three types: cells of rank α, contiguity
+subdiagrams Πi and connected components of the complement ∆α−1 −� Πi. We name these
+subdiagrams tiles of index 2, 1 and 0 respectively and refer to them also as internal tiles.
+We consider also external 2-cells of ∆ as tiles of index 2, so with these extra tiles we obtain
+a tiling of the 2-sphere. Boundary loops of all tiles carry naturally induced partitions into
+subpaths (allowed to be whole loops) called tiling sides, defined precisely as follows (see
+Figure 8):
+• The boundary loop δΠi of each contiguity subdiagram Πi is partitioned as P·u·Q·v
+where P and Q are the contiguity arcs; thus δΠi consists of four tiling sides.
+20
+
+1
+1
+1
+1
+1
+1
+1
+1
+1
+0
+0
+0
+0
+0
+0
+2
+2
+2
+Figure 8.
+• A component Θ of ∆α−1 − �
+i Πi has the induced boundary marking of rank α − 1
+(in this case, a tiling side can be a cyclic side of Θ).
+• The boundary loop of a cell of rank α either has no nontrivial partition (in this case
+it is considered as a cyclic tiling side) or is partitioned as an alternating product of
+contiguity arcs of subdiagrams Πi and paths S where S−1 is a side of a component
+of ∆α−1 − �
+i Πi.
+• The partition of the boundary loop L of an external cell is defined as follows: we
+take the partition of L induced by the boundary marking of rank α − 1 of ∆α−1 and
+additionally subdivide sides of rank α−1 into alternating products of contiguity arcs
+of subdiagrams Πi and paths S where S−1 is a side of a component of ∆α−1 − �
+i Πi.
+Note that we view on tiling sides as paths, i.e. they are considered with direction. By
+construction, the set of all tiling sides is closed under inversion, and each tiling side occurs
+in a unique way in a boundary loop of a tile.
+7.2. Definition. Let S be the set of tiling sides associated with {Πi}. For every tile T, we
+denote S(T) the set of tiling sides occurring in the boundary loops of T.
+A discrete connection on a pair (∆, {Πi}) is a function w : S → R such that w(s−1) = −w(s)
+for any s. Given w, we define the curvature κ(T) of each internal tile T:
+κ(T) = (−1)index(T)χ(T) +
+�
+s∈S(T)
+w(s).
+(Note that inequality χ(T) ̸= 1 is possible only if T has index 0.) For an external tile T, by
+definition,
+κ(T) =
+�
+s∈S(T)
+w(s).
+By definition, the total curvature κ(∆) of ∆ is the sum of curvatures of all internal tiles
+of ∆. The total curvature of external tiles of ∆ is the curvature along the boundary of ∆,
+denoted κ(∂∆).
+7.3. Proposition (A discrete version of the Gauss–Bonnet theorem). For any diagram ∆ of
+rank α ≥ 1 and any set {Πi} of disjoint contiguity subdiagrams of ∆,
+κ(∆) + κ(∂∆) = χ(∆).
+In particular, if κ(T) is non-positive for any internal tile T then κ(∂∆) ≥ χ(∆).
+21
+
+Proof. Let t be the number of cells of rank α of ∆. It follows from the second equality of
+Lemma 6.10 that
+�
+T
+(−1)index(T)χ(T) = χ(∆α−1) + t = χ(∆)
+where the sum is taken over all internal tiles T of ∆. In the expansion of κ(∆) + κ(∂∆) all
+summands w(s) are canceled because of the assumption w(s−1) = −w(s).
+□
+7.4. Proposition (bounding the number of cells). Let ∆ be a reduced diagram of rank α ≥ 1
+with c(∆α−1) > 0. Denote
+(7-1)
+ν =
+ζ
+1 − 2ζ = 1
+18,
+θ = 1
+6(5 − 22ν) = 17
+27.
+Let T be a tight set of contiguity subdiagrams of ∆. We assume that the following extra
+condition is satisfied:
+(*) Each cell of rank α of ∆ has at most one contiguity subdiagram Π ∈ T to sides of ∆.
+Let M be the number of cells of rank α of ∆. Then
+(7-2)
+θM ≤ 2
+3(1 + ν)b(∆) − χ(∆).
+For the proof, we define a discrete connection w on the pair (∆, {Πi}). Note that w(S−1) =
+−w(S) by Definition 7.2 and thus defining w(S) automatically defines w(S−1).
+Recall that sides of ∆α−1 are divided into three types: sides of ∆, central arcs of bridges of
+native rank α and the boundary loops of cells of rank α. If S is a side of ∆α−1 or a subpath
+of a side of ∆α−1 then we assign to S type I, II or III respectively.
+Before defining w, we perform on ∆ the following “cleaning” procedure: if a bridge of ∆α−1
+occurs in the boundary of some contiguity subdiagram Πi then we cut off Πi from ∆ taking
+the bond in the boundary of Πi as a new bridge of the resulting ∆α−1. Thus we may assume
+that
+(**) every bridge of ∆α−1 occurs in the boundary of a tile of index 0 (i.e. a connected
+component of ∆α−1 − �
+Π∈T Π).
+We define w as follows:
+(i) Let Θ be a connected component of ∆α−1 − �
+Π∈T Π. For each bond or bridge u of rank
+α − 1 occurring in the boundary of Θ, define
+w(u) = −1
+3(1 + ν).
+For each side S of Θ,
+w(S) = ζθ|S|α−1.
+(ii) Let Π ∈ T and let δΠ = Pu1Qu2 as in Definition 6.5. By (**), for each i = 1, 2 the tiling
+side u−1
+i
+occurs in the boundary of a connected component of ∆α−1 − �
+Π∈T Π. By (i), we
+already have
+w(ui) = −w(u−1
+i ) = 1
+3(1 + ν).
+22
+
+We define w(P) (the definition of w(Q) is similar):
+(7-3)
+w(P) =
+
+
+
+
+
+0
+if P has type I or II
+1
+3(1 − 2ν)
+if P has type III and Q has type I
+1
+6(1 − 2ν)
+if P has type III and Q has type II or III
+(iii) Let D be a cell of rank α of ∆ and S be a tiling side occurring in δD. The value of w(S)
+is already defined by (i) and (ii). We have:
+• If S−1 is the contiguity arc of a contiguity subdiagram Π ∈ T of D to a side of ∆α−1
+of type I or II then w(S) = −1
+3(1 − 2ν).
+• If S−1 is the contiguity arc of a contiguity subdiagram Π ∈ T of D to a side of ∆α−1
+of type III then w(S) = −1
+6(1 − 2ν).
+• If S−1 occurs in the boundary of a connected component of ∆α−1 − �
+Π∈T Π then
+w(S) = −ζθ|S|α−1.
+We provide an upper bound for the curvature of any internal tile. For contiguity subdia-
+grams Π ∈ T we immediately have κ(Π) ≤ 0 by (ii).
+Let Θ be a connected component of ∆α−1 − �
+Π∈T Π. We have
+κ(Θ) = χ(Θ) − 1
+3(1 + ν)b(Θ) + ζθ
+�
+S
+|S|α−1
+where the sum is taken over the sides S of Θ.
+If α = 1 then � |S|α−1 = 0 (Proposition 6.4(iii)). If α ≥ 2 then by Proposition 7.8α−1,
+θ
+�
+|S|α−1 ≤ 2
+3(1 + ν)b(Θ) − χ(Θ)
+Using the fact that c(Θ) > 0 it is easy to check that κ(Θ) ≤ 0 in both cases α = 1 and
+α ≥ 2. (The critical case is when b(Θ) = 3 and χ(Θ) = 1; in this case we have κ(Θ) = −ν if
+α = 1 and κ(Θ) = 0 if α ≥ 2 by definition (7-1) of ν).
+Finally, let D be a cell of rank α of ∆. We prove that κ(D) ≤ −θ. By (*), D has at most
+one contiguity subdiagram to sides of ∆α−1 of type I. We consider first the case when D
+has one. Let r be the number of contiguity subdiagrams of D to sides of types II and III.
+The remaining r + 1 subpaths S1, S2, . . . Sr+1 of δD are tiling sides such that S−1
+i
+belong to
+boundary loops of connected components of ∆α−1 − �
+Π∈T Π; so we have
+κ(D) ≤ 1 − 1
+3(1 − 2ν) − r
+�1
+6(1 − 2ν)
+�
+− ζθ
+r+1
+�
+i=1
+|Si|α−1.
+By condition (S1) in 2.8 and Lemmas 6.6, 6.8,
+r+1
+�
+i=1
+|Si|α−1 ≥ (1 − ρ − rλ)Ω = (9 − r)λΩ.
+Hence
+(7-4)
+κ(D) ≤ 2
+3(1 + ν) − r
+�1
+6(1 − 2ν)
+�
+− ζθλΩ max(0, 9 − r).
+23
+
+If r ≥ 9 then the coefficient before r in the right-hand side of (7-4) is negative. If r ≤ 9 then
+the coefficient is
+−1
+6(1 − 2ν) + ζθλΩ
+which is positive since by the second inequality (2-3) we have ζθλΩ ≥ 20ζθ = θ > 1
+6. Hence
+the maximal value of the expression in (7-4) is when r = 9. Substituting r = 9 into the
+right-hand side of (7-4) we obtain the expression
+2
+3(1 + ν) − 9
+6(1 − 2ν)
+which is equal −θ by (7-1). This shows that κ(D) ≤ −θ.
+Assume that D has no contiguity subdiagrams to sides of type I. Let, as above, r be the
+number of contiguity subdiagrams of D to sides of types II and III and S1, S2, . . . Sr be the
+remaining r tiling sides occurring in δD such that S−1
+i
+belong to boundary loops of connected
+components of ∆α−1 − �
+Π∈T Π. Instead of (7-4) we have
+(7-5)
+κ(D) ≤ 1 − r
+�1
+6(1 − 2ν)
+�
+− ζθN max(0, 1 − rλ).
+If we allow r to be a non-negative real then the maximal value of the right-hand side is when
+1 − rλ = 0.
+Substituting r = 1
+λ into the left-hand side of (7-5) we obtain the expression
+1 − 1 − 2ν
+6λ
+which is less then −θ since λ ≤
+1
+24.
+Finally, we compute an upper bound for κ(∂∆).
+For a tiling side S occurring in the
+boundary loop of an external cell of ∆ (the loop has the form L−1 where L is a boundary
+loop of ∆) we have three possibilities: either S−1 is a contiguity arc of a subdiagram Π ∈ T,
+S−1 is a side of a component of ∆α−1 − �
+Π∈T Π, or S−1 is a bridge of ∆α−1 In the first two
+cases we have w(S) ≤ 0 according to (ii) or (i) respectively. If S−1 is a bridge of ∆α−1 then
+by (**), S−1 is also a bridge of some component of ∆α−1 − �
+Π∈T Π and by (i),
+w(S) = 1
+3(1 + ν).
+Note that each bridge of ∆ produces at most two bridges of ∆α−1. Hence b(∆α−1) ≤ 2b(∆).
+We obtain
+(7-6)
+κ(∂∆) ≤ 1
+3(1 + ν)b(∆α−1) ≤ 2
+3(1 + ν)b(∆)
+Application of Proposition 7.3 gives
+2
+3(1 + ν)b(∆) − θM ≥ χ(∆)
+as required. The proof of Proposition 7.4 is finished.
+7.5. Lemma. Let ∆ be a reduced disk diagram of rank α ≥ 1. If ∆ has a single (cyclic or
+non-cyclic) side then ∆ has no cells of rank α.
+24
+
+Proof. Let ∆ be a reduced disk diagram of rank α with a single side, i.e. ∆ is of monogon or
+nullgon type. Assume that ∆ has a cell of rank α. We choose such ∆ with minimal possible
+non-zero number M of cells of rank α. We then have χ(∆α−1) ≤ 0 and hence c(∆α−1) > 0.
+We can assume that ∆ is given a tight set T of contiguity subdiagrams. If each cell of rank α
+of ∆ has at most one contiguity subdiagram Π ∈ T to the side of ∆ then application of
+Proposition 7.4 would give
+θM ≤ 2
+3(1 + ν) − 1 < 0.
+Therefore, ∆ has a cell D of rank α having two contiguity subdiagram Π1, Π2 ∈ T to the
+side of ∆. The union D ∪ Π1 ∪ Π2 cuts off from ∆ a disk diagram ∆′ of rank α with a single
+side and a single bridge (Figure 9). The assumption that ∆ is reduced implies that ∆′ is
+D
+∆′
+Π1
+Π2
+Figure 9.
+reduced as well. By the choice of ∆, ∆′ has no cells of rank α. Then for some component Θ
+of ∆α−1 − �
+Π∈T Π we have c(Θ) = 0 contrary to the choice of a tight set T of contiguity
+subdiagrams of ∆ (Definition 6.12).
+□
+7.6. Proposition. If a non-empty word X is reduced in Gα then X ̸= 1 in Gα.
+Proof. Let α ≥ 1. Let X be reduced in Gα and X = 1 in Gα. Consider a reduced disk
+diagram ∆ of rank α with one side labeled X and one bridge labeled by the empty word.
+Lemma 7.5 says that ∆ has no cells of rank α and hence we have X = 1 in Gα−1. Since
+Rα ⊆ Rα−1, arguing by induction we conclude that X = 1 in the free group G0. Since X is
+freely reduced (definition 2.5) we conclude that X is empty.
+□
+7.7. Lemma. Let ∆ be a reduced diagram of rank α ≥ 1 and let u be a simple path in ∆
+homotopic rel endpoints to a subpath S of a side of ∆. Assume, moreover, that the label of u
+is equal in Gα−1 to a word in Hα−1. Then the subdiagram of ∆ with boundary loop Su−1 has
+no cells of rank α.
+Proof. Let ∆′ be the subdiagram of ∆ with boundary loop Su−1 and let w ∈ Hα−1 be a word
+such that label(u) = w in Gα−1. We attach to ∆′ a diagram Θ over the presentation of Gα−1
+with boundary loop uw−1 where label(w) = w. We consider ∆′ ∪ Θ as a diagram of rank α
+with one side S and one bridge w−1. Note that any simple path in ∆′ ∪ Θ with endpoints
+in ∆′ is homotopic rel endpoints to a simple path in ∆′. Moreover, this holds also if ∆′ ∪ Θ
+is refined to a diagram Σ and we take a refinement of ∆′ in Σ instead of ∆′. This implies
+that ∆′ ∪ Θ is a reduced diagram of rank α. Then by Lemma 7.5, ∆′ ∪ Θ has no cells of
+rank α.
+□
+25
+
+7.8. Proposition (bounding sides of a small diagram, raw form). Let ∆ be a small diagram
+of rank α ≥ 1. Assume that ∆ is not of bigon type and c(∆α−1) > 0. Then
+(7-7)
+θ
+�
+S
+|S|α ≤ 2
+3(1 + ν)b(∆) − χ(∆)
+where the sum is taken over all sides S of ∆.
+Proof. We make ∆ reduced and endow it with a tight set T of contiguity subdiagrams. We
+assign to subpaths of sides of ∆α−1 type I, II and III as in the proof of Proposition 7.4 and
+make several observations about T.
+Claim 1: There are no contiguity subdiagrams Π ∈ T between two (not necessarily distinct)
+sides of type I of ∆α−1.
+Assume Π is such a contiguity subdiagram. Let δΠ = Pu1Qu2 where P and Q are the
+contiguity arcs of Π.
+According to Definition 6.5 at least one of ui’s, say u1, is a bond
+in ∆α−1 (otherwise Π = ∆α−1 contrary to the assumption c(∆α−1) > 0). Checking with
+Definition 6.1 we see that u1 is also a bond in ∆ (condition (iii) of Definition 6.1 holds due
+to Lemma 7.7). This contradicts the assumption that ∆ is small.
+Claim 2: Up to inessential change of ∆ we may assume that condition (*) of Proposition 7.4
+is satisfied, i.e. each cell of rank α of ∆ has at most one contiguity subdiagram Π ∈ T to
+sides of type I of ∆α−1.
+Assume that a cell D of rank α has two contiguity subdiagrams Πi ∈ T (i = 1, 2) to
+sides Si of type I. Let Pi be the contiguity arc of Πi that occurs in Si. The boundary loop
+of D ∪ Π1 ∪ Π2 has the form P1u1P2u2 where labels of ui are in Hα. Since ∆ is small, at
+least one of the conditions (iii) or (iv) of Definition 6.1 should be violated for each of the
+paths ui. If S1 = S2 and some ui (and hence both u1 and u2) are homotopic rel endpoints
+to a subpath of S1 then D ∪ Π1 ∪ Π2 cuts off a reduced disk subdiagram ∆′ of ∆ with one
+bridge u−1
+1
+or u−1
+2 . By Lemma 7.5, ∆′ has no cells of rank α. Then either ∆′ is a component
+of ∆α−1 −�
+Π∈T Π or ∆′ contains a component Θ of ∆α−1 −�
+Π∈T Π with c(Θ) = 0. We come
+to a contradiction with the choice of a tight set T of contiguity subdiagrams of ∆.
+Assume that condition (iv) of Definition 6.1 fails for both u1 and u2. Then, up to renumer-
+ation of Π1 and Π2, D ∪ Π1 ∪ Π2 cuts off a simply connected subdiagram ∆′ with boundary
+loop u−1
+1 T1vT2 where P1T1 is an ending subpath of S1, v is a bridge of ∆, T2P2 is a starting
+subpath of S2 and labels of P1T1 and T2P2 are empty, see Figure 10a. In this case, we cut
+off the subdiagram D ∪ Π1 ∪ Π2 ∪ ∆′ from ∆. The operation does not change the values
+of � |S|α, b(∆) and χ(∆) in (7-7) and preserves the assumption that ∆ is small. We have
+also c(∆α−1) > 0 for the modified ∆ (otherwise ∆ would be a monogon type contradicting
+Lemma 7.5).
+Claim 3: Up to inessential change of ∆ we may assume that there are no contiguity subdia-
+grams Π ∈ T between sides of type I and II of ∆α−1.
+Assume that Π ∈ T is a contiguity subdiagram between sides of type I and II. Let δΠ =
+Pu1Qu2 where P occurs in a side S of ∆ and Q occurs in the central arc R of a bridge
+v = v1Rv2. Observe that any of the endpoints of P can be joined with any of the endpoints
+of v by a path labeled with a word in Hα in a graph composed from paths u1, u2 and v, see
+26
+
+P1
+T1
+v
+D u1
+P2
+T2
+u2
+S1
+S2
+∆′
+Π1
+Π2
+v1
+S1
+P
+S2
+u1
+Q
+R1
+u2
+Π
+v2
+R2
+a
+b
+Figure 10.
+Figure 10b. Since ∆ is small, this easily implies that v and S are adjacent in the boundary
+of ∆. Up to symmetry, assume that vS occurs in a boundary loop of ∆. so R = R1QR2
+and S = S1PS2. Note that label(S1P) is empty (otherwise v1R1u−1
+1
+would give a bond in ∆
+after refinement) and label(QR2) is nonempty (because u1 is a bond in ∆α−1). We cut off
+the subdiagram of ∆ bounded by QR2v2S1Pu1. As in the proof of the previous claim, the
+operation does not change the values of terms in (7-7), the value of c(∆α−1) and keeps the
+assumption that ∆ is small. On the other hand, we decrease the total length of labels of
+sides ∆α−1. The claim is proved.
+We now define a discrete connection w∗ on (∆, T) by changing the function w defined in
+the proof of Proposition 7.4. The new function w∗ differs from w only on contiguity arcs of
+contiguity subdiagrams Π ∈ T as follows. Let δΠ = Pu1Qu2 where P and Q are the contiguity
+arcs of Π. By Claims 1 and 3, if P has type I then Q has necessarily type III. Instead of
+(7-3) we define
+w∗(P) =
+
+
+
+
+
+θ
+if P has type I
+1
+3(1 − 2ν) − θ
+if P has type III and Q has type I
+1
+6(1 − 2ν)
+in all other cases
+For contiguity subdiagrams Π ∈ T we immediately have κ∗(Π) ≤ 0 where κ∗ denotes the
+curvature function defined from w∗. If Θ is a connected component of ∆α−1 − �
+Π∈T Π then
+κ∗(Θ) = κ(Θ) ≤ 0. Let D be a cell of rank α of ∆. In view of Claim 2
+κ∗(D) ≤ κ(D) + θ ≤ 0.
+We provide a bound for κ∗(∂∆). Let t be the number of all contiguity subdiagrams Π ∈ T
+between sides of type I and sides of type III. Then
+κ∗(∂∆) ≤ 1
+3(1 + ν)b(∆α−1) − θt − ζθ
+�
+S∈sides(Θ)
+|S|α−1
+≤ 2
+3(1 + ν)b(∆) − θ
+�
+S∈sides(∆)
+|S|α
+27
+
+where Θ runs over all connected components of ∆α−1 − �
+Π∈T Π. Applying Proposition 7.3
+we obtain
+2
+3(1 + ν)b(∆) − θ
+�
+S∈sides(∆)
+|S|α ≥ χ(∆)
+as required.
+□
+Below we will often use Proposition 7.8 in a slightly simplified form. We introduce yet
+another numerical parameter
+η = 1 + 2ν
+θ
+= 30
+17.
+7.9. Proposition (bounding sides of a small diagram, simplified form). If ∆ is a small
+diagram of rank α of positive complexity then
+(7-8)
+�
+S∈ sides(∆)
+|S|α ≤ η c(∆).
+Proof. By Proposition 6.4(iii) we may assume that α ≥ 1.
+It remains to notice that if
+c(∆) ≥ 1 then
+1
+θ
+�2
+3(1 + ν)b(∆) − χ(∆)
+�
+≤ η c(∆).
+(The critical case is when b(∆) = 3 and χ(∆) = 1. In this case we have the equality.)
+□
+7.10. Lemma. Let ∆ be a reduced diagram of rank α ≥ 1 and let T be a tight set of contiguity
+subdiagrams of ∆. Let D be a cell of rank α of ∆. Then the following is true.
+(i) Let Π1 and Π2 be two contiguity subdiagrams of D to a side S of ∆α−1. Then a
+subdiagram Θ of ∆ bounded by δD, Π1, Π2 and S (there are two of them if S is a
+cyclic side) is not simply connected (see Figure 11a).
+(ii) Let Π be a contiguity subdiagram of D to itself. Then the subdiagram Θ′ of ∆ bounded
+by δD and Π (see Figure 11b) is not simply connected.
+(iii) If ∆ is simply connected then any cell of rank α has at most one contiguity subdia-
+gram to each side of ∆α−1 and has no contiguity subdiagrams to itself.
+S
+Π1
+Π2
+Θ
+R
+u
+Θ′
+Π
+D
+D
+a
+b
+Figure 11.
+28
+
+Proof. (i) Assume that Θ is simply connected. We consider Θ as a diagram of rank α with
+a single side that is a subpath of S. The assumption that ∆ is reduced implies that Θ is
+reduced. By Lemma 7.5 Θ has no cells of rank α. Then we obtain a contradiction with the
+choice of a tight set T of contiguity subdiagrams of ∆.
+(ii) Assume that Θ′ is simply connected. Let ∂Θ′ = Ru where R−1 occurs in the boundary
+loop of D and u−1 is the bond in ∆α−1 that occurs in ∂Π. We consider Θ′ as a a diagram
+of rank α with one side S labeled by the empty word and one bridge Ru (formally, to fit
+the definition in 5.1 we have to take a copy of Θ′ and perform a refinement to make S a
+non-empty path). By Lemma 7.5 Θ′ has no cells of rank α and we come to a contradiction
+since in this case u−1 cannot be a bond in ∆α−1 due to condition (iii) of Definition 6.1.
+(iii) follows from (i) and (ii).
+□
+7.11. Proposition (diagrams of small complexity are single layered). Let ∆ be a reduced
+diagram of rank α ≥ 1 and let T be a tight set of contiguity subdiagrams of ∆.
+(i) If ∆ is a disk diagram of bigon type then every cell of rank α of ∆ has a contiguity
+subdiagram Π ∈ T to each of the two sides of ∆.
+(ii) If ∆ is a disk diagram of trigon or tetragon type then every cell of rank α of ∆ has
+contiguity subdiagrams Π ∈ T to at least two sides of ∆.
+(iii) If ∆ is an annular diagram with two cyclic sides then every cell of rank α of ∆ has
+a contiguity subdiagram Π ∈ T to each of the sides of ∆.
+(iv) If ∆ is an annular diagram with one cyclic side and one non-cyclic side then every
+cell D of rank α of ∆ has at least two contiguity subdiagrams Π, Π′ ∈ T to sides
+of ∆. Here we admit the possibility that both Π and Π′ are contiguity subdiagrams
+between D and the non-cyclic side of ∆.
+Proof. Let ∆ be a reduced diagram of rank α of a type listed in (i)–(iv). We call a cell D of
+rank α of ∆ regular if it satisfies the conclusion of the corresponding statement (i)–(iv) and
+exceptional otherwise. We need to prove that ∆ has no exceptional cells. Observe that by
+Lemma 7.10, an exceptional cell has at most one contiguity subdiagram to sides of ∆, i.e.
+such a cell satisfies condition (*) of Proposition 7.4. We use induction on the number M of
+cells of rank α of ∆.
+(i) Let ∆ be of bigon type, i.e. a disk diagram with two sides. If ∆ has no regular cells
+of rank α but has at least one exceptional cell then application of Proposition 7.4 gives a
+contradiction.
+Assume that D is a regular cell of ∆. Let Πi (i = 1, 2) be the contiguity subdiagram
+of D to Xi. The complement of D ∪ Π1 ∪ Π2 in ∆ consists of two components ∆1 and ∆2
+of bigon type with the induced boundary marking of rank α (see Figure 12a). The set of
+subdiagrams Π ∈ T contained in ∆i is a tight set of contiguity subdiagrams of ∆i. Each
+of the subdiagrams ∆i has a smaller number of cells of rank α, so the statement follows by
+induction.
+(ii) Let ∆ be of trigon or tetragon type. Assume that ∆ has a regular cell D. Let Πi (i =
+1, 2) be contiguity subdiagrams of D to sides of ∆. The complement of ∆−D∪Π1∪Π2 consists
+of two components ∆1 and ∆2 with the induced boundary marking of rank α (Figure 12b)
+making them diagrams of rank α. If ∆ is of trigon type then ∆1 and ∆2 are of trigon and
+bigon types. If ∆ is of tetragon type then either ∆1 and ∆2 are of tetragon and bigon types,
+or both ∆i are of trigon type. Then we can refer to (i) and the inductive hypothesis.
+29
+
+D
+D
+∆1
+∆2
+∆1
+∆2
+a
+b
+Π2
+Π1
+Π1
+Π2
+Figure 12.
+Assume that all cells of rank α of ∆ are exceptional. Then by Proposition 7.4
+(7-9)
+θM ≤ 8
+3(1 + ν) − 1
+which implies M ≤ 2. Following the proof of Proposition 7.4 we compute a better bound
+for M and conclude that M = 0.
+Assume that M ≥ 1 and let D be a cell of rank α of ∆. Consider the discrete connection w
+on (∆, T) defined in the proof of Proposition 7.4. An upper bound for κ(D) is given by (7-4).
+The right-hand side of (7-4) is a linear expression on r and, as we have seen in the proof of
+Proposition 7.4, in the case r ≤ 9 the coefficient before r is positive. To get a value for the
+upper bound, we compute the maximal possible value of r. Observe that by Lemma 7.10, D
+has no contiguity subdiagrams to itself, has at most one contiguity subdiagram to another
+cell of rank α of ∆ (if that cell exists) and the number of contiguity subdiagrams of D to
+sides of type II is at most 4; so r ≤ 5. Then the maximal value of the right-hand side of
+(7-4) is achieved when r = 5. Substituting r = 5 into (7-4) and using (2-3) we obtain
+κ(D) ≤ 2
+3(1 + ν) − 5
+6(1 − 2ν) − 4ζθλΩ
+≤ −1
+6 + 7
+3ν − 4θ = −138
+54 .
+By (7-6)
+κ(∂∆) ≤ 8
+3(1 + ν) = 152
+54 .
+Proposition 7.3 gives
+1 = κ(∆) + κ(∂∆) ≤ 14
+54.
+The contradiction shows that the assumption M ≥ 1 is impossible.
+(iii): Similarly to the proof of (ii), assume first that ∆ has a regular cell D of rank α with
+two contiguity subdiagrams Π1 and Π2 to sides of ∆. By Lemma 7.10(i) these are contiguity
+subdiagrams to distinct sides of ∆. Then the complement ∆ − (D ∪ Π1 ∪ Π2) is a diagram
+of bigon type and the statement follows directly from (i).
+If all cells of rank α of ∆ are exceptional and there is at least one cell of rank α then
+application of Proposition 7.4 gives an immediate contradiction.
+(iv): Assume that ∆ has a regular cell D of rank α with two contiguity subdiagrams Πi
+(i = 1, 2) to sides of ∆.
+There are two cases depending on whether or not Π1 and Π2
+are contiguity subdiagrams to distinct sides of ∆ (see Figure 13).
+In the first case, the
+30
+
+D
+D
+Π1
+Π2
+Π1
+Π2
+∆1
+∆2
+Figure 13.
+complement ∆ − (D ∪ Π1 ∪ Π2) is a diagram of trigon type and the statement follows from
+the already proved part (ii). In the second case, ∆ − (D ∪ Π1 ∪ Π2) consists of a simply
+connected component ∆1 and and an annular component ∆2 with one non-cyclic side. For
+cells of rank α in ∆1 the statement follows by (i) and for cells of rank α in ∆2 we can apply
+induction since ∆2 has a strictly smaller number of cells of rank α than ∆.
+If all cells of rank α of ∆ are exceptional then application of Proposition 7.4 gives M =
+0.
+□
+7.12. Proposition (small diagrams of trigon or tetragon type). Let ∆ be a small diagram
+of rank α of trigon or tetragon type with sides Si (1 ≤ i ≤ k, k = 3 or k = 4). Then
+3
+�
+i=1
+|Si|α ≤ 4ζη
+or
+4
+�
+i=1
+|Si|α ≤ 6ζη
+in the trigon and tetragon cases, respectively.
+Proof. By Proposition 6.4(iii) we may assume that α ≥ 1.
+We assume that ∆ is reduced and is given a tight set T of contiguity subdiagrams. Fol-
+lowing arguments from the proof of Proposition 7.8 we can assume that Claims 1–3 from
+that proof hold in our case. By Claim 2 and Proposition 7.11(ii), ∆ has no cells of rank α.
+By Claims 1 and 3, T has only contiguity subdiagrams between sides of ∆α−1 of type II.
+Hence any side of ∆ occurs entirely in a boundary loop of a connected component Θ of
+∆α−1 − �
+Π∈T Π. By Lemma 6.10, �
+Θ c(Θ) = c(∆α−1). Applying Proposition 7.9α−1 to
+components Θ of ∆α−1 − �
+Π∈T Π we obtain
+�
+i
+|Si|α−1 ≤ ηc(∆α−1) ≤ (b(∆α−1) − 2)η
+which gives the required inequality by 4.14(iii).
+□
+7.13. Proposition (cell in a diagram of small complexity). Let ∆ be a reduced diagram of
+rank α ≥ 1 of one of the types listed in Proposition 7.11. Let T be a tight set of contiguity
+subdiagrams on ∆ and let D be a cell of rank α of ∆. Let Pi, i = 1, 2, . . . , r be the contiguity
+arcs of contiguity subdiagrams of D to sides of ∆ that occur in δD. Then:
+(i) If ∆ has bigon type or is an annular diagram with two cyclic sides then r = 2 and
+µ(P1) + µ(P2) ≥ 1 − 2λ − 16ζηω.
+31
+
+(ii) If ∆ has trigon type then 2 ≤ k ≤ 3 and
+k
+�
+i=1
+µ(Pi) ≥ 1 − 3λ − 24ζηω.
+(iii) If ∆ is an annular diagram with one cyclic side and one non-cyclic side then 2 ≤
+k ≤ 3 and
+k
+�
+i=1
+µ(Pi) ≥ 1 − 4λ − 24ζηω.
+Proof. Assume that C is another cell of rank α of ∆. By Proposition 7.11, C has at least
+two contiguity subdiagrams Π1, Π2 to sides of ∆. Let ∆′ be the connected component of
+∆−C−Π1 −Π2 containing D. Then ∆′ inherits from ∆ the boundary marking of rank α and
+the tight set of contiguity subdiagrams. Observe also that ∆′ is also a diagram of rank α of
+one of the types in cases (i)–(iii); moreover, it is of the same type (i)—(iii) or has a smaller
+complexity. In this case the statement is reduced by induction to the case of a diagram with
+a smaller number of cells of rank α.
+It remains to consider the case when D is a single cell of rank α of ∆. The equality r = 2
+in (i) and the bound 2 ≤ r ≤ 3 in (ii) and (iii) follow from Lemma 7.10. With bounds from
+Lemmas 6.6, 6.8, Propositions 7.9, 7.12 for α := α − 1 and inequality (4-2), an easy analysis
+shows that the worst cases for the lower bound on �
+i µ(Pi) are as shown in Figure 14. We
+Figure 14.
+then get the corresponding inequality in (i)–(iii).
+□
+32
+
+8. Fragments
+In this section we establish several properties of fragments of rank α ≥ 1. Most of them
+are proved using facts about relations in Gα−1. Starting from this point we use extensively
+statements from subsequent Sections 9–13 for values of rank β < α. We also switch our main
+action scene to Cayley graphs Γα−1 and Γα.
+All statements in this section are formulated and proved under assumption α ≥ 1.
+The following observation is a consequence of the assumption that the graded presentation
+of Gα is normalized, condition (S3) and the fact that centralizers of non-torsion elements
+of Gα−1 are cyclic (Proposition 13.8α−1). Recall that two periodic lines L1 and L2 in Γα−1
+are called parallel if sP1,L1 = sP2,L2 where Pi is the period of Li (see 4.2).
+8.1. Lemma. If L1 and L2 are two parallel periodic lines in Γα−1 whose periods are relators
+of rank α then L1 = L2.
+Proof. Let Li (i = 1, 2) be two parallel periodic lines in Γα−1 whose periods Ri are relators
+of rank α. Up to cyclic shift of Ri we can assume that Ri ∈ X±1
+α
+where Xα is the set of
+defining relators of rank α in the presentation (2-1). Let vi be a vertex on Li such that the
+label of Li starts at vi with Ri. Let g = v−1
+1 v2 ∈ Gα (recall that we identify vertices of Γα
+with elements of Gα). Since L1 and L2 are parallel we have gR2g−1 = R1. By (S3) we have
+either R1, R2 ∈ Xα or R−1
+1 , R−1
+2
+∈ Xα, so according to Definition 2.10, we get R1 ≖ R2 and
+R1 ≖ Rt
+0 where R0 it the root of R1. Since the centralizer of R1 is cyclic, we have g = Rk
+0
+for some integer k. This implies L1 = L2.
+□
+8.2. Corollary (Small cancellation in the Cayley graph). Let L1 and L2 be periodic lines
+in Γα−1 with periods R1 and R2, respectively, where both Ri are relators of rank α. Assume
+that L1 and L2 have close subpaths S1 and S2 such that |S1|α−1 ≥ λ|R1|α−1. Then L1 = L2.
+Proof. If |Si| ≤ |Ri| for i = 1, 2 then the statement follows directly from condition (S2-
+Cayley) in 4.12. Let |S1| > |R1| or |S2| > |R2|. Using Proposition 9.21α−1 and condition (S1)
+we find close subpaths S′
+1 and S′
+2 of S1 and S2 with |Si| ≤ |Ri|, i = 1, 2 and |Sj|α−1 ≥ λ|Rj|α−1
+for j = 1 or j = 2. This reduces the statement to the previous case.
+□
+8.3. Proposition. A relator of rank α is strongly cyclically reduced in Gα−1.
+Proof. Let R be a relator of rank α. Assume that some power Rt is not reduced in Gα−1.
+According to definition 2.5, for some 1 ≤ β ≤ α − 1 there exists a subword S of Rt which is
+close in Gβ−1 to a piece P of rank β with µ(P) > ρ. Since R is cyclically reduced in Gα−1
+we have |S| > |R|. Then according to the definition in 2.6 we have |R◦|β ≤ 1 and hence
+|R◦|α−1 ≤ ζα−β−1|R◦|β ≤ 1
+contradicting (S1) and (2-3).
+□
+8.4.
+A fragment path of rank α in Γα−1 is a path F labeled by a fragment of rank α. We
+assume that F has an associated R-periodic segment P with R ∈ Xα which is close to F. We
+call P the base for F.
+Note that this agrees with the definition in 2.6. If F is a fragment of rank α with asso-
+ciated triple (P, u, v) and F is a path in Γα−1 with label(F) ≖ F then the loop F−1uPv with
+label(uPv) ≖ uPv gives a base P for F. Conversely, if F is a fragment of rank α in Γα−1
+with base P then choosing a loop F−1uPv with label(u), label(v) ∈ Hα−1 and denoting F, P,
+33
+
+u and v the corresponding labels we obtain a fragment F of rank α with associated triple
+(P, u, v).
+If β ≥ α and paths F and P in Γβ are obtained by mapping a fragment ¯F of rank α with
+base ¯P in Γα−1 then, by definition, we consider F as a fragment of rank α with base P in Γβ.
+Abusing the language we will use the term ‘fragment’ for both fragment words and frag-
+ment paths in Γβ.
+Recall that by a convention in 4.2, a base P for a fragment F of rank α in Γβ has an
+associated relator R of rank α and the unique infinite R-periodic extension L. If β = α − 1
+then L is a bi-infinite path (which is simple by Proposition 8.3) that we call the base axis
+for F. If β > α then L is winding over a relator loop labeled R that we call the base relator
+loop for F.
+8.5.
+We describe a way to measure fragments of rank α. If P is a subword of a word Rk
+where R is a relator of rank α then we define
+(8-1)
+µ(P) = |P|α−1
+|R◦|α−1
+.
+Note that this agrees with the definition in 4.11 of the function µ(S) on the set of pieces S
+of rank α. If F is a fragment of rank α ≥ 1 then the size µf(F) of F is defined to be equal
+to µ(P) where P is the associated subword of Rk and R is the associated relator of rank α.
+Thus, for example, µf(F) = 1
+2 means approximately that F is close in rank α − 1 to a “half”
+of its associated relator of rank α.
+If F is a fragment of rank α in Γβ then we set µf(F) = µf(label(F)). This means that µf(F)
+is given by the formula
+µf(F) = |P|α−1
+|R◦|α−1
+.
+where P is the base for F and R is the relator associated with P.
+Using Proposition 9.21<α we can easily reformulate the definition of a reduced in Gα word
+in 2.5 in the following way: a word X is reduced in Gα if and only if X is freely reduced and
+contains no fragments F of rank 1 ≤ β ≤ α with µf(F) > ρ.
+8.6. Definition. Two fragments F and G of rank α in Γα−1 are compatible if their base axes
+are parallel. Note that by Lemma 8.1, the base axes of fragments of rank α are parallel if
+and only if they coincide.
+In the case β ≥ α, two fragments F and G of rank α in Γβ are defined to be compatible if
+they have compatible lifts in Γα−1, or, equivalently, F and G have the same base relator loop.
+It will be convenient to extend compatibility relation to fragments of rank 0. Recall that
+according to the definition in 2.6 fragments of rank 0 are letters in A±1. Thus, fragments
+of rank 0 in Γβ are paths of length 1. By definition, fragments F and G of rank 0 in Γβ are
+compatible if and only if F = G.
+We write compatibility of fragments as F ∼ G. Note that we have in fact a family of
+relations with two parameters α ≥ 0 and β ≥ max(0, α − 1): compatibility of fragments of
+rank α in Γβ. The values of β and α will be always clear from the context. Below we will use
+also “compatibility up to invertion” relation on the set of fragments of rank α in Γβ, denoted
+F ∼ G±1 and meaning that F ∼ G or F ∼ G−1. Both are obviously equivalence relations.
+34
+
+8.7. Proposition (fragment stability in bigon of the previous rank). Let α ≥ 1. Let X and Y
+be reduced close paths in Γα−1. Let K be a fragment of rank α in X with µf(K) ≥ 2.3ω. Then
+there exists a fragment M of rank α in Y such that M ∼ K and
+µf(M) > µf(K) − 2.6ω.
+Proof. Let P be the base for K. By (4-2) and Proposition 10.16α−1 we have P = z1P′z2 where
+P′ is close to a subpath M of Y and |zi|α−1 < 1.3 (i = 1, 2). Then M is a fragment of rank α
+with base P′, so µf(M) = µ(P′). By (4-2)
+µ(z1) + µ(z2) < 2.6ω
+and hence
+µ(P′) > µ(P) − 2.6ω = µf(K) − 2.6ω.
+□
+8.8. Proposition (fragment stability in trigon of the previous rank). Let X−1∗Y1∗Y2∗ be a
+coarse trigon in Γα−1. Let K be a fragment of rank α in X such that µf(K) ≥ 2.5ω. Then at
+least one of the following statements holds:
+• For i = 1 or i = 2 there is a fragment Mi of rank α in Yi such that Mi ∼ K and
+µf(Mi) > µf(K) − 2.8ω.
+• For each i = 1, 2 there is a fragments Mi of rank α in Yi such that Mi ∼ K and
+µf(M1) + µf(M2) > µf(K) − 3ω.
+Proof. This follows from Proposition 10.18α−1 in a similar way as in the proof of Proposi-
+tion 8.7.
+□
+8.9. Proposition (fragment stability in conjugacy relations of the previous rank). Let
+X be a word cyclically reduced in Gα−1.
+Let Y be a word reduced in Gα−1, u ∈ Hα−1
+and Y u = z−1Xz in Gα−1 for some z. We represent the conjugacy relation by two lines
+. . . Y−1u−1Y0u0Y1u1 . . . and ¯X = . . . X−1X0X1 . . . in Γα−1 where label(Xi) ≖ X, label(Yi) ≖ Y
+and label(ui) ≖ u (see 4.3).
+Let K be a fragment of rank α in ¯X with |K| ≤ |X| and
+µf(K) ≥ 2.5ω. Then at least one of the following statements is true:
+• For some i, there is a fragment M of rank α in Yi such that M ∼ K and
+µf(M) > µf(K) − 2.9ω.
+• For some i, there are fragments M1 and M2 of rank α in Yi and Yi+1 respectively
+such that Mi ∼ K (i = 1, 2) and
+µf(M1) + µf(M2) > µf(K) − 3ω.
+Proof. Follows from Proposition 10.19α−1.
+□
+8.10. Proposition (inclusion implies compatibility). Let K and M be fragments of rank α
+in Γβ, β ≥ α − 1. Assume that K is contained in M and µf(K) ≥ λ + 2.6ω. Then K ∼ M.
+Proof. First consider the case β = α − 1. Let P and Q be bases for K and M, respectively.
+By Proposition 10.16α−1, there are close subpaths P′ of P and Q′ of Q such that µ(P′) ≥ λ.
+Then by Corollary 8.2 P and Q have the same infinite periodic extension and we conclude
+that K and M are compatible.
+35
+
+If β ≥ α then we consider lifts ˜K and ˜M of K and M in Γα−1 such that ˜K is contained in
+˜M and apply the already proved part.
+□
+8.11. Proposition (dividing a fragment). Let K be a fragment of rank α in Γβ, β ≥ α − 1.
+If K = K1K2 then either K1 or K2 contains a fragment F of rank α with F ∼ K and µf(F) >
+µf(K) − ζω, or K can be represented as K = F1uF2 where Fi are fragments of rank α, F1 is a
+start of K1, F2 is an end of K2, F1 ∼ F2 ∼ K and
+µf(F1) + µf(F2) > µf(K) − ζω.
+Proof. If α = 1 then u can be taken empty and the statement is trivial. If β = α − 1 ≥ 1
+then the statement follows from Proposition 9.21α−1. The case β > α − 1 follows from the
+case β = α − 1.
+□
+As an immediate consequence of Propositions 8.10 and 8.11 we get:
+8.12. Proposition (overlapping fragments). Let X be a reduced path in Γβ, β ≥ α−1. Let K
+and M be non-compatible fragments of rank α in X. Assume that K ≤ M and µf(K), µf(M) ≥
+λ+2.7ω. Then there are a start K1 of K disjoint from M and an end M1 of M disjoint from K
+such that K1 and M1 are fragments of rank α, K1 ∼ K, M1 ∼ M, µf(K) − µf(K1) < λ + 2.7ω
+and µf(M) − µf(M1) < λ + 2.7ω.
+8.13. Proposition (union of fragments). Let X be a reduced path in Γα−1 and let Ki (i = 1, 2)
+be compatible fragments of rank α in X. Assume that µf(Ki) ≥ 5.7ω for i = 1 or i = 2. Then
+the union of K1 and K2 is a fragment of rank α with the same base axis. Moreover, if K1
+and K2 are disjoint then µf(K1 ∪ K2) ≥ µf(K1) + µf(K2) − 5.7ω.
+Proof. By Lemma 8.1, K1 and K2 have a common base axis. If some of the Ki’s is contained
+in the other then there is nothing to prove. Otherwise the statement easily follows from
+Proposition 10.21α−1.
+□
+8.14. Corollary (compatibility preserves order). Let X be a reduced path in Γα−1, let Ki, Mi
+(i = 1, 2) be fragments of rank α in X and let µf(Ki), µf(Mi) ≥ λ + 2.6ω. Assume that
+K1 ∼ K2, M1 ∼ M2 and K1 ̸∼ M1. Then K1 < M1 if and only if K2 < M2.
+Proof. By Proposition 8.10, for each i = 1, 2 neither of Ki or Mi can be contained in the
+other, so we have either Ki < Mi or Mi < Ki. It is enough to prove the statement in the case
+K1 = K2. Assume, for example, that M1 < K1 < M2. Then by Proposition 8.13 M1 ∪ M2 is a
+fragment of rank α with M1 ∪M2 ̸∼ K1 and we get a contradiction with Proposition 8.10.
+□
+8.15. Proposition (no inverse compatibility). Let K and M be fragments of rank α in a
+reduced path X in Γα−1. Let µf(K), µf(M) ≥ 5.7ω. Then K ̸∼ M−1.
+Proof. Follows from Lemma 8.1 and Proposition 10.21α−1.
+□
+8.16. Proposition. Let K be a fragment of rank β in Γα where 1 ≤ β ≤ α.
+(i) Let R be the base loop for K labeled by a relator R of rank β and let R0 be the root
+of R. Then the subgroup {g ∈ Gα | gK ∼ K} is finite cyclic and conjugate to ⟨R0⟩.
+(ii) Let X be a word representing an element of Gα which is not conjugate to a power
+of R0. Let ¯X be an X-periodic line in Γα labeled X∞. Then sX, ¯
+XK ̸∼ K.
+(iii) Under hypothesis of (ii), if K is a subpath of ¯X and µf(K) ≥ 2λ+5.3ω then |K| < 2|X|.
+36
+
+Proof. (i) It follows from Lemma 8.1β that gK ∼ K if and only if gR = R. Since label(R) ≖ Rt
+0
+and R0 is a non-power, the stabilizer of K in Gα is a subgroup conjugate to ⟨R0⟩.
+(ii) follows immediately from (i).
+(iii) If K is a subpath of ¯X, µf(K) ≥ 2λ+5.3ω and |K| ≥ 2|X| then using Propositions 8.11β
+and 8.10β we conclude that either s−1
+X,¯XK ∼ K or sX,¯XK ∼ K, a contradiction with (ii).
+□
+9. Consequences of diagram analysis
+Following the terminology introduced in 4.16, a coarse r-gon in Γα is a loop of the form
+P = X1u1X2u2, . . . , Xrur
+where paths Xi are reduced and ui are bridges of rank α.
+Let us assume that each bridge ui of P is given an associate bridge partition of rank α (see
+4.13) and consider a filling φ : ∆(1) → Γα of P by a disk diagram ∆ over the presentation
+of Gα, i.e. ∆ has boundary loop ˜X1˜u1˜X2˜u2, . . . , ˜Xr˜ur where φ(˜Xi) ≖ Xi and φ(˜ui) ≖ ui. We
+can assume that ∆ has a boundary marking of rank α with sides ˜Xi and bridges ˜ui (see 5.1)
+and that each ˜ui has an induced bridge partition of rank α. Applying to ∆ the reduction
+process described in 5.4 we get a reduced diagram. Note that during the process, bridges ˜ui
+of ∆ can be changed by switching. To keep the equality φ(˜ui) ≖ ui we have to perform
+appropriated switching of bridges ui (see 4.13). As a consequence we obtain:
+9.1. Proposition (filling coarse polygons by diagrams). Let α ≥ 1 and P = X1u1X2u2, . . . , Xrur
+be a coarse r-gon in Γα with fixed bridge partitions of all bridges ui. Then, after possible
+switching of bridges ui, there exists a reduced disk diagram ∆ of rank α which fills P.
+9.2. Definition. The α-area of P, denoted Areaα(P), is the number of cells of rank α of a
+filling diagram ∆ as in Proposition 9.1. To avoid correctness issues, we assume formally that
+Areaα(P) is defined with respect to a particular choice of ∆.
+The image φ(δD) in Γα of the boundary loop of a cell of rank α of ∆ is an active relator
+loop for P for a particular choice ∆. Thus Areaα(P) is the number of active relator loops
+for P. Abusing the language, we call the inverse loop φ(δD)−1 an active relator loop for P
+as well.
+9.3. Remark. Equality Areaα(X1u1X2u2, . . . , Xrur) = 0 is equivalent to the assertion that
+X1u1X2u2, . . . , Xrur lifts to Γα−1 after possible switching of bridges ui.
+9.4.
+As a special case of a coarse polygon, consider a coarse bigon X−1uYv in Γα, α ≥ 1. Up
+to switching of bridges u and v we can assume that there is a reduced diagram ∆ of rank α
+which fills X−1uYv via a map φ : ∆(1) → Γα. We can assume also that ∆ is given a tight set T
+of contiguity subdiagrams. The boundary loop of ∆ has the form ˜X−1˜u˜Y˜v with sides ˜X−1
+and ˜Y which are mapped onto X−1 and Y respectively. By Proposition 7.11(i) each cell of
+rank α of ∆ has a contiguity subdiagram to each of the sides ˜X−1 and ˜Y. The boundary
+loops of cells of rank α and the bridges of these contiguity subdiagrams form a graph mapped
+in Γα as in Figure 15. Let Ri be images in Γα of boundary loops of cells of rank α of ∆
+and let Ki, Mi, Qi and Si be subpaths of X, Y and Ri, respectively, that are images of the
+corresponding contiguity arcs of contiguity subdiagrams of cells of rank α to ˜X−1 and ˜Y, as
+shown in the figure. According to the definition in 8.4, Ki and Mi are fragments of rank α
+37
+
+u
+Y
+M1
+M2
+M3
+v
+R1
+R2
+R3
+X
+K1
+K2
+K3
+S1
+S2
+S3
+Q1
+Q2
+Q3
+Figure 15.
+with bases Q−1
+i
+and Si and base relator loops R−1
+i
+and Ri respectively. We call Ki and Mi
+active fragments of rank α of the coarse bigon X−1uYv.
+Thus, if Areaα(X−1uYv) = t then there are precisely t disjoint active fragments of rank α
+in each of the paths X and Y. Note again that the set of active relator loops and the set of
+active fragments formally depend on the choice of particular ∆ and T.
+9.5.
+Let, as above, P = X−1uYv be a coarse bigon in Γα and ∆ a reduced diagram of
+rank α with δ∆ = ˜X−1˜u˜Y˜v filling P via a map φ : ∆(1) → Γα (we assume that the switching
+operation is already applied to u and v if needed). We assume that ∆ has a tight set T of
+contiguity subdiagrams. Let R = φ(δD) be an active relator loop of P and let Q−1w1K−1w2
+and S−1w3Mw4 be images of boundary loop of contiguity subdiagrams in T of the cell D to
+sides ˜X−1 and ˜Y respectively as in Figure 16. Then two loops P1 and P2 as shown in the figure
+can be considered as coarse bigons in Γα with sides that are subpaths of X and Y. They are
+X
+Y
+u
+v
+K
+Q
+S
+w1
+w2
+w3
+M
+w4
+P1
+R
+P2
+Figure 16.
+filled by reduced subdiagrams of ∆, so we have Areaα(P1) + Areaα(P2) = Areaα(P) − 1. We
+will use this simple observation in inductive arguments.
+38
+
+9.6.
+In a similar way, let P = X1u1X2u2X3u3 be a coarse trigon in Γα. After possible switching
+of bridges ui, we can find a reduced diagram ∆ of rank α with boundary loop ˜X1˜u1˜X2˜u2˜X3˜u3
+which fills P via a map φ : ∆(1) → Γα of P where φ(˜Xi) = Xi and φ(˜ui) = ui. We can also
+assume that ∆ has a tight set T of contiguity subdiagrams. By Proposition 7.11(ii) each cell
+of rank α of ∆ has contiguity subdiagrams in T to at least two sides ˜Xi. This implies that
+for any active relator loop R of P there are two or three fragments Ki (i = 1, 2 or i = 1, 2, 3)
+of rank α with base loop R that occur in distinct paths Xj. Similarly to the bigon case, we
+call them active fragments of rank α of P.
+As in the bigon case, for any active relator loop R of P we can consider a coarse bigon P1
+and a coarse trigon P2 respectively, as shown in Figure 17, with Areaα(P1) + Areaα(P2) =
+Areaα(P) − 1.
+X1
+K1
+K2
+X2
+X3
+R
+P1
+P2
+Figure 17.
+9.7. Proposition (active fragments in bigon). Let P = X−1uYv be a coarse bigon in Γα,
+α ≥ 1.
+(i) Let K and M be active fragments of rank α of P in X and Y, respectively, with
+mutually inverse base active relator loops. Then K ∼ M−1,
+µf(K) + µf(M) > 1 − 2λ − 1.5ω
+and
+µf(K), µf(M) > 7λ − 1.5ω.
+(ii) Let K and K′ be two distinct active fragments of rank α in X. Then K ̸∼ K′.
+Proof. (i): It follows directly from the construction that K ∼ M−1. The first inequality follows
+from Proposition 7.13(i). Since X and Y are reduced we have µf(K) ≤ ρ and µf(M) ≤ ρ which
+implies the lower bound on µf(K) and µf(M).
+(ii): Assume that K ∼ K′. Let M and M′ be the corresponding active fragments of rank α
+in Y. By (i), we have M ∼ M′. Then by Proposition 8.13 and the first inequality of (i),
+µf(K ∪ K′) + µf(M ∪ M′) ≥ 2 − 4λ − 17.4ω > 2ρ
+which contradicts the hypothesis that X and Y are reduced.
+□
+We introduce the notation for the lower bound on the size of active fragments in (i):
+ξ0 = 7λ − 1.5ω.
+39
+
+9.8. Definition. We say that paths X and Y in Γα are close in rank β ≤ α if there exist
+bridges u and v of rank β such that X−1uYv is a loop that can be lifted to Γβ. (So ‘being
+close’ for paths in Γα means the same as ‘being close in rank α’.)
+9.9. Remark. If X and Y are labeled with freely reduced words then X and Y are close in
+rank 0 if and only if X = Y.
+9.10. Proposition (lifting bigon). Let 0 ≤ β < α and X−1uYv be a coarse bigon in Γα where
+u and v are bridges of rank β. Assume that for all γ in the interval β + 1 ≤ γ ≤ α either X
+or Y has no fragments K of rank γ with µf(K) ≥ ξ0. Then X−1uYv can be lifted to Γβ and,
+consequently, X and Y are close in rank β.
+Proof. This is a consequence of Proposition 9.7 and Remark 9.3.
+□
+9.11. Proposition (no active relators). Let α ≥ 1, X−1uYv be a coarse bigon in Γα and
+Areaα(X−1uYv) = 0. Assume that |X|α > 2 + 6ζ2η. Then X and Y can be represented as
+X = w1X1w2 and Y = z1Y1z2 where X1 and Y1 are close in rank α−1 and |wi|α, |zi|α ≤ 1+4ζ2η
+(i = 1, 2).
+Proof. By Remark 9.3 we can assume that X−1uYv lifts to Γα−1. To simplify notations, we
+assume that X−1uYv is already in Γα−1. Let u = u1Pu2 and v = v1Qv2 where ui, vi are
+bridges of rank α − 1 and P, Q are paths labeled by pieces of rank α. We apply Proposition
+9.19(ii)α−1 to the coarse tetragon X−1u1Pu2Yv1Qv2. Observe that if a subpath of P or Q is
+close (in Γα−1) to a subpath S of X then |S|α ≤ 1. Since |X|α > 2 + 6ζ2η we cannot get the
+first case of the conclusion of Proposition 9.19(ii)α−1. Therefore, the second case holds: we
+have X = X1z1X2z2X3 where X1 is close to a start of P, X2 is close to a subpath of Y, X3 is
+close to an end of Q and |zi|α−1 ≤ 4ζη (i = 1, 2). Then |X1z1|α ≤ 1+4ζ2η, |z2X3|α ≤ 1+4ζ2η
+and we get the required bound.
+□
+9.12. Corollary (no active fragments). Let X and Y be close reduced paths in Γα, α ≥ 1.
+Assume that either X or Y has no fragments K of rank α with µf(K) ≥ ξ0. Assume also that
+|X|α > 2 + 6ζ2η. Then X and Y can be represented as X = w1X1w2 and Y = z1Y1z2 where X1
+and Y1 are close in rank α − 1 and |wi|α, |zi|α ≤ 1 + 4ζ2η (i = 1, 2).
+9.13. Corollary (no active fragments, iterated). Let X and Y be close reduced paths in Γα.
+Let 0 ≤ β < α and assume that for all γ in the interval β + 1 ≤ γ ≤ α either X or Y has no
+fragments K of rank γ with µf(K) ≥ ξ0. Let |X|α ≥ 2 + 3ζ. Then X and Y can be represented
+as X = w1X1w2 and Y = z1Y1z2 where X1 and Y1 are close in rank β and |wi|α < 1 + 5ζ2η
+(i = 1, 2).
+9.14. Proposition. Let X be a nonempty freely reduced word equal 1 in Gα. Then X has a
+subword P which is a piece of rank β where 1 ≤ β ≤ α and µ(P) > 136ω.
+Proof. By Proposition 7.6, X is not reduced in Gα and therefore contains a fragment K of
+rank β where 1 ≤ β ≤ α and µf(K) ≥ ρ. Let β ≥ 1 be the minimal rank such that X contains
+a fragment K of rank β with µf(K) ≥ ξ0. If β = 1 then K is already a piece of rank 1 with
+µ(K) ≥ ξ0 > 138ω by (4-1). Let β > 1. Let K be a fragment in Γβ−1 with label(K) ≖ K and
+S a base for K. By Corollary 9.13β−1 we have S = w1Pw2 where |wi|β−1 < 1.03 (i = 1, 2) and
+P = label(P) occurs in K. By (4-1), µ(P) ≥ ξ0 − 2.06ω = 7λ − 3.56ω > 136ω.
+□
+40
+
+9.15. Proposition (active fragments in trigon). Let P = X1u1X2u2X3u3 be a coarse trigon
+in Γα, let R be an active relator loop for P and let Ki (i = 1, 2 or i = 1, 2, 3) be active
+fragments of rank α with base loop R. Then Ki ∼ Kj for all i, j,
+�
+i
+µf(Ki) > 1 − 3λ − 2.2ω
+and
+µf(Ki) > 3λ − 1.1ω
+for at least two indices i.
+Proof. We have Ki ∼ Kj by construction.
+The first inequality follows from Proposition
+7.13(ii). Since Xi is reduced in Gα we have µ(Ki) ≤ ρ = 1 − 9λ. This implies the second
+inequality.
+□
+9.16. Proposition (no active fragments in conjugacy relations). Let X and Y be words
+cyclically reduced in Gα, α ≥ 1. Let X = Z−1Y Z in Gα for some Z. Assume that no cyclic
+shift of X contains a fragment K of rank α with µf(K) ≥ ξ0. Then there exists a word Z1
+such that Z1 = Z in Gα and X = Z−1
+1 Y Z1 in Gα−1.
+Proof. Let ∆0 be a disk diagram of rank α with boundary label X−1Z−1Y Z. We produce
+an annular diagram ∆1 by gluing two boundary segments of ∆0 labeled Z−1 and Z. The
+diagram ∆1 can be assigned a boundary marking of rank α with two cyclic sides X−1 and Y.
+We denote Z the path in ∆ with label(Z) ≖ Z that joins starting vertices of Y and X. Let
+∆2 be a reduced diagram of rank α obtained from ∆1 by reduction process. According to
+the remark in 5.7, ∆1 and ∆2 have the same frame type. It follows from Lemma 4.8 that
+there exists a path Z1 in ∆2 joining starting vertices of boundary loops Y1 and X−1
+1
+such
+that label(X1) ≖ X, label(Y1) ≖ Y and Z1 = Z in Gα where Z1 ≖ label(Z1). By Proposition
+7.13(i), ∆2 has no cells of rank α. Then X = Z−1
+1 Y Z1 in Gα−1.
+□
+9.17. Proposition (no active fragments in conjugacy relations, iterated). Let X and Y
+be cyclically reduced in Gα words which represent conjugate elements of Gα, α ≥ 1. Let
+β ≤ α. Assume that at least one of the words X or Y has the property that no its cyclic shift
+contains a fragment K of rank γ with µf(K) ≥ ξ0 and β < γ ≤ α. Let ¯X = . . . X−1X0X1 . . .
+and ¯Y = . . . Y−1Y0Y1 . . . be parallel periodic lines in Γα with label(Xi) ≖ X and label(Yi) ≖ Y
+representing the conjugacy relation. Then some vertices on ¯X and ¯Y are joined by a bridge
+of rank β.
+Moreover, for any subpath Z of ¯X there exists a loop S−1uTv which can lifted to Γβ such
+that S and T are subpaths of ¯X and ¯Y respectively, u and v are bridges of rank β and Z is
+contained in S.
+Proof. Since ¯X and ¯Y are parallel, if vertices a on ¯X and b on ¯Y are joined by a path labeled Z
+then the same is true for all their translates sk
+X,¯Xa and sk
+Y,¯Yb. Then the second statement
+follows from the first.
+Let ∆ be an annular diagram of rank α with boundary loops ˆX−1 and ˆY and φ : ˜∆(1) → Γα
+a combinatorially continuous map of the 1-skeleton of the universal cover ˜∆ of ∆ to Γα
+sending lifts ˜X of ˆX and ˜Y of ˆY to ¯X and ¯Y respectively. We can assume that ∆ is reduced
+and has a tight set of contiguity subdiagrams. If β = α and ∆ has a cell of rank α then the
+statement follows from Proposition 7.11(iii). If ∆ has no cells of rank α then we can lift ¯X
+and ¯Y to Γα−1 and use induction on α. If β < α and at least one of the words X or Y has no
+41
+
+cyclic shift containing a fragment K of rank α with µf(K) > ξ0 then by Proposition 7.13(i),
+∆ has no cells of rank α and, again, the statement follows by induction.
+□
+9.18. Proposition (small coarse polygons). Let P = X1∗X2∗ . . . Xr∗ be a coarse r-gon in Γα
+where r ≥ 3 and Xi are sides of P. Assume that there are no pairs of close vertices lying on
+distinct paths Xi and Xj except pairs {τ(Xi), ι(Xi+1)} and {τ(Xr), ι(X1)}. Then
+�
+i
+|Xi|α ≤ (r − 2)η.
+If r = 3 or r = 4 then we have a stronger bound
+�
+i
+|Xi|α ≤ 2(r − 1)ζη.
+Proof. Consider a filling φ : ∆(1) → Γα of P by a reduced disk diagram ∆ of rank α. Let
+δ∆ = ¯X1u1¯X2u2 . . . ¯Xrur where ui are bridges and Xi are sides of ∆ with φ(¯Xi) = Xi. The
+hypothesis of the proposition implies that ∆ is small.
+Then the statement follows from
+Propositions 7.9 and 7.12.
+□
+9.19. Proposition (trigons and tetragons are thin).
+(i) Let X−1∗Y1∗Y2∗ be a coarse trigon in Γα. Then X can be represented as X = X1zX2
+where X1 is close to a start of Y1, X2 is close to an end of Y2 and |z|α ≤ 4ζη.
+(ii) Let X−1∗Y1∗Y2∗Y3∗ be a coarse tetragon in Γα. Then at least one of the following
+possibilities holds:
+• X can be represented as X = X1zX2 where X1 is close to a start of Y1, X2 is close
+to an end of Y3 and |z|α ≤ 6ζη; or
+• X can be represented as X = X1z1X2z2X3 where X1 is close to a start of Y1, X2 is
+close to a subpath of Y2, X3 is close to an end of Y3 and |zi|α ≤ 4ζη (i = 1, 2).
+Proof. (i) We can represent X1 = X1zX2, Yi = Yi1wiYi2 (i = 1, 2) with close pairs (X1, Y11),
+(Y12, Y−1
+21 ) and (Y22, X2) where no vertices lying on distinct paths z, w1 and w2 are close
+except appropriate endpoints (Figure 18a). Then the statement follows by application of
+Proposition 9.18 to z−1∗w1∗w2∗.
+X1
+z
+X2
+Y11
+w1
+Y12
+Y21
+w2
+Y22
+X1
+X2
+Y1
+Y21
+Y22
+Y3
+a
+b
+Figure 18.
+(ii) If there is a pair of close vertices on Y1 and Y3 then the statement follows from (i) giving
+the first alternative. If there is a pair of close vertices on X and on Y2 then we represent
+X and Y2 as X = X1X2, Y2 = Y21Y22 where τ(X1) and τ(Y21) are close, and apply (i) to
+42
+
+X−1
+1 ∗Y1∗Y21∗ and X−1
+2 ∗Y22∗Y3∗ (Figure 18b). We then come to the second alternative to
+the statement. Otherwise we use an argument similar to the proof of (i) coming to the first
+alternative.
+□
+9.20. Proposition (small cyclic monogon). Let X be a word cyclically reduced in Gα and let
+X be conjugate in Gα to a word Y u where Y is reduced in Gα and u is a bridge of rank α.
+Let ¯X = �
+i∈Z Xi and �
+i∈Z Yiui be lines in Γα representing the conjugacy relation. Assume
+that no vertex on X0 is close to a vertex on Yi. Then |X|α ≤ η.
+Proof. Let ∆ be an annular diagram of rank α with boundary loops ˆX and ˆY−1ˆu−1 represent-
+ing the conjugacy relation. We consider ∆ as having a cyclic side ˆX, a non-cyclic side ˆY−1
+and a bridge ˆu−1. Up to switching of ˆu−1 we can assume that ∆ is reduced. The hypothesis
+implies that ∆ cannot have a bond between ˆX and ˆY−1 after any refinement. Assume that
+∆ has a bond v (possibly after refinement) joining two vertices on the same side ˆY−1. Then
+v cuts off from ∆ a simply connected subdiagram Σ with boundary loop Z1ˆu−1Z2v±1 where
+ˆY−1 = Z2WZ1 for some W. According to Definition 6.1, at least one of the words label(Zi)
+(i = 1, 2) is nonempty. Removing Σ from ∆ we obtain a diagram ∆′ with a shorter total label
+of its two sides. Hence, by induction, we can assume that ∆′ is small. Then |X|α = |ˆX|α ≤ η
+by Proposition 7.9.
+□
+9.21. Proposition (closeness fellow traveling). Let X and Y be close reduced paths in Γα,
+α ≥ 1. Then X and Y can be represented as X = U1U2 . . . Uk and Y = V1V2 . . . Vk (Ui and Vi
+can be empty) where the starting vertex of each Ui is close to the starting vertex of Vi and
+|Ui|α, |Vi|α ≤ ζ for all i.
+Proof. Observe that the statement of the lemma holds in the case α = 0 with |Ui|0, |Vi|0 = 1.
+Thus we may refer to the statement of the lemma in rank α−1 with bounds |Ui|α−1, |Vi|α−1 ≤
+1 which imply |Ui|α, |Vi|α ≤ ζ. Observe also that if X = X1X2 . . . Xr and Y = Y1Y2 . . . Yr
+where for each i, Xi and Yi are close then the statement of the lemma for each pair (Xi, Yi)
+implies the statement of the lemma for X and Y. By 9.5 we represent X and Y as X =
+X1X2 . . . Xr and Y = Y1Y2 . . . Yr where pairs (Xi, Yi) satisfy the following conditions (1)
+or (2) in the alternate way: (1) for some bridges ui and vi of rank α the loop X−1
+i uiYivi lifts
+to Γα−1 or (2) there are loops X−1
+i wi1Riwi2 and Yiwi3Siwi4 which can be lifted to Γα−1 such
+that Si and Ri occur in one relation loop of rank α and wij are bridges of rank α − 1 (see
+Figure 19). We can assume that pairs (X1, Y1) and (Xr, Yr) satisfy (2) and that in the case
+Xi
+wi1
+wi2
+wi3
+wi4
+Ri
+Si
+Yi
+Figure 19.
+43
+
+of (2), subpaths Xi, Yi of X, Y and Si, Ri of the appropriate relation loop cannot be extended.
+We prove the statement for each of the pair (Xi, Yi).
+Case of (1): Omitting the index i for Xi and Yi, assume that a loop X−1w1Pw2Yw3Qw4
+lifts to Γα−1 where wi are bridges of rank α − 1 and P and Q are labeled by pieces of
+rank α. Without changing notations, we assume that X−1w1Pw2Yw3Qw4 is already in Γα−1.
+By the maximal choice of Xi, Yi, Si and Ri in the case of (2), there are no close vertices
+on pairs (X, P), (X, Q), (Y, P) and (Y, Q) except appropriate endpoints (i.e. except ι(X) and
+ι(P) for (X, P) etc.). Depending on existence of close vertices on pairs (P, Q) and (X, Y) we
+consider three cases (a)–(c) as in Figure 20. In case (a) we have |X|α, |Y|α ≤ 6ζ2η < ζ by
+X
+Y
+w1
+P
+w2
+w3
+Q
+w4
+X1
+X2
+X3
+Y1
+Y2
+Y3
+(a)
+(b)
+(c)
+Figure 20.
+Proposition 9.18α−1. In case (b) taking the maximal pair of close subpaths of P and Q we
+get |X|α, |Y|α ≤ 4ζ2η < ζ again by Proposition 9.18α−1. In case (c) we have X = X1X2X3
+and Y = Y1Y2Y3 where X2 and Y2 are close. Taking X2 and Y2 maximal possible we get
+|Xi|α, |Yi|α ≤ 4ζ2η for i = 1, 3 by Proposition 9.18α−1. For X2 and Y2 we can apply the
+statement for α := α − 1.
+Case of (2): In the second case by the statement of the lemma for α := α − 1 we have
+X = U1U2 . . . Uk and Y = W1W2 . . . Wl where |Ui|α, |Wi|α ≤ ζ, the starting vertex of each Ui
+can be joined by a bridge of rank α − 1 with a vertex on R and the starting vertex of each
+Wi can be joined by a bridge of rank α − 1 with a vertex on S. Then each ι(Ui) is close to
+ι(Y) and each ι(Wi) is close to τ(X). We take X = U1U2 . . . Uk+l and Y = V1V2 . . . Vk+l where
+Uk+1, . . . , Uk+l, V1, . . . , Vk are empty and Vj = Wj−k for k + 1 ≤ j ≤ k + l.
+□
+9.22. Lemma. Let X be a reduced path and R a relation loop of rank α in Γα, α ≥ 1. Let ui
+(i = 1, 2) be a path labeled by a word in Hα−1 and joining vertices ai on X and bi on R. Let
+Y be the subpath of X±1 that starts at a1 and ends at a2, and let R = R1R2 where Ri starts
+at bi (Figure 21). Then one of the two loops Yu2R−1
+1 u−1
+1
+or Yu2R2u−1
+1
+lifts to Γα−1.
+Proof. We fill the loop Yu2R−1
+1 u−1
+1
+by a disk diagram ∆ of rank α with boundary loop ¯Y¯u2S¯u−1
+1
+where label(S) ≖ label(R−1
+1 ). We take ¯Y as a side and ¯u2S¯u−1
+1
+as a bridge of ∆ with bridge
+partition ¯u2·S·¯u−1
+1 . Then we apply the reduction process making ∆ reduced. After reduction,
+we get either label(S) ≖ label(R−1
+1 ) or label(S) ≖ label(R2). By Lemma 7.5, ∆ has no cells
+of rank α.
+Depending on the case, this implies that either Yu2R−1
+1 u−1
+1
+or Yu2R2u−1
+1
+lifts
+to Γα−1.
+□
+44
+
+X
+a1
+Y
+a2
+u1
+u2
+R1
+R2
+b1
+b2
+Figure 21.
+9.23. Proposition (compatibility lifting). Let 1 ≤ β ≤ α. Let K and M be fragments of
+rank β which occur in a reduced path X in Γα. Let ˆX be a lift of X in Γβ−1 and ˆK and ˆM
+be the subpaths of ˆX which are projected onto K and M respectively. Then K ∼ M implies
+ˆK ∼ ˆM and K ∼ M−1 implies ˆK ∼ ˆM−1.
+Proof. Assume that K ∼ Mε where ε = ±1. Let R be the common base loop for K and Mε.
+Lemma 9.22 implies that R can be lifted to a line ˆR which is the common base axis for both
+ˆK and ˆMε. This implies ˆK ∼ ˆMε.
+□
+9.24. Corollary. Let 1 ≤ β ≤ α. Then statements of Proposition 8.13, Corollary 8.14 and
+Proposition 8.15 hold for fragments of rank β in a reduced path X in Gα.
+More precisely, let X be a reduced path in Γα. Then the following is true.
+(i) Let Ki (i = 1, 2) be fragments of rank β in X, K1 ∼ K2 and µf(Ki) ≥ 5.7ω for i = 1
+or i = 2. Then K1 ∪ K2 is a fragment of rank β with K1 ∪ K2 ∼ K1. If K1 and K2 are
+disjoint then µf(K1 ∪ K2) ≥ µf(K1) + µf(K2) − 5.7ω.
+(ii) Let Ki, Mi (i = 1, 2) be fragments of rank β in X with µf(Ki), µf(Mi) ≥ γ + 2.6ω.
+Assume that K1 ∼ K2, M1 ∼ M2 and K1 ̸∼ M1. Then K1 < M1 if and only if
+K2 < M2.
+(iii) If K and M are fragments of rank β in X and µf(K), µf(M) ≥ 5.7ω then K ̸∼ M−1.
+10. Stability
+Let FA be a free group with basis A and let X−1Y1Y2 . . . Yk+1 = 1 be a relation in FA where
+X, Y1, . . . , Yk are freely reduced words in the generators A. Then for any occurrence of a
+letter aε ∈ A±1 in X there is a unique occurrence of the same letter aε in some Yi which cancels
+with a−ε in X−1Y1Y2 . . . Yk+1. The main goal of this section is to establish an analog of this
+statement for relations in Gα. The role of letters aε will be played by fragments of rank α and
+instead of relation X−1Y1Y2 . . . Yk+1 = 1 we consider coarse polygons X−1∗Y1∗ . . . Yk∗ in Γα
+(for our considerations, it is enough to consider cases k = 1, 2, 3). The role of correspondence
+of canceled letters will be played by equivalence relation ‘K ∼ L±1’.
+There are two essential differences of the case of groups Gα from the case of a free group FA.
+One is a “fading effect”: a fragment in Yi can be of a “smaller size” than an initial fragment
+in X. Another difference is that bridges of the coarse polygon can produce exceptions for
+stability (to describe such situations we introduce a special relation between fragments and
+bridges of the same rank β, see Definition 10.4).
+We start with a statement which shows how closeness is propagated in coarse tetragons
+in Γα−1. This is essentially a consequence of inductive hypotheses.
+45
+
+10.1. Definition (uniformly close). For α ≥ 1, we say that vertices a1, a2, . . . , ar of Γα are
+uniformly close if at least one of the following is true:
+• they are pairwise close in rank α − 1; or
+• there exists a relator loop R of rank α such that each ai is close in rank α − 1 to a
+vertex on R.
+We cover also the case α = 0: vertices a1, a2, . . . , ar of Γ0 are said to be uniformly close if
+a1 = a2 = · · · = ar.
+Note that uniformly close vertices are pairwise close. If r = 2 then being uniformly close
+and being close is equivalent.
+10.2. Lemma. Let α ≥ 1, X and Y be close reduced paths in Γα−1, and let S−1∗T1∗T2∗T3∗
+be a coarse tetragon in Γα−1 such that Y is a subpath of S. Assume that |X|α−1 ≥ 5.2. Then
+X can be represented as z0X1z1 . . . Xrzr (1 ≤ r ≤ 3) where Xi is close to a subpath Wi of
+some Tji, j1 < · · · < jr and
+(10-1)
+�
+i
+|Xi|α−1 > |X|α−1 − 5.8.
+Moreover:
+(i) if r = 3 then we have a stronger bound
+�
+i
+|Xi|α−1 > |X|α−1 − 3.4.
+(ii) There is a subpath Y1 of Y such that the starting vertices ι(X1), ι(Y1) and ι(W1) are
+uniformly close and the same is true for the ending vertices ι(Xr), ι(Y1) and ι(Wr).
+Proof. If α = 1 the statement is obvious (see Remark 10.3 below). Let α > 1. Let Z be
+a reduced path joining ι(S) and τ(T2) which exists by Proposition 11.1α−1 (see Figure 22).
+We apply Proposition 10.18α−1 first to the coarse trigon S−1∗Z∗T3∗ and then, possibly, to
+X
+Y
+S
+Z
+T1
+T2
+T3
+Figure 22.
+the coarse trigon Z−1∗T1∗T2. Since |X|α−1 ≥ 5.2, after the first application of Proposition
+10.18α−1, we find either a subpath X3 of X that is close to a subpath of T3 or a subpath X′
+of X that is close to a subpath of Z with |X′|α−1 > |X|α−1 − 2.75 > 2.45. In the latter
+case, the second application of 10.18α−1 gives the remaining X1 and/or X2. If r < 3 then
+for the bound (10-1), the worst cases are when we get two Xi’s after double application of
+10.18α−1. In those cases we have once case (iii) of 10.18α−1 and another time case (i) or (ii).
+46
+
+Hence �
+i |Xi|α−1 > |X|α−1 − 3 − 2.75. Statement (ii) follows from the appropriate part of
+Proposition 10.18α−1.
+Assume that r = 3 and therefore X = z0X1z1X2z2X3z3 where each Xi is close to a subpath
+of Ti. From application of Proposition 10.18α−1 we have |z0|α−1, |z3|α−1 < 1.3. Then using
+Proposition 9.19(i)α−1 we extend all Xi to get |z1|α−1, |z2|α−1 ≤ 4ζη < 0.4. This proves (i).
+□
+10.3. Remark. If α = 1 then hypotheses of Lemma 10.2 say that X = Y and S−1T1T2T3 is a
+loop in the Cayley graph Γ0 of the free group G0. Then the statement of the lemma holds
+without the assumption |X|α−1 ≥ 5.2. Furthermore, in the conclusion we have �
+i |Xi|α−1 =
+|X|α−1.
+10.4. Definition (independence). Let 1 ≤ β ≤ α, K be a fragment of rank β in Γα and u
+be a bridge of rank β in Γα. Recall that K is considered with the associated base loop R of
+rank β. We say that K is independent of u if either label(u) ∈ Hβ−1 or u possesses a bridge
+partition u = v·S·w of rank β where S occurs in a relator loop L of rank β such that L ̸= R±1.
+It follows from the definition that if K is independent of u and M ∼ K±1 then M is also
+independent of u.
+10.5. Proposition (non-active fragment in bigon). Let α ≥ 1, X−1uYv be a coarse bigon
+in Γα and let X = F0K1F1 . . . KrFr where Ki are the associated active fragments of rank α.
+Let K be a fragment of rank α in X with µf(K) ≥ 2λ + 5.8ω. Assume that K ̸∼ Ki for all i
+and that K is independent of u and v. Then there exists a fragment of rank α in Y such that
+M ∼ K and
+µf(M) ≥ µf(K) − 2λ − 3.4ω.
+Proof. By Proposition 8.10 K is a subpath of one of the paths F0K1, K1F1K2, . . . , KrFr. We
+consider the case when K is a subpath of some KiFiKi+1 (the cases when K is a subpath of F0K1
+or KrFr are similar; see also the remark in the end of the proof). Let Y = H0M0H1 . . . MrHr
+where Mi are the corresponding active fragments of rank α in Y.
+As we can see from 9.4, there is a loop T = (KiFiKi+1)−1w1S1w2Hiw3S2w4 which can be
+lifted to Γα−1 and where wj are bridges of rank α − 1 and S1 and S2 occur in base loops
+for Ki and Ki+1 respectively (see Figure 23). Abusing notation we assume that T is already
+S
+L
+Ki
+Fi
+Ki+1
+X
+K
+w1
+S1
+w2
+Hi
+w3
+S2
+w4
+Y
+Figure 23.
+47
+
+in Γα−1. Then, instead of base loops, S1 and S2 occur in base axes L1 and L2 for Ki and Ki+1
+respectively.
+Let L be the base axis for K and S the base for K (which is contained in L by definition).
+Assumptions K ̸∼ Ki and K ̸∼ Ki+1 imply L ̸= Li (i = 1, 2). By Corollary 8.2, if a subpath P
+of S is close to a subpath of Si then µ(P) < λ. Then by Lemma 10.2 we find a subpath Q
+of S which is close to a subpath M of Hi and satisfies
+µ(Q) > µ(S) − 2λ − 3.4ω.
+Then M is a fragment of rank α with base Q. Clearly, M satisfies the conclusion of the
+proposition.
+If K is a subpath of F0K1 or KrFr, a similar argument applies. For example, assume that
+K is a subpath of F0K1. As above, we assume that all paths are in Γα−1 not changing their
+notations. Let L be a base axis for K. By hypothesis, either label(u) ∈ Hα−1 or u = u1Vu2
+where V occurs in a line L1 labeled by the infinite power R∞ of a relator R of rank α and
+L1 is distinct from L. In the case label(u) ∈ Hα−1 we apply Proposition 10.18α−1. Otherwise
+the argument is the same as in the case when K is a subpath of KiFiKi+1. The case when K
+is a subpath of KrFr is similar.
+Finally, there is a “degenerate” case when Areaα(X−1uYv) = 0 and both u and v are bridges
+of rank α − 1. In this case, the statement follows directly from Proposition 8.7.
+□
+10.6. Proposition (fragment stability in bigon). Let α ≥ 1, X−1uYv be a coarse bigon in Γα
+and let K be a fragment of rank α in X with µf(K) ≥ 2λ+5.8ω. Assume that K is independent
+of u and v. Then there exists a fragment M of rank α in Y such that M ∼ K±1 and
+µf(M) ≥ min{µf(K) − 2λ − 3.4ω, ξ0}
+Proof. Let X = F0K1F1 . . . KrFr and Y = H0M0H1 . . . MrHr where Ki and Mi are the associ-
+ated active fragments of rank α. If K ∼ Ki for some i then we can take M = Mi due to
+Proposition 9.7. Otherwise we apply Proposition 10.5.
+□
+10.7. Proposition (fragment stability in trigon). Let α ≥ 1, X−1u1Y1u2Y2u3 be a coarse
+trigon in Γα and let K be a fragment of rank α in X with µf(K) ≥ 3λ + 10ω. Assume that
+K is independent of any of ui. Then there is a fragment M of rank α in Y1 or Y2 such that
+M ∼ K±1 and
+µf(M) > min
+�
+3λ − 1.1ω, 1
+2(µf(K) − 3λ − 6.8ω)
+�
+.
+Proof. The idea of the proof is the same as in the proof of Proposition 10.5.
+To avoid
+complicated notations, we proceed by induction on the α-area of P = X−1u1Y1u2Y2u3 as
+described in 9.6.
+Assume that R is an active relator loop of rank α of P.
+As observed
+in 9.6, there are two or three fragments Ni (i = 1, 2 or i = 1, 2, 3) of rank α with base
+loop R that occur in distinct paths X−1, Y1 or Y2. By Proposition 9.15 we can assume that
+µf(Ni) ≥ 3λ−1.1ω for i = 1, 2. If K ∼ N±1
+1
+then we for the required M we take that Ni which
+occurs in Y1 or Y2. Let K ̸∼ N±1
+1 .
+If N1 and N2 occur in Y1 and Y2 then we can replace P by a coarse trigon with smaller
+α-area and use induction (see Figure 24a). (In this case u2 is replaced by a new bridge u′
+2
+and the assumption K ̸∼ N±1
+1
+implies that K is independent of u′
+2.) Otherwise, assume that
+N1 occurs in X−1 and N2 occurs in Y1 (the case when N2 occurs in Y2 is symmetric).
+48
+
+K
+X
+Y1
+Y2
+N1
+N2
+N1
+N2
+K
+a
+b
+Figure 24.
+Since K ̸∼ N−1
+1
+we have either K < N−1
+1
+or K > N−1
+1 . In the first case, we reduce the
+statement to the case of a coarse bigon as in Figure 24b and apply Proposition 10.5. In the
+second case, the statement follows by inductive hypothesis.
+It remains to consider the case Areaα(P) = 0. Then the loop P can be lifted to Γα−1 and
+we assume that P is already in Γα−1. Let L be the base axis for K and S the base for K. Since
+K is independent of ui (when viewed in Γα), we have either label(ui) ∈ Hα−1 or ui = viQiwi
+where label(vi), label(wi) ∈ Hα−1 and Qi occurs in a line Li labeled by the infinite power R∞
+i
+of a relator Ri of rank α such that Li ̸= L. We obtain a coarse r-gon with sides X−1, Y1, Y2
+and Qi where 3 ≤ r ≤ 6 (see Figure 25). We consider the “worst” case r = 6 (the other cases
+are similar, with application of Propositions 10.18α−1 or 8.7α−1 where needed). Let Z be a
+K
+X
+v1
+Q1
+w1
+Y1
+v2
+Q2
+w2
+Y2
+v3
+Q3
+w3
+Z
+Figure 25.
+reduced path joining τ(u1) and ι(u3) existing by Proposition 11.1α−1. By Corollary 8.2, if a
+subpath P of S is close to a subpath of Qi then µ(P) < λ. Then the statement easily follows by
+applying Lemma 10.2 twice to coarse tetragons X−1v1Q1w1Zv3Q3w3 and Z−1Y1v2Q2w2Y2.
+□
+10.8. Lemma. Let α ≥ 1, X be a piece of rank 1 ≤ β < α or a fragment of rank β < α.
+Then X contains no fragment K of rank α with µf(K) ≥ 3.2ω.
+In particular, any fragment K of rank α with µf(K) ≥ 3.2ω is a nonempty word (since
+otherwise it would occur in a fragment of rank 0).
+Proof. We consider the case when X is a fragment of rank β < α. We represent X by
+a path X in Γα−1. Assume that X contains a fragment K of rank α with µf(K) ≥ 3.2ω.
+49
+
+Let S be a base for K with |S|α−1 ≥ 3.2.
+By Lemma 10.8≤α−1 and Corollary 9.13 we
+have S = w1S1w2 and K = z1K1z2 where S1 and K1 are close in rank max(0, β − 1) and
+|S1|α−1 > |S|α−1 −2−10ζ2η > 1.15. If β = 0 we already get a contradiction since in this case
+|K1| ≤ 1 but |S1| ≥ |S1|α−1 > 1. Let β ≥ 1. Up to change of notation, we assume that X,
+K1 and S1 are lifted to Γβ−1. Let T be a base for X. By Proposition 10.16β−1 a subpath T1
+of T is close to a subpath S2 of S with |S2|α−1 > |S1|α−1 − 2.6ζ > 1. Then S2 is a fragment
+of rank β with base T1 and we should have |S2|α−1 ≤ 1, a contradiction.
+In the case when X is a piece of rank α a similar argument works with skipping application
+of Proposition 10.16β−1.
+□
+10.9. Lemma. Let α ≥ 1 and X be a word cyclically reduced in Gα−1. Assume that a cyclic
+shift of X contains a fragment K of rank α with µf(K) ≥ 6.5ω. Then X is strongly cyclically
+reduced in Gα−1.
+Proof. Let F be a fragment of rank 1 ≤ β ≤ α − 1 in a word Xt. Assume that |F| > |X|.
+Using Proposition 8.11 represent K as K ≖ K1uK2 where µf(K1), µf(K2) > 3.2ω. Since
+|K| ≤ |X|, F should contain a translate of K1 or K2. But this is impossible by Lemma 10.8.
+Hence |F| ≤ |X| and then µf(F) ≤ ρ since X is cyclically reduced in Gα−1. This shows that
+any power Xt is reduced in Gα−1, i.e. X is strongly cyclically reduced in Gα−1.
+□
+10.10. Proposition (fragment stability in conjugacy relations with cyclic sides). Let α ≥ 1
+and X and Y be words which are cyclically reduced in Gα and represent conjugate elements
+of Gα. Let ¯X = �
+i∈Z Xi and ¯Y = �
+i∈Z Yi be parallel lines in Γα representing the conjugacy
+relation. Let K be a fragment of rank α in ¯X with µf(K) ≥ 2λ + 5.8ω and |K| ≤ |X|. Then
+there is a fragment M of rank α in ¯Y such that M ∼ K±1 and
+µf(M) ≥ min{µf(K) − 2λ − 3.4ω, ξ0}
+Proof. By Lemma 10.9 X is strongly cyclically reduced in Gα−1. We claim that a cyclic shift
+of Y also contains a fragment F of rank α with µf(F) ≥ 6.5 and thus Y is strongly cyclically
+reduced in Gα−1 as well. Indeed, by Proposition 9.17 with β := α − 1 we may assume for
+some cyclic shifts X′ and Y ′ of X and Y we have Y ′ = w−1X′w in Gα−1 where w ∈ Hα−1.
+Then existence of F easily follows by Propositions 8.11 and 8.7.
+Consider a reduced annular diagram ∆ of rank α with boundary loops ˆX and ˆY−1 repre-
+senting the conjugacy relation given in the proposition. Let ˜∆ be the universal cover of ∆
+and let φ : ˜∆(1) → Γα be a combinatorially continuous map which sends lifts of ˆX and ˆY to
+¯X and ¯Y respectively.
+Assume that ∆ has a cell of rank α. Let D be some lift of this cell in ˜∆. By Proposition
+7.13(i), φ(δD) and φ(δD)−1 are base loops for fragments Ni (i = 1, 2) of rank α in ¯X and ¯Y
+respectively, such that µf(N1)+µf(N2) ≥ 1−2λ−1.5ω. Since X and Y are cyclically reduced
+in Gα we have µf(Ni) ≤ ρ and hence µf(Ni) ≥ 1−ρ−2λ−1.5ω = ξ0. By construction, we have
+N1 ∼ N−1
+2 . Since ¯X and ¯Y are parallel, we have sk
+X,¯XN1 ∼ sk
+Y,¯YN−1
+2
+for any k ∈ Z. If K ∼ sk
+X,¯XN1
+for some k then we can take sk
+Y,¯YN2 for M. Otherwise we have sk
+X,¯X,N1 < K < sk+1
+X,¯XN1 for
+some k and the rest of the argument is the same as in the proof of Proposition 10.5.
+Now assume that ∆ has no cells of rank α. We can assume that ∆ is a reduced diagram
+of rank β for some β ≤ α − 1 and in case β ≥ 1, ∆ has at least one cell of rank β. If β = 0
+then ¯X = ¯Y and there is nothing to prove. Let β ≥ 1. Up to change of notations, we assume
+that K, ¯X and ¯Y are lifted to Γα−1. Proposition 7.13(i)β implies that some vertices a on ¯X
+50
+
+and b on ¯Y are joined by a bridge of rank β. This is true also for any translates si
+X,¯Xa and
+si
+Y,¯Yb. Then the statement follows by Proposition 8.7 (here we use that X and Y are strongly
+cyclically reduced in Gα−1).
+□
+10.11. Lemma. Let α ≥ 1 and S be a word cyclically reduced in Gα−1. Assume that S is
+conjugate in Gα−1 to a word T1v1T2v2 where Ti are reduced in Gα−1 and vi are bridges of
+rank α. Let ¯S = �
+i∈Z Si and �
+i∈Z T(i)
+1 v(i)
+1 T(i)
+2 v(i)
+2
+be parallel lines in Γα−1 representing the
+conjugacy relation. Denote U2i = T(i)
+1
+and U2i+1 = T(i)
+2 .
+Assume that a reduced path X in Γα−1 is close to a subpath Y of ¯S with |Y| ≤ |S|. Let
+|X|α−1 ≥ 8. Then X can be represented as z0X1z1 . . . Xrzr (1 ≤ r ≤ 4) where each Xi is close
+to a subpath of some Uji, j1 < · · · < jr, jr − j1 ≤ 3 and
+�
+i
+|Xi|α−1 ≥ |X|α−1 − 9.
+Proof. Let Z be a word reduced in Gα−1 such that Z = T1v1T2 in Gα−1. We join ι(T(i)
+1 ) and
+τ(T(i)
+2 ) with the path Zi labeled Z. Since |X|α−1 ≥ 8, application of Propositions 10.19α−1
+gives X = w1X′w2 or X = w1X′w2X′′w3 where, respectively, X′ is close to a subpath of some Zi
+and |X′|α−1 ≥ |X|α−1 − 2.9 or for some i, X′ is close to a subpath of Zi, X′′ is close to a
+subpath of Zi+1 and |X′|α−1 + |X′′|α−1 ≥ |X|α−1 − 3. Then a single or double application of
+Proposition 10.18α−1 gives the required Xi’s.
+□
+10.12. Proposition (fragment stability in conjugacy relations with non-cyclic side). Let
+α ≥ 1 and X be a word cyclically reduced in Gα. Assume that X is conjugate in Gα to
+a word Y u where Y is reduced in Gα and u is a bridge of rank α. Let ¯X = �
+i∈Z Xi and
+�
+i∈Z Yiui be parallel lines in Γα representing the conjugacy relation. Let K be a fragment of
+rank α in ¯X with µf(K) ≥ 3λ + 9ω and |K| ≤ |X|. Assume that K is independent of any of
+the bridges ui. Then there is a fragment M of rank α in some Yk such that M ∼ K±1 and
+µf(M) > min
+�5
+2λ − 1.1ω, 1
+2(µf(K) − 3λ − 6.8ω)
+�
+.
+Proof. Let ∆ be an annular diagram of rank α with boundary loops ˆX−1 and ˆYˆu representing
+the conjugacy relation. Let ˜∆ be the universal cover of ∆ and φ : ˜∆(1) → Γα a combinatorially
+continuous map sending lifts ˜Xi, ˜Yi and ˜ui of ˆX, ˆY and ˆu to Xi, Yi and ui respectively. Up to
+switching of ˆu, we assume that ∆ is reduced and has a tight set T of contiguity subdiagrams.
+Case 1: ∆ has no cells of rank α. Then parallel lines ¯X = �
+i∈Z Xi and �
+i∈Z Yiui can be
+lifted to Γα−1; we assume that they and the subpath K of ¯X are already lifted to Γα−1. If
+u ∈ Hα−1 then the statement follows by Proposition 10.19α−1, so we assume that u /∈ Hα−1.
+Let L be the base axis for K and S the base for K. Since K is independent of ui (when viewed
+in Γα) we have ui = w(i)
+1 Qiw(i)
+2
+where label(w(i)
+j ) ∈ Hα−1 and Qi occurs in a line Li labeled
+by the infinite power R∞
+i
+of a relator Ri of rank α such that Li ̸= L. By Corollary 8.2, if
+a subpath P of S is close to a subpath of Qi then µ(P) < λ. Applying Lemma 10.11 we
+conclude that either there exists a fragment M of rank α in some Yk such that M ∼ ¯K and
+µf(M) > µf(K) −2λ −9ω or there exist fragments M1 and M2 of rank α in some Yk and Yk+1
+respectively such that M1 ∼ M2 ∼ K and
+µf(M1) + µf(M2) > µf(K) − 2λ − 9ω.
+51
+
+In the latter case, for at least one Mi we have µf(Mi) > 1
+2(µf(K) − 2λ − 9ω) and we can take
+its image in Γα for the required M.
+Case 2: ∆ has at least one cell of rank α. Let D be such a cell and let ˜D be a lift of D
+in ˜∆. By Proposition 7.11(iv) and Lemma 7.10(i), D has two or three contiguity subdiagrams
+Πi ∈ T to sides of ∆, at most two to ˆY and at most one to ˆX−1. By Proposition 7.13(iii),
+φ(δ˜D) is the base loop for two or three fragments Ni (i = 1, 2 or i = 1, 2, 3) of rank α in two
+or three of the paths ¯X−1, Yj and Yj+1 for some j, respectively, with
+(10-2)
+�
+i
+µf(Ni) > 1 − 4λ − 2.2ω.
+Since µf(Ni) ≤ ρ for each i, for at least two indices i we have
+µf(Ni) > 1
+2(1 − 4λ − 2.2ω − ρ) = 5
+2λ − 1.1ρ.
+Note that all Ni are pairwise compatible. If K ∼ N±1
+1
+then for the required M we can take
+that Ni which occurs in Yi or in Yj+1 and has a larger µf(Ni). Hence we can assume that
+K ̸∼ N±1
+i
+for all Ni produced by all lifts ˜D of all cells D of rank α of ∆.
+Assume that D has two contiguity subdiagrams Πi ∈ T (i = 1, 2) to ˆY, i.e. the corre-
+sponding fragments N1 and N2 of rank α occur in Yk and Yk+1 respectively. Then we cut
+off from ∆ the subdiagram ∆ ∪ Π1 ∪ Π2 and the remaining simply connected component.
+This replaces ∆ with a new diagram ∆′ with a smaller number of cells of rank α, Yi with a
+subpath of Yi, bridges ui with another bridges u′
+i and the assumption that K ̸∼ N±1
+i
+for Ni
+produced by all lifts ˜D of D implies that K is independent of all new bridges u′
+i. In this case
+we can apply induction on the number of the cells of rank α of ∆.
+We may assume now that each cell D of rank α of ∆ has precisely two contiguity subdi-
+agrams Πi ∈ T to sides of ∆, one to ˆX−1 and another one to ˆY. This implies that each lift
+of D produces two fragments Ni, one in ¯X−1 and one in some Yj. Let {D1, D2, . . . , Dk} be
+the set of all cells of rank α of ∆. For each lift ˜D(j)
+i
+(t ∈ Z) of Di, denote N(j)
+i,1 and N(j)
+i,2 the
+corresponding fragments of rank α that occurs in ¯X−1 and Yj respectively (the requirement
+that N(j)
+i,2 occurs in Yj determines uniquely the lift ˜D(j)
+i
+and the fragment N(j)
+i,1). Note that
+(10-2) implies
+µf(N(j)
+i,k) > 1 − 4λ − 2.2ω − ρ = 5λ − 2.2ω.
+We order cells Di to get N(j)
+i,2 ordered in Yj as N(j)
+1,2 ≪ · · · ≪ N(j)
+k,2. Consequently, in ¯X we
+have · · ·N(j)
+1,1
+−1 ≪ · · · ≪ N(j)
+k,1
+−1 ≪ N(j+1)
+1,1
+−1 ≪ · · · ≪ N(j+1)
+k,1
+−1 · · · (Figure 26).
+By the
+N(i)
+11
+N(i)
+12
+N(i)
+13
+N(i)
+21
+N(i)
+22
+N(i)
+23
+Yi
+N(i+1)
+11
+N(i+1)
+12
+N(i+1)
+13
+N(i+1)
+21
+N(i+1)
+22
+N(i+1)
+23
+Yi+1
+¯X
+ui
+ui−1
+Figure 26.
+assumption above, we have K ̸∼ N(j)
+i,1
+−1 for all i, j. Then by Proposition 8.10 we have either
+52
+
+N(j)
+i,1
+−1 < K < N(j)
+i+1,1
+−1 for some i, j or N(j)
+k,1
+−1 < K < N(j+1)
+1,1
+−1 for some i. In each of these
+cases, we find the required M by applying an appropriate part of the proof of Proposition 10.5
+or Proposition 10.7.
+□
+We will use the following observation.
+10.13. Lemma.
+(i) Let K be a fragment of rank 1 ≤ β ≤ α in Γα. Let M be either
+another fragment of rank β in Γα such that K ∼ M±1 or a bridge of rank β such that
+K is not independent of M. Then any of the endpoints of K can be joined with any
+of the endpoints of M by a bridge w of rank β.
+Moreover, w can be chosen with the following property. If N is any other fragment
+of rank β such that N ̸∼ M±1 then N is independent of w.
+(ii) Let K1, K2, . . . , Kr be fragments of rank β ≤ α in Γα such that K1 ∼ K±1
+i
+for all i.
+Then all endpoints of all Ki are uniformly close.
+Proof. Follows from definitions in 8.4 and Definition 10.4.
+□
+10.14. Lemma. Let (Xi, Yi) (i = 1, 2) be two pairs of close reduced paths in Γα where X1
+and X2 are subpaths of a reduced path ¯X. Assume that for the common subpath Z of X1 and X2
+we have |Z|α ≥ 2.2. Then there exists a triple ai (i = 1, 2, 3) of uniformly close vertices on
+Z, Y1 and Y2 respectively.
+Proof. If α = 0 there is nothing to prove. Let α ≥ 1. Let X−1
+i uiYivi (i = 1, 2) be a coarse
+bigon where ui and vi are bridges of rank α.
+Case 1: Areaα(X−1
+i uiYivi) = 0 for both i = 1, 2. We apply Proposition 9.11 and find
+loops X′−1
+i
+u′
+iY′
+iv′
+i that can be lifted to Γα−1 where X′
+i and Y′
+i are subpaths of Xi and Yi
+respectively. For the common part Z′ of X′
+1 and Z′
+2 we have |Z′|α ≥ |Z|α − 2.04 ≥ 0.16 and
+hence |Z′|α−1 ≥ 3.2. Then the statement follows by induction.
+Case 2: Areaα(X−1
+i uiYivi) > 0 for i = 1 or i = 2. Without loss of generality, assume that
+K and M are active fragments of rank α in X1 and in Y1, respectively, such that K ∼ M−1.
+Let X1 = S1KS2 and Y1 = T1MT2. If S1K contains Z then we shorten X1 and Y1 replacing
+them with S1K and T1 thereby decreasing Areaα(X−1
+1 u1Y1v1) as described in 9.5. Similarly,
+if KS2 contains Z then we can replace X1 and Y1 with KS2 and T2. Therefore, we can assume
+that K is contained in Z. We take a1 = ι(K) and a2 = ι(M). If K is not independent of u2 or
+from v2 then for a3 we can take ι(Y2) or τ(Y2) respectively. Otherwise by Proposition 10.6
+there exists a fragment N of rank α in Y2 such that N ∼ K±1 and we can take a3 = ι(N).
+□
+10.15. Lemma. Let (S, T) and (X, Y) be pairs of close reduced paths in Γα where Y is an end
+of S and the ending vertices τ(X), τ(Y) = τ(S) and τ(T) are uniformly close. Then there
+exists a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T respectively, such that
+a1 cuts off a start X1 of X with |X1|α < 1.3 and a2 cuts off a start Y1 of Y with |Y1|α < 1.15.
+Proof. We can assume α ≥ 1. We use induction on |X|+|Y|+|T|. If |X|α < 1.3 and |Y|α < 1.2
+there is nothing to prove. We assume that |X|α ≥ 1.3 or |Y|α ≥ 1.15. It is enough to find
+a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T respectively, such that at
+least one ai cuts off a proper start of appropriate path X, Y or T.
+Let X−1u1Yu2 and S−1v1Tv2 be coarse bigons in Γα where ui and vi are bridges of rank α.
+Case 1: Areaα(X−1u1Yu2) = Areaα(S−1v1Tv2) = 0. We assume that u2 and v2 are defined
+from the condition that τ(X), τ(Y) and τ(T) are uniformly close; that is, either u2 and v2 are
+53
+
+bridges of rank α − 1 or have the form u2 = w1P1w2 and v2 = w3P2w4 where wi are bridges
+of rank α − 1 and P±1
+i
+are subpaths of a relator loop R of rank α. We consider the second
+case (the case when u2 and v2 are bridges of rank α − 1 is treated in a similar manner).
+Without changing notations, we assume that loops X−1u1Yu2 and S−1v1Tv2 are lifted
+to Γα−1 and, consequently, all paths introduced are in Γα−1 (the only change is that P±1
+i
+become subpaths of an R-periodic line ˜R where R is a relator of rank α). After choosing ai
+(i = 1, 2, 3) in Γα−1 we pass on to their images in Γα.
+Case 1a: |X|α ≥ 1.3. If a vertex b1 ̸= τ(X) on X is close in rank α − 1 to a vertex b2 on P1
+then we can take a1 := b1, a2 := τ(Y) and a3 := τ(T). We assume that no such b1 and b2
+exist. Then application of Proposition 9.19(ii)α−1 shows that X = z1X′z2 where X′ is close to
+a subpath Y′ of Y, |z1|α ≤ 1 + 4ζ2η, |z2|α ≤ 4ζ2η and hence |X′|α ≥ 0.3 − 8ζ2η.
+Assume first that α ≥ 2. Then shortening X′ from the end by Proposition 9.21α−1 we can
+assume that z1X′ is a proper start of X (and that X′ is still close to a subpath Y′ of Y). For
+the shortened X′, we have |X′|α > 0.3−8ζ2η−ζ2 > 0.26 which implies |X′|α−1 ≥ 1
+ζ|X′|α > 5.2.
+Let v1 = w5Qw6 where w5, w6 are bridges of rank α −1 and Q is labeled by a piece of rank α.
+Application of Lemma 10.2 gives a triple of uniformly close vertices ai (i = 1, 2, 3) where
+a1 lies on X′, a2 lies on Y′ and a3 lies either on Q or T. If a3 lies on Q then we replace
+it with ι(T). In the case α = 1 we shorten X′ by one edge and for the new X′ we have
+|X′|α > 0.3 − 8ζ2η − ζ > 0. We can still apply Lemma 10.2 due to Remark 10.3, so the
+argument remains the same.
+Case 1b: |Y|α ≥ 1.15. Similarly to Case 1, we can assume that there is no vertex b ̸= τ(Y)
+on Y (and hence on S since |Y|α−1 ≥ 1.15
+ζ
+= 23) close in rank α − 1 to a vertex on P1 or
+on P2. Applying Proposition 9.19(ii)α−1 we represent Y and S as Y = z1Y′z2, S = z3S′z4
+where Y′ is close (in rank α − 1) to a subpath X′ of X, S′ is close to a subpath T′ of
+T and |z1|α, |z3|α < 1 + 4ζ2η, |z2|α, |z4|α < 4ζ2η. In the case α = 1 there is a common
+subpath Z of X′, Y′, S′ and T′ of size |Z|α ≥ |Y|α − 1 − 8ζ2η > 0 and we can take ι(Z) for
+all ai. In the case α ≥ 2, shortening Y′ from the end by Proposition 9.21α−1 we can assume
+that z1Y′ is a proper start of Y. Let Z be the common subpath of Y′ and S′. We have
+|Z|α > |Y|α − 1 − 8ζ2η − ζ2 > 0.11 and hence |Z|α−1 > 2.2. Then the statement follows by
+Lemma 10.14α−1.
+Case 2: Areaα(S−1v1Tv2) > 0. Let K and M be active fragments of rank α in S and in T,
+respectively, such that K ∼ M−1. Let S = G1KG2 and T = H1MH2. Note that |K|, |M| > 0 by
+Lemma 10.8. If K is not contained in Y then we replace S and T with KG2 and H2 respectively
+and use induction. Assume that K is contained in Y. We first take a2 := ι(K), a3 := ι(M).
+If M is not independent on u1 or from u2 then we take a1 := ι(X) or a1 := τ(X) respectively.
+Otherwise by Proposition 10.6 there exits a fragment N of rank α in X such that N ∼ M±1.
+In this case we take a1 := ι(N) by Lemma 10.13(ii).
+Case 3: Areaα(X−1u1Yu2) > 0. Let K and M be active fragments of rank α in X and Y
+respectively such that K ∼ M−1. Then take a1 := ι(K), a2 := ι(M). Depending on whether
+M is not independent of v1 or v2 we find a3 similarly to the case 2 using Proposition 10.6
+and Lemma 10.13(ii).
+□
+10.16. Proposition (closeness transition in bigon). Let (X, Y) and (S, T) be pairs of close
+reduced paths in Γα where Y is a subpath of S. Assume that |X|α ≥ 2.3. Then X = z1X′z2
+where X′ is close to a subpath W of T and |zi|α < 1.3 (i = 1, 2).
+54
+
+Moreover, we have Y = t1Y′t2 where |t1|α, |t2|α < 1.15 and triples (ι(X′), ι(Y′), ι(W)) and
+(τ(X′), τ(Y′), τ(W)) are uniformly close.
+Proof. We can assume that α ≥ 1. Let X−1u1Yu2 and S−1v1Tv2 be coarse bigons in Γα where
+ui and vi are bridges of rank α. By Lemma 10.15 it is enough to find a triple ai (i = 1, 2, 3)
+of uniformly close vertices on X, Y and T respectively. An easy analysis involving Proposi-
+tion 10.6 shows how to do this in the case when Areaα(X−1u1Yu2) > 0 or Areaα(S−1v1Tv2) >
+0. It remains to consider the case when Areaα(X−1u1Yu2) = Areaα(S−1v1Tv2) = 0. Let
+vi = vi1Rivi2 (i = 1, 2) where vij is a bridge of rank α − 1 and Ri is labeled by a piece of
+rank α. By Proposition 9.11 we have X = w1X1w2 where endpoints of X1 and a subpath Y1
+of Y can be joined by bridges u′
+1 and u′
+2 of rank α − 1, so that the loop X−1
+1 u′
+1Y1u′
+2 can be
+lifted to Γα−1 and |wi|α ≤ 1 + 4ζ2η (i = 1, 2). Without changing notations, we assume that
+loops X−1
+1 u′
+1Y1u′
+2 and S−1v1Tv2 are already lifted to Γα−1 (and Y1 is still a subpath of S). We
+have
+|X1|α ≥ |X|α − |w1|α − |w2|α > 0.3 − 8ζ2η > 0.26
+and, consequently, |X1|α−1 > 5.2. By Lemma 10.2 there is a triple of uniformly close ver-
+tices b1 on X, b2 on Y and b3 on one of the paths R1, T or R2. For a1 and a2 we take images
+of b1 and b2 in Γα. Depending on the location of b3 we take for a3 the image of either ι(T),
+b3 or τ(T) as shown in Figure 27.
+□
+a1 X1
+X
+a2 Y1
+S
+u′
+1
+u′
+2
+v11
+v12
+v21
+v22
+T
+R1
+R2
+b
+a3
+a3 = b
+b
+a3
+Figure 27.
+10.17. Lemma. Let (X, Y) be a pair of close reduced paths in Γα, and let S−1∗T1∗T2∗ be a
+coarse trigon in Γα where Y is an end of S and ending vertices τ(X), τ(Y) and τ(T2) are
+uniformly close. Then either
+(i) there exists a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T1
+respectively, such that a1 cuts off a start X1 of X with |X1|α < 1.3;
+(ii) there exists a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T2
+respectively, such that a1 cuts off a start X1 of X with |X1|α ≤ 1.45.
+Proof. We can assume α ≥ 1. We use the same strategy as in the proof of Lemma 10.15 and
+proceed by induction on |X| + |Y| + |T2|. In view of Lemma 10.15, it is enough to prove that
+if |X| ≥ 1.45 then there exists a triple ai of uniformly close vertices on X, Y and some Ti
+respectively such that a1 or a2 cuts off a proper start of the appropriate path X or Y.
+Let ui (i = 1, 2) and vj (j = 1, 2, 3) be bridges of rank α in Γα such that u1Xu2Y−1 is a
+coarse bigon and S−1v1T1v2T2v3 is a coarse trigon.
+55
+
+Case 1: Areaα(X−1u1Yu2) = Areaα(S−1v1T1v2T2v3) = 0. We assume that u2 and v3 are
+defined from the condition that τ(X), τ(Y) and τ(T2) are uniformly close; that is, either u2
+and v3 are bridges of rank α − 1 or have the form u2 = u21Qu22 and v3 = v31P3v32 where
+u2i, v3i are bridges of rank α − 1 and Q±1, P±1
+3
+are subpaths of a relator loop R of rank α.
+We consider the second case (in the first case the argument is similar). Let vi = vi1Pivi2
+(i = 1, 2) where vij is a bridge of rank α − 1 and label(Pi) is a piece of rank α.
+We can assume that there is no vertex on X other than τ(X) which is close in rank α−1 to
+a vertex on R (otherwise we can take those for a1 and a2 as in the proof of Lemma 10.15). By
+Remark 9.3, we can assume that loops X−1u1Yu2 and S−1v1T1v2T2v3 can be lifted to Γα−1.
+Abusing notations, we assume that they are already in Γα−1. Application of Proposition
+9.19(ii)α−1 shows that X = w1X′w2 where X′ is close to a subpath Y′ of Y, |w1|α ≤ 1 + 4ηζ2,
+|w2|α ≤ 4ηζ2 and hence |X′|α ≥ 0.45 − 8ηζ2.
+As in the proof of Lemma 10.15 the proof slightly differs in cases α ≥ 2 and α ≥ 1. In
+the case α ≥ 2, shortening X′ from the end by Proposition 9.21α−1 we can assume that w1X′
+is a proper start of X, with a new bound |X′|α > 0.45 − 8ηζ2 − ζ2 > 0.41 which implies
+|X′|α−1 > 8.2. If there is a triple of uniformly close vertices on X′, Y′ and some Pi then we
+are done. We assume that no such triple exists. Let S1 be a reduced path joining ι(T1) and
+τ(T2) (see Figure 28). By Lemma 10.2 we have X′ = z1X′′z2 where X′′ is close to a subpath
+X′
+w1
+w2
+S
+Y′
+u22
+R
+v11
+P1
+v12
+T1
+v21
+P2
+v22
+T2
+v31
+P3
+v32
+S1
+Figure 28.
+of S1. Moreover, the lemma says that there exists a triple of uniformly close vertices on X′,
+Y′ and S1 and then applying Lemma 10.17α−1 we may assume that |zi|α−1 < 1.45. Then
+|X′′|α−1 ≥ |X′|α−1 − |z1|α−1 − |z2|α−1 > 5.3.
+Another application of Lemma 10.2 gives a triple of uniformly close vertices bi (i = 1, 2, 3)
+where b1 lies on X′, b2 lies on Y′ and b3 lies either on T1 or on T2. For ai we take the images
+of bi in Γα.
+In the case α = 1 the argument is similar (see Case 1a in the proof of Lemma 10.15) with
+no need for a lower bound on |X′′|α−1 for application of Lemma 10.2.
+Case 2: r = Areaα(S−1v1T1v2T2v3) > 0. Let L be an active relator loop for S−1v1T1v2T2v3
+and Ki (i = 1, 2 or i = 1, 2, 3) be the associated active fragments of rank α occurring in
+S, T1 or T2. If some Ki occurs in T1 and some Kj occur in T2 then we can shorten T1
+56
+
+and T2 decreasing r as described in 9.6. A similar inductive argument works in the case
+when some Ki occurs in S and is not contained in Y. Thus we may assume that there are
+only K1 and K2„ K1 is contained in Y and K2 occurs in T1 or T2. By Proposition 9.15,
+µf(Ki) ≥ 3λ − 1.1ω. The rest of the argument is the same as in the Case 2 of the proof of
+Lemma 10.15.
+Case 3: Areaα(X−1u1Yu2) > 0. Let K and M be active fragments of rank α in X and in Y
+respectively such that K ∼ M−1. We take a1 := ι(K), a2 := ι(M) and define a3 according to
+the following cases:
+• If M is not independent of v1 then a3 := ι(T1);
+• If M is not independent of v2 then a3 := τ(T1);
+• If M is not independent of v3 then a3 := τ(T2);
+• Otherwise by Proposition 10.7 applied to M there exists a fragment N or rank α in
+T1 or T2 such that M ∼ N±1. Then a3 := ι(N).
+□
+10.18. Proposition (closeness transition in trigon). Let (X, Y) be a pair of close reduced
+paths in Γα, and let S−1∗T1∗T2∗ be a coarse trigon in Γα where Y is a subpath of S. Assume
+that |X|α ≥ 2.45. Then X can be represented as in one of the following three cases:
+(i) X = z1X1z2 where X1 is close to a subpath W1 of T1 and |z1|α < 1.3, |z2|α < 1.45.
+(ii) X = z1X2z2 where X2 is close to a subpath W2 of T2 and |z1|α < 1.45, |z2|α < 1.3.
+(iii) X = z1X1z3X2z2 where Xi is close to a subpath Wi of Ti (i = 1, 2), |z1|α, |z2|α < 1.3
+and |z3|α < 0.4.
+Moreover, we can assume that there exists a subpath Y′ of Y such that triples (ι(Xp), ι(Y′), ι(Wp))
+and (τ(Xq), τ(Y′), τ(Wq)) are uniformly close where p and q are the smallest and the greatest
+indices of Xi in (i)–(iii), i.e. p = q = 1 in (i), p = q = 2 in (ii) and p = 1, q = 2 in (iii).
+Proof. Let ui (i = 1, 2) and vj (j = 1, 2, 3) be bridges of rank α such that u1Xu2Y−1 is a
+coarse bigon and S−1v1T1v2T2v3 is a coarse trigon. In view of Lemmas 10.15 and 10.17,
+finding a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and some Ti implies the
+conclusion of the proposition except the bound |z3|α < 0.4 in (iii). The latter follows from
+Proposition 9.19(i). An easy analysis as in Cases 2 and 3 of the proof of Lemma 10.17 shows
+how to find the vertices ai in the case when Areaα(X−1u1Yu2) > 0 or Areaα(S−1v1Tv2T2v3) >
+0. It remains to consider the case when Areaα(X−1u1Yu2) = Areaα(S−1v1Tv2T2v3) = 0. Let
+vi = wi1Riwi2 (i = 1, 2, 3) where label(wij) ∈ Hα−1 and the label of Ri is a piece of rank α.
+By Proposition 9.11 we have X = w1X1w2 where endpoints of X1 and a subpath Y1 of Y
+can be joined by bridges u′
+1 and u′
+2 of rank α − 1 and the loop X1u′
+1Y−1
+1 u′−1
+2
+can be lifted
+to Γα−1 and |wi|α ≤ 1 + 4ζ2η (i = 1, 2). Without changing notations, we assume that loops
+X−1
+1 u′
+1Y1u′
+2 and S−1v1Tv2 are already in Γα−1 (and Y1 is still a subpath of S). We have
+|X1|α ≥ |X|α − |w1|α − |w2|α > 0.41
+and, consequently, |X1|α−1 > 8.2. Then we find ai applying Lemmas 10.17α−1 and 10.2 as in
+the proof of Lemma 10.17.
+□
+10.19. Proposition (closeness transition in conjugacy relations). Let S be a word cyclically
+reduced in Gα. Assume that S is conjugate in Gα to a word Tv where T ∈ Rα and v ∈ Hα.
+Let ¯S = �
+i∈Z Si and �
+i∈Z Tivi be lines in Γα representing the conjugacy relation.
+57
+
+Assume that a reduced path X in Γα is close to a subpath Y of ¯S with |Y| ≤ |S|. Let
+|X|α ≥ 2.45. Then either:
+(i) X can be represented as X = z1X1z2 where X1 is close to a subpath W1 of Ti for
+some i and |z1|α, |z2|α < 1.45.
+(ii) X can be represented as X = z1X1z3X2z2 where for some i, X1 is close to a subpath
+W1 of Ti, X2 is close to a subpath W2 of Ti+1, |z1|α, |z2|α < 1.3 and |z3|α ≤ 0.4.
+Moreover, we can assume that there exists a subpath Y′ of Y such that triples (ι(X1), ι(Y′), ι(W1))
+and (τ(Xq), τ(Y′), τ(Wq)) are uniformly close where q = 1 in (i) and q = 2 in (ii).
+Proof. It is enough to find a uniformly close triple of vertices ai (i = 1, 2, 3) on X, Y and
+some Ti and then use Lemmas 10.17 or 10.15. Let X−1u1Yu2 be a coarse bigon where u1 and u2
+are bridges of rank α. If Areaα(X−1u1Yu2) > 0 then we reach the goal using Proposition 10.12
+and Lemma 10.13(ii). Assume that Areaα(X−1u1Yu2) = 0.
+Let ∆ be an annular diagram of rank α with boundary loops ˆS−1 and ˆTˆv representing the
+conjugacy relation. Let ˜∆ be the universal cover of ∆ and φ : ˜∆(1) → Γα the combinatorially
+continuous map sending lifts ˜Si, ˜Ti and ˜vi to Si, Ti and vi respectively. We assume that ∆
+is reduced and has a tight set of contiguity subdiagrams. Let r be the number of cells of
+rank α of ∆.
+Assume that r > 0 and let D be a cell of rank α of ∆. By Proposition 7.11(iv) and Lemma
+7.10(i), D has two or three contiguity subdiagrams Πi ∈ T to sides of ∆, at most two to ˆT
+and at most one to ˆS−1. If there are two contiguity subdiagrams Πi (i = 1, 2) of D to ˆT then
+we consider a new annular diagram ∆′ obtained by cutting off D∪Π1 ∪Π2 and the remaining
+simply connected component from ∆, and new words T ′ and v′ where T ′ is a subword of T.
+In this case, the statement follows by induction on r.
+We can assume now that D has one contiguity subdiagram to ˆS−1 and one to ˆT. Let ˜Di
+(i ∈ Z) be the lifts of D in ˜∆. With an appropriate numeration of ˜Di’s, each relation loop
+φ(δ˜Di) is a base loop for a fragment Ki in ¯S−1 and a fragment Mi in Ti. By Proposition
+7.13(iii),
+µf(K−1
+i ) + µf(Mi) > 1 − 4λ − 2.2ω.
+Since T is reduced in Gα, we have µf(Mi) ≤ ρ and hence
+µf(K−1
+i ) > 5λ − 2.2ω.
+If none of K−1
+i ’s is contained in Y then we can apply Proposition 10.18. Otherwise we use
+an argument similar to one in Case 2 of the proof of Lemma 10.15.
+Now assume that ∆ has no cells of rank α. Without changing notations, we assume that
+parallel lines ¯S = �
+i∈Z Si, �
+i∈Z Tivi and paths X and Y are lifted to Γα−1 so that Y is still a
+subpath of ¯S. Let v ≖ w1Rw2 where wi ∈ Hα−1 and R is a piece of rank α. We represent vi
+accordingly as vi = w(i)
+1 Riw(i)
+2 . Let Z be a word reduced in Gα−1 such that Z = Tw1R and
+let Zi (i ∈ Z) be appropriate paths in Γα−1 with label(Zi) ≖ Z (Figure 29). Since |X|α ≥ 2.45
+we have |X|α−1 ≥ 1
+ζ |X|α ≥ 49. By Proposition 10.19α−1, a subpath X′ of X with |X′|α−1 > 23
+is close to a subpath of some Zi. Then using Proposition 10.18α−1 we find a triple bi of
+uniformly close vertices on X′, Y and Ti or Ri respectively. If b3 lies on Ti then for the
+desired ai we take images of bi in Γα. If b3 lies on Ri then for ai (i = 1, 2, 3) we take images
+of b1, b2 and τ(Ti), respectively.
+□
+58
+
+X
+u1
+Y
+u2
+Zi
+Zi+1
+Ti
+Ti+1
+w(i)
+1
+w(i)
+2
+w(i+1)
+1
+w(i+1)
+2
+Ri
+Ri+1
+¯S
+Figure 29.
+10.20. Lemma. Let 1 ≤ β ≤ α and X be a reduced path in Γα. Let K1 and K2 be fragments
+of rank β in X such that µf(Ki) ≥ λ + 2.6ω (i = 1, 2), K1 < K2 and K1 ̸∼ K2. If a bridge of
+rank β starts or ends at ι(X) then K2 is independent of u. Similarly, if a bridge of rank β
+starts or ends at τ(X) then K1 is independent of u.
+Proof. We consider the case when ι(u) = ι(X) (all other cases are similar). Assume that K2
+is not independent of u. By Definition 10.4, u = vSw where S occurs in a relation loop R of
+rank β, v and w are bridges of rank β − 1 and R±1 is the base relation loop for K. Let ˜R
+and ˜X be lifts of R and X in Γβ−1 so that ˜R±1 is the base axis for ˜K2. Lemma 9.22 implies
+that the starting vertex of ˜X is close to a vertex on ˜R. Then using Proposition 10.21α−1 we
+conclude that the starting segment ˜X1˜K2 of ˜X is a fragment of rank α with base axis ˜R. Since
+K1 is contained in ˜X1˜K2, Proposition 8.10 gives K1 ∼ K2, a contradiction.
+□
+10.21. Proposition (closeness preserves order). Let X1X2 and Y1Y2 be reduced paths in Γα
+such that endpoints of Xi and Yi are close in the order as in Figure 30. Then |X1|α, |Y2|α <
+5.7.
+X1
+X2
+Y1
+Y2
+u1
+u2
+u3
+Figure 30.
+Proof. We can assume that α ≥ 1. Due to symmetry, it is enough to show that |X1|α < 5.7.
+Denote ui (i = 1, 2, 3) bridges of rank α joining endpoints of Xi and Yi as shown in Figure 30.
+Claim 1: Areaα(X−1
+1 u1Y2u−1
+2 ) ≤ 1.
+Proof of Claim 1. Assume that Areaα(X−1
+1 u1Y2u−1
+2 ) ≥ 2. Let Ki and Mi (i = 1, 2) be active
+fragments of rank α in X1 and Y2, respectively, such that K1 < K2 and Ki ∼ M−1
+i . By
+Proposition 9.7(ii) and Lemma 10.20, K2 is independent of u1. Similarly, M2 and hence K2,
+are independent of u3. By Propositions 9.7 and 10.5 applied to (X1X2)−1u1Y−1
+1 u−1
+3 , there is
+59
+
+a fragment N of rank α in Y1 such that N ∼ K±1
+2
+and µf(N) ≥ 5λ − 4.9ω. We obtain a
+contradiction with Corollary 9.24(ii),(iii).
+□
+Claim 2: If Areaα(X−1
+1 u1Y2u−1
+2 ) = 0 and label(u1), label(u2) ∈ Hα−1 then |X1|α < 1 + 6.1ζ.
+Proof of Claim 2. If r = Areaα(X2u3Y1Y2u−1
+2 ) > 0 then we can reduce the statement to the
+case of a smaller r as explained in 9.4. So we can assume that Areaα(X2u3Y1Y2u−1
+2 ) = 0. Then
+loops X−1
+1 u1Y2u−1
+2
+and X2u3Y1Y2u−1
+2
+can be lifted to Γα−1 (up to possible switching of u3).
+To simplify notations, we assume that these loops are already in Γα−1. Let u3 = v1Qv2
+where label(vi) ∈ Hα−1 and label(Q) is a piece of rank α. We obtain a coarse trigon in Γα−1
+with sides X1X2, Q and Y1, see Figure 31. Applying Propositions 9.19(i)α−1 and 10.21α−1 we
+obtain
+|X1X2|α < 1 + 4ζ2η + 5.7ζ < 1 + 6.1ζ.
+□
+u1
+u2
+X1
+X2
+Y1
+Y2
+v1
+Q
+v2
+Figure 31.
+The rest of the proof: If Areaα(X−1
+1 u1Y2u−1
+2 ) = 0 then the statement follows from Claim 2
+and Proposition 9.11. By Claim 1, it remains to consider the case Areaα(X−1
+1 u1Y2u−1
+2 ) = 1.
+Then X1 can be represented as R1S1R2S2R3 (see Figure 32) where each Ri is a fragment of
+rank α and by Claim 2 and Proposition 9.19(ii)α−1 each Si satisfies |Si|α < 1 + 6.1ζ + 8ζ2η.
+We obtain
+|X1|α < 3 + 2(1 + 6.1ζ + 8ζ2η) < 5.7.
+The proof is completed.
+□
+R1
+S1
+R2
+S2
+R3
+X2
+Y1
+Y2
+u1
+u3
+u2
+Figure 32.
+In the end of the section we formulate several statements about stability of fragments in
+a more general setup when fragments have arbitrary rank β in the interval 0 ≤ β ≤ α.
+60
+
+10.22. Proposition. Let S and T be close reduced paths in Γα. Let 0 ≤ β < α and let X
+and Y be close in rank β reduced paths in Γα such that Y is a subpath of S. Assume that
+|X|α ≥ 2.3 and Y contains no fragments K of rank γ with β < γ ≤ α and µf(K) ≥ ξ0.
+Then X can be represented as X = w1X′w2 where X′ is close in rank β to a subpath of T and
+|wi|α < 1.2 (i = 1, 2).
+Proof. Let S−1u1Tu2 and X−1v1Yv2 be corresponding coarse bigons. If Areaα(S−1u1Tu2) >
+0 then by the argument from 9.5 we reduce the statement to a new pair (S, T) and a
+coarse bigon S−1u1Tu2 with a smaller value of Areaα(S−1u1Tu2). Hence we can assume that
+Areaα(S−1u1Tu2) = 0. Without changing notations, we assume that both loops S−1u1Tu2
+and X−1v1Yv2 are in Γα−1. Let ui = ui1Piui2 where label(uij) ∈ Hα−1 and label(Pi) is a piece
+of rank α. Observe that if a subpath X′ is close to a subpath of P1 or P2 then |X′|α ≤ 1.
+Since |X|α ≥ 2.3 applying Lemma 10.2 we find a subpath of X close to a subpath of T. We
+consider the case when X = z0X1z1X2z2X3z3 where Xi (i = 1, 2, 3) are close to subpaths of P1,
+T and P2 respectively (the other cases from Lemma 10.2 give a better lower bound on |X2|α).
+By Lemma 10.15 we can assume that |z0|α−1, |z3|α−1 < 1.3 and by Proposition 9.19(i)α−1 we
+can assume that |z1|α−1, |z2|α−1 < 0.4. We have |X1|α, |X3|α ≤ 1, so |X2|α > 2.3−2−3ζ = 0.15
+and hence |X2|α−1 > 3. Then by Corollary 9.13α−1 we have X2 = t1X′t2 where X′ is close
+in rank β to a subpath of T and |ti|α−1 < 1.03. We have X = z1X1z2t1X′t2z3X3z4 where
+|z1X1z2t1|α < 1 + 2.73ζ < 1.2 and a similar bound holds for |t2z3X3z4|α.
+□
+10.23. Proposition. Let X and Y be reduced paths in Γα. Let 1 ≤ β ≤ α and assume that
+either X or Y contains no fragments N of rank γ with β < γ ≤ α and µf(N) ≥ ξ0.
+Let Ki (i = 1, 2) be fragments of rank β in X such that K1 ̸∼ K2 and K1 < K2. Assume
+that at least one of the following conditions holds:
+(*) there exist fragments Mi (i = 1, 2) of rank β in Y such that µf(Mi) ≥ λ + 2.7ω,
+Ki ∼ M±1
+i
+and M1 < M2; or
+(**) X and Y are close in rank β.
+Then the following is true:
+(i) Let N be a fragment of rank β in X with µf(N) ≥ 2λ + 9.1ω such that K1 < N < K2
+and N ̸∼ Ki for i = 1, 2. Then there exists a fragment N′ of rank β in Y such that
+N′ ∼ N±1, M1 < N′ < M2 in case (*) and
+(10-3)
+µf(N′) ≥ min{µf(Ni) − 2λ − 3.4ω, ξ0}
+In case (*), if M1 and M2 are disjoint then we can assume that M1 ≪ N′ ≪ M2.
+This is the case (that is, M1 and M2 are necessarily disjoint) if µf(N) ≥ 4λ + 9ω.
+(ii) Assume that µf(Ki) ≥ 2λ + 9.1ω and in case (*), µf(Mi) ≥ 2λ + 9.1ω.
+Let K′
+i
+(i = 1, 2) be a pair of another fragments of rank β in X and M′
+i (i = 1, 2) a pair of
+another fragments of rank β in Y such that µf(K′
+i), µf(M′
+i) ≥ 2λ + 9.1ω, K′
+i ∼ M′±1
+i
+(i = 1, 2) and K′
+1 ̸∼ K′
+2. Then K′
+1 < K′
+2 if and only if M′
+1 < M′
+2.
+Furthermore, the statement of the proposition is true also in the case β = 0 if we drop all
+conditions of the form µf(·) ≥ . . . for fragments of rank β.
+Proof. If β = 0 then by Proposition 9.10 we have Mi = Ki (i = 1, 2), M1 ∪ M2 = K1 ∪ K2 in
+case (*) and X = Y in case (**). Then the statement is trivial. We assume that β ≥ 1.
+(i): Assume that (*) holds. First assume that M1 and M2 are disjoint. Let X1 = K1 ∪ K2
+and Y1 be the subpath of Y between M1 and M2, i.e. Y = ∗M1Y1M2∗. By Lemma 10.13(i)
+61
+
+and Proposition 9.10 we have a loop X−1
+1 uY1v that can be lifted to Γβ where u and v are
+bridges of rank β. Up to change of notation, we assume that X−1
+1 uY1v is already in Γβ. Again
+by Lemma 10.13(i)β, N is independent of u and v. By Proposition 10.6β, there exists N′ in
+Y1 satisfying (10-3) such that N′ ∼ N±1, i.e. we have M1 ≪ N′ ≪ M2 as required.
+Assume that M1 and M2 have a nonempty intersection. By Proposition 8.12β there exist
+fragments M′
+1 and M′
+2 of rank β such that M′
+i ∼ Mi, M′
+1 is a start of M1 disjoint from M2
+and M′
+2 is an end of M2 disjoint from M1. Let Y2 = M1 ∪ M2. Using the argument above
+with Y2 instead of Y1 and M′
+1 instead of M1 we find N1 in Y2 disjoint from M2 such that
+µf(N1) > 5.7ω and N1 ∼ N±1. Similarly, using Y2 instead of Y1 and M′
+2 instead of M2 we
+find N2 in Y2 disjoint from M1 such that µf(N2) > 5.7ω and N2 ∼ N±1. Then we can take
+N′ = N1 ∪ N2 by Corollary 9.24(i), (iii).
+If µf(N) ≥ 4λ + 9ω then µf(N′) > 2λ + 5.6ω and using Propositions 8.11β and 8.10β we
+conclude that M1 and M2 cannot cover N′ together, i.e. M1 ≪ M2.
+In case (**) we already have a loop X−1uYv with bridges u and v of rank β. We lift it
+to Γβ and then apply Lemma 10.20β to see that the lift of N is independent of the lifts of u
+and v. Then application of Proposition 10.6β gives the required N′.
+(ii): An easy analysis with a help of Propositions 9.24(ii) and 8.10β shows that it is enough
+to prove the following: Let X and Y be reduced paths in Γα. Let Ki (i = 1, 2, 3) be fragments of
+rank β in X, Mi (i = 1, 2, 3) be fragments of rank β in Y, µf(Ki), µf(Mi) ≥ λ+9.1ω, Ki ∼ M±1
+i
+for all i and Ki ̸∼ Kj for i ̸= j. If K1 < K2 < K3 and M1 < M3 then M1 < M2 < M3.
+Assume that this is not the case, that is, we have K1 < K2 < K3, M1 < M3 and either
+M2 < M1 or M3 < M2. By (i), there exists a fragment N of rank α in Y such that K2 ∼ N±1
+and M1 < N < M3. Then by Propositions 9.24(i) and 8.10β we obtain M1 ∼ N or M3 ∼ N, a
+contradiction.
+□
+10.24. Proposition. Let X and Y be words strongly cyclically reduced in Gα, representing
+conjugate elements of Gα. Let ¯X and ¯Y be lines in Γα representing the conjugacy relation.
+Let 1 ≤ β ≤ α. Assume that at least one of the words X or Y has the property that no
+its cyclic shift contains a fragment K of rank γ with µf(K) > ξ0 and β < γ ≤ α. Let
+¯X = . . . X−1X0X1 . . . and ¯Y = . . . Y−1Y0Y1 . . . be lines in Γα representing the conjugacy
+relation.
+(i) Then for any fragment K of rank β in ¯X with µf(K) ≥ 2λ + 9.1ω there exists a
+fragment M of rank β in ¯Y such that M ∼ K±1 and
+µf(M) ≥ min{µf(K) − 2λ − 3.4ω, ξ0}
+(ii) If X and Y are strongly cyclically reduced in Gα then the correspondence between
+fragments of rank β in ¯X and in ¯Y preserves the ordering in the following sense: if
+Ki (i = 1, 2) are fragments of rank β in ¯X, Mi (i = 1, 2) are fragments of rank β
+in ¯Y, µf(Ki), µf(Mi) ≥ 2λ + 9.1ω, Ki ∼ M±1
+i
+(i = 1, 2) and K1 ̸∼ K2. Then K1 < K2
+if and only if M1 < M2.
+Furthermore, the statement of the proposition is true also in the case β = 0 if we drop all
+conditions of the form µf(·) ≥ . . . for fragments of rank β.
+Proof. By Proposition 9.17 every subpath of ¯X can be extended to be close in rank β to
+a subpath of ¯Y. Then (i) follows from Proposition 8.16(ii) and Proposition 10.23(i) with
+K1 = s−1
+X,¯XK and K2 = sX,¯XK. Statement (ii) follows by Proposition 10.23(ii). In the case
+β = 0 the statement becomes trivial after application of Proposition 9.17.
+□
+62
+
+11. Reduced representatives
+The main goal of this section is to prove that any element of Gα can be represented by a
+reduced word and to prove a cyclic analog of this statement (Proposition 11.5).
+11.1. Proposition (reduced representative). Every element of Gα can be represented by
+a reduced in Gα word which contains no fragments F of rank 1 ≤ β ≤ α with µf(F) ≥
+1
+2 + 2λ + 15ω.
+11.2. Lemma. Let m ≥ 3 and X−1∗Y1∗Y2∗ · · ·∗Ym∗ be a coarse (m+1)-gon in Γα−1. Assume
+that there are indices 1 ≤ t1 < t2 < · · · < tk ≤ m (k ≥ 1) such that
+t1 ≤ 3,
+tk ≥ m − 2,
+tj − tj−1 ≤ 2 for all j
+and
+|Ytj|α−1 > 4η
+for all j.
+Assume further that there are no close vertices in each of the pairs (Yi, Yi+1), (Y1, Yt1),
+(Ytj, Ytj+1), (Ytk, Ym) except appropriate endpoints (i.e. except τ(Yi) and ι(Yi+1)). Then
+each of the paths Ytj has a vertex close to a vertex aj on X and these vertices aj are in X in
+the (non-strict) order from start to end.
+Proof. We first claim that there are no close vertices in pairs (Yi, Yj) for j − i > 1. Assume
+there are. We choose such a pair with minimal possible j − i. Then an ending segment Y′
+i
+of Yi, paths Yi+1, . . . , Yj−1 and a starting segment Y′
+j of Yj form a coarse r-gon with
+r = j − i + 1 ≥ 3. Applying Proposition 9.18α−1 we get
+j−1
+�
+k=i+1
+|Yi|α−1 ≤ (r − 2)η.
+On the other hand, it follows from the hypothesis of the lemma that there are at least
+min(1, 1
+2(r − 3)) paths Ytk among Yi+1, . . . , Yj−1 and hence
+j−1
+�
+k=i+1
+|Yi|α−1 > 4η min
+�
+1, 1
+2(r − 3)
+�
+.
+We get a contradiction since the right hand side of the inequality is at least (r − 2)η. This
+proves the claim.
+Shortening if necessary Y1 and X we can assume that there is no pair of close vertices
+on Y1 and X other that (ι(Y1), ι(X)).
+Similarly, we can assume that there is no pair of
+close vertices on Ym and X other than (τ(Ym), τ(X)). Now we claim that there is a pair
+of close vertices on Yi and X for some 2 ≤ i ≤ m − 1. Indeed, otherwise we can apply
+Proposition 9.18α−1 to the whole coarse (m + 1)-gon X−1∗Y1∗Y2∗ · · ·∗Ym∗ and obtain a
+contradiction since 4kη ≥ (m − 1)η.
+Let (b, c) be a pair of close vertices on X and Yi0 where 2 ≤ i0 ≤ m − 1. Let b divide X as
+X1X2 and c divide Yi0 as Z1Z2 If there is at least one index tj in the interval 2 ≤ tj ≤ i0 − 1
+then the conditions of the lemma are satisfied for the coarse (i0+1)-gon X−1
+1 ∗Y1∗ . . . Yi0−1∗Z1∗
+and we conclude by induction that every Ytj with tj < i0 has a vertex close to a vertex aj on
+X and the vertices aj occur in X in the appropriate order. Similarly, we conclude the same
+for every path Ytj with tj > i0. This implies the statement for all Ytj.
+□
+63
+
+11.3. Lemma. Let X be a word reduced in Gα−1. Assume that for any fragment K of rank α
+in X we have
+µf(K) ≤ 1 − 3λ − 5ω.
+Then there exists a word Y equal to X in Gα which is reduced in Gα−1 and such that for any
+fragment M of rank α in Y we have
+µf(M) < 1
+2 + 2λ + 15ω.
+In particular, Y is reduced in Gα (note that 1
+2 + 2λ + 15ω < ρ = 1 − 9λ by (2-3) and (4-1).)
+Proof. We represent X by a reduced path X in Γα−1. Denote
+t = 1
+2 + 11ω.
+Let K1, . . . , Kr be a maximal set of pairwise non-compatible fragments of rank α in X with
+µf(Ki) ≥ t. We assume that each Ki has maximal size µf(Ki) in its equivalence class of
+compatible fragments of rank α occurring in X. Using Proposition 8.12 we shorten each Ki
+from the start obtaining a fragment ¯Ki of rank α so that ¯Ki do not intersect pairwise; we
+have µf(¯Ki) > µf(Ki) − λ − 2.7ω. Let
+X = S0¯K1S1 . . . ¯KrSr.
+Let Pi be a base for ¯Ki; for each i, we have a coarse bigon ¯K−1
+i uiPivi with bridges ui and vi.
+Let Pi ≖ label(Pi) and PiQ−1
+i
+be the associated relator of rank α. We consider a path in Γα−1
+Z = S∗
+0u∗
+1Q1v∗
+1S∗
+1 . . . u∗
+rQrv∗
+rS∗
+r
+where labels of S∗
+i , u∗
+i and v∗
+i are equal to corresponding labels of Si, ui and vi and label(Qi) ≖
+Qi. Note that label(Z) = X in Gα. We perform the following procedure:
+(i) if a pair of vertices on Qi and S∗
+i are close and is distinct from (τ(Qi), ι(S∗
+i )) then
+we choose a bridge w of rank α − 1 joining these vertices, replace v∗
+i with w and
+shorten Qi from the end and S∗
+i from the start; similarly, if a pair of vertices on Qi
+and S∗
+i−1 are close and is distinct from (ι(Qi), τ(S∗
+i−1)) then we choose a bridge w of
+rank α−1 joining them and replace u∗
+i with w shortening Qi from the start and S∗
+i−1
+from the end; we apply recursively the operation until possible;
+(ii) if a vertex on Qi is close to a vertex on Q∗
+i+1 then we choose a bridge w of rank α −1
+joining these vertices, shorten Qi from the end and Qi+1 from the end and join then
+by w (so S∗
+i is eliminated and v∗
+i S∗
+i u∗
+i is replaced with a bridge w of rank α − 1); we
+apply recursively the operation until possible;
+After the procedure, we obtain a path
+Z1 = T0U0R1U1 . . . RrUrTr
+where for each i, Ri is a subpath of Qi and Ui either is a bridge of rank α − 1 or has the
+form wiTizi where Ti is a subpath of S∗
+i and wi and zi are bridges of rank α − 1. Let Y be
+a reduced path with the same endpoints as Z1. Our goal is to prove that the label Y of Y
+satisfies the requirement of the lemma, that is, for any fragment N of rank α in Y we have
+µf(N) < 1
+2 + 2λ + 15ω.
+We compute a lower bound for µ(Ri). Fix i and let Qi = Q′RiQ′′. At step (i) of the
+procedure, we do not shorten Qi more than this would give a fragment of rank α in X with
+a base that is a proper extension of Pi, so we get µ(Qi) ≥ 1 − µf(Ki) ≥ 3λ + 5ω. At step (ii)
+64
+
+we shorten Qi from each side by less than λ + 0.4ω (this follows from Proposition 9.19(i)α−1,
+Proposition 8.15 and Corollary 8.2). This implies µ(Ri) > λ+4ω and, in particular, |Ri|α−1 >
+4η.
+We apply Lemma 11.2 with X := Y where Ri and Ti play the role of Yi’s and Ri are taken
+as Yti. The lemma says that each path Ri has a vertex close to a vertex on Y and these
+vertices on Y are appropriately ordered. We can write
+Y = V0M1V1 . . . MrVr
+where each Mi is close to a subpath of Qi (at the moment each Mi is empty because it is
+represented by a vertex on Y). Extending Mi’s we make them maximal so that no vertex
+on Wi except ι(Vi) is close to a vertex on Qi and no vertex on Vi except τ(Vi) is close to a
+vertex on Qi+1. Up to location of Z in Γα−1 we can assume that it starts at ι(X). Combining
+the two graphs shown in Figure 33a and mapping them to Γα we obtain a graph as shown
+in Figure 33b. This graph is similar to one obtained from a single-layer diagram (as in Fig-
+u1
+v1
+u2
+v2
+ur
+vr
+S0
+S1
+S2
+S∗
+r−1
+Sr
+u∗
+1
+v∗
+1
+u∗
+2
+v∗
+2
+u∗
+r
+v∗
+r
+S∗
+0
+S∗
+1
+S∗
+2
+S∗
+r−1
+S∗
+r
+Q1
+Q2
+Y
+T0
+U0
+R1
+U1
+R2
+U2
+Ur−1
+Rr
+Ur
+Tr
+a
+b
+Figure 33.
+ure 15). An easy analysis with use of Proposition 9.19α−1, Proposition 8.15 and Corollary 8.2
+shows that Mi and some extension ˜Ki of ¯Ki satisfy the bound as in Proposition 9.7, i.e.
+µf(Mi) + µf(˜Ki) > 1 − 2λ − 1.5ω.
+Since µf(˜Ki) ≤ µf(Ki) ≤ 1 − 3λ − 5ω we obtain that for all i,
+µf(Mi) > λ + 3.5ω.
+65
+
+Let N be a fragment of rank α in Y. By Proposition 8.10, we have either N ∼ Mi or
+N ⊆ Mi ∪ Mi+1 for some i. In the case when N ⊆ Mi ∪ Mi+1, N ̸∼ Mi and N ̸∼ Mi+1 we can
+apply the argument from the proof of Proposition 10.5 and find a fragment N′ in X such that
+µf(N′) > µf(N) − 2λ − 3.4ω.
+We have also N′ ̸∼ Ki, Ki+1 and hence N′ ̸∼ Kj for all j. By the choice of the Ki’s, we have
+µf(K′) < t and hence
+µf(N) < t + 2λ + 3.4ω < 1
+2 + 2λ + 15ω.
+Assume that N ∼ Mi for some i. Let ¯Q and ¯P be bases for N and Ki respectively. Images
+of ¯Q−1 and ¯P in Γα are subpaths of a relator loop and have at most two overlapping parts.
+We give an upper bound for µ(¯Q) + µ(¯P) by finding an upper bound for the size of each
+overlapping part. Assume, for example, that an end of the image of ¯P in Γα overlaps with a
+start of the image of ¯Q−1. Changing the location of Z in Γα−1 we can assume that ¯P and ¯Q−1
+overlap on a subpath W of the same size already in Γα−1.
+We consider the case i < r (see Figure 34; the case i = r is similar with a better upper
+bound on µ(W)). We apply Proposition 9.19(ii)α−1 to a coarse tetragon with one side W and
+Ki
+¯Ki+1
+S
+X
+¯P
+¯Q
+W
+V
+N
+L
+Y
+Pi+1
+Mi+1
+Figure 34.
+other sides which are an end S of Si¯Ki+1, a start V of M−1
+i+1V−1
+i
+and a subpath of a common
+base axis L for K−1
+i+1 and Ni+1. In the worst case we have W = W1z1W2z2W3 where W1 is
+close to a subpath of V−1, W2 is close to a subpath of L−1, W3 is close to a subpath of S−1
+and |zi|α−1 ≤ 4ηζ. Proposition 10.21α−1 implies |W1|α−1 < 5.7 and |W3|α−1 < 5.7. Since
+Ki ̸∼ Ki+1 we obtain µ(W2) < λ. Hence
+µ(W) < λ + 2ω(5.7 + 4ηζ) < λ + 13ω.
+We obtain
+µf(N) + µf(Ki) < 1 + 2λ + 26ω.
+Since µf(Ki) ≥ t this implies the required bound µf(N) < 1
+2 + 2λ + 15ω.
+□
+11.4. Lemma. Let α ≥ 1 and X be a word reduced in Gα and a ∈ A±1 a letter in the
+generators of Gα. Let Y be a word reduced in Gα−1 such that Y = Xa in Gα−1. Then Y has
+no fragments K of rank α with µf(K) ≥ ρ + 6.2ω.
+Proof. Follows from Lemma 10.8 and Proposition 8.8.
+□
+66
+
+Proof of Proposition 11.1. It is trivial if α = 0. In the case α ≥ 1 Proposition 11.1 follows
+by induction from Lemmas 11.3 and 11.4 since ρ + 6.2ω < 1 − 3λ − 5ω.
+□
+We turn to the cyclic analogue of Proposition 11.1:
+11.5. Proposition (cyclically reduced representative). Every element of Gα of finite order
+is conjugate to a cyclically reduced word of the form Rk
+0 where R0 is the root of a relator of
+rank β, 1 ≤ β ≤ α.
+Every element of Gα of infinite order is conjugate to a strongly cyclically reduced word
+in Gα.
+11.6. Lemma (a cyclic version of Lemma 11.2). Let X be a word cyclically reduced in Gα−1
+representing an element of Gα−1 of infinite order. Let m ≥ 2, Y1, . . . , Ym be words reduced in
+Gα−1, u1, . . . , um be bridges of rank α−1 and let X be conjugate to Y1u1 . . . Ymum in Gα−1. Let
+�
+i∈Z Y(i)
+1 u(i)
+1 . . . Y(i)
+m u(i)
+m and ¯X = �
+i∈Z X(i) be lines in Γα−1 labeled (Y1u1 . . . Ymum)∞ and X∞
+respectively representing the conjugacy relation.
+Assume that there are indices 1 ≤ t1 < t2 < · · · < tk ≤ m (k ≥ 1) such that
+m + t1 − tm ≤ 2,
+tj − tj−1 ≤ 2
+for all j,
+and
+|Ytj|α−1 > 4η
+for all j.
+Assume that there are no close vertices in each of the pairs (Y(0)
+i , Y(0)
+i+1), (Y(0)
+m , Y(1)
+1 ), (Y(0)
+tj , Y(0)
+tj+1),
+(Y(0)
+tk , Y(1)
+t1 ) except appropriate endpoints (i.e. except pairs (τ(Y(0)
+i ), ι(Y(0)
+i+1)) and (τ(Y(0)
+m ), ι(Y(1)
+1 ))).
+Then each of the paths Y(0)
+tj , j = 1, . . . , k has a vertex close to a vertex aj on ¯X and these
+vertices aj are in the (non-strict) order corresponding to the order of the Y(0)
+j ’s (and ak is
+located non-strictly before sX,¯Xa0).
+Proof. The proof follows the proof of Lemma 11.2 with appropriate changes.
+Claim 1: There are no close vertices in pairs (Y(0)
+i , Y(0)
+j ) with j − i > 1 and (Y(0)
+i , Y(1)
+j ) with
+j + m − i > 1.
+The proof repeats the argument from the proof of Lemma 11.2.
+Claim 2: For some i, there are close vertices in the pair (Y(0)
+i , ¯X).
+Assume this is not true. Consider an annular diagram ∆ of rank α − 1 with boundary
+loops ˆX−1 and ˆY1ˆu1 . . . ˆYmˆum and a combinatorially continuous map φ : ˜∆ → Γα−1 such that
+φ maps the boundary of ˜∆ to ¯X−1 and �
+i Y(i)
+1 u(i)
+1 . . . Y(i)
+m u(i)
+m . The assumption, Claim 1 and
+the hypothesis of the lemma imply that ∆ is small. Application of Proposition 7.9α−1 gives
+�
+i
+|Yi|α−1 ≤ ηm.
+On the other hand, from the hypothesis of the lemma we have �
+i |Yi|α−1 ≥ 4kη > ηm, a
+contradiction. This proves the claim.
+By Claim 2, assume without loss of generality that there is a vertex b on Y(0)
+1
+which is
+close to a vertex c on ¯X. Let b divide Y(0)
+1
+as Y(0)
+1
+= Z1Z2 and up to cyclic shift of X, assume
+67
+
+that X(0) starts at c. Now we can directly apply Lemma 11.2 to the coarse (m + 2)-gon
+(X(0))−1∗Z2u(0)
+1 Y(0)
+2 . . . u(0)
+m−1Y(0)
+m u(0)
+m Z1∗
+and get the required conclusion.
+□
+11.7. Lemma (a cyclic version of Lemma 11.3). Let X be a word strongly cyclically reduced
+in Gα−1. Assume that X is not conjugate in Gα to a power of the root of a relator of rank
+β ≤ α. Next, assume that for any fragment K of rank α in a cyclic shift of X we have
+µf(K) ≤ 1 − 4λ − 8ω.
+Then there exists a word Z conjugate to X in Gα which is strongly cyclically reduced in Gα−1
+and such that no power Zk contains a fragment L of rank α with
+µf(L) < 1
+2 + 2λ + 15ω.
+In particular, Z is strongly cyclically reduced in Gα.
+Proof. The general scheme is the same as in the proof of Lemma 11.3. Let ¯X = �
+i∈Z Xi be
+a line in Γα−1 labeled X∞. First we note that for any fragment K of rank α in ¯X we have
+sX,¯XK ̸∼ K by Proposition 8.16(ii). By Propositions 8.10 and 8.11 there exists a starting
+segment K′ of K that is a fragment of rank α with µf(K′) > µf(K) − λ − 3ω and |K′| ≤ |X|,
+i.e. label(K′) occurs in a cyclic shift of X. Then the hypothesis of the lemma implies that ¯X
+contains no fragments K of rank α with µf(K) ≥ 1 − 3λ − 5ω.
+Denote t =
+1
+2 + 11ω. We can assume that there is at least one fragment K of rank α
+in ¯X with µf(K) ≥ t (otherwise we can take Z := X).
+We choose a maximal set K1,
+. . . , Kr of pairwise non-compatible fragments of rank α in ¯X with µf(Ki) ≥ t such that
+K1 < · · · < Kr < sX,¯XK1 and Kr ̸∼ sX,¯XK1 (after choosing K1 we use Proposition 8.16(ii) to
+get sX,¯XK1 ̸∼ K1). We assume that each Ki has maximal size µf(Ki) in its class of compatible
+fragments of rank α in ¯X. Using Proposition 8.12 we shorten each Ki from its start obtaining
+a fragment ¯Ki of rank α so that all ¯Ki do not intersect pairwise and |K1 ∪Kr| ≤ |X|; we have
+µf(¯Ki) > µf(Ki) − λ − 2.7ω. Passing to a cyclic shift of X (and changing all Xi accordingly)
+we may assume also that
+X0 = ¯K1S1 . . . ¯KrSr.
+Let Pi be the base for ¯Ki and ¯K−1
+i uiPivi a loop in Γα−1 with bridges ui and vi. Denote
+Si ≖ label(Si), Pi ≖ label(Pi), ui ≖ label(ui), vi ≖ label(vi) and let PiQ−1
+i
+be the associated
+relator of rank α. Let
+Z = u1Q1v1S1u2Q1v2S2 . . . urQrvrSr.
+Let Y be a word strongly cyclically reduced in Gα−1 that is conjugate to Z in Gα−1. We
+prove that Y satisfies the requirements of the lemma. Note that Y and hence Z are conjugate
+to X in Gα.
+We transform Z using a procedure analogous to the procedure described in the proof of
+Lemma 11.3. At any moment, we will have a word Z1 of the form
+Z1 = R1U1 . . . RrUr,
+conjugate to Z in Gα−1 where each Ri is a subword of Qi and each Ui either is a bridge of
+rank α − 1 or has the form wiTizi where wi, zi are bridges of rank α − 1 and Ti is a subword
+of Si. At the start, we have Ri = Qi and Ui = viSiui+1 (here and below i + 1 is taken
+68
+
+modulo r). The transformation procedure consists of the following steps applied recursively
+until possible.
+(i) Suppose that Ui has the form wiTizi above. If Ri = R′R′′, Ti = T ′T ′′ where |R′′| +
+|T ′| > 0 and R′′wiT ′ is equal in Gα−1 to a bridge w of rank α −1 then replace Ri, wi
+and Ti with R′, w and T ′′ respectively; similarly, if Ti = T ′T ′′, Ri+1 = R′R′′ where
+|T ′′| + |R′| > 0 and T ′′ziR′ is equal in Gα−1 to a bridge w of rank α − 1 then replace
+Ti, zi and Ri+1 with T ′, w and R′′ respectively.
+(ii) If Ri = R′R′′ and Ri+1 = R∗R∗∗ where |R′′| + |R∗| > 0 and R′′UiR∗ is equal in
+Gα−1 to a bridge w of rank α − 1 then replace Ri, Ui and Ri+1 with R′, w and R∗∗
+respectively.
+Similar to the proof of Lemma 11.3, after performing the procedure we obtain |Ri|α−1 > 4η
+for all i.
+Let ¯Z = �
+i∈Z Z(i) be a line in Gα−1 labeled Z∞ and let Q(i)
+j denote the appropriate subpath
+of Z(i) labeled Qj. We can implement the procedure above on the line ¯Z instead of a word Z
+by changing appropriate paths instead of words (to each change of words in (i) or (ii) there
+corresponds infinitely many changes of paths translated by sX,¯X). As a result, we get a line
+�
+i∈Z Z(i)
+1
+so that the corresponding subpath R(i)
+j
+of Z(i)
+1
+is also a subpath of Q(i)
+j . Denote
+also T(i)
+j
+the appropriate subpath of Z(i)
+1
+labeled Tj. Let ¯Y = �
+i∈Z Y(i) be the line in Gα−1
+such that ¯Z and ¯Y are associated with conjugate words Z and Y . We apply Lemma 11.6
+with ¯X := ¯Y where R(i)
+j
+and T(i)
+j
+play the role of Y(i)
+j ’s and R(i)
+j
+are taken as Y(i)
+tj . According
+to the lemma, each path R(0)
+j
+has a vertex close to a vertex on ¯Y, these vertices on ¯Y are
+ordered along ¯Y in the increasing order of the index j, and the length of the segment of ¯Y
+between the first and the last one is not more that |Y |. Up to cyclic shift of Y , we can write
+Y(0) = W0M1W1 . . . MrWr
+where each Mj is close to a subpath of Q(0)
+j . Taking Mj maximal with these properties we
+obtain, as in the proof of Lemma 11.3,
+µf(Mi) > λ + 3.5ω
+for all j.
+The rest of the proof is similar to the proof of Lemma 11.3.
+□
+11.8. Lemma. If X is a reduced path in Γα and the endpoints of X are close then |X|α ≤ 1.
+Proof. For α ≥ 1 this follows from Lemma 9.22.
+□
+11.9. Lemma. If P is a piece of rank α then for any fragment K of rank α in P we have
+µf(K) ≤ max{λ, µ(P) + 2ω}.
+Proof. Let P be a path in Γα−1 with label(P) ≖ P, let R be the associated relator of rank α
+and let L be the line labeled R∞ extending P.
+Assume that K is a fragment of rank α
+contained in P. If the base axis for K is distinct from L then µf(K) < λ by Corollary 8.2.
+Otherwise the base Q for K is contained in L and Lemma 11.8α−1 implies
+µf(K) = µ(Q) ≤ µ(K) + 2ω ≤ µ(P) + 2ω.
+□
+69
+
+11.10. Proposition. Let P be a piece of rank 1 ≤ β ≤ α with µ(P) ≤ ρ − 2ω. Then P
+is reduced in Gα. If R ≖ QS where R is a relator of rank β then either Q or S is reduced
+in Gα.
+Proof. The first statement follows from Lemmas 10.8 and 11.9. If R is a relator of rank β
+and R ≖ QS then by 4.14(ii), we have either µ(Q) ≤ 1
+2 + ω or µ(S) ≤ 1
+2 + ω. It remains to
+note that 1
+2 + ω < ρ − 2ω.
+□
+Proof of Proposition 11.5. Let X be a word representing an element of Gα. We may assume
+that X is reduced in Gα as a non-cyclic word. We perform a “coarse cyclic cancellation”
+in X: represent X as UX1V where V U is equal in Gα to a bridge u of rank α and X1 has
+the minimal possible length. Let u ≖ v1Pv2 where P is a piece of rank α. We can assume
+that µ(P) ≤
+1
+2 + ω. Let Y be a word cyclically reduced in Gα−1 and conjugate to X1u
+in Gα−1. Note that X1u and hence Y are conjugate to X in Gα. We show that either Y
+is conjugate in Gα−1 to a power Rt
+0 of the root R0 of a relator of rank β ≤ α or no cyclic
+shift of Y contains a fragment K of rank α with µf(K) ≥ ρ + 2λ + 16ω. In the first case, by
+Proposition 11.10 we can assume that Rk
+0 is cyclically reduced in Gα and we come to the first
+alternative of Proposition 11.5. Otherwise, according to Proposition 11.5α−1 we can assume
+that Y is strongly cyclically reduced in Gα−1. Then we apply Lemma 11.7 to find a strongly
+cyclically reduced in Gα word Z conjugate to Y in Gα (note that ρ+2λ+16ω < 1−4λ−8ω),
+coming to the second alternative.
+Let ¯Y = �
+i∈Z Yi and �
+i∈Z X(i)
+1 v(i)
+1 Piv(i)
+2 be lines in Γα−1 representing the conjugacy relation.
+We observe that
+(i) The base axis of any fragment N of rank α in Pi with µf(N) ≥ λ is the infinite
+periodic extension of Pi. In particular, If N1 and N2 are fragments of rank α in Pi
+with µf(Nj) ≥ λ then N1 ∼ N2. (This follows from Corollary 8.2.)
+Now formulate some consequences of the choice of X1 of minimal possible length:
+(ii) There exist no fragments N1 and N2 of rank α in X(i)
+1 and in X(i+1)
+1
+, respectively, such
+that N1 ∼ N2 and µf(Ni) ≥ 3.2ω.
+Indeed, assume that such N1 and N2 do exist. Note that both N1 and N2 are nonempty
+by Lemma 10.8. By Lemma 10.13(i), any two of the endpoints of the images of N1 and N2
+in Γα are close. Then we can shorten X1 to its subword X2 so that X2u′ is conjugate to X
+in Gα for some u′ ∈ Hα contrary to the choice of X1 (see Figure 35a; in the figure we have
+N2 ≪ sY,¯YN1 in X(i+1)
+1
+but in all other cases we can easily find an appropriate path X2 with
+|X2| < X1 and take X2 := label(X2)).
+(iii) There exist no fragments N1 and N2 of rank α in X(i)
+1 and in Pi or Pi−1, respectively,
+such that N1 ∼ N2, µf(N1) ≥ 3.2ω and µf(N2) ≥ λ. (Otherwise using (i) we can
+shorten X1 to X2 := label(X2) as shown in Figure 35b.)
+Let Q be a word reduced in Gα−1 which is equal to X1v1P in Gα−1. We denote Qi the corre-
+sponding path in Γα−1 joining ι(X(i)
+1 ) with τ(Pi). Using (iii), Proposition 8.8 and Lemma 11.9
+we conclude that
+(iv) There are no fragments M of rank α in Qi with µf(M) ≥ ρ + λ + 6.2ω.
+Assume that K is a fragment of rank α in ¯Y with µf(K) ≥ ρ + 2λ + 16ω and |K| ≤ |Y |. By
+(iv) and Proposition 8.9, for some i there are fragments M1 and M2 of rank α in Qi and Qi+1
+70
+
+X(i+1)
+1
+N1
+N2
+sY,¯YN1
+X2
+X(i)
+1
+X(i)
+1
+X(i+1)
+1
+N1
+N2
+Pi
+Pi
+v(i)
+1
+v(i)
+2
+v(i)
+1
+v(i)
+2
+a
+b
+Figure 35.
+respectively such that Mj ∼ K (i = 1, 2) and µf(Mj) > λ + 6.8ω. By Proposition 8.8 there is
+a fragment N1 of rank α such that M1 ∼ N1 and either N1 occurs in X(i)
+1 and µf(N1) > 3.2ω or
+N1 occurs in Pi and µf(N1) > λ. Similarly, there is a fragment N2 of rank α such that M2 ∼ N2
+and either N2 occurs in X(i+1)
+1
+and µf(N2) > 3.2ω or N2 occurs in Pi+1 and µf(N2) > λ. If
+N1 occurs in X(i)
+1
+and N2 occurs in X(i+1)
+1
+we get a contradiction with (ii). If N1 occurs in
+Pi and N2 occurs in X(i+1)
+1
+or N1 occurs in X(i)
+1
+and N2 occurs in Pi+1 we get a contradiction
+with (iii). Finally, if N1 occurs in Pi and N2 occurs in Pi+1 then by (i), we have sY,¯YN1 ∼ N2
+and hence K ∼ sY,¯YK. By Proposition 8.16(i)α−1 this implies that Y is conjugate in Gα−1 to
+a power of the root of a relator of rank α. This finishes the proof.
+□
+11.11. Proposition. Let R be a relator of rank β ≤ α and let R ≖ Rn
+0 where R0 is the root
+of R. Then R0 has order n in Gα.
+Proof. Let k be a proper divisor of n. By Lemma 10.8, Rk
+0 contains no fragments K of rank γ
+with µf(K) ≥ 3.2ω, for all γ = β + 1, . . . , α. By Proposition 11.10β, Rk
+0 is cyclically reduced
+in Gβ and hence also in rank α. Hence Rk
+0 ̸= 1 in Gα.
+□
+11.12. Proposition (conjugate powers of relator roots). Let R be a relator of rank 1 ≤ β ≤ α
+and let R ≖ Rn
+0 where R0 is the root of R. If Rk
+0 = g−1Rl
+0g in Gα for some k, l ̸≡ 0 (mod n)
+then g ∈ ⟨R0⟩ and k ≡ l (mod n).
+Proof. By Proposition 11.11, if Rk
+0 = g−1Rl
+0g in Gα and g ∈ ⟨R0⟩ then k ≡ l (mod n). It
+remains to prove that equality Rk
+0 = g−1Rl
+0g for k, l ̸≡ 0 (mod n) implies g ∈ ⟨R0⟩.
+By Proposition 11.10 we can assume that Rk
+0 and Rl
+0 are cyclically reduced in Gα. We
+represent g by a word Z and consider an annular diagram ∆ of rank α with two cyclic
+sides X1 and X2 labeled R−k
+0
+and Rl
+0 which is obtained from a disk diagram with boundary
+label R−k
+0 Z−1Rl
+0Z by gluing two boundary segments labeled Z−1 and Z. Let Z be the path
+in ∆ with label(Z) ≖ Z that joins starting vertices of X2 and X1.
+We apply to ∆ the reduction process 5.7. By Lemma 4.8, we can replace Z by a new
+path Z1 with the same endpoints such that label(Z1) = Z in Gα (so label(Z1) represents g
+in Gα). We can assume also that ∆ has a tight set T of contiguity subdiagrams.
+Case 1: ∆ has a cell D of rank α. By Proposition 7.13(i), D has a contiguity subdiagram
+Πi ∈ T to each of the sides Xi of ∆. Moreover, if δΠi = SiuiQivi where S−1
+i
+is a contiguity
+arc occurring in δD then µ(Si) > λ. By Lemma 10.8 this implies β = α. Let label(δ∆) ≖ R′
+71
+
+where R′ is a relator of rank α. Consider lines ¯X1, ¯X2 and ¯R in Γα−1 labeled R±∞, R±∞
+and R′∞ which are obtained by mapping the universal cover of the subgraph of ∆ shown
+in Figure 36. By Corollary 8.2 we get ¯X1 = ¯X2 = ¯R. This implies that label(Z1) is equal
+X1
+X2
+Z1
+S1
+Q1
+u1
+v1
+S2
+u2
+Q2
+v2
+R
+¯X1
+¯X2
+D
+Figure 36.
+in Gα−1 to a power of R0, as required.
+Case 2: ∆ has no cells of rank α. Then we have equality Rk
+0 = Z−1
+1 Rl
+0Z1 in Gα−1. If
+β < α then the statement follows from Proposition 11.12α−1. Let β = α. If kl > 0 then
+the statement follows from Proposition 13.8α−1. If kl < 0 then by Corollary 13.10(i)α−1 we
+obtain R0 = g−1R−1
+0 g which contradicts our condition (S3) on the presentation of Gα.
+□
+11.13. Proposition. Every element of Gα of infinite order has the form hm where h is a
+non-power.
+Proof. We need to prove this only in the case α ≥ 1. Let g ∈ Gα be an element of infinite
+order. It is enough to find an upper bound on |m| in the equality of the form g = hm. Up
+to conjugation, we represent g and h by a strongly cyclically reduced in Gα words X and Y
+by Proposition 11.5. Let β be the maximal rank with 1 ≤ β ≤ α such that a cyclic shift
+of X contains a fragment K of rank β with µf(K) ≥ ξ0. (It there is no such K then by
+Proposition 9.16 X in conjugate to Y m in the free group G0 and then |m| ≤ |X|.) Using
+Propositions 10.24(i) and 8.16(ii) we find m pairwise non-compatible fragments M of rank β
+with µf(M) ≥ ξ0 − 2λ − 3.4ω in a cyclic shift of X. This again implies |m| ≤ |X|.
+□
+12. Coarsely periodic words and segments over Gα
+In this section we analyze words which are “geometrically close” in Gα to periodic words.
+In Sections 12 and 13 we use the following notation for numeric parameters:
+ξ1 = ξ0 − 2.6ω,
+ξ2 = ξ1 − 2λ − 3.4ω.
+12.1. Definition. A simple period over Gα is a strongly cyclically reduced word representing
+a non-power element of Gα.
+According to 2.5, if A is a simple period over Gα then any word An is reduced over Gα.
+Proposition 7.6 implies that A has infinite order in Gα.
+72
+
+12.2. Definition. Let A be a simple period over Gα. The activity rank of A is the maximal
+rank β such that an A-periodic word contains a fragment K of rank β ≥ 1 with µf(K) ≥ ξ1
+or it is 0 if no such fragments exist.
+12.3. Case of activity rank 0. The arguments below differ depending on whether the activity
+rank β of a simple period over Gα is positive or 0. However, the difference is only that in
+the case β ≥ 1 we use various conditions on the size µf(F) of fragments F of rank β. All
+definitions, statements and proofs in Sections 12 and 13 apply in cases when the activity
+rank β of a simple period over Gα is 0 simply ignoring conditions of the form µf(·) ≥ . . . for
+fragments of rank β (i.e. assuming that these conditions are all formally true in case β = 0).
+Below we do not distinguish this special case β = 0.
+We will use the following notations. If K and M are fragments of the same rank 0 ≤ β ≤ α
+occurring in a reduced path X in Γγ then K ≲ M means K < M or K ∼ M; similarly, K � M
+means K < M and K ̸∼ M . Note that by Corollary 9.24(ii), for fragments K, M of rank
+β ≥ 1 with µf(K), µf(L) ≥ γ + 2.6ω the relation ‘K ≲ M’ depends only on their equivalence
+classes with respect to compatibility. Thus, for fixed X and β it induces the linear order on
+the set of equivalence classes of ‘∼’ of fragments N of rank β in X with µf(N) ≥ γ +2.6ω. (In
+case β = 0 relation K ≲ M is defined on subpaths on length 1 and means K ≪ M or K = M.)
+12.4. Definition. Let A be a simple period over Gα and β the activity rank of A. A reduced
+path S in Γα is a coarsely periodic segment with period A (or a coarsely A-periodic segment
+for short) if there exists a path P labeled by an A-periodic word, fragments K0, K1 of rank β
+in P and fragments M0, M1 of rank β in S such that:
+• P starts with K0 and ends with K1; S starts with M0 and ends with M1;
+• K0 ∼ M±1
+0 , K1 ∼ M±1
+1
+and K0 ̸∼ K1;
+• µf(Ki) ≥ ξ1, µf(Mi) ≥ ξ2 (i = 0, 1);
+• sA,PK0 ≲ K1 (informally, P “contains at least one period A”).
+The path P is a periodic base for S. The infinite A-periodic extension of P is an axis for S.
+Note that the starting fragment M0 and the ending fragment M1 of S are defined up to
+compatibility.
+Note also that by Lemma 10.13(i) and Proposition 9.10, P and S are close in rank β. In
+particular, if β = 0 then P = Q and thus P is an A-periodic segment.
+We will be assuming that a coarsely A-periodic segment is always considered with a fixed
+associated axis. (In fact, we prove later that the axis of a coarsely A-periodic segment is
+defined in a unique way, see Corollary 13.9). Note that under this assumption, the periodic
+base P for S is defined up to changing the starting and the ending fragments K0 and K1 of
+rank β with compatible ones.
+The label of a coarsely A-periodic segment in Γα is a coarsely A-periodic word over Gα.
+Note that a simple period A over G0 is any cyclically freely reduced word that is not a
+proper power. A coarsely A-periodic word over G0 is simply any A-periodic word P with
+|P| > |A|.
+12.5. Definition. We measure the size of a coarsely A-periodic segment S, which roughly
+corresponds to the number of periods A, in the following way. Let P be the periodic base
+for S and K0, K1 as in Definition 12.4. Then we write ℓA(S) = t where t is the maximal
+integer such that st
+A,PK0 ≲ K1. Thus, we always have ℓA(S) ≥ 1.
+73
+
+Since we consider a fixed associated axis for S, the number ℓA(S) does not depend on the
+choice of a periodic base P.
+If S is a coarsely A-periodic word over Gα then we formally define ℓA(S) to be the maximal
+possible value of ℓA(S) where S is a coarsely A-periodic segment labeled S.
+12.6. Remark. (i) It immediately follows from the definition that t is also the maximal integer
+such that K0 ≲ s−t
+A,PK1. Thus, ℓA(S) = ℓA−1(S−1).
+(ii) To compute ℓA(S) we have to take a path S in Γα with label(S) ≖ S and then choose a
+periodic base P for S so that ℓA(S) is maximal possible; it will follow from Proposition 13.7
+that any choice of P gives in fact the same value ℓA(S).
+12.7. Remark. Up to changing the periodic base P, we can always assume in Definition 12.5
+that both K0 and its translation st
+A,PK0 occur in P. In this case we have |P| ≥ ℓA(S)|A|.
+12.8. Definition. Let S1 and S2 be coarsely A-periodic segments in Γα.
+We say that S1 and S2 are compatible if they have the same axis and strongly compatible
+if they share a common periodic base.
+We use notations S1 ∼ S2 and S1 ≈ S2 for compatibility and strong compatibility respec-
+tively.
+Note that in the case S1 ≈ S2 any periodic base for S1 is a periodic base for S2 and vice
+versa. This easily follows from Definition 12.4.
+If S1 and S2 are coarsely A-periodic segments in Γ0 then S1 ∼ S2 if and only if they have
+a common periodic extension and S1 ≈ S2 if and only if S1 = S2.
+12.9. Proposition. Let S1 and S2 be coarsely A-periodic segments in Γα.
+(i) If S1 ≈ S2 then ℓA(S1) = ℓA(S2).
+(ii) Assume that S1 and S2 occur in a reduced path X in Γα and S1 ∼ S2. Then the union
+of S1 and S2 in X is an A-coarsely periodic segment where a periodic base for S1 ∪S2
+is the union of periodic bases f or S1 and S2 in their common infinite A-periodic
+extension.
+Proof. (i) is immediate consequence of Definition 12.8.
+(ii) follows from Proposition 10.23(ii).
+□
+12.10.
+We describe a procedure of shortening a coarsely A-periodic segment S by a “given
+number k of periods”. Let k ≥ 1 and ℓA(S) ≥ k + 1. Let β be the activity rank of S, let
+P a periodic base for S and let Ki and Mi (i = 0, 1) be starting and ending fragments of
+rank β of P and S respectively as in Definition 13.3. We have K0 < sk
+A,PK0 ≲ s−1
+A,PK1 < K1
+and it follows from Proposition 8.16(ii) that sk
+A,PK0 ̸∼ K0 and sk
+A,PK0 ̸∼ K1. By Proposition
+10.23(i) there exists a fragment N of rank β in S with µf(N′) ≥ ξ2 such that sk
+A,PK0 ∼ N±1.
+Then S1 = N ∪ M1 is an end of S which is a coarsely A-periodic segment with periodic base
+P1 = sk
+A,PK0 ∪ K1 and ℓA(S1) = ℓA(S) − k. We note that:
+(i) The result of the operation is defined up to the strict compatibility.
+(ii) We have P = XP1 where |X| = k|A|.
+(iii) If k ≥ 2 then by Proposition 10.23(i) we can find also a fragment N′ of rank β in S
+with µf(N′) ≥ ξ2 such that sk−1
+A,P K0 ∼ N′±1 and N′ and N are disjoint. Then S = S0uS1
+where S0 = M0 ∪ N′ is a coarsely A-periodic segment with periodic base K0 ∪ sk−1
+A,PK0
+and ℓA(S0) = k − 1.
+74
+
+(iv) The starting position of S1 depends only on the starting position of S; more precisely,
+if S′ is a start of S and S1 and S′
+1 are obtained from S and S′ as above then S′
+1 is a
+start of S1 up to strict compatibility of S′
+1; if S ≈ S′ then S1 ≈ S′
+1.
+12.11. Definition. If S1 is obtained from S by the procedure in 12.10 then we say that S1
+is obtained by shortening of S by t periods from the start. In the symmetric way, we define
+shortening of S by t periods from the end.
+If ℓA(S) ≥ 2t+1 and S′ is obtained from S by applying the operation from both sides then
+S′ is the result of truncation of S by t periods.
+12.12. Definition. We define two numeric parameters associated with a simple period A
+over Gα: the stable size [A]α of A in rank α,
+[A]α = inf
+m≥1
+|(Am)◦|α
+m
+and the stability decrement hα(A):
+hα(A) =
+� 1.2
+[A]α
+�
++ 1.
+If ℓA(S) ≥ 2hα(A) + 1 then the result of truncation of S by hα(A) periods is the stable
+part of S. By claim 12.10(iv) and its symmetric version, the function ‘S → stable part of S’
+respects strict compatibility: if S1 ≈ S2 and S∗
+i is the stable part of Si then S∗
+1 ≈ S∗
+2.
+The basic fact about [A]α and hα(A) is the following observation.
+12.13. Lemma. If X is an A-periodic word and |X| ≥ m|A| then |X|α ≥ m[A]α. In partic-
+ular, if |X| ≥ (hα(A) − 1)|A| then |X|α ≥ 1.2.
+Proof. We have
+|X|α ≥ |Am
+1 |α ≥ |(Am)◦|α ≥ m[A]α
+where A1 is the cyclic shift of A at which X starts. The second statement follows from the
+first.
+□
+The principal role of the stable part is described by the following proposition.
+12.14. Proposition (stability of coarsely periodic words). Let S be a coarsely A-periodic
+segment in Γα with ℓA(S) ≥ 2hα(A) + 1 and let S∗ be the stable part of S. If X and Y are
+close reduced paths in Γα and S is a subpath of X then Y contains a coarsely A-periodic
+segment T such that T ≈ S∗.
+Proof. Let P and P∗ be periodic bases for S and S∗ respectively. Let β be the activity rank
+of A and let Ki and Mi (i = 0, 1) be fragments of rank β in P and in S, respectively, from
+Definition 13.3 applied to P and S. Denote t = hα(A).
+Let X and Y be as in the proposition. If α = 0 then X = Y and there is nothing to prove.
+Let α > 0. We claim that P = z1P′z2 where P′ is close in rank β to a subpath of Y and
+|zi|α < 1.2. Indeed, if β = α then it easily follows from Proposition 10.6 and Lemma 10.13(i)
+that P is already close to a subpath of Y. If β < α then we observe that S contains no
+fragments K of rank γ with β < γ ≤ α and µf(K) ≥ ξ0 due to the definition of the activity
+rank and Proposition 8.7≤α. Then the claim follows by Proposition 10.22.
+75
+
+By Lemma 12.13 we have |zi| < (t−1)|A|. This implies that st−1
+A,PK0 ∪s−t+1
+A,P K1 is contained
+in P′. Note that P∗ = st
+A,PK0 ∪ s−t
+A,PK1 where µf(K0), µf(K1) ≥ ξ1. Then by Proposition
+10.23(i) we find a subpath T which is a coarsely A-periodic segment with periodic base P∗
+and, consequently, we have T ≈ S∗.
+□
+We use parameter hα(A) also in several other situations.
+12.15. Proposition. Let P be a periodic segment in Γα with a simple period A over Gα.
+Assume that |P| ≥ m|A| where m ≥ 2hα(A) + 3. Let X be a reduced path in Γα such that P
+and X are close. Then there exist a subpath P1 of P and a subpath X1 of X such that X1 is a
+coarsely A-periodic segment with periodic base P1 and ℓA(X1) = m − 2hα(A) − 2.
+Proof. Let β be the activity rank of A. Using Corollary 9.13 and Lemma 12.13 we find close
+in rank β subpaths P2 of P and X2 of X with |P2| ≥ m−2hα(A)+2. By Proposition 8.16(iii)
+any fragment K of rank β in P with µf(K) ≥ 2λ + 5.3ω satisfies |K| < 2|A|, so according
+to Definition 12.4 there exists a fragment K of rank β in P with µf(K) ≥ ξ1. Shortening K
+from the end by Proposition 8.12 if β ≥ 1 and using again Proposition 8.16(ii) we find a
+fragment K1 of rank β with µf(K1) > ξ1 − λ − 2.7ω that is a start of K disjoint from sA,PK;
+hence |K1| ≤ |A|. We can assume that K occurs in P2 and is closest to the start of P2. Then
+P2 contains m−2hα(A) translates si
+A,PK of K for i = 0, . . . , m−2hα(A)−1 and contains also
+sm−2hα(A)
+A,P
+K1. Applying Proposition 10.23(i) we find fragments Mi (i = 1, . . . , m−2hα(A)−1)
+of rank β in X2 with µf(Mi) ≥ ξ2 such that si
+A,PK ∼ M±1
+i . Then X1 = M1 ∪ Mm−2hα(A)−1 is a
+coarsely A-periodic segment with periodic base sA,PK ∪ sm−2hα(A)−1
+A,P
+K and we have ℓA(X1) =
+m − 2hα(A) − 2.
+□
+12.16. Proposition. Let S be a coarsely A-periodic word over Gα and B a simple period
+over Gα conjugate to A. Let ℓA(S) ≥ 2hα(A) + 3. Then a subword T of S is a coarsely
+B-periodic word over Gα with ℓB(T) ≥ ℓA(S) − 2hα(A) − 2.
+Proof. We represent S by a coarsely A-periodic segment S in Γα. Let P a periodic base for S,
+let L1 be the axis of S and let L2 be the B-periodic line parallel to L1. Denote β1 and β2
+activity ranks of A and B respectively.
+According to Definition 12.2, either L1 or L2 contains no fragments K of rank γ with
+β1 < γ ≤ α and µf(K) ≥ ξ1. Let K0 and K1 be fragments of rank β1 with µf(Ki) ≥ ξ1 that are
+a start and an end of P respectively. We have sℓA(S)
+A,L1 K0 ≲ K1. By Proposition 10.24(i), there
+exist fragments M0 and M1 of rank β1 in L2 with µf(Mi) ≥ ξ2 such that Ki ∼ M±1
+i . Since L1
+and L2 are parallel, we have sA,L1 = sB,L2 and hence sℓA(S)
+B,L2 M0 ≲ M1 by Proposition 10.24(ii).
+Then Q = M0 ∪ sℓA(S)
+B,L2 M0 ∪ M1 is close in rank β1 to P, |Q| ≥ ℓA(S) and the statement follows
+by Proposition 12.15.
+□
+13. Overlapped coarse periodicity
+The main result of this section is Proposition 13.4 which can be thought as an analog
+of a well known property of periodic words: if two periodic words have a sufficiently large
+overlapping then they have a common period. We need such an analog in a more general
+context where closeness plays the role of overlapping. As a main technical tool, instead of
+coincidence of letters in the overlapping case we use correspondence of fragments of rank β ≤
+76
+
+α in strictly close in rank β segments in Γα given by Proposition 10.23. A difficulty is caused
+by the “fading effect” of this correspondence: a fragment size can decrease when passing
+from one segment to the other. To overcome this difficulty, we use a special combinatorial
+argument [9, Lemma 6.4].
+13.1. Lemma (penetration lemma, [9, Lemma 6.4]). Let S0, S1, . . . , Sk be a finite collection
+of disjoint sets. Assume that the following assertions hold:
+(i) Each Si is pre-ordered, i.e. endowed with a transitive relation ‘<i’.
+(ii) There is an equivalence relation a ∼ b on the union �
+i Si such that for any a, b in
+the same set Si we have either a <i b, b <i a or a ∼ b; in other words, we have an
+induced linear ordering on the set of equivalence classes on each Si.
+(iii) We assume that the equivalence preserves the pre-ordering in neighboring sets: if
+a, b ∈ Si, a′, b′ ∈ Si+1, a ∼ a′ and b ∼ b′ then a <i b ⇔ a′ <i+1 b′.
+If c ∈ Si, a, b ∈ Sj and a ≲j b (where a ≲j b denotes ‘a <j b or a ∼ b’) then we
+say that c penetrates between a and b if there exists c′ ∼ c such that a ≲j c′ ≲j b.
+(iv) There is a subset of �
+i Si of stable elements that have the following property: if
+c ∈ Si is stable, a ≲i c ≲i b, a′, b′ ∈ Sj, a′ ≲j b′, a ∼ a′ and b ∼ b′ then c penetrates
+between a′ and b′.
+(v) For each i ≤ k − 1, there are stable elements ai, bi ∈ Si and a′
+i, b′
+i ∈ Si+1 such that
+ai ∼ a′
+i, bi ∼ b′
+i and ai <i bi.
+Finally, let c0 ∈ S0 be stable and a0 ≲0 c0 ≲0 b0. Assume that c0 penetrates between ai
+and bi for each i = 1, 2, . . . , k − 1. Then c0 penetrates between ak and bk.
+The following observation is a special case of [9, Lemma 6.2].
+13.2. Lemma. Suppose a group G acts on set X. Let g, h ∈ G, x0, x1, . . . , xt ∈ X and for
+some r, s ≥ 0 with gcd(r, s) = 1 and r + s ≤ t,
+gxi = xi+r (i = 0, 1, . . . , t − r),
+hxi = xi+s (i = 0, 1, . . . , t − s).
+Assume that the stabilizer H of x0 is malnormal in G. Then either g, h ∈ H (and hence
+x0 = x1 = · · · = xt) or there exists d ∈ G such that g = dr and h = ds.
+Proof. Induction on r +s. We can assume that r ≤ s. If r > 0 then we have g−1hxi = xi+s−r
+for 0 ≤ i ≤ t − s and the statement follows from the inductive hypothesis with h := g−1h,
+s := s−r and t := t−r. Otherwise we have r = 0 and s = 1. Then h−1ghx0 = gx0 = x0 and
+by malnormality of H, we have either g, h ∈ H or g = 1 (and then g = h0 and h = h1).
+□
+13.3. Definition. Let X and Y be reduced paths in Γα. We say that X and Y are strictly
+close in rank β ≤ α if there are fragments K0, K1 of rank β in X and fragments M0, M1 of
+rank β in Y such that:
+• µf(Ki), µf(Mi) ≥ ξ2 (i = 0, 1).
+• X starts with K0 and ends with K1; Y starts with M0 and ends with M1;
+• K0 ∼ M±1
+0 , K1 ∼ M±1
+1
+and K0 ̸∼ K1.
+By Lemma 10.13(i), paths which are strictly close in rank β are also close in rank β. One
+of the advantages of strict closeness is that this relation is transitive (this follows immediately
+from Definition 13.3). Note that a coarsely periodic segment P in Γα and its periodic base S
+are strictly close according to Definition 12.4 (and the condition in Definition 12.4 is slightly
+77
+
+stronger because of the lower bound on the size of the starting and the ending fragments
+of S).
+13.4. Proposition. Let A be a simple period over Gα, β the activity rank of A and Pi
+(i = 0, 1) be two A-periodic segments in Γα. Let Si (i = 0, 1) be a reduced path in Γα which
+is strictly close to Pi. Assume that S0 is contained in S1. Assume also that P0 contains at
+least one period A in the sense that there exist fragments K and K′ of rank β in P0 such that
+µf(K), µf(K′) ≥ ξ2 and K′ ∼ sA,P0K. Then P0 and P1 have a common periodic extension.
+Proof. Denote
+ξ3 = ξ2 − 2λ − 3.4ω = 3λ − 10.9ω.
+Throughout the proof, “fragment M” means “fragment M of rank β with µf(M) ≥ ζ3” (or
+simply “fragment M of rank 0” if β = 0, see 12.3).
+Let a line Li be the infinite periodic extension of Pi and let g be an element of Gα such
+that L1 = gL0, so sA,P1 = gsA,P0g−1. Our argument relies on establishing a correspondence
+between fragments of rank β in Pi and Si. It will be convenient to consider fragments of
+rank β in four paths Pi and Si as four disjoint sets, i.e. we will formally consider pairs (M, X)
+where X ∈ {P0, P1, S0, S1} and M is a fragment occurring in X. We will refer to M as a
+“fragment belonging to X” or simply as a “fragment in X”.
+We introduce two operations on fragments in Pi and Si. Let M and N be fragments each
+belonging to some Pi or Si.
+(i) If M belongs to Pi, N belongs to Si and M ∼ N±1 then either of M and N jumps to
+the other.
+(ii) M translates to N in the following cases (a)–(d):
+(a) M and N belong to the same Pi and N ∼ sk
+A,PiM for some k ∈ Z; or
+(b) M belongs to P0, N belongs to P1 and N ∼ gsk
+A,P0M for some k ∈ Z; or
+(c) M belongs to P1, N belongs to P0 and N ∼ g−1sk
+A,P1M for some k ∈ Z.
+(In other words, M translates to N in cases (a)–(c) if they have the same position in
+their corresponding periodic lines Li with respect to the period A up to compatibil-
+ity.)
+(d) An “identical” case: M ∼ N and they belong to some Si and Sj respectively.
+Note that the two operations are reversible and are defined up to compatibility.
+Let K and K′ be fragments in P0 such that µf(K), µf(K′) ≥ ξ1 and K′ ∼ sA,P0K, as assumed
+in the proposition. Let M be a maximal set of pairwise non-compatible fragments which can
+be obtained by operations (i) and (ii) starting from K. By Proposition 8.10, neither of any
+two fragments in M is contained in the other, so M is a finite set.
+The following assertion is the principal step of the proof.
+Claim: The jump operation is always possible inside M; that is, for any M ∈ M in Pi or
+in Si, i ∈ {0, 1}, there exists a fragment N of rank α in Si or, respectively, in Pi such that
+M ∼ N±1.
+Proof of the claim. We assume that some M ∈ M is given and prove existence of the
+required N. The proof will consist of application of Lemma 13.1. We do a necessary prepa-
+ration.
+According to the definition of M, there is a sequence T0 = K, T1, . . . , Tl = M of fragments
+Tj ∈ M such that Tj+1 is obtained from Tj by one of the operations (i) or (ii). We can
+78
+
+assume that the sequence has no two translations in a row (otherwise we can replace them
+by a single translation) and has no two jumps in a row (otherwise they eliminate). Assume
+also for convenience that T0 → T1 is a translation (by inserting a trivial translation if
+needed). Thus for each i, T2j translates to T2j+1 and T2j+1 jumps to T2j+2. We can assume
+that the last step Tl−1 → Tl is a translation, so l = 2k − 1 for some k.
+Now roughly speaking, we move all fragments Tj along with the corresponding paths Pi
+or Si belonging them, to the same location up to compatibility. We define a sequence Y0,
+Y1, . . . , Yk of paths in Γα and a sequence Wj of fragments in Yj for j = 0, 1, . . . , k − 1. For
+each j we will have Wj = fjT2j+1 for some fj ∈ Gα. The definition of Yj and fj goes as
+follows.
+Denote (X1, X2, X3, X4) = (P0, S0, P1, S1) and let J(i) denote the index such that a fragment
+in Xi jumps to a fragment in XJ(i) (i.e. (J(1), J(2), J(3), J(4)) = (2, 1, 4, 3)). Denote also I(j)
+the index such that T2j−1 belongs to XI(j). Thus, T2j belongs to XJ(I(j)).
+We start with Y0 = XI(0) and W0 = T1, so f0 = 1. Assume that j < k−1 and Yj and fj are
+already defined. If T2j → T2j+1 is a translation by (a)–(c) then there exists fj+1 ∈ Gα such
+that fj+1XI(j+1) and fjXJ(I(j)) belong to the same A-periodic line and fj+1T2j+1 ∼ fjT2j.
+We take Yj+1 = fj+1XI(j+1) ∪ fjXJ(I(j)). Otherwise T2j → T2j+1 is a translation by (d), i.e.
+XJ(I(j)) is either S0 or S1. In this case we take fj+1 = fj and Yj+1 = fjS1. Finally, define
+Yk = fkXJ(I(k−1)). We have fj+1T±1
+2j+2 ∼ fj+1T2j+1 ∼ fjT2j for all j = 0, 1, . . . , k − 2 and
+hence W0 ∼ W±1
+1
+∼ · · · ∼ W±1
+k−1. Figure 37 illustrates the construction.
+W0
+W1
+W2
+W3
+Figure 37.
+By strict closeness of pairs (P0, S0) and (P1, S1), each Xi starts with a fragment Ui and
+ends with a fragment Vi such that µf(Ui), µf(Vi) ≥ ξ2, Ui ̸∼ Vi and we have Ui ∼ U±1
+J(i) and
+Vi ∼ V±1
+J(i)
+We now apply Lemma 13.1 where:
+• Sj is the set of all fragments N in Sj with µf(N) ≥ ξ3.
+• N <i N′ is defined as ‘N ̸∼ N′ and N < N′ in Sj’.
+• Equivalence of N, N′ ∈ �
+j Sj is defined as N ∼ N′±1.
+• N ∈ �
+j Sj is defined to be stable iff µf(N) ≥ ξ2.
+• For aj, bj, a′
+j and b′
+j we take appropriate translates of Ui and Vi, namely, fjUI(j),
+fjVI(j), fjUJ(I(j)) and fjVJ(I(j)) respectively.
+We have conditions (i)–(v) of Lemma 13.1 satisfied: condition (i) holds in case β ≥ 1 by
+Corollary 9.24(ii), condition (ii) holds by Proposition 8.10β, conditions (iii) and (iv) hold by
+79
+
+Proposition 10.23 in view of the inequality ξ3 ≥ 2λ + 9.1ω and, finally, condition (v) holds
+immediately by construction.
+For c0 in Lemma 13.1 we take T1. Note that up to compatibility, we can assume that
+µf(T1) ≥ ξ2, so T1 is stable. (By construction, T1 is obtained from T0 = K by translation
+to XI(0); if T1 is compatible with the starting or the ending fragment of XI(0) then we can
+assume µf(T1) ≥ ξ2 due to Definition 13.3; otherwise we can assume that T1 is a literal
+translation of K and then µf(T1) = µf(K) ≥ ξ1.) Since T1 = W0 ∼ W±1
+1
+∼ · · · ∼ W±1
+k−1
+and each Wj occurs in fjXI(j), T1 penetrates between each pair fjUI(j) and fjVI(j) for j =
+0, 1, . . . , k − 1. All the hypotheses of Lemma 13.1 are satisfied and applying it we find a
+fragment Wk in fk−1XJ(I(k−1)) such that W±1
+k
+∼ Wk−1 = fk−1M. Then M → f −1
+k−1Wk is the
+required jump. This finishes the proof of the claim.
+We finish the proof of the proposition. Let K0 = K, K1, . . . , Km ∼ sA,P0K be all fragments
+in M between K and sA,P0K in their natural order, i.e. we have K0 < K1 < · · · < Km. Let
+M0, . . . , Mm ∈ M be fragments in P1 such that Mi ∼ K±1
+i
+for all i (each Mi is obtained
+from Ki by two jumps). Note that M0 < M1 < · · · < Mm by Proposition 10.23. Since M
+is closed under translations, the number of fragments in M between M0 and sA,P1M0 is the
+same as the number of fragments in M between K and sA,P0K, i.e. we have Mm ∼ sA,P1M0.
+This implies that K0 translates to some Mq, i.e. Mq ∼ gst
+A,P0K±1
+0
+for some t and hence
+Mi+q ∼ gst
+A,P0K±1
+i
+for i = 0, 1, . . . , m − q,
+Mi+q−m ∼ gst−1
+A,P0K±1
+i
+for i = m − q + 1, . . . , m.
+Note that gcd(m.q) = 1 since M is generated by a single fragment K. By Propositions 8.16(i),
+11.12 and Corollary 9.24(iii), the subgroup {g ∈ Gα | gM0 ∼ M±1
+0 } is malnormal in Gα. We
+now apply Lemma 13.2 where for xi we take the equivalence class of Mi in the set of fragments
+of rank β in Γα under compatibility up to invertion. By the lemma, ⟨g, sA,P0⟩ is cyclic. Since
+A is a non-power, we get g ∈ ⟨sA,P0⟩ which means that L1 = L2.
+□
+As an immediate consequence of Proposition 13.4 we get:
+13.5. Corollary (overlapping coarse periodicity). Let S0 and S1 be coarsely periodic segments
+in Γα with the same simple period A over Gα. If S0 is contained in S1 then S0 ∼ S1.
+13.6. Corollary. Let S and T be non-compatible coarse periodic segments in Γα with the
+same simple period A which occur in a reduced path X. Let ℓA(S) ≥ 3. Assume that S1 is
+obtained from S by shortening by 2 periods from the end if S < T or by shortening by 2
+periods from the start if S > T. Then S1 and T are disjoint.
+Proof. Without loss of generality, we assume that S < T and S1 is obtained from S by
+shortening by 2 periods from the end. By 12.10(iii) we have S = S1uS2 where S2 is a coarsely
+A-periodic segment with S2 ∼ S. B y hypothesis we have S2 ̸∼ T and then by Corollary 13.5,
+neither of S2 or T is contained in the other. This implies that S1 and T are disjoint.
+□
+13.7. Proposition (strictly close periodic paths with one period). Let A be a simple period
+over Gα and β the activity rank of A. Let P0 and P1 be strictly close in rank β paths in Γα
+labeled by periodic words with period A. Assume that there exist fragments K, K′ of rank β
+in P0 such that µf(K), µf(K′) ≥ ξ2 and sA,P0K ∼ K′. Then P0 and P1 have a common periodic
+extension.
+Proof. This is a special case of Proposition 13.4 with S0 = S1 = P1.
+□
+80
+
+13.8. Proposition. Let g ∈ Gα be a non-power of infinite order and let h ∈ Gα. If gk =
+h−1glh for some k, l > 0 then h ∈ ⟨g⟩ and k = l.
+Proof. By Proposition 11.5, up to conjugation we can assume that g is represented by a
+simple period A over Gα. It is enough to prove that h ∈ ⟨A⟩.
+Consider two periodic lines L0 and L1 in Gα with period A which represent the conjugacy
+relation. We have h ∈ ⟨A⟩ if and only if L0 = L1. Let β be the activity rank of A. By
+Proposition 10.24 we find strictly close in rank β subpaths Pi of Li with any desired bound
+|P0| ≥ t|A|. Then the statement follows from Proposition 13.7.
+□
+As an immediate consequence we get:
+13.9. Corollary. Let S0 and S1 be coarsely A-periodic segments in Γα and Li (i = 1, 2) be an
+axis for Si. If S0 ∼ S1 then L1 = L2.
+13.10. Corollary. Let g ∈ Gα be an element of infinite order. Then the following is true.
+(i) g has the unique root; i.e. there exists a unique non-power element g0 ∈ Gα such
+that g = gt
+0 for some t ≥ 1.
+(ii) If hr ∈ ⟨g⟩ and hr ̸= 1 then h ∈ ⟨g0⟩ where g0 is the root of g.
+(iii) If g is conjugate to g−1 then g is the product of two involutions.
+Proof. (i) is direct consequence of Propositions 11.13 and 13.8.
+(ii) follows from (i) and Proposition 13.8 because gt
+0 = hr implies gt
+0 = h−1gt
+0h.
+(iii) Assume that g = h−1g−1h.
+From g = h−2gh2 we conclude that h2 = 1 by (ii).
+Similarly, we have (hg)2 = 1 and then g = h · hg.
+□
+13.11. Corollary. Assume that each relator R of each rank β ≤ α has the form R = Rn
+0
+where R0 is the root of R and n is odd (n can vary for different relators R). Then Gα has
+no involutions and no element of Gα is conjugate to its inverse.
+Proof. By Proposition 11.5, any element of finite order of Gα is conjugate to some power
+Rt
+0 of the root R0 of a relator R of rank β ≤ α. By Proposition 11.11, Rt
+0 has an odd
+order and cannot be an involution. The second statement follows from the first by Corollary
+13.10(iii).
+□
+13.12. Lemma. Let P be an A-periodic segment in Γα with a simple period A over Gα. Let
+S be a coarsely periodic segment in P with another simple period B over Gα and assume that
+A and B are not conjugate in Gα. Then the following is true.
+(i) S ̸∼ st
+A,PS for any t ̸= 0.
+(ii) If ℓB(S) ≥ 3 then |S| < 2|A|.
+Proof. (i) Assume that S ∼ st
+A,PS for some t ̸= 0. Let L1 be the infinite periodic extension
+of P, and let L2 be the axis for K. By Corollary 13.9 we have L2 = st
+A,PL2, so st
+A,P = sr
+B,Q for
+some r ̸= 0. Since A and B are non-powers, by Corollary 13.10(ii) sε
+A,P = sB,Q for ε = ±1
+and hence Lε
+1 and L2 are parallel. From the fact that S is a subpath of P we easily deduce by
+Proposition 10.23 (taking for β the activity rank of A) that ε = 1. We obtain a contradiction
+with the assumption that A and B are not conjugate in Gα.
+(ii) By 12.10(iii) we represent S as S = S1uS2 where S1 and S2 are coarsely periodic
+segments with period B and ℓB(S1) ≥ ℓB(S) − 2. By (i) and Corollary 13.5, s−1
+A,PS does not
+contain S1 and sA,PS does not contain S2. This implies |S| < 2|A|.
+□
+81
+
+13.13. Proposition. Let P and Q be close periodic segments in Γα with the same simple
+period A over Gα. If |P| ≥ (2hα(A) + 1)|A| (where hα(A) is defined in 12.12) then P and Q
+belong to the same A-periodic line.
+Proof. Follows from Propositions 12.15 and 13.7.
+□
+We finish the section by formulating technical statements which we will need in the con-
+struction of relations of Burnside groups. We use notation S ⪅ T for ‘S < T or S ≈ T’.
+13.14. Lemma. Let S and T be coarsely A-periodic segments occurring in a reduced path X
+in Γα. Assume that some periodic bases for S and T have the same label. If S is contained
+in T then S ≈ T.
+Proof. Assume that S is contained in T. Let Pi (i = 1, 2) be periodic bases for S and T
+respectively, with label(P1) ≖ label(P2). Let β be the activity rank of A. By Proposition 13.4,
+P1 and P2 have a common periodic extension. Let Ki and Mi (i = 0, 1, 2, 3) be fragments
+of rank β with µf(Ki), µf(Mi) ≥ ξ2 such that P1 = K0 ∪ K1, P2 = K2 ∪ K3, S = M0 ∪ M1,
+T = M2 ∪M3 and Ki ∼ Mi for all i. We have M2 ≲ M0 � M1 ≲ M3 which by Proposition 13.4
+implies K2 ≲ K0 and K1 ≲ K3. Now from label(P1) ≖ label(P2) we conclude that K2 ∼ K0
+and K1 ∼ K3, i.e. S ≈ T.
+□
+13.15. Lemma. Let X and Y be close reduced paths in Γα. Let S0, S1 be coarsely A-periodic
+segments in X and T0, T1 be coarsely A-periodic segments in Y such that ℓ(Si) ≥ 2hα(A) + 1,
+Si ≈ Ti for i = 0, 1 and S0 ̸∼ S1. Then S0 < S1 if and only if T0 < T1.
+Proof. By Corollary 13.5, none of S0 and S1 is contained in the other and the same is true
+for T0 and T1. Assume, for example, that S0 < S1 and T1 < T0. Let X1 and Y1 be the
+starting segments of X and Y ending with S1 and S2 respectively. By Proposition 12.14 with
+X := X1 and Y := Y1 there exists U in Y1 such that U ≈ S∗
+0 where S∗
+0 is the stable part of S0.
+Then U ∪ T0 is a coarsely A-periodic segment containing T1 and we get a contradiction with
+Corollary 13.5.
+□
+13.16. Lemma. Let X and Y be reduced paths in Γα. Let S0, S1 be coarsely A-periodic seg-
+ments in X and T0, T1 be coarsely A-periodic segments in Y such that S0 ⪅ S1, T0 ⪅ T1 and
+Si ≈ Ti, i = 0, 1.
+(i) Let U be a coarsely A-periodic segment in X such that S0 ⪅ U ⪅ S1, ℓA(U) ≥ hα(A)+1
+and U is the stable part of some other coarsely A-periodic segment in X. Then there
+exists a coarsely A-periodic segment V in Y such that T0 ⪅ V ⪅ T1 and U ≈ V.
+(ii) Let Ui (i = 1, 2) be coarsely A-periodic segments in X and Vi (i = 1, 2) be coarsely
+A-periodic segments in Y such that ℓA(Ui) ≥ 2hα(A) + 1 (i = 1, 2), S0 ⪅ Ui ⪅ S1,
+T0 ⪅ Vi ⪅ T1 and Ui ≈ Vi for i = 1, 2. Assume that U2 ≈ gU1 for some g ∈ Gα,
+i.e. U1 and U2 have periodic bases with the same label. Then U1 ⪅ U2 if and only if
+V1 ⪅ V2.
+Proof. Let β be the activity rank of A.
+(i): Let U be the stable part of ¯U and ¯U = Z1UZ2. We consider several cases.
+Case 1: U ̸∼ Si for i = 0, 1. Then by Corollary 13.5 we have S0 < ¯U < S1. Since S0 ∪ S1
+and T0 ∪ T1 are close, existence of V follows from Proposition 12.14.
+Case 2: Exactly one of the relations U ∼ Si (i = 0, 1) holds. Without loss of generality,
+assume that U ∼ S0 and U ̸∼ S1. By Corollary 13.5 we have ¯U < S1. If U ≈ S0 there is
+nothing to prove. Assume that U ̸≈ S0 and hence UZ2 is contained in S0 ∪ S1.
+82
+
+By the construction of the stable part, UZ2 is a coarsely A-periodic segment with ℓA(UZ2) =
+ℓ(U) + hα(A) ≥ 2hα(A) + 1. Let W be the stable part of UZ2. Using Proposition 12.14 with
+X := S0 ∪S1 and Y := T0 ∪T1 we find a coarsely A-periodic segment W′ in T0 ∪T1 such that
+W ≈ W′. By Proposition 12.9(ii),
+S0∪U is a coarsely A-periodic segment and since W′ ∼ T0, T0∪W′ is a coarsely A-periodic
+segment as well. By 12.10(iv) (more formally, by the symmetric version of 12.10(iv)) W is
+an end of U which implies S0 ∪ U ≈ T0 ∪ W′.
+Now let P be a periodic base for U. By the construction of the stable part, P starts with
+a fragment N of rank β with µf(N) ≥ ξ1. Since P is contained in a periodic base for T0 ∪ W′,
+by Proposition 10.23 we find a fragment N′ of rank β in T0 ∪ W′ such that µf(N′) ≥ ξ2 and
+N′ ∼ N. Then for the desired V we can take the end of T0 ∪ W′ starting with N′.
+Case 3: U ∼ S0 ∼ S1. Then a periodic base P for U is contained in a periodic base for
+S0 ∪ S1. By the construction of the stable part, P starts and ends with fragments N0 and N1
+of rank β with µf(Ni) ≥ ξ1. Then using Proposition 10.23 we find fragments N′
+i (i = 0, 1) of
+rank β in T0 ∪ T1 such that µf(N′
+i) ≥ ξ2 and N′
+i ∼ Ni (i = 1, 2). We can take V = N′
+0 ∪ N′
+1.
+(ii): We consider two cases.
+Case 1: U1 ∼ U2. Let P1 and P2 be periodic bases for U1 and U2 with label(P1) ≖ label(P2)
+which have a common periodic extension. It easily follows from Proposition 10.23(ii) that
+U1 < U2 ⇔ P1 < P2 and U1 ≈ U2 ⇔ P1 = P2. Since Pi is also a periodic base for Vi, a
+similar statement holds for Vi’s which clearly implies the required conclusion.
+Case 2: U1 ̸∼ U2. Without loss of generality, we assume that U1 < U2, V1 > V2 and
+come to a contradiction. We can assume also that X = S0 ∪ S1, Y = T0 ∪ T1 and hence X
+and Y are close in rank α. Let U∗
+i and V∗
+i be stable parts of Ui and Vi. By Corollary 13.6,
+U1 is disjoint from U∗
+2. Let X = X1U1X2U∗
+2X3 and Y = Y1V∗
+2Y2V1Y3. By Proposition 12.14
+with X = X1U1X2 and Y := Y1 there exists a coarsely A-periodic segment W in Y1 such
+that W ≈ U∗
+1. Then W ∼ U1 ∼ V1 and by Proposition 12.9(ii) and Corollary 13.5 we get
+U1 ∼ W ∼ W ∪ V1 ∼ V2 ∼ U2, the desired contradiction.
+□
+14. Comparing α-length of close words
+In this section, we prove the following proposition.
+14.1. Proposition. Let X, Y ∈ Rα be close in rank α. Then
+|Y |α < 1.3|X|α + 2.2.
+Recall that a fragment word F of rank α is considered with fixed associated words S, u, v
+and a relator R of rank α such that F = uSv in Gα−1, u, v ∈ Hα−1 and S is a subword of Rk
+for some k > 0. If F is a path in Γα−1 labeled F then this uniquely defines the base S for F.
+Let F and G be fragments of rank α in a word X. Let X be a path in Γα−1 labeled X and
+F, G the corresponding subpaths of X. We write F ∼ G if F ∼ G (so the relation is formally
+defined for the occurrences of F and G in X).
+Recall that the size |X|α of a word X in rank α is the minimal possible value of weightα(F)
+of a fragmentation F of rank α of X. A fragmentation F of rank α of X is a partition
+X ≖ F1 · F2 · · ·Fk where Fi is a nonempty subword of a fragment of rank β ≤ α. Assuming
+that each Fi is assigned a unique value of β, the weight in rank α of F is defined by formula
+weightα(F) = mα + ζmα−1 + ζ2mα−2 + · · · + ζαm0
+83
+
+where mβ is the number of subwords of fragments of rank β in F.
+We call a fragmentation F of X minimal if weightα(F) = |X|α.
+We call a subword F of a fragment of rank β ≥ 1 a truncated fragment of rank β. We
+will be assuming that with a truncated fragment F of rank α there is an associated genuine
+fragment ¯F of rank β such that F is a subword of ¯F. If F is a path in Γα with label(F) ≖ F
+then we have the associated fragment ¯F in Γα such that F is a subpath of ¯F. Note a truncated
+fragment of rank 1 is simply a fragment of rank 1.
+We extend the compatibility relation to truncated fragments of rank β in a word X in the
+following natural way. If F and G are truncated fragments of rank β in X and ¯F and ¯G
+their associated fragments of rank β in Γα then F ∼ G if and only if ¯F ∼ ¯G.
+14.2.
+Let F = F1 · F2 · . . . · Fk be a fragmentation of rank α of a word X. Let Fi be a
+truncated fragment of rank β ≥ 1 in F. Assume that Fi can be extended in X to a larger
+truncated fragment G of rank β, i.e.
+X ≖ F1F2 . . . F ′
+pF ′′
+p . . . Fi . . . F ′
+qF ′′
+q . . . Fk
+where Fp ≖ F ′
+pF ′′
+p , Fq ≖ F ′
+qF ′′
+q and G ≖ F ′′
+p . . . Fi . . . F ′
+q (here we consider the case 1 < i < k;
+cases i = 1 and i = k differ only in notation). Then we can produce a new fragmentation F′
+of rank α, X ≖ F1 · · ·Fp−1 · [F ′
+p] · G · [F ′′
+q ] · Fq+1 · · · Fk where square brackets mean that F ′
+p
+and F ′′
+q are absent if empty. We say that F′ is obtained from F by extending Fi to G. Note
+that if F is minimal then in the case i > 1, we necessarily have p = i − 1 and nonempty F ′
+p
+and in the case i < k we necesarily have q = i + 1 and nonempty F ′′
+q .
+14.3. Lemma. Let F = F1 · F2 · . . . · Fk be a minimal fragmentation of rank α ≥ 1 of a word
+X ∈ Rα.
+(i) Let Fi be a truncated fragment of rank α in F. Then |Fi|α−1 ≥ 1
+ζ and Fi = uFv
+where F is a fragment of rank α, Fi ∼ F, |u|α−1, |v|α−1 < ζ and the base P for the
+corresponding fragment F in Γα−1 satisfies |P|α−1 > 13.
+(ii) If K is a fragment of rank α in X and µf(K) ≥ 3λ + 15ω then Fi ∼ K for some i.
+(iii) Let X = P0K1P1 . . . KrPr where Ki are fragments of rank α with µf(Ki) ≥ 3λ + 13ω
+for all i. Then there exists another minimal fragmentation F′ of rank α of X such
+that each Ki is contained in a compatible truncated fragment of rank α in F′.
+Proof. (i) If |Fi|α−1 < 1
+ζ then we could replace Fi by its fragmentation of rank α − 1 which
+would decrease the weight of F. By Proposition 9.21α−1 in the case α ≥ 2 (in the case α = 1
+we take u and v empty) we have Fi = uFv where F is a fragment of rank α, Fi ∼ F and
+|u|α−1, |v|α−1 < ζ. If F is the corresponding fragment of rank α in Γα−1 and P is the base
+for F then by Proposition 14.1α−1
+|P|α−1 > 1
+1.3
+�1
+ζ − 2ζ − 2.2
+�
+> 13.
+(ii) Let K be a fragment of rank α in X and µf(K) ≥ 3λ + 15ω. We assume that there is
+no truncated fragment Fi of rank α such that Fi ∼ K.
+By Proposition 8.10 and the assumption, if H is a common part of K and some Fi of
+rank α then H contains no fragment K′ of rank α with µf(K′) ≥ λ + 2.6ω. By Lemma 10.8,
+if H is a common part of K and some Fi of rank β < α then H contains no fragment K′ of
+84
+
+rank α with µf(K′) ≥ 3.2ω. In particular, K is not contained in any Fi . Let
+X ≖ F1F2 . . . F ′
+pF ′′
+p . . . F ′
+qF ′′
+q . . . Fk
+where
+Fp ≖ F ′
+pF ′′
+p ,
+Fq ≖ F ′
+qF ′′
+q ,
+K ≖ F ′′
+p Fp+1 . . . F ′
+q.
+If some Fi is contained in K and has rank α then by the remark above and 14.2, K is covered
+by at most three of the Fj’s. In this case, by Proposition 8.11 we would have
+µf(K) ≤ 3(λ + 2.6ω) + 2ζω < 3λ + 15ω
+contrary to the hypothesis. Therefore, each Fi that contained in K has rank β < α. Now
+by Proposition 8.11, FpFp+1 . . . Fq contains a fragment K′ of rank α with
+µf(K′) ≥ µf(K) − 2(λ + 2.6ω) − 2ζω > 29ω.
+For a base P of K′ we have |P|α−1 > 29 and by Proposition 14.1α−1, |K′|α−1 > 20. This
+implies that weightα(Fp · Fp+1 · . . . · Fq) > 1 and we get a contradiction with minimality
+of F since we can replace FpFp+1 . . . Fq in F by a single truncated fragment of rank α. This
+finishes the proof.
+(iii) By (ii), for each i = 1, 2, . . . , r there exists a truncated fragment Fti of rank α in F
+such that Ki ∼ Fti. Proposition 8.13 easily implies that Fti ∪ Ki is a truncated fragment of
+rank α. For each i = 1, 2, . . . , r we consequently replace Fti in F by Fti ∪ Ki. Since we do
+not increase weightα(F), the resulting fragmentation F′ of X is also minimal.
+□
+14.4. Lemma. Let α ≥ 1 and X, Y ∈ Rα be close in rank α − 1. Then
+|Y |α < 1.3|X|α + 2.2ζ.
+Proof. Let F be a minimal fragmentation of X. We represent X and Y by close paths X and Y
+in Γα−1. Then F induces the partition of X, denoted ¯F, into (path) truncated fragments of
+ranks ≤ α.
+Let
+X = P0H1P1 . . . HrPr
+where H1, . . . , Hr are all truncated fragments of rank α in ¯F. If r = 0 then |X|α = ζ|X|α−1,
+|Y |α ≤ ζ|X|α−1 and the statement simply follows from Proposition 14.1α−1. We assume
+r > 0. By Lemma 14.3(i), for each i we have Hi = uiH′
+ivi where H′
+i is a fragment of rank α,
+H′
+i ∼ Hi, |u|α−1, |v|α−1 < ζ, and the base Si for Hi satisfies |Si|α−1 > 13. Using Proposition
+10.16α−1 we find fragments H′′
+i and Gi of rank α in X and Y respectively where H′
+i = wiH′′
+i zi,
+|wi|α−1, |zi|α−1 < 1.15, Hi ∼ H′′
+i ∼ Gi and H′′
+i and Gi are close in rank α − 1. Using Lemma
+10.13(i)α−1 after each application of Proposition 10.16α−1 we can assume that Gi are disjoint,
+i.e.
+Y = Q0G1Q1 . . . GrQr.
+By Proposition 14.1α−1 we have
+|Q0|α−1 < 1.3|P0u1w1|α−1 + 2.2,
+|Qi|α−1 < 1.3|ziviPiui+1wi+1|α−1 + 2.2
+(i = 1, . . . , r − 1),
+|Qk|α−1 < 1.3|zkvkPk|α−1 + 2.2.
+We have also
+|X|α = r + ζ
+r
+�
+i=1
+|Pi|α−1
+and
+|Y |α ≤ r + ζ
+r
+�
+i=1
+|Qi|α−1.
+85
+
+Then
+|Y |α < r + 1.3ζ
+r
+�
+i=1
+|Pi|α−1 + 1.3rζ(2.3 + 2ζ) + 2.2ζ(r + 1)
+= (1 + 1.3ζ(4.5 + 2ζ))r + 1.3ζ
+r
+�
+i=1
+|Pi|α−1 + 2.2ζ
+< 1.3|X|α + 2.2ζ.
+□
+Proof of Proposition 14.1. Let X, Y ∈ Rα be close in rank α. Let F be a minimal fragmen-
+tation of X. We consider close paths X and Y in Γα labeled X and Y respectively. Then F
+induces the partitions of X into (path) truncated fragments of ranks ≤ α,
+X = F1 · F2 · . . . · Fk.
+Let X−1uYv be a coarse bigon. We fix some bridge partitions of u and v. Let ∆ be a filling
+diagram of rank α with boundary loop ˜X−1˜u˜Y˜v. Up to switching of u and v we can assume
+that ∆ is reduced and has a tight set T of contiguity subdiagrams. Let D1, . . . , Dr be all
+cells of rank α of ∆. In the process of forming T we assume that we pick first the contiguity
+subdiagrams of Di to ˜X−1 choosing them with maximal possible contiguity segment occurring
+in ˜X−1. Let
+X = P0K1P1 . . . KrPr
+and
+Y = Q0M1Q1 . . . MrQr.
+where Ki and Mi are the corresponding active fragments of rank α in X and Y. By the way
+we produce T and by Proposition 9.21α−1 in the case α ≥ 2 we have the following:
+(*) For all i, the fragment Ki cannot be extended in Pi−1KiPi. In particular, if F is a
+truncated fragment of rank α contained in Pi−1KiPi and containing Ki then F = w1Kiw2
+where |wi|α−1 < ζ (i = 1, 2)
+By Lemma 14.3(iii) we can assume that each Ki is contained in a compatible truncated
+fragment Fti of rank α. Let
+X = P′
+0Ft1P′
+1 . . . FtrP′
+r.
+Note that
+|X|α = r +
+�
+i
+|P′
+i|α
+and
+|Y |α ≤ r +
+�
+i
+|Qi|α.
+By (*),
+|P′
+i|α ≥ |Pi|α − ζ2 for i = 0, r,
+|P′
+i|α ≥ |Pi|α − 2ζ2 for 1 ≤ i ≤ r − 1.
+Hence
+(14-1)
+|X|α ≥ r +
+�
+i
+|Pi|α − 2rζ2.
+We give an upper bound on |Qi|α in terms of |Pi|α. First we consider the case 1 ≤ i ≤
+r − 1. There are three possibilities for the subdiagram of ∆ surrounded by Di and Di+1
+and contiguity subdiagrams of Di and Di+1 to ˜X−1 and ˜Y, depending on the presence of
+contiguity subdiagrams from T (see Figure 38). Note that according to Definition 6.12, all
+the components of ∆ − ∪Π∈T are small diagrams of rank α − 1, so we can use bounds from
+Proposition 7.12α−1. In cases (a) and (b) we have |Qi|α ≤ 6ζ2η < 0.6ζ and |Qi|α ≤ 4ζ2η <
+86
+
+Di
+Di+1
+˜X
+˜Y
+(a)
+(b)
+(c)
+Figure 38.
+0.4ζ respectively. Assume that case (c) holds. Then Pi = u1Su2 and Qi = v1Tv2 where S
+and T are close in rank α − 1 and |ui|α, |vi|α ≤ 4ζ2η < 0.4ζ. Using Lemma 14.4, we get
+|Qi|α < 1.3|Pi|α + 3ζ
+Note that this inequality holds also in cases (a) and (b).
+Now let i = 0 or i = r. If r > 0 then the difference of the case i = 0 from the case
+1 ≤ i ≤ r − 1 is that we can have an extra contiguity subdiagram between Y and the central
+arc of ˜u (see Figure 39). We then have
+˜X
+˜Y
+˜u
+D1
+Figure 39.
+|Q0|α < 1 + 1.3|P0|α + 3ζ
+and, similarly,
+|Qr|α < 1 + 1.3|Pr|α + 3ζ.
+If r = 0 we have a single bound instead,
+|Q0|α < 2 + 1.3|P0|α + 3ζ.
+Summarizing, with (14-1) we get
+|Y |α ≤ r + γ
+�
+i
+|Pi|α + 2 + 3ζ(r + 1)
+= (1 + 3ζ)r + 1.3
+�
+i
+|Pi|α + 2 + 3ζ
+< 1.3|X|α + 2.2.
+□
+14.5. Corollary. If F is a fragment of rank α and µf(F) ≥ tω then |F|α−1 >
+1
+1.3(t − 2.2).
+In particular, |F| >
+1
+1.3ζ1−α(t − 2.2).
+14.6. Corollary. Let Y = u1X1u2X2u3 in Γα where Xi, Y ∈ Rα and ui ∈ Hα.
+Then
+|Y |α ≤ 1.3(|X1|α + |X2|α) + 4.8.
+87
+
+Proof. Follows from Propositions 9.19(i) and 14.1.
+□
+The following two statements are proved under the assumption that a normalized presen-
+tation (2-1) of G satisfies the iterated small cancellation condition (S0)–(S3) for all α ≥ 1.
+We therefore will be assuming that all statements starting from Section 5 hold for all values
+of α.
+14.7. Proposition. Let W be a word with |W| ≤ α and let W = X in Gα where X ∈ Rα.
+Then |X|α < 0.3, X contains no fragments F of rank β > α with µf(F) ≥ 3ω and, in
+particular, X ∈ ∩α≥1Rα.
+By Corollary 14.5 it is enough to prove that |X|α < 0.3. We proceed by induction on α.
+If α = 1 then X is the freely reduced form of W and |X|1 ≤ ζ|X| < 0.3. Let α > 1.
+Let W ≖ W1a, a ∈ A±1 and W1 = X1 in Gα−1 where X1 ∈ Rα−1. By Corollary 14.5, the
+inductive hypothesis and Proposition 9.15, equality X = X1a holds already in Gα−1. By
+Corollary 14.6α−1
+|X|α ≤ ζ|X|α−1 ≤ ζ(1.3(0.3 + 0.3) + 4.8) < 0.3.
+14.8. Corollary. Every element of G can be represented by a word X reduced in G such that
+for some α ≥ 1, X contains no fragments F of rank β ≥ α with µf(F) ≥ 3ω.
+15. A graded presentation for the Burnside group
+In this section we show that for sufficiently large odd n the Burnside group B(m, n) has
+a graded presentation which satisfies the iterated small cancellation condition formulated in
+Section 2.
+We fix an odd number n > 2000. We are going to construct a graded presentation of the
+form
+(15-1)
+�
+A
+�� Cn = 1 (C ∈
+�
+α≥1
+Eα)
+�
+where all relators of all ranks α are n-th powers. We assume that values of the parameters λ
+and Ω are chosen as in Theorem 3, i.e.
+λ = 80
+n ,
+Ω = 0.25n.
+We will use also the following extra parameters:
+p0 = 39,
+p1 = p0 + 26 = 65.
+In what follows, we define the set Eα+1 under the assumption that sets Eβ are already
+defined for all β ≤ α. We fix the value of rank α ≥ 0 and assume that the presentation
+(15-1) satisfies small cancellation conditions (S0)–(S3) in 2.8, 2.9 and in normalized in the
+sense Definition 2.10 for all values of the rank up to α.
+We can therefore assume that all statements in Sections 5–13 are true for the current value
+of α and below.
+According to Propositions 11.5 and 11.13 each element of infinite order of Gα is conjugate
+to a power of a simple period over Gα. We will define Eα+1 as a certain set of simple periods
+over Gα. This will automatically imply condition (S0) with α := α + 1.
+Since n is odd, by Corollary 13.11 we obtain also that (S3) holds with α := α + 1.
+88
+
+Before going to the chain of definitions in the next section, we formulate the following two
+conditions (P1) and (P2) on Eα+1 (which can be viewed as “periodic” versions of (S1) and
+(S2) for the value of rank α := α + 1).
+(P1) For each A ∈ Eα+1, [A]α ≥ 0.25.
+(P2) Let L1 and L2 be periodic lines in Γα with periods A, B ∈ Eα+1 respectively. Assume
+that a subpath P of L1 and a subpath of Q of L2 are close and |P| ≥ p1|A|. Then L1
+and L2 are parallel.
+The main goal of the construction of Eα+1 will be to satisfy (P1) and (P2). Note that (P1)
+immediately implies (S1) for α := α + 1 because of the assumption n > 2000. Later we
+prove that (P2) implies (S2)α+1. (The difference between (P2) and (S2)α+1 is that in (P2)
+we measure periodic words by the number of periods while in (S2)α+1 we use the length
+function | · |α. An appropriate bound will be given in Proposition 16.6.)
+Our first step is to define a set of simple periods over Gα which potentially violate (P2)
+(they will be excluded in the definition of Eα+1).
+15.1. Definition. A simple period A over Gα is suspended of level 0 if there exist a simple
+period B not conjugate in Gα to A and words P ∈ Per(A) and Q ∈ Per(B) such that P
+and Q are close in Gα and |Q| ≥ p1|B|.
+At first sight, we could simply define Eα+1 by excluding periods A as in Definition 15.1
+from the set of all simple periods over Gα. However, in this case we cannot guarantee a
+necessary lower bound on [A]α for A ∈ Eα+1 in (P1). Roughly speaking, we need to claim
+that a fragment of rank β ≤ α can cover only a “small” part of a periodic word with a period
+A ∈ Eα+1; moreover, we need an exponentially decreasing upper bound on the size of this
+part when β decreases (compare with the definition of the function | · |α in 2.7). To achieve
+this, we enlarge the set of excluded simple periods over Gα+1 by adding potentially “bad”
+examples of this sort.
+15.2. Definition. A simple period A over Gα is suspended of level m ≥ 1 if there exist a
+suspended period B of level m − 1 not conjugate to A in Gα, and a reduced in Gα word
+of the form XQY such that Q ∈ Per(B), |Q| ≥ 4|B| and XQY is close in Gα to a word
+P ∈ Per(A).
+15.3. Definition. Let Pα denote the set of all simple periods over Gα and Sα denote the set
+of all suspended simple periods over Gα of all levels m ≥ 0. For Eα+1 we take any set of
+representatives of equivalence classes in Pα \ Sα with respect to the equivalence
+A ∼ B ⇔ A is conjugate to B±1 in Gα.
+The definition implies that any simple period over Gα in Pα \ Sα has finite order in Gα+1.
+Since Pα+1 ⊆ Pα, it follows that any simple period over Gα+1 and, in particular, any word
+in Eβ for β ≥ α + 1 belongs to Sα. As a consequence, we prove now that a fragment of rank
+α + 1 cannot cover a large periodic word with a simple period A over Gα+1. (So here is the
+trick: the definition of the set of suspended periods over Gα of levels m ≥ 1 serves condition
+(P1) for the future rank α + 1.)
+15.4. Remark. By construction, we obtain a normalized presentation (15-1) (see Definition
+2.10).
+89
+
+15.5. Proposition. Let A be a simple period over Gα+1.
+If an A-periodic word P is a
+subword of a fragment of rank α + 1 then |P| < 4|A|.
+Proof. As observed above, A ∈ Sα. Let UPV be a fragment of rank α+1 where P ∈ Per(A).
+Then UPV is close in Gα to a word Q ∈ Per(B) where B ∈ Eα+1. Since A is of infinite order
+in Gα+1, it is not conjugate to B in Gα. In this case, Definition 15.2 says that if |P| ≥ 4|A|
+then B ∈ Sα which would contradict Definition 15.3.
+□
+Proposition 15.5 with α := α −1 is an important but not sufficient ingredient in the proof
+of (P1). We need also to ensure that if a subword of fragment of rank β < α is a subword
+of an A-periodic word with A ∈ Eα+1 then its length compared to |A| is “exponentially
+decreasing when β decreases”. We prove a precise form of this statement in the next section
+by showing that coarsely periodic words have a certain property of hierarchical containment:
+a coarsely A-periodic word S over Gα has t disjoint occurrences of coarsely periodic words
+over Gα−1 with sufficiently large number of periods where t is approximately the number of
+periods A in S.
+16. Hierarchical containment of coarsely periodic words
+Starting from this point, all statements are formulated and proved under assumption that
+the group G has a specific presentation (15-1) defined in Section 15. The goal of this section
+is to prove the following property of suspended periods over Gα and to finalize the proof of the
+fact that the presentation (15-1) satisfies conditions (S0)–(S3). As in Section 15 we assume
+fixed the value of rank α ≥ 0 and assume that the normalized presentation (15-1) satisfies
+conditions (S0)–(S3) for ranks less or equal α; so we can use all statements in Sections 5–15
+for any rank up to α.
+16.1. Proposition. Let A be a suspended period over Gα. Then there exists a simple period B
+over Gα such that:
+(i) A cyclic shift of A contains a coarsely B-periodic word T over Gα with ℓB(T) ≥ p0.
+(ii) Moreover, this subword T has the following property. Let S be a coarsely A-periodic
+segment in Γα with ℓA(S) ≥ 4. Then there are an A-periodic base P for S, ℓA(S) − 3
+translates T, sA,PT, . . . , sℓ(S)−4
+A,P
+T of a coarsely B-periodic segment T in P with
+label(T) ≖ T and ℓA(S)−3 disjoint coarsely B-periodic segments Vi (i = 0, 1, . . . , ℓ(S)−
+4) in S such that Vi ≈ si
+A,PT for all i.
+We start with showing how Proposition 16.1α−1 implies (P1) in the case α ≥ 1.
+16.2. Lemma. Let A be a simple period over Gα and let S and Vi (i = 0, 1, . . . , ℓA(S) − 4)
+be as in Proposition 16.1α−1. Then for any i, Vi ∪ Vi+4 is not contained in a fragment of
+rank α.
+Proof. As in Proposition 16.1α−1, let P be an A-periodic base for S in Γα−1 containing t − 3
+translates T, sA,PT, . . . , st−4
+A,PT where T is a coarsely periodic segment with another period B
+and ℓB(T) ≥ p0. Assume that a fragment K of rank α in Γα−1 contains Vi and Vi+4. Let L
+be the base axis for K, so L is a C-periodic line with C ∈ Eα. Denoting V∗
+i the stable part
+of Vi, by Proposition 12.14α−1 we find W and W′ in L such that W ≈ V∗
+i and W′ ≈ V∗
+i+4.
+Then W ∪ W′ is close to si
+A,PT∗ ∪ si+4
+A,PT∗. Since A ∈ Sα−1, according to Definition 15.2α−1
+this should imply C ∈ Sα−1, a contradiction.
+□
+90
+
+16.3. Lemma. Let α ≥ 1. Assume that a (linear or cyclic) word X has r disjoint occurrences
+of coarsely A-periodic words Ui (i = 1, . . . , r) over Gα−1 with ℓA(Ui) ≥ p0. Then |X|α−1 ≥ 5r.
+Proof. The statement is immediate if α = 1. Assume that α > 1.
+Consider a fragmentation F of rank α − 1 of X (definition 2.7). Let S1, . . . , Sk be the
+subwords of fragments of rank α−1 in F. By Proposition 16.1α−1 each Ui contains p0−3 = 36
+disjoint coarsely B-periodic words Vi,j (j = 1, . . . , 36) over Gα−2 with ℓB(Vi,j) ≥ p0. We can
+assume that Ui and Vi,j are indexed in their natural order from the start to the end in X.
+By Lemma 16.2, each Si intersects at most 6 consequent subwords Vi,j, Vi,j+1, . . . , Vi,j+5.
+Excluding Vi,j with 1 ≤ j ≤ 6, we obtain that each Si intersects at most 6 of all the
+remaining Vi,j. By induction, we conclude that
+|X|α−1 ≥ k + 5ζ max{0, 30r − 6k}
+With fixed r, the minimal value of the right-hand side is achieved when 30r − 6k = 0. This
+gives the bound |X|α−1 ≥ 5r.
+□
+We prove the following stronger form of (P1):
+16.4. Proposition. For any simple period A over Gα we have [A]α ≥ 0.25 and, consequently,
+hα(A) ≤ 6.
+Proof. If α = 0 then [A]0 ≥ 1 by the definition of [·]0. Let α ≥ 1. Take any r ≥ 1. Consider
+a fragmentation F of rank α of the cyclic word (Ar)◦. Assume that F consists of words Si,
+i = 1, 2, . . . , N where the first k are subwords of fragments of rank α. By Proposition 15.5α−1
+we have |Si| < 4|A| for i = 1, 2, . . . , k. This implies that the cyclic word (Ar−4k)◦ can be
+partitioned into subwords of words in some subset of the remaining Si, i = k+1, k+2, . . . , N.
+Therefore,
+|(Ar)◦|α ≥ k + ζ|(Ar−4k)◦|α−1.
+Proposition 16.1α−1 says that (Ar−4k)◦ has at least r − 4k disjoint occurrences of a coarsely
+B-periodic word K over Gα−1 with ℓB(K) ≥ p0. Then by Lemma 16.3,
+|(Ar)◦|α ≥ k + 5ζ(r − 4k) = 0.25r.
+This holds for all r ≥ 1, so by Definition 12.12 we get [A]α ≥ 0.25 and hence hα(A) ≤ 6.
+□
+The following lemma is a key tool in the proof of Proposition 16.1.
+Very roughly, it
+corresponds to the statement “if a word W is periodic with two simple periods A and B at
+the same time, and if |W| ≥ 2|A|, |W| ≥ 2|B| then B is a cyclic shift of A”.
+16.5. Lemma. Let L0 and L1 be periodic lines in Γα with simple periods A and B over Gα,
+respectively. Let S be a coarsely C-periodic segment in L0 where C is another simple period
+over Gα, ℓC(S) ≥ 25. Assume that there exist coarsely C-periodic segments T0, T1, T2 in L1
+such that T0 < T1 < T2 and Ti ≈ si
+A,L0S, i = 0, 1, 2.
+If T0 ≲ s−1
+B,L1T1 or sB,L1T1 ≲ T2 then, if fact, T0 ≈ s−1
+B,L1T1, sB,L1T1 ≈ T2, words A and B
+represent conjugate elements of Gα and periodic lines L0 and L1 are parallel.
+Proof. Denote P0 = S ∪ s2
+A,L0S and P1 = T0 ∪ T2. Let S∗ and T∗
+i be stable parts of S and Ti.
+The crucial argument is similar to one in the proof of Proposition 13.4. Denote P the
+set of all coarsely C-periodic segments U in Γα such that U ≈ gS∗ for some g ∈ Gα (i.e. U
+and S∗ have the same labels of their periodic bases). We introduce translations and jumps
+on the set of coarsely C-periodic segments U ∈ P which occur in P0 or P1. As in the proof
+91
+
+of Proposition 13.4, it will be convenient to consider two disjoint sets of those U ∈ P which
+occur in P0 and in P1. (So formally we introduce the set Pi (i = 0, 1) of pairs (U, Pi) where
+U occurs in Pi; thus si
+A,L0S∗ belongs to P0 and T∗
+i belongs to P1 for i = 0, 1, 2. For a coarsely
+C-periodic segment U ∈ P, saying ‘U occurs in Pi’ we mean the corresponding element of Pi.)
+Let U, V ∈ P be coarsely C-periodic segments each occurring in some Pi.
+(i) If U and V occur in different paths Pi and U ≈ V then U jumps to V.
+(ii) U translates to V in the following cases:
+U and V occur in P0 and U ≈ sk
+A,L0V for some k ∈ Z; or
+U and V occur in P1 and U ≈ sk
+B,L1V for some k ∈ Z.
+Let M be a maximal set of pairwise non-(strictly compatible) segments which can be obtained
+by these two operations from S∗. Lemma 13.14 implies that M is a finite set. As in the proof
+of Proposition 13.4 we prove the following claim.
+Claim: The jump operation is always possible inside M; that is, for any U ∈ M in Pi,
+i ∈ {0, 1}, there exists V ∈ P in P1−i such that V ≈ U.
+To prove the claim, we will apply Lemma 13.1 and do a necessary preparatory work.
+Assume that U ∈ M belongs to P0 (the other case differs only in notation). Let V0 = S∗, V1,
+. . . , Vl = U be a sequence of coarsely C-periodic segments Vi ∈ M such that Vi+1 is obtained
+from Vi by one of the operations (i) or (ii). We can assume that V2j → V2j+1 are translations
+and V2j+1 → V2j+2 are jumps, so l = 2k −1 for some k. Under this assumption, V2j → V2j+1
+is a translation inside P0 if j is even and inside P1 if j is odd. We then define a sequence Y0,
+Y1, . . . , Yk of paths in Γα (Yj will be periodic segments with alternating periods A and B)
+and a sequence Wj ∈ P of coarsely C-periodic segments in Yj for j = 0, 1, . . . , k−1 such that
+W0 = V1 and Wi ≈ W0 for all i. For each j we will have Wj = fjV2j+1 for some fj ∈ Gα.
+The definition of Yj and fj goes as follows.
+We start with Y0 = P0 and W0 = V1, so f0 = 1. Assume that j < k − 1 and Yj and fj
+are already defined. For even j, V2j translates to V2j+1 inside P0, so there exists fj+1 ∈ Gα
+of the form fjst
+A,P0 such that fj+1V2j+1 ≈ fjV2j. Thus, fiP0 and fj+1P0 have a common
+A-periodic extension and we take Yj+1 = fiP0 ∪ fj+1P0. Similarly, for odd j V2j translates
+to V2j+1 inside P1. We take fj+1 ∈ Gα of the form fjst
+B,P1 such that fj+1V2j+1 ≈ fjV2j and
+take Yj+1 = fiP0 ∪ fj+1P0 inside a common B-periodic extension of fiP0 and fj+1P0. Note
+that k is odd because V2k+1 = U is assumed to occur in P0. We finally set Yk = fk−1P1.
+We now apply Lemma 13.1 where:
+• Sj is the set of all coarsely C-periodic segments V ∈ P in Yj.
+• Sj is pre-ordered by ‘≨’.
+• Equivalence is strict compatibility.
+• A segment V ∈ �
+j Sj is defined to be stable if V is the stable part of some coarsely
+C-periodic segment in Yj.
+• For aj, bj, a′
+j and b′
+j we take appropriate translates of S∗ and T∗
+i; namely, fjS∗,
+fjs2
+A,L0S∗, fjT∗
+0 and fjT∗
+2 if j is even or fjT∗
+0, fjT∗
+2, fjS∗ and fjs2
+A,L0S∗ if j is odd,
+respectively.
+• c0 is V1.
+Note that by Proposition 16.4 we have hα(C) ≤ 6. Hence the hypothesis ℓC(S) ≥ 25 implies
+ℓC(V) ≥ 13 ≥ 2hα(C)+1 for any V ∈ P. Condition (ii) of Lemma 13.1 holds by Lemma 13.14.
+Conditions (iii) and (iv) of Lemma 13.1 hold by Lemma 13.16. By the lemma, there exists a
+92
+
+coarsely C-periodic segment Vk ∈ P in fk−1P1 such that Vk ≈ fk−1U. This gives the required
+jump U → f −1
+k−1Vk. The claim is proved.
+Let r be the number of coarsely C-periodic segments V ∈ M such that and K∗ ≨ V ⪅
+sA,L0K∗ and let q be the number of coarsely C-periodic segments V ∈ M such that T∗
+1 ≨ N ⪅
+sB,L1T∗
+1 (in other words, r and q are the numbers of coarsely C-periodic segments V ∈ M
+in one period A and in one period B, respectively). Note that gcd(r, q) = 1 because M is
+generated by a single segment S∗.
+We assume first that either T0 ⪅ s−1
+B,L1T1 or sB,L1T1 ⪅ T2.
+Since M is closed under
+translations modulo equivalence ‘≈’, each of these relations implies q ≤ r and hence implies
+the other one. Let U0, U1, . . . , Ut be all coarsely C-periodic segments in M belonging to P0
+arranged in their order in P0 (so Ui form a set of representatives of coarsely C-periodic
+segments in M modulo ‘≈’). The group Gα acts on the set P/≈. It follows from Corollary 13.9
+that the action is free. For equivalence classes [Ui] of Ui we have
+sA,L0[Ui] = [Ui+r], i = 0, 1, . . . , t − r
+sB,L1[Ui] = [Ui+q], i = 0, 1, . . . , t − q.
+Note also that t ≥ 2r + 1. Applying Lemma 13.2 we get sA,L0 = dq and sB,L1 = dr for some
+d ∈ Gα. Since A and B are non-powers we get q = r = 1 which immediately implies the
+conclusion of the proposition.
+For the proof, it remains to consider cases T0 ∼ s−1
+B,L1T1 and sB,L1T1 ∼ T2. We consider
+the case sB,L1T1 ∼ T2 (the case T0 ∼ s−1
+B,L1T1 is symmetric). By the already proved part, we
+can assume that T2 ≨ sB,L1T1. We show that the assumption leads to a contradiction.
+We have T0 ≨ s−1
+B,L1T2 ≨ T1, so there exists T3 ∈ M such that T3 ≈ s−1
+B,L1T2. T3 jumps to
+some S3 ∈ M in L0 such that S3 ∼ S and S3 ≨ S. Then S3 translates to S4 ≈ sA,L0S3 and we
+have S4 ∼ S2 and S4 ≨ S2. Then S4 jumps to some T4 in L1 and we can continue the process
+infinitely (see Figure 40).
+□
+T0
+T3
+T1
+T4
+T2 sB,L0T1
+s−1
+A,L0S
+S3
+S
+S4
+sA,L0S
+L1
+L0
+Figure 40.
+Proof of Proposition 16.1. Let A be a suspended period of level m over Gα,
+Assume first that m = 0. Then by Definition 15.1 and Proposition 12.15 an A-periodic
+segment R in Gα contains a coarsely B-periodic segment ˆT with ℓB(ˆT) ≥ p1−2hα(B)−2 ≥ 51
+where B is not conjugate to A in Gα. By Lemma 13.12 we have ˆT ̸∼ sA,RˆT and |ˆT| < 2|A|.
+Let T be the stable part of ˆT. Since hα(B) ≥ 2 by Definition 12.12, we have |T| < |A| by
+Corollary 13.6. Note also that ℓB(T) ≥ ℓB(ˆT) − 2hα(B) ≥ p0. Let T ≖ label(T). We show
+that T has the required property (ii) formulated in Proposition 16.1
+Let S be a coarsely A-periodic segment in Γα with ℓA(S) ≥ 4 and let P be a periodic base
+for S. Denote t = ℓ(S). By Remark 12.7 we can assume that |P| ≥ t|A|. Up to placing ˆT
+93
+
+in Γα we can assume that P contains t−2 translates ˆT, sA,PˆT, . . . , st−3
+A,PˆT of ˆT. Using Lemma
+10.13(i) (which implies that strictly compatible coarsely periodic segments are close) and
+Proposition 12.14 we find disjoint Vi (i = 0, . . . , t − 3) in S such that Vi ≈ si
+A,PT. This
+proves the proposition in the case m = 0.
+Let m ≥ 1. The proof consists of two parts. First we provide a construction of a coarsely
+B-periodic segment T satisfying condition (i) of Proposition 16.1 and then we prove (ii).
+Construction of T. According to Definition 15.2, there exists a sequence A0, A1, . . . ,
+Am = A of simple periods over Gα where A0 is suspended of level 0, for each i ≤ m − 1 Ai is
+not conjugate to Ai+1 and there are reduced in Gα close words XiQiYi and Pi+1 ∈ Per(Ai+1)
+where Qi ∈ Per(Ai) and |Qi| ≥ 4|Ai|. For each i, we consider corresponding close paths
+XiQiYi and Pi+1 in Γα and place then in such a way that Qi and Pi have the common infinite
+Ai-periodic extension Li. We denote also L0 the infinite A0-periodic extension of Q0.
+As we proved above, there is a coarsely B-periodic segment ˆT0 in Q0 with ℓ(ˆT0) ≥ 51 and
+the stable part T0 satisfying ℓ(T0) ≥ p0 and |T0| < |A|. Up to positioning ˆT0 in L0 we can
+assume that Q0 contains translates s−1
+A0,L0T0 and sA0,L0T0 of T0. In what follows, if Z is a
+coarsely B-periodic segment in Γα then Z∗ denotes the stable part of Z. By Lemma 13.12,
+st
+A0,L0T0 ̸∼ T0 for any t ̸= 0 and hence s−1
+A0,L0T0 � ˆT0 � sA0,L0T0. By Proposition 12.14
+there are T1, U1,1 and W1,1 in P1 such that T1 ≈ T0, U1,1 ≈ s−1
+A0,L0T∗
+0 and W1,1 ≈ sA0,L0T∗
+0.
+Application of Lemma 16.5 with S := T∗
+0 (note that ℓB(T∗
+0) ≥ p0 − 12 ≥ 27) gives s−1
+A1,L1T1 �
+U1,1 and W1,1 � sA1,L1T1. In particular, we have |T1| ≤ |A1|. In the case m = 1 we take
+T := label(T1).
+Assume that m ≥ 2. We continue a procedure of finding coarsely B-periodic segments Ti
+in Pi. Up to positioning Q1 in L1 we can assume that Q1 contains both s−1
+A1,L1T1 and sA1,L1T1.
+Using Proposition 12.14 we find U2,2, U2,1, W2,1 and W2,2 in P2 such that U2,2 ≈ s−1
+A1,L1T∗
+1,
+U2,1 ≈ U∗
+1,1, W2,1 ≈ W∗
+1,1 and W2,2 ≈ sA1,L1T∗
+1. By Lemma 13.15, U2,2 � U2,1 � W2,1 � W2,2.
+We have U2,1 ≈ s−1
+A0,L0T∗∗
+0 , W2,1 ≈ sA0,L0T∗∗
+0
+and using Proposition 12.14 once more with
+X := s−1
+A0,L0T∗∗
+0 ∪sA0,L0T∗∗
+0 and Y := U2,1∪W2,1 we find T2 in P2 such that T2 ≈ T0. Application
+of Lemma 16.5 gives s−1
+A2,L2T2 � U2,2 and W2,2 � sA2,L2T2. In particular, |T2| ≤ |A2|.
+Repeating in a similar manner, we find Um,m, Um,m−1, Wm,m−1 and Wm,m in Pm such that
+Um,m ≈ s−1
+Am−1,Lm−1T∗
+m−1, Um,m−1 ≈ U∗
+m−1,m−1, Wm,m−1 ≈ W∗
+m−1,m−1, Wm,m ≈ sAm−1,Lm−1T∗
+m−1
+and Um,m � Um,m−1 � Wm,m−1 � Wm,m.
+Then we successively find Um,m−2, Wm,m−2,
+Um,m−3, Wm,m−3, . . . , Um,1, Wm,1 such that Um,i ≈ U∗
+i,i ≈ s−1
+Ai−1,Li−1T∗∗
+i−1 and Wm,i ≈ V∗
+i,i ≈
+sAi−1,Li−1T∗∗
+i−1. Finally, we find Tm in Pm such that Tm ≈ T0. Application of Lemma 16.5
+gives s−1
+Am,LmTm � Um,m and Wm,m � sAm,LmTm which implies |Tm| ≤ |Am|.
+We take
+T := label(Tm). This completes the construction, The whole procedure is schematically
+shown in Figure 41. Note that in Pm we have
+s−1
+Am,LmTm � Um,m � Um,m−1 � · · · � Um,1 � Tm � Wm,1 � · · · � Wm,m � sAm,LmTm.
+Proof of (ii). Let S be a coarsely Am-periodic segment in Γα and let P be a periodic base
+for S. Denote t = ℓAm(S). By Remark 12.7 we can assume that |P| ≥ t|A|, so P contains
+t − 1 translates T, sA,PT, . . . , st−2
+A,PT of a coarsely B-periodic segment T which is a translate
+of Tm constructed above. By Proposition 12.14, S contains coarsely B-periodic segments Z0,
+Z1, . . . , Zt−2 such that Zi ≈ si
+A,PT∗. We claim, moreover, that for 1 ≤ i ≤ t−3 there exist Vi
+94
+
+s−1
+A0,L0T0
+ˆT0
+sA0,L0T0
+s−1
+A1,L1T1
+U11
+T1
+V11 sA1,L1T1
+s−1
+A2,L2T2
+U22
+U21
+T2
+V21
+V22 sA2,L2T2
+s−1
+A3,L3T3
+U33
+U32
+U31
+T3
+V31
+V32
+V33 sA3,L3T3
+L0
+L1
+L2
+L3
+Figure 41.
+in S such that Vi ≈ si
+A,PT and Vi are all disjoint. Since ℓB(Vi) = ℓB(Tm) ≥ p0 this will finish
+the proof.
+Fix an index k in the interval 1 ≤ i ≤ t − 3. Up to positioning P and S in Γα we can
+assume that P and Pm have the common Am-periodic extension Lm and sk
+A,PT = Tm. By
+Lemma 16.5, s−1
+Am,LmT � Um,m and Wm,m � sAm,LmT. Then using Proposition 12.14 as in
+the procedure above, we successively find pairs (Ui, Wi) for i = m, m − 1, . . . , 1 such that
+Zk−1 � Um � Um−1 � · · · � U1 � Zk � W1 � · · · � Wm � Zk+1 and Ui ≈ U∗
+i,i, Wi ≈ W∗
+i,i
+for i = m, m − 1, . . . , 1. Then using Proposition 12.14 again with X := s−1
+A0,L0T∗∗
+0 ∪ sA0,L0T∗∗
+0 ,
+Y := U1 ∪ W1 and S = ˆT0 gives Vk with U1 � Vk � W1 and Vk ≈ T0 ≈ sk
+A,PT. The proof is
+finished.
+□
+16.6. Proposition. Let A ∈ Eα+1 and t ≥ 1 be an integer. Let P be an A-periodic word with
+|P| = t|A|. Then
+t
+n + t < µ(P) <
+t
+n − t + ω.
+Moreover, for t ≥ 200 we have also
+0.89 t
+n < µ(P) < 1.12 t
+n.
+Proof. Denote N = |(An)◦|α. Recall that µ(P) = |P|α/N. Up to cyclic shift of A, we assume
+that P ≖ At. For the lower bound on µ(P) in the first inequality, we observe that the cyclic
+word (An)◦ can be covered with ⌈n
+t ⌉ copies of P. By 4.14, this implies
+N <
+�n
+t + 1
+�
+|P|α
+which is equivalent to
+t
+n+t < µ(P). Similarly, for the upper bound we observe that ⌊n
+t ⌋
+disjoint copies of P can be placed inside (An)◦. Then again by 4.14,
+N ≥
+�n
+t
+�
+(|P|α − 1) >
+�n
+t − 1
+�
+(|P|α − 1)
+which implies by (S1) with α := α + 1
+µ(P) <
+t
+n − t + 1
+N ≤
+t
+n − t + ω.
+95
+
+If t ≥ 200 then we partition At into k subwords Ati with 80 ≤ ti ≤ 120. We have
+�
+i
+|Ati|α − (k − 1) ≤ |P|α ≤
+�
+i
+|Ati|α.
+and by the already proved bounds on µ(Ati), for each i we have
+0.94ti
+n < µ(Ati) < 1.07ti
+n + 1
+N .
+Then
+µ(P) ≥
+�
+i
+µ(Ati) − k − 1
+N
+> 0.94 t
+n − k
+N .
+By Proposition 16.4, N ≥ 0.25n. Hence
+k
+N ≤ t
+80
+� n
+N
+� 1
+n ≤ 0.05 t
+n
+and we obtain the required bound µ(P) > 0.89 t
+n. Similarly, for the upper bound on µ(P)
+we get
+µ(P) ≤
+�
+i
+µ(Ati) < 1.07 t
+n + k
+N ≤ 1.12 t
+n.
+□
+16.7. Corollary. (P2) implies (S2)α+1.
+Proof. By Proposition 16.6, if P is a subword of An with A ∈ Eα+1 and µ(P) ≥ λ then
+|P| ≥ t|A| where t satisfies
+t
+n − t ≥ λ − ω ≥ 1
+24 −
+1
+480
+and hence t > 76. Since 76 > p1, the required implication is straightforward.
+□
+16.8. Proposition. Presentation (15-1) satisfies (P2) and therefore satisfies the iterated
+small cancellation condition (S0)–(S3) for all α ≥ 1.
+Proof. Indeed, assume that L1 and L2 are periodic lines in Γα with periods A, B ∈ Eα+1
+respectively. Let P and Q be close subpath of L1 and L2, respectively, such that |P| ≥ p1|A|.
+If A is conjugate to B in Gα then A = B according to Definition 15.3 and the statement
+follows from Proposition 13.13. If A is not conjugate to B in Gα then B is suspended of
+level 0 as a simple period over Gα and hence cannot belong to Eα+1.
+□
+From this point, we may assume that all statements in Sections 5–16 are true for all values
+of rank α.
+16.9. Proposition. Every element of G is conjugate to a power of some C ∈ �
+α≥1 Eα.
+Proof. Let g ∈ G. If g has finite order then by Proposition 11.5, g is conjugate to a power
+of some C ∈ �
+α≥1 Eα. We assume that g has infinite order and come to a contradiction.
+By Corollary 14.8 we represent g by a word X reduced in G such that for some α ≥ 1,
+X contains no fragments F of rank β ≥ α with µf(F) ≥ 3ω. By our assumption, X has
+infinite order in all Gβ for β ≥ α. By Propositions 11.13 and 11.5, X is conjugate in Gα to
+a word of the form At where A is a simple period over Gα. Using Proposition 7.13(iii) we
+conclude that X is conjugate to At already in Gα−1. Then applying Proposition 8.9 with
+96
+
+β := α, α + 1, . . . we see that no cyclic shift of A contains a fragment K of rank β ≥ α with
+µf(K) ≥ 9ω and that A is cyclically reduced in Gβ for all β > α. Moreover, by Propositions
+8.16(iii) and 8.11, A is strongly cyclically reduced in Gβ for all β > α.
+Assume that for some β ≥ α, A is conjugate in Gβ to a power Br of a simple period
+over Gβ. By Proposition 9.16, A and Br are conjugate already in Gα. Since A is a non-
+power in Gα, we have r = 1 and then by Propositions 11.13 and 11.5, A is a non-power
+in Gβ. We showed that A is a simple period over Gβ for any β ≥ α. But this is impossible
+because by Proposition 16.4 we should have |A|β ≥ 0.25 and hence |A| ≥ 0.25ζ−β for any
+β ≥ α.
+□
+As an immediate consequence we get:
+16.10. Corollary. G satisfies the identity xn = 1 and therefore is isomorphic to the free
+Burnside group B(m, n).
+References
+[1] W. Burnside. On an unsettled question in the theory of discontinuous groups. Quart. J. Math.,
+33:230–238, 1902.
+[2] R. Coulon. On the geometry of Burnside quotients of torsion free hyperbolic groups. Int. J. Algebra
+Comput., 24(3):251–345, 2014.
+[3] T. Delzant and M. Gromov. Courbure m´esoscopique et th´eorie de la toute petite simplification. J. of
+Topology, 1:894–836, 2008.
+[4] S. V. Ivanov. The free Burnside groups of sufficiently large exponents. Internat. J. Algebra Comput.,
+4(1-2):1–308, 1994.
+[5] S. V. Ivanov and A. Yu. Ol’shanskii. Hyperbolic groups and their quotients of bounded exponents.
+Trans. Amer. Math. Soc., 348:2091–2138, 1996.
+[6] R. Lyndon and P. Schupp. Combinatorial Group Theory. Classics in Mathematics. Springer, 2001.
+[7] Jr M. Hall. Solution of the Burnside problem for exponent six. Ill. J. Math., 2:764–786, 1958.
+[8] С. И. Адян. Проблема Бернсайда и тождества в группах. Наука, Москва, 1975. Engl. transl.: S.
+I. Adian, The Burnside problem and identities in groups, Ergeb. Math. Grenzgeb., 95, Springer-Verlag,
+Berlin–New York, 1979, ISBN: 3-540-08728-1.
+[9] И. Г. Лысёнок. Бесконечные бернсайдовы группы четного периода. Изв. РАН. Сер. матем.,
+60(3):3–224, 1996. Engl. transl.: I. G. Lysenok, Infinite Burnside groups of even exponent, Izvestiya:
+Mathematics, 60:3 (1996), 453–654.
+[10] И. Г. Лысёнок. Подход к изучению конечно определенных групп, основанный на понятии
+дискретной кривизны. Мат. заметки, 103(4):568–575, 2018. Engl. transl.: Math. Notes, 103:4 (2018),
+610–615.
+[11] П. С. Новиков and С. И. Адян. О бесконечных периодических группах. I-III. Известия АН СССР.
+сер. матем., 32:212–244; 251–524; 709–731, 1968. Engl. transl.: P. S. Novikov and S. I. Adian, Infinite
+periodic groups, I-III, Math. USSR-Izv. 2 (1968), 209–236; 241–479; 665–685.
+[12] А. Ю. Ольшанский. О теореме Новикова–Адяна. Матем. сб., 118(160)(2(6)):203–235, 1982. Engl.
+transl.: A. Yu. Ol’shanski˘ı, On the Novikov–Adyan theorem, Math. USSR-Sb. 46:2 (1983), 203–236.
+[13] А. Ю. Ольшанский. Геометрия определяющих соотношений в группах. Наука, Москва, 1989. Engl.
+transl.: A. Yu. Ol’shanski˘ı, Geometry of defining relations in groups, Kluwer, 1991.
+[14] И. Н. Санов. Решение проблемы Бернсайда для показателя 4. Учен. зап. ЛГУ. Сер. мат. наук.,
+10:166–170, 1940. [I. N. Sanov, Solution of Burnside’s problem for exponent 4, Uchenye Zapiski
+Leningrad. Gos. Univ. Ser. Mat., 1940, no. 10, 166–170].
+[15] В. А. Тартаковский. Решение проблемы тождества для групп с k-сократимым базисом при k > 6.
+Изв. АН СССР. Сер. матем., 13(6):483–494, 1949. [V. A. Tartakovskii, Solution of the word problem
+for groups with a k-reduced basis for k > 6, Izv. Akad. Nauk SSSR Ser. Mat. 13:6 (1949), 483–494].
+97
+
diff --git a/idE4T4oBgHgl3EQfSgzP/content/tmp_files/load_file.txt b/idE4T4oBgHgl3EQfSgzP/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c62493dad10349b48bec257e38e13ebcf25b1f5d
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@@ -0,0 +1,4928 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf,len=4927
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='05000v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='GR]  12 Jan 2023  A SAMPLE ITERATED SMALL CANCELLATION THEORY FOR  GROUPS OF BURNSIDE TYPE  IGOR LYSENOK  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We develop yet another technique to present the free Burnside group B(m, n)  of odd exponent n with m ≥ 2 generators as a group satisfying a certain iterated small  cancellation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using the approach, we provide a reasonably accessible proof that  B(m, n) is infinite with a moderate bound n > 2000 on the odd exponent n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Introduction  The free m-generated Burnside group B(m, n) of exponent n is, by definition, the relatively  free group in the variety of groups satisfying the identity xn = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' B(m, n) ≃ Fm/F n  m where  Fm is the free group of rank m and F n  m is the subgroup of Fm generated by all n-th powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Obtaining a structural information about groups B(m, n) is known to be a difficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The primary question of this sort is whether B(m, n) is finite for given m, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The  question is known as the Burnside problem [1] and it is still not completely answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The  group is shown to be finite for exponents n = 2, 3 [1], n = 4 [14] and n = 6 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A negative  solution to the Burnside problem is given by the Novikov–Adian theorem [11, 8] stating that  the Burnside group B(m, n) of odd exponent n ≥ 665 with m ≥ 2 generators is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As  for now, infiniteness of B(m, n) is established for exponents of the form n = 665r or n ≥ 8000  and any number m ≥ 2 of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that B(m, r) is a homomorphic image of B(m, n)  if n is a multiple of r, so in this case infiniteness of B(m, r) implies infiniteness of B(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The case when the exponent n does not have a large odd divisor was treated in [4, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Although it is believable that free Burnside groups B(m, n) are infinite for considerably  lower values of n (and there are several announcements of results of this sort) the lowest  published and carefully checked bound is still 665, obtained by Adian [8] for the case of odd  exponent n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A principal step in understanding the structure of the group B(m, n) in the infinite case  was made in the fundamental work by Novikov and Adian [11] and its improved version [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  One of the ingredients of the proof was a tightly interweaved version of the small cancellation  theory similar to one developed by Tartakovski˘ı [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It was also shown in [8] that for m ≥ 2  and odd n ≥ 665 the group B(m, n) has several properties similar to key properties of small  cancellation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A basic one is layered Dehn’s property: a freely reduced nonempty word  representing the identity in the group contains a large part of a defining relator modulo  relations of the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This easily implies that any such word should contain a  subword of the form Xt for sufficiently large t which in turn implies that B(m, n) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Unfortunately, the approach due to Novikov–Adian, even in its polished and improved form  in [8], is extremely technical and has a complicated logical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Several later works  [12, 13, 3, 2] pursued the goal to find a more conceptually explicit and technically simpler  This research was supported by the Russian Science Foundation (project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 21-11-00318).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  1  approach to infinite Burnside groups, and more generally, to “infinite quotient of bounded  exponent” phenomena in wider classes of groups as in [5, 3, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As an underlying basic  idea, all these approaches utilize a small cancellation theory in a more or less explicit form  though based on different implementation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It was eventually realized that iterated  small cancellation theory is indeed a relevant framework to present Burnside groups of large  exponents as well as many other examples of infinitely presented groups of a “monster”  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In an explicit form, a relevant version of the theory was formulated by Gromov and  Delzant [3] and Coulon [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' However, both approaches need extremely large exponents to be  applied to Burnside groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (In fact, the both incorporate “non-constructive” tools so that  the proof does not provide any explicit lower bound on the exponent n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  Two questions naturally arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' What is the lower bound on the exponent n for which the  iterated small cancellation approach can be applied to Burnside groups B(m, n)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Do we  need a sophisticated technical framework to use the approach for reasonably small values of  the exponent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for example, for values which are about several hundreds or less?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The main goal of the present paper is to develop a sample version of the iterated small  cancellation theory specially designed for free Burnside groups B(m, n) with a “moderate”  lower bound on the exponent n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' More precisely, our technique works for odd exponents  n > 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We consider our approach as a first approximation and an introduction to a considerably  more technical result on infiniteness of Burnside groups with substantially smaller bounds  on the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The iterated small cancellation condition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We fix a group G given by a graded presentation  (2-1)  �  A  �� R = 1 (R ∈  �  α≥1  Xα)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Here we assume that the set of defining relators is partitioned into the union of subsets Xα  indexed by a positive integer α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We call cyclic shifts of words R ∈ X±1  α  relators of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Thus, the set of all relators of rank α is symmetrized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' closed under cyclic shifts and  taking inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  With the presentation of G, there are naturally associated level groups Gα defined by all  relations of rank up to α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (2-2)  Gα =  �  A  �� R = 1 (R ∈  �  β≤α  Xβ)  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Our small cancellation condition depends on two positive real-valued parameters λ  and Ω satisfying  (2-3)  λ ≤ 1  24,  λΩ ≥ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We introduce also two other parameters with fixed value:  ρ = 1 − 9λ,  ζ = 1  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The role of λ, Ω, ρ and ζ can be described as follows:  λ is an analog of the small cancellation parameter in the classical condition C′(λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2  Ω is the lower bound on the size of a relator R of rank α in terms of the length  function | · |α−1 associated with Gα−1 (defined below in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' see condition (S1)  in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  ρ is the reduction threshold used in the definition of a reduced in Gα word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In-  formally, a reduced in Gα word cannot contain more that ρ-th part of a relator of  rank α up to closeness in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  ζ is the rank scaling factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' it determines how the function | · |α rescales when  incrementing the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For any α ≥ 0, we introduce the set Hα of bridge words of rank α recursively by setting  H0 = {the empty word},  Hα = {uSv | u, v ∈ Hα−1, S is a subword of a relator of rank α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The definition immediately implies that Hα−1 ⊆ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note also that all sets Hα are closed  under taking inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We call two elements x, y ∈ Gα close if x = uyv for some u, v ∈ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This relation will  be often used in the case when x and y are represented by words in the generators A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In  that case we say that words X and Y are close in rank α if they represent close elements  of Gα, or, equivalently, X = uY v in Gα for some u, v ∈ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For α ≥ 0, the set Rα of words reduced in Gα, the set of fragments of rank α and the  length function | · |α are defined by joint recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A word X in the generators A is reduced in G0 if X is freely reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A word X is reduced  in Gα for α ≥ 1 if it is reduced in Gα−1 and the following is true: if a subword S of a  relator R of rank α is close in rank α − 1 to a subword of X then  |S|α−1 ≤ ρ|R|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A word X is cyclically reduced in Gα if any cyclic shift of X is reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A nonempty word F is a fragment of rank α ≥ 1 if F is reduced in Gα−1 and is close  in rank α − 1 to a subword P of a word of the form Rk where R is a relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (In  almost all situations P will be a subword of a cyclic shift of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=') A fragment of rank 0 is a  word of length 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' a single letter of the alphabet A±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  It is convenient to assume that each fragment F of rank α ≥ 1 is considered with fixed  associated words P, u, v and a relator R of rank α such that F = uPv in Gα−1, u, v ∈ Hα−1  and P is a subword of Rk for some k > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' a fragment is formally a quintuple (F, P, u, v, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A fragmentation of rank α of a (linear or cyclic) word X is a partition of X into  nonempty subwords of fragments of ranks β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F is a fragmentation of rank α of X  then by definition, the weight of F in rank α is defined by  weightα(F) = mα + ζmα−1 + ζ2mα−2 + · · · + ζαm0  where mβ is the number of subwords of fragments of rank β in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Here we assume that each  subword in F is assigned a unique rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We now define a semi-additive length function | · |α on words in the generators A:  |X|α = min{weightα(F) | F is a fragmentation of rank α of X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that |X|0 is the usual length |X| of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The iterated small cancellation condition consists of the following three conditions  (S0)–(S3) where the quantifier ‘for all α ≥ 1’ is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (S0) (“Relators are reduced”) Any relator of rank α is cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (S1) (“Relators are large”) Any relator R of rank α satisfies  |R|α−1 ≥ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (S2) (“Small overlapping”) For i = 1, 2, let Si be a starting segment of a relator Ri of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that S1 = uS2v in Gα−1 for some u, v ∈ Hα−1 and |S1|α−1 ≥  λ|R1|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then R1 = uR2u−1 in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  It can be proved that a group G satisfying conditions (S0)–(S2) possesses core properties  of small cancellation groups, in particular, a version of Dehn’s property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We will impose,  however, an extra condition on the graded presentation of G which implies cyclicity of all  finite subgroups of groups Gα and avoids difficulties caused by existence of non-cyclic finite  subgroups in the case of Burnside groups B(m, n) of even exponent n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (S3) (“No inverse conjugate relators”) No relator of rank α is conjugate in Gα−1 to its  inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As we see below, this condition is satisfied if each relator R of rank α has the form Rn  0  where the exponent n (which can vary for different R) is odd and R0 is a non-power in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  See Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Starting from Section 8, we will use a mild extra assumption on the graded presentation  (2-1) by requiring it to be normalized in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The assumption is not essential  and just makes arguments simpler (mainly due to Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1) slightly improving bounds on  the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We call a graded presentation (2-1) normalized if the following assertions  hold:  (i) Every relator R ∈ Xα has the form R ≖ Rt  0 where R0 represents a non-power element  of Gα−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' R0 does not represent in Gα−1 an element of the form gk for k ≥ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we  call R0 the root of a relator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) If R, S ∈ Xα and R ̸= S then R and S are not conjugate in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that the condition to be normalized is not restrictive: every graded presentation can  be replaced with a normalized one (although formally speaking, this replacement could affect  the iterated small cancellation condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' however, in real applications this would hardly be  the case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Checking conditions (S0)–(S3) requires knowledge about groups Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus pre-  senting a group by relations satisfying the iterated small cancellation condition already re-  quires a proof of properties of groups Gα by induction on the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Main results  As in the case of classical small cancellation, the iterated small cancellation condition  has strong consequences on the presented group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A basic one is an analog of the Dehn  property: every non-empty freely reduced word representing the trivial element of the group  “contains a large part” of a relator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4  In what follows, we assume that a group G is given by a normalized graded presentation  satisfying conditions (S0)–(S3) above and for any α ≥ 0, Gα denotes the group defined by  all relations of ranks up to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that a word X is reduced in G if it is reduced in Gα  for all α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The following theorem is an immediate consequence of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a non-empty word in the generators A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X reduced in Gα then  X ̸= 1 in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X is reduced in G then X ̸= 1 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By expanding the definition of a reduced word in G we get an equivalent formulation  which is more in the spirit of the small cancellation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a freely reduced non-empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X = 1 in G then for some α ≥ 1, X  has a subword close in Gα−1 to a subword P of a relator R of rank α with |P|α−1 ≥ ρ|R|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In the classical small cancellation theory, existence of a Dehn reduced representatives for  group elements is a simple consequence of the fact that a word containing more than a half of  a relator can be shortened by applying the corresponding relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This approach does not  work in our version of the iterated small cancellation and existence of reduced representatives  is a nontrivial fact proved below and formulated in Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 and Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Every element of Gα can be represented by a word reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Every  element of G can be represented by a word reduced in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Many other properties of groups Gα and G are established in Sections 5–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Our principal  result shows that our version of the iterated small cancellation theory can be applied to  free Burnside groups of odd exponent n with a moderate lower bound on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The following  theorem is a consequence of Propositions 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 and Corollary 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 (see also Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For odd n > 2000 and m ≥ 2, the free Burnside group B(n, m) has a normalized  graded presentation  �  A  �� Cn = 1 (C ∈  �  α≥1  Eα)  �  satisfying conditions (S0)–(S3) with λ = 80  n , Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following theorem is a well known property of Burnside groups of sufficiently large  odd exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is direct consequence of Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6 (the definition of ω is  given in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let n > 2000 be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a non-empty freely reduced word that is equal 1  in B(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X has a subword of the form C480 where C ∈ �  α≥1 Eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that, with existence of infinite aperiodic words in the 2-letter alphabet (see for  example [8, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3]) this implies infiniteness of B(n, m) for odd n > 2000 and m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The present approach has much in common with paper [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' However, the  approach in [9] was based on the assumption that defining relations of the group under  consideration are of the form xn = 1 for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Although the general scheme of  a large portion of our proofs is the same as in [9], our arguments are in different technical  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We tried to make the iterated small cancellation condition as simple possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular,  we use a simple version of closeness in groups Gα (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' However, when presenting  5  the free Burnside group as an iterated small cancellation group, this version is not optimal  for the bound on the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A more refined version would significantly lower the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Nevertheless, we consider the bound n > 2000 on the exponent as a reasonable balance  between its optimality and the complexity of definitions and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The whole approach relies essentially on the simultaneous induction on the rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since  the proof of required statements about groups Gα needs a comprehensive analysis of certain  types of relations in groups of previous ranks, the number of inductive hypotheses in quite  large (several tens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We think that a large number of inductive hypotheses is an unavoidable  feature of any “small cancellation” approach to infinite Burnside groups with a reasonably  small lower bound on the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that in the “basic” small cancellation theory in  Sections 5–7 we use Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 (with its immediate consequence Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9) as the  only inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We briefly mention essential ingredients of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Sections 5–7 are devoted to analysis of van Kampen diagrams over the presentation (2-2)  of the group Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 we introduce diagrams with a special marking of the boundary so  that the boundary loops of a diagram are divided into sides and bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The label of a side  is a word reduced in Gα and bridges are “small” sections between sides labeled by bridge  words of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' According to the marking, there are diagrams of bigon, trigon, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We then analyze a global structure of a diagram with marked boundary using the notion of  contiguity subdiagram (see 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For the quantitative analysis, we use a version of discrete  connection in the spirit of [10] and the corresponding discrete analog of the Gauss-Bonnet  formula (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The main outcomes are the bound on the total size of sides of  a diagram with no bonds (Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12) and the “single layered” structure of  diagrams of small complexity (Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The results of Sections 5–7 serve as a background for further analysis of relations in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The most important type of relations under consideration are “closeness” relations in Gα of  the form X = uY v where X, Y ∈ Rα and u, v ∈ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The structural description of diagrams  over the presentation of Gα transfers naturally to the language of the Cayley graph Γα of Gα,  see 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In Γα, words in the generators of the group are represented by paths and relations  in Gα are represented by loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The relation X = uY v becomes a loop X−1uYv in Γα  which can be viewed as a coarse bigon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we say also that paths X and Y are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The single  layered structure of the filling diagram implies one-to-one correspondence between fragments  of rank α in X and in Y that come from the 2-cells of the diagram, called active fragments  of rank α with respect to the coarse bigon X−1uYv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To express the correspondence, we use  the compatibility relation, defined in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, on the set of fragments of rank α in Γα (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' paths  in Γα labeled by fragments of rank α): if K and M are the corresponding active fragments  of rank α in X and Y, respectively, then K and M−1 are compatible (Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In Section 9 we perform this passage from diagrams over the presentation of Gα to the  Cayley graph Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We establish several properties of coarse bigons, trigons and more generally,  coarse polygons in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider also conjugacy relations in Gα which are represented by  parallel infinite lines in Γα (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A fundamental property of close paths X and Y in Γα with label(X), label(Y) ∈ Rα is that  the correspondence between fragments of rank α in X and Y extends to non-active ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If  K is a fragment in X of sufficiently large size then there exists a fragment of M of rank α  in Y such that K is compatible with either M or M−1, with possible exceptions of extreme  6  positions of K in X (Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Speaking informally, fragments of rank α play the  role of letters when coincidence of words is replaced by closeness in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This property of  close paths X and Y in Γα and its analogs for coarse trigons in Gα (Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7) and for  conjugacy relations in Gα (Propositions 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12) provide a technical base to analyze  further properties of groups Gα and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, the correspondence between fragments  of rank α in coarse bigons, under an appropriate adaptation, is crucial when we consider in  Section 13 close in Gα periodic words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In Section 11 we prove that any element of Gα can be represented by a reduced word  (Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1) and is conjugate to an element represented by a cyclically reduced word  and, moreover, by a strongly cyclically reduced word if it has infinite order (definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15,  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Sections 12 and 13 are preparatory for analysis of periodic relations over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In Section 12  we introduce the set of coarsely periodic words over Gα which are close (in a stronger sense  then defined in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4) to periodic words with a strongly reduced in Gα period (Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The main result of Section 13, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, is an analog of a well known property of  periodic words stating that if two periodic words have a sufficiently large overlapping (for  example, if they overlap for at least two of each of the periods) then they have a common  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In the last two Sections 15 and 16 we define a set of defining relations of the form Cn = 1  (C ∈ �  α≥1 Eα) for the Burnside group B(m, n) and prove that this set satisfies the iterated  small cancellation condition (S0)–(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' More precisely, in Definitions 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 we describe  the recursive step to define Eα+1 given Eβ for β ≤ α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' given the presentation of Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The  principal idea to build sets Eα can be roughly described as “classification of periodic words  by depth of periodicity” and is similar to one used in [11, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that other approaches  [12, 13, 4, 5, 3, 2] to groups of “Burnside type” use construction of periodic relations Cn = 1  where for the next rank, C are chosen to be “short in size” with respect to the current group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We believe that the “depth of periodicity” approach, allthough more technical in several  aspects, gives a more optimal lower bound on the exponent n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Preliminaries  Starting from Section 5 we assume fixed a value of rank α ≥ 0 and a presentation (2-2) of a  group Gα with relators R ∈ Xβ defined for all ranks β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that the presentation  of Gα is normalized and satisfies conditions (S0)–(S3) and inequalities (2-3) for all ranks up  to the fixed value α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the proofs we will use forward references to statements for smaller  values of rank, as already established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We will use references like “Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3α−1” or  “Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4<α” etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' which mean “statement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 for rank α − 1” or “statement  of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 for all ranks β < α” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' With a few exceptions, statements whose  formulation includes the case α = 0, are trivial or follow directly from definitions in that  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We fix a set A of generators for a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By a word we always mean a group  word over the alphabet A±1 = A ∪ {a−1 | a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We use notation X ≖ Y for identical  equality of words X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By X◦ we denote the cyclic word represented by a plain word X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A subword Y of a word X is always considered with an associated occurrence of Y in X  that is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To make it formal, we associate with a subword Y of X a pair  of words (U, V ) such that UY V ≖ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If Y is a subword of X with an associated pair (U, V )  7  then writing Y ≖ WZ we mean that W and Z are viewed as subwords of X with associated  pairs (U, ZV ) and (UW, V ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that ‘subword Y of X1’ and ‘subword Y of  X2’ are formally two distinct objects if X1 ̸= X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It will be always clear from the context  which ambient word is assumed for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A periodic word with period A, or an A-periodic word for short, is any subword of At  for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' According to the convention about subwords, an A-periodic word P is always  considered with an associated occurrence of P in a word At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A partition of a word X is a representation of X as concatenation X = X1 · X2 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' · Xk of  some subwords Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A word X is covered by a collection of words (Yi)i if X admits a partition  X = X1 · X2 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' · Xk such that Xi is a subword of some Yti and ti ̸= tj for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We use the term ‘graph’ as a synonym for ‘combinatorial 1-complex’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Edges of  a graph are considered as having one of the two possible directions, so formally all our graphs  are directed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By ι(e) and τ(e) we denote the starting and the ending vertices of an edge e,  respectively, and e−1 denotes the inverse edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An A-labeling on a graph Γ is a function from  the set of edges of Γ with values in A±1 ∪ {1} such that label(e−1) = label(e)−1 for any e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  here 1 denotes the empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An A-labeling naturally transfers to paths in Γ, so the label  of a path P is a word in A±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If P is a path in Γ then ι(P) and τ(P) denote the starting and  the ending vertices of P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For any vertex a of Γ, there is the unique empty path  at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We identify this empty path with vertex a itself, so ι(a) = τ(a) = a and label(a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A path is simple if it visits no vertex twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Two paths are disjoint if they have no common  and no mutually inverse edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A line in Γ is a bi-infinite path (we do not assume that lines  have no loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If X and Y are subpaths of a simple path Z then we write X ≪ Y if Z = Z1XZ2YZ3 for  some Zi and X < Y if Z = Z1XuZ2 = Z1vYZ2 for some Zi and non-empty u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Although  both relations depend on Z, it will be always clear from the context which Z is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Clearly, if neither X and Y is contained in the other then either X < Y or Y < X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The union  X ∪ Y of subpaths X and Y of Z is the shortest subpath of Z containing both X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The Cayley graph Γ(G, A) of a group G with a generating set A is naturally viewed as  an A-labeled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We identify vertices of Γ(G, A) with elements of G, so if ι(P) = a and  τ(P) = b then label(P) is a word representing a−1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The group G acts on Γ(G, A) by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A path P in Γ(G, A) labeled by an A-periodic word is an A-periodic segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An A-  periodic line is a bi-infinite path labeled by A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since an A-periodic word is assumed to  have an associated occurrence in some At, an A-periodic segment P can be uniquely extended  to an A-periodic line called the infinite periodic extension of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If P and Q are A-periodic  segments, P is a subpath of Q and the both have the same infinite periodic extension then  Q is a periodic extension of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We define also the translation element sA,P ∈ G that shifts the infinite periodic extension L  of P forward by a period A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By definition, sA,P can be computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Take any vertex a  on L such that the label of L at a starts with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then sA,P = aAa−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If L1 and L2 are two periodic lines with periods A1 and A2 respectively then L1 and L2 are  parallel if sA1,L1 = sA2,L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Mapping relations in the Cayley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It follows from the definition of the Cayley  graph that a word X in the generators A represents the identity of G if and only if some  8  (and therefore, any) path X in Γ(G, A) with label(X) ≖ X is a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus relations in G  are represented by loops in Γ(G, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This representation will be our basic tool to analyze  relations in a group using geometric properties of its Cayley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will often use the following notational convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X1X2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xn = 1 is a relation in  a group G then we represent it by a loop X1X2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xn in the Cayley graph of G typed with  the same letters in sans serif where, by default, label(Xi) ≖ Xi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We represent also conjugacy relations in G by parallel periodic lines in Γ(G, A) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let X = Z−1Y Z in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consider a loop X−1Z−1YZ′ in Γ(G, A) with label(X) ≖ X, label(Y) ≖  Y and label(Z) ≖ label(Z′) ≖ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We extend X to an X-periodic line L1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' X−1X0X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  with label(Xi) ≖ X and X0 = X and, in a similar way, extend Y to a Y -periodic line  L2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y−1Y0Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' with label(Yi) ≖ Y and Y0 = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we get a pair of parallel lines L1  and L2 that represents conjugacy of X and Y in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will be freely switch between the language of paths in Cayley graphs and word relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Van Kampen diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let G be a group with a presentation P = ⟨A | R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A diagram ∆  over P is a finite 2-complex ∆ embedded in R2 with a given A-labeling of the 1-skeleton ∆(1)  such that the label of the boundary loop of every 2-cell of ∆ is either empty, has the form  a±1a∓1 for a ∈ A or is a relator in R±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that here we use an extended version of the  widely used definition by allowing boundary loops of 2-cells labeled with empty word or freely  cancellable pair of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This allows us to avoid technical issues related to singularities  (see [13, §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5] or [9, §4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By default, all diagrams are assumed to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We refer to 2-cells of a diagram ∆ simply as to cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 1-cells and 0-cells are edges and  vertices as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By δD we denote the boundary loop of a cell D and by δ∆ we denote  the unique boundary loop of ∆ in case when ∆ is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We fix an orientation  of R2 and assume that boundary loops of cells of ∆ and boundary loops of ∆ are positively  oriented with respect to the cell or to the diagram, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This means, for example,  that (δD)−1 is a boundary loop of the diagram ∆−D obtained by removal of a cell D from ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that boundary loops of ∆ and of its cells are defined up to cyclic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to van Kampen lemma ([6, Theorem V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1] and [13, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1]) a word X  in the generators A represents the identity in G if and only if there exists a simply connected  diagram ∆ over P with label(δ∆) ≖ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Words X and Y represent conjugate elements of G if  and only if there exists an annular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' homotopy equivalent to an annulus) diagram over P  with boundary loops X and Z such that label(X) ≖ X and label(Z) ≖ Y −1 ([6, Lemma V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2]  and [13, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If Σ is a subdiagram of ∆ then ∆ − Σ denotes the subdiagram of ∆ obtained as the  topological closure of the complement ∆ \\ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ∆ and ∆′ be diagrams over P such that ∆′ is obtained from ∆ by either  contracting an edge e with label(e) ≖ 1 to a vertex,  contracting a cell D with label(δD) ≖ 1 to a vertex, or  contracting a cell D with label(δD) ≖ a±1a∓1, a ∈ A, to an edge labeled a±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We call the inverse transition from ∆′ to ∆ an elementary refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A sequence of ele-  mentary refinements is a refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  There are several common use cases for refinement:  9  Any diagram can be made by refinement non-singular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' homeomorphic to a punc-  tured disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, any simply connected diagram can be refined to a non-  singular disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If C is a boundary loop of ∆ represented as a product C = X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xk of paths Xi then,  after refinement, the corresponding boundary loop of a new diagram ∆′ becomes  X′  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' X′  k where each Xi refines to a nonempty path X′  i (see the definition in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Combinatorially continuous maps of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider the class of maps between  A-labeled graphs which are label preserving and can be realized as continuous maps of  topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' More precisely, a map φ : Λ → Λ′ between A-labeled graphs Λ and Λ′ is  combinatorially continuous if  φ sends vertices to vertices and edges to edges or vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for any edge e of Λ, φ(e) is  a vertex only if e has the empty label;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' if φ(e) is an edge then label(φ(e)) = label(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  if φ(e) is an edge then φ preserves the starting and the ending vertices of e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' if φ(e)  is a vertex then φ(e) = φ(ι(e)) = φ(τ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A combinatorially continuous map φ : Λ → Λ′ extends in a natural way to the map  denoted also by φ, from the set of paths in Λ to the set of paths in Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Clearly, φ preserves  path labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If a diagram ∆′ is obtained from a diagram ∆ by refinement then we have a combinatorially  continuous map φ : ∆′(1) → ∆(1) induced by the sequence of contractions ∆′ → ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If P is a  path in ∆ and P′ = φ(P) then P refines to P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Mapping diagrams in Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is well known that simply connected diagrams  can be viewed as combinatorial surfaces in the Cayley complex of a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since we do not  make use of two-dimensional structure, we adapt this view to the case of Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If ∆ is a simply connected diagram over P then there exists a combinatorially continuous  map φ : ∆(1) → Γ(G, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Any two such maps φ1, φ2 : ∆(1) → Γ(G, A) differ by translation  by some element g ∈ G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' φ1 = tgφ2 where tg : x �→ gx is the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In particular, if X is a loop in Γ(G, A) and for the boundary loop ¯X of ∆ we have label(¯X) =  label(X) then there is a map φ : ∆(1) → Γ(G, A) such that φ(¯X) ≖ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case we say  that ∆ fills X via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If ∆ is not simply connected then we can consider a combinatorially continuous map  φ : ˜∆(1) → Γ(G, A) where ˜∆ is the universal cover of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Again, any two such maps  φ1, φ2 : ˜∆(1) → Γ(G, A) differ by translation by an element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The set {Li}i of boundary  loops of ∆ lifts to a (possibly infinite) set of bi-infinite boundary lines {˜Lj  i}i,j of ˜∆ and thus  produces a set of lines {φ(˜Lj  i)}i,j in Γ(G, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Each φ(˜Lj  i) can be viewed as an Pi-periodic line  with period Pi = label(Li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We will be interested mainly in the case when ∆ is an annular  diagram, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' homotopy equivalent to a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case, boundary loops L1 and L2 of ∆  produce two Pi-periodic lines φ(˜Li) (i = 1, 2) in Γ(G, A) such that φ(˜L1) and φ(˜L2)−1 are  parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ and ∆′ be diagrams of the same homotopy type over a presentation  of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that a label preserving bijection Li �→ L′  i is given between boundary  loops of ∆ and ∆′ (which is usually clear from the context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that ∆ and ∆′ have  the same frame type if there exist combinatorially continuous maps φ : ˜∆(1) → Γ(G, A) and  10  ψ : ˜∆′(1) → Γ(G, A) such that for each i we have the same sets of lines (or loops if ∆ and ∆′  are simply connected) {φ(˜Lj  i)}j = {ψ(˜L′j  i )}j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following two observations follow easily from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Two simply connected diagrams ∆ and ∆′ have the same frame type if and  only if the labels of their boundary loops are equal words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ∆ and ∆′ be annular diagrams with boundary loops {L1, L2} and {L′  1, L′  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then ∆  and ∆′ have the same frame type if and only if the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Take any vertices ai  on Li (i = 1, 2) and let p be a path from a1 to a2 in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exist vertices a′  i on L′  i  (i = 1, 2) and a path p′ from a′  1 to a′  2 in ∆′ such that the label of Li read at ai and the label  of L′  i read at a′  i are equal words and label(p) = label(p′) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Diagrams ∆ and ∆′ have the same frame type in the following two cases:  ∆′ is obtained from ∆ by refinement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  ∆′ is obtained from ∆ by cutting off a simply connected subdiagram and replacing it  with another simply connected subdiagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Groups Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Throughout the paper we will study a fixed family of groups Gα given by  a presentation (2-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consequently, most of the related terminology will involve rank α as  a parameter (though in some cases, it is not mentioned explicitly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for example, the already  introduced measure µf(F) of fragments of rank α formally depends on α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Diagrams over the presentation of Gα are referred simply as diagrams over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For  1 ≤ β ≤ α, a cell of a diagram D over Gα with label(δD) ∈ Xβ is a cell of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Cells with  trivial boundary labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' empty or of the form aa−1) are cells of rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The Cayley graph of Gα is denoted Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that if β > α then we have a natural covering  map Γβ → Γα of labeled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A loop L in Γα lifts to Γβ as a loop if and only if label(L) = 1  in Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By a piece of rank α we call any (including empty) subword of a relator of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S is a subword of a cyclic shift of a relator R then we say also that S is a piece  of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We admit that a piece of rank α be the empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that our definition differs  from the traditional view on a piece in the small cancellation theory as a common starting  segment of two distinct relators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We assume that a piece S of rank α always has an associated relator R of rank α such  that S is a start of R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' so formally a piece of rank α should be viewed as a pair of the form  (S, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Associated relators are naturally inherited under taking subwords and inversion: if S  is a piece of rank α with associated relator R = ST and S = S1S2 then S1 and S2 are viewed  as pieces of rank α with associated relators R and S2TS1 respectively and S−1 is viewed as  a piece of rank α with associated relator S−1T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For pieces of rank α we use a “measure” µ(S) ∈ [0, 1] defined by µ(S) =  |S|α−1  |R◦|α−1 as in (8-1)  where R is the associated relator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (Recall that R◦ denotes the cyclic word represented by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  If for some β, S is a path in Γβ or in a diagram over the presentation of Gβ and S is labeled  by a piece of a relator of rank α (or by an R-periodic word where R is a relator of rank α)  then we abbreviate µ(label(S)) simply as µ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Reformulation of conditions (S2) and (S3) in terms of Cayley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The following  conditions on the presentation (2-1) are equivalent to (S2) and (S3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  11  (S2-Cayley) Let Li (i = 1, 2) be an Ri-periodic line in Γα−1 where Ri is a relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If L1 and L2 have close subpaths P1 and P2 with |Pi| ≤ |Ri| and µ(P) ≥ γ then L1 and L2 are  parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (S3-Cayley) There are no parallel R-periodic and R−1-periodic lines in Γα−1 where R is a  relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Bridge partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We define also a bridge partition of rank α of a word w ∈ Hα as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A bridge partition of rank 0 is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A bridge partition of rank α ≥ 1 either  has the form w1 ·S ·w2 where wi ∈ Hα−1 and S is a piece of rank α called the central  piece of w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' or  is a single factor w itself in the case w ∈ Hα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If w is a bridge word of rank α endowed with a bridge partition u · S · v and ST is the  relator of rank α associated with S then w′ = uT −1v is a bridge word of rank α equal to w  in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that w′ is obtained from w by switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case we assume also that  w′ is endowed with the bridge partition u · T −1 · v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, applying the switching operation  twice results in the initial word w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will be considering paths in Cayley graphs Γβ labeled by bridge words of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  call them bridges of rank α (with a slight abuse of terminology, we will also use this term  in Section 5 for boundary paths with appropriate label in diagrams over the presentation  of Gα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If w is bridge of rank α in Γβ then a bridge partition of rank α of w is either a  factorization w = u · S · v where u and v are bridges of rank α − 1 and label(S) is a piece  of rank α or a trivial factorization with the single factor w if w is bridge of rank α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In  the former case, if also β ≥ α, we define the switching operation on w in a similar way as  in the case of words;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' namely, we take the word w′ obtained from w ≖ label(w) by switching  and consider the path w′ with label(w′) ≖ w′ starting at the same vertex as w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since w = w′  in Γβ, bridges w and w′ have the same endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following properties of the function | · |α follow from the definition:  (i) |X|α + |Y |α − 1 ≤ |XY |α ≤ |X|α + |Y |α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' in particular, if Y is a subword of X then  |Y |α ≤ |X|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) More generally, if a collection of words (Xi)i covers a (plain or cyclic) word X then  |X|α ≤  �  i  |Xi|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If (Xi)1≤i≤k is a collection k of disjoint subwords of X then  �  i  |Xi|α ≤ |X|α + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) |X|α ≤ ζ|X|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iv) |X◦|α = min{|Y |α | Y is a cyclic shift of X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If X is a path in Γβ or in a diagram over the presentation of Gβ then we use abbreviation  |X|α = |label(X)|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Reduced words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The set of words reduced in Gα is denoted Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The definition imme-  diately implies that Rα is closed under taking subwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A word X is strongly cyclically reduced in Gα if any power Xt is reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Coarse polygon relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A relation in Gα of the form X1u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xmum = 1 where  words Xi are reduced in Gα and ui are bridge words of rank α, is called a coarse m-gon  relation in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can write coarse polygon relations in different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For example, a  coarse bigon relation can be written as X = uY v where X and Y are reduced in Gα and  u, v ∈ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this form, the relation represents closeness of words X and Y in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We transfer some terminology from words to paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We call paths in Γα with label reduced in Gα simply reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that, according to  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, a reduced path X in Γα is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies that we can correctly treat  the ordering of subpaths of X, intersections of subpaths, unions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Two vertices of Γα are close if they can be joined by a bridge of rank α (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Two  paths X and Y in Γα are close if their starting vertices and their ending vertices are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We say that a loop P = X1u1X2u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Xrur in Γα is a coarse r-gon if each Xi is reduced  and each ui is a bridge of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Paths Xi are sides of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that paths X and Y in Γα are close if and only if X−1uYv is a coarse bigon for some u  and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' All concepts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' relations, functions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=') and statements involving paths  in the Cayley graphs Γα are invariant under the action of Gα in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For example,  if paths X and Y in Γα are close then paths gX and gY are also close for any g ∈ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  adopt a convention (which is essential for the invariance) that the action of Gα is extended  onto extra data associated with paths in Γα: for example, if F is a fragment of rank β with  base P then then gF is considered as a fragment of rank β with base gP and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This  implies, for example, that µf(F) = µf(gF) for any g ∈ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will implicitly use symmetry with respect to inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For example, if F is a fragment  of rank β with base P then F−1 is a fragment of rank β with base P−1 and µf(F−1) = µf(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If a statement admits two symmetric forms then only one of them is formulated (as in case  of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15, for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Numerical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In many cases, it will be notationally more convenient to use  instead of Ω its inverse:  ω = 1  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that by (2-3),  (4-1)  ω ≤  1  480  and  λ ≥ 20ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will extensively use ω as a unit to measure pieces and fragments of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Condition (S1) in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 will be often used in the following form: if P is a piece of a relator R  of rank α then  (4-2)  µ(P) ≤ ω|P|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For reader’s convenience, we list our other global numerical parameters indicating the  places where they first appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  ν =  ζ  1 − 2ζ = 1  18,  θ = 1  6(5 − 22ν) = 17  27  (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4),  13  η = 1 + 2ν  θ  = 30  17  (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9),  ξ0 = 7λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω  (Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7),  ξ1 = ξ0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω  (Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2),  ξ2 = ξ1 − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω  (Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Diagrams with marked boundary  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Boundary marking of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We start with introducing a class of diagrams over the  presentation (2-2) of Gα with extra data which, in particular, represent coarse polygon  relations in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ∆ be a non-singular diagram over the presentation (2-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We say that ∆ has a  boundary marking of rank α if for each boundary loop L of ∆, there is fixed a representation  as a product L = X1u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xmum of nonempty paths Xi and ui where labels of Xi are reduced  in Gα and the label of each ui belongs to Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Paths Xi are called sides and paths ui are  called bridges of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We allow also that the whole boundary loop L of ∆ is viewed a side  called a cyclic side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case we require that the label of L is cyclically reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If X1u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xmum = 1 is a coarse polygon relation in Gα then there exists a disk diagram  with boundary label X1u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xmum such that label(Xi) ≖ Xi and label(ui) ≖ ui for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Refining ∆ if necessary (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4) we can assume that ∆ is non-singular and all paths Xi  and ui are nonempty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ∆ satisfies the definition above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In a similar way, we can associate  with a conjugacy relation in Gα an annular diagram over the presentation of Gα with an  appropriate boundary marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Unless otherwise stated, “a diagram of rank α” will always mean “a non-singular diagram  over the presentation (2-2) with a fixed boundary marking of rank α”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We use terms “di-  agrams of monogon, bigon, trigon type etc.”  to name disk diagrams of rank α with the  appropriate number of sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If ∆ is a diagram of rank α then by b(∆) we denote the number of bridges  of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We define the complexity c(∆) of ∆ by  c(∆) = b(∆) − 2χ(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Decrementing the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a diagram of rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By ∆α−1 we denote the  diagram over the presentation of Gα−1 obtained by removal from ∆ of all cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up  to refinement of ∆, we assume that ∆α−1 is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We assume that every bridge w of ∆ is given a bridge partition of rank α as defined in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for some bridges w a factorization w = u · S · v is fixed where label(u), label(v) ∈ Hα−1  and label(S) is a piece of rank α, and for all other w we have label(w) ∈ Hα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case  when w has a nontrivial bridge partition u · S · v we say that w has native rank α and call S  the central arc of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will be always assuming that all factors u, v and S are nonempty paths (this can be  achieved by refinement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We then define a naturally induced boundary marking of rank α−1 of ∆α−1 (see Figure 1):  Sides of ∆ become sides of ∆α−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we have also extra sides of ∆α−1 defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If D is a cell of rank α of ∆ then boundary loop (δD)−1 of ∆α−1 becomes a cyclic  side of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  14  For each bridge w of rank α of ∆ we do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If the bridge partition of w  is of the form u = v · S · w then we take v and w as bridges of ∆α−1 and the central  arc S as a side of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise we have label(w) ∈ Hα−1 and we take w as a  bridge of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  cells of rank α  ∆  ∆α−1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Producing ∆α−1 from ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Sides of ∆ and ∆α−1 are drawn by thicker lines  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Cell cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We introduce two types of elementary reductions of a diagram ∆ of  rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In both cases, we reduce the number of cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3, we assume  that a bridge partition is fixed for each bridge ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let C and D be two cells of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that C and D form a cell-cell cancellable  pair if there exists a simple path p joining two vertices a and b in the boundaries of C and D  respectively, so that the label of the path QpRp−1 is equal 1 in Gα−1 where Q and R are  boundary loops of C and D starting at a and b respectively see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case,  a  b  p  C  D  Q  R  Θ  D  C  v  S  w  ∆  ∆  ∆  a  b  c  T  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  we can perform the procedure of cell-cell cancellation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We remove cells C and D  from ∆, cut the remaining diagram along p and fill in the resulting region by a diagram Θ  over the presentation of Gα−1 (see Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The boundary marking of the new diagram  naturally inherits the boundary marking of ∆ and the labels of sides and bridges are not  changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Now let u be a bridge of native rank α of ∆ with bridge partition u = v ·S·w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The label S  of S has an associated relator R of rank α such that R ≖ ST for some T (according to the  convention in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We attach a cell C of rank α to ∆ along S so that (ST)−1 becomes  the label of the boundary loop (ST)−1 of C (see Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For the new diagram ∆ ∪ C we  15  define the boundary marking of rank α with a new bridge vT−1w instead of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We call this  operation switching of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If C and another cell D of rank α of ∆ form a cell-cell cancellation pair in ∆ ∪ C then we  say that u and D form a bridge-cell cancellable pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case, after performing a cell-cell  cancellation in ∆ ∪ C we obtain a diagram ∆′ having one cell of rank α less than ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We will  refer to this reduction step as bridge-cell cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition (Reduced diagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a diagram of rank α ≥ 1 with fixed bridge  partitions for all bridges of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that ∆ is reduced if it has no cancellable pairs after  any refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In what follows, we will be assuming that a diagram ∆ of rank α ≥ 1 has fixed  bridge partitions of all bridges of ∆ if it is required by context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, this applies  when we consider the subdiagram ∆α−1 and the property of ∆ to be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If a diagram ∆ of rank α is not reduced then, after possible re-  finement, we obtain a cancellable pair which can be removed by performing the reduction  procedure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, any diagram of rank α ≥ 1 can be transformed to a re-  duced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that we use a sequence of transformations of the following two types in the  reduction process:  transformations preserving the frame type (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  bridge switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Thus, after reduction the new diagram ¯∆ has the same frame type as ∆ up to bridge  switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following observation follows from definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 and will be used without  explicit reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Σ be a subdiagram of a reduced diagram ∆ of rank α ≥ 1 such that  the central arc of any bridge of Σ is either a subpath of the central arc of a bridge of ∆ or a  subpath of (δD)−1 where D is a cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then Σ is reduced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Reduction to the previous rank  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a diagram of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A bond in ∆ is a simple path u satisfying  the following conditions:  (i) u joins two vertices on sides of ∆ and intersects the boundary of ∆ only at the  endpoints of u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) label(u) is equal in Gα to a word in Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) u is not homotopic in ∆ (rel endpoints) to a subpath of a side of ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iv) u does not cut off from ∆ a simply connected subdiagram with boundary loop u±1pvq  where p is an end of a side of ∆, v is a bridge of ∆, q is a start of a side of ∆ and  labels of p and q are empty words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In most cases, we will assume that the label of a bond u already belongs to Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note  that this condition can always be achieved by cutting ∆ along u and attaching a subdiagram  with boundary loop u±1v where label(v) ∈ Hα and its mirror copy, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A diagram of rank α is small if it has no bonds after any refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  16  u  u  v  p  q  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Excluded cases in (iii) and (iv)  u  u  v  u′  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following observation is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) The property of a diagram ∆ of rank α to be small depends only on the frame type  of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) The property of a diagram of rank α to be small is preserved under switching of  bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If ∆ is a small diagram of rank 0 with c(∆) > 0 then labels of all sides of ∆ are  empty words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a diagram of rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A disk subdiagram Π of ∆α−1 is a  contiguity subdiagram of ∆ if the boundary loop of Π has the form Pu1Qu2 where P−1 and Q−1  are nonempty subpaths of sides of ∆α−1 and each of the two paths ui is either a bond in ∆α−1  with label(ui) ∈ Hα−1 or a bridge of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that here we use Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 with rank  α − 1 instead of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The paths P±1 and Q±1 are contiguity arcs of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If P−1 and Q−1 occur, respectively, in  sides S and T of ∆α−1 then we say that Π is a contiguity subdiagram of S to T (or between S  and T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, if P and Q are contiguity arcs of a contiguity subdiagram with  boundary loop Pu1Qu2 then labels of P−1 and Q are close in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma (small cancellation in reduced diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced diagram of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Π be a contiguity subdiagram of ∆ with boundary loop δΠ = PuQv where P and Q are  the contiguity arcs of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that P−1 occurs in the boundary loop of a cell D of rank α  and Q−1 occurs in a side S of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then:  (i) If S is a side of ∆ then µ(P) < ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  17  (ii) If S is the boundary loop of a cell D′ distinct from D then µ(P) < λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If S is the central arc of a bridge of ∆ then µ(P) < λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S is a side of ∆ then the label of S is reduced in Gα (or cyclically reduced in Gα  if S is a cyclic side), as defined in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then µ(P) < ρ by the definition of a reduced word  in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that µ(P) ≥ γ and S = δD′ where D′ is a cell distinct from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let R and R′ be  boundary loops of D and D′ starting at the initial and terminal vertices of u, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By the small cancellation condition (S2) we have label(R) = label(uR′u−1) in Gα−1, hence D  and D′ form a cell-cell cancellable pair contrary to the hypothesis that ∆ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If µ(label(P)) ≥ λ and S is the central arc of a bridge of ∆ then in a similar way we see  that D and S form a cell-bridge cancellable pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  Note that the lemma leaves uncovered a possibility when S = δD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' when Π is a contiguity  subdiagram of D to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This case needs a special consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A cell D of rank α in a diagram ∆ of rank α ≥ 1 is folded if there exists a  simple path u joining two vertices a and b in the boundary of D so that label(PQuQPu−1) = 1  in Gα−1 where P and Q are subpaths of δD from a to b and from b to a respectively (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Q  P  u  a  b  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma (no folded cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that no relator of rank α is conjugate in Gα−1 to its  inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then folded cells do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consequently, if Π is a contiguity subdiagram of a  cell of rank α to itself then for a contiguity arc P of Π we have µ(label(P)) < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The first statement is an immediate consequence of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If Π is a contiguity  subdiagram of a cell D of rank α to itself and P is a contiguity arc of Π with µ(label(P)) ≥ λ  then, as in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, we conclude that D is a folded cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will be considering finite sets of disjoint contiguity subdiagrams of a diagram ∆ of  rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Our goal is to produce a maximal, in an appropriate sense, such a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let {Πi} be a finite set of pairwise disjoint contiguity subdiagrams of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Each connected  component Θ of the complement ∆α−1 − � Πi is a diagram of rank α − 1 with a naturally  induced boundary marking of rank α − 1 defined as follows:  18  Bridges of ∆α−1 occurring in the boundary of Θ become bridges of Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If u is a bond of ∆α−1 occurring in the boundary of some contiguity subdiagram Πi  and u−1 occurs in the boundary of Θ then u−1 becomes a bridge of Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The rest of the boundary of Θ consists of subpaths of sides of ∆α−1, or possibly  cyclic sides of ∆α−1, which are viewed as sides of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following observation follows easily by induction on the number of contiguity subdi-  agrams in a set {Πi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let {Πi} be a set of r pairwise disjoint contiguity subdiagrams of a diagram ∆  of rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let {Θj} be the set of all connected components of the complement ∆α−1 −  �  i Πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  �  j  c(Θj) = c(∆α−1),  �  j  χ(Θj) = χ(∆α−1) + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a diagram of rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists another diagram ∆′  of rank α and a finite set {Πi} of pairwise disjoint contiguity subdiagrams of ∆′ such that:  (i) ∆′ is obtained from ∆ by replacing its subdiagram ∆α−1 with another subdiagram  over the presentation of Gα−1 of the same frame type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' in particular, ∆ and ∆′ have  the same boundary marking and the same frame type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) any connected component Θ of ∆′  α−1 − �  i Πi is a small diagram of rank α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) if c(∆α−1) > 0 then c(Θ) > 0 for each connected component Θ of ∆′  α−1 − �  i Πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a diagram of rank α and let {Πi} be a finite set of pairwise disjoint contiguity  subdiagrams of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that a connected component Θ of ∆α−1 −�  i Πi has a bond, pos-  sibly after refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We describe how to obtain from {Πi} a new set of disjoint contiguity  subdiagrams by either increasing the set or increasing the part of ∆ covered by {Πi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  track on two inductive parameters: the number N of connected components of ∆α−1 −�  i Πi  and the total length L of sides of these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Refining Θ inside ∆ we may assume that Θ has a bond u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An easy analysis shows that any  bond in Θ is also a bond in ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Performing surgery as described in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 we may assume  that the label of u belongs to Hα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Observe that u cuts Θ into a subdiagram Θ1 or two subdiagrams Θ1 and Θ2 which inherit  the boundary marking of rank α −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' From the definition of complexity c(∗) we immediately  see that c(Θ) = �  i c(Θi) in either of the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since u is not homotopic to a subpath  of a side of Θ we have c(Θi) ≥ 0 for each Θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We change the set {Πi} depending on the  following two cases:  Case 1: u cuts Θ into two subdiagrams Θ1 and Θ2 and at least one of them, say Θ1,  satisfies c(Θ1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then Θ1 is a simply connected subdiagram with two bridges, and hence  a contiguity subdiagram of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that if for both Θ1 and Θ2 we have c(Θ1) = c(Θ2) = 0  then ∆ has no cells of rank α and is itself a contiguity subdiagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We then can take  {Πi} = {∆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let v be the other bridge of Θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If u is a bridge of ∆α−1 then we simply add Θ1 to the  set {Πi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise v−1 is a bond of ∆α−1 occurring in the boundary loop of some Πi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  then we attach Θ1 to Πi (see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that the label of at least one side of Θ1 is  19  nonempty (by condition (iv) of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 applied to Θ and u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence after performing  this operation, L is strictly decreased and N is not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Θ2  Θ1  Π  v  u  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: Case 1 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We refine ∆ so that u “bifurcates” into two paths u′ and u′′  (Figure 7) and obtain a “degenerate” contiguity subdiagram Π of ∆ between u′ and u′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  then add Π to the set {Πi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The operation strictly increases N not changing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  ∆α−1  u  u′  u′′  Π  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Starting from the empty set of contiguity subdiagrams Πi, we perform recursively the  procedure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Each step we either decrease L not changing N or increase N  not changing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Furthermore, each time there is at most one connected component Θ of  ∆α−1 − �  i Πi with c(Θ) ≤ 0 and it exists only if c(∆α−1) ≤ 0 for the initial diagram ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, N is bounded from above, so the procedure terminates after finitely many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Upon termination, all connected components of ∆α−1 −�  i Πi become small by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that a set {Πi} satisfying the conclusion of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 is a  tight set of contiguity subdiagrams of ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Global bounds on diagrams  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ∆ be a diagram of rank α ≥ 1 and {Πj} a set of disjoint contiguity subdiagrams  of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have a tiling of ∆ by subdiagrams of three types: cells of rank α, contiguity  subdiagrams Πi and connected components of the complement ∆α−1 −� Πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We name these  subdiagrams tiles of index 2, 1 and 0 respectively and refer to them also as internal tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We consider also external 2-cells of ∆ as tiles of index 2, so with these extra tiles we obtain  a tiling of the 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Boundary loops of all tiles carry naturally induced partitions into  subpaths (allowed to be whole loops) called tiling sides, defined precisely as follows (see  Figure 8):  The boundary loop δΠi of each contiguity subdiagram Πi is partitioned as P·u·Q·v  where P and Q are the contiguity arcs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' thus δΠi consists of four tiling sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  20  1  1  1  1  1  1  1  1  1  0  0  0  0  0  0  2  2  2  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A component Θ of ∆α−1 − �  i Πi has the induced boundary marking of rank α − 1  (in this case, a tiling side can be a cyclic side of Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The boundary loop of a cell of rank α either has no nontrivial partition (in this case  it is considered as a cyclic tiling side) or is partitioned as an alternating product of  contiguity arcs of subdiagrams Πi and paths S where S−1 is a side of a component  of ∆α−1 − �  i Πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The partition of the boundary loop L of an external cell is defined as follows: we  take the partition of L induced by the boundary marking of rank α − 1 of ∆α−1 and  additionally subdivide sides of rank α−1 into alternating products of contiguity arcs  of subdiagrams Πi and paths S where S−1 is a side of a component of ∆α−1 − �  i Πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that we view on tiling sides as paths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' they are considered with direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  construction, the set of all tiling sides is closed under inversion, and each tiling side occurs  in a unique way in a boundary loop of a tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S be the set of tiling sides associated with {Πi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For every tile T, we  denote S(T) the set of tiling sides occurring in the boundary loops of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A discrete connection on a pair (∆, {Πi}) is a function w : S → R such that w(s−1) = −w(s)  for any s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Given w, we define the curvature κ(T) of each internal tile T:  κ(T) = (−1)index(T)χ(T) +  �  s∈S(T)  w(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (Note that inequality χ(T) ̸= 1 is possible only if T has index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=') For an external tile T, by  definition,  κ(T) =  �  s∈S(T)  w(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By definition, the total curvature κ(∆) of ∆ is the sum of curvatures of all internal tiles  of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The total curvature of external tiles of ∆ is the curvature along the boundary of ∆,  denoted κ(∂∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (A discrete version of the Gauss–Bonnet theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For any diagram ∆ of  rank α ≥ 1 and any set {Πi} of disjoint contiguity subdiagrams of ∆,  κ(∆) + κ(∂∆) = χ(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In particular, if κ(T) is non-positive for any internal tile T then κ(∂∆) ≥ χ(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  21  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let t be the number of cells of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It follows from the second equality of  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 that  �  T  (−1)index(T)χ(T) = χ(∆α−1) + t = χ(∆)  where the sum is taken over all internal tiles T of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the expansion of κ(∆) + κ(∂∆) all  summands w(s) are canceled because of the assumption w(s−1) = −w(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (bounding the number of cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced diagram of rank α ≥ 1  with c(∆α−1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote  (7-1)  ν =  ζ  1 − 2ζ = 1  18,  θ = 1  6(5 − 22ν) = 17  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let T be a tight set of contiguity subdiagrams of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that the following extra  condition is satisfied:  (*) Each cell of rank α of ∆ has at most one contiguity subdiagram Π ∈ T to sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let M be the number of cells of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  (7-2)  θM ≤ 2  3(1 + ν)b(∆) − χ(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For the proof, we define a discrete connection w on the pair (∆, {Πi}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that w(S−1) =  −w(S) by Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 and thus defining w(S) automatically defines w(S−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Recall that sides of ∆α−1 are divided into three types: sides of ∆, central arcs of bridges of  native rank α and the boundary loops of cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S is a side of ∆α−1 or a subpath  of a side of ∆α−1 then we assign to S type I, II or III respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Before defining w, we perform on ∆ the following “cleaning” procedure: if a bridge of ∆α−1  occurs in the boundary of some contiguity subdiagram Πi then we cut off Πi from ∆ taking  the bond in the boundary of Πi as a new bridge of the resulting ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus we may assume  that  (**) every bridge of ∆α−1 occurs in the boundary of a tile of index 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' a connected  component of ∆α−1 − �  Π∈T Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We define w as follows:  (i) Let Θ be a connected component of ∆α−1 − �  Π∈T Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For each bond or bridge u of rank  α − 1 occurring in the boundary of Θ, define  w(u) = −1  3(1 + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For each side S of Θ,  w(S) = ζθ|S|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let Π ∈ T and let δΠ = Pu1Qu2 as in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (**), for each i = 1, 2 the tiling  side u−1  i  occurs in the boundary of a connected component of ∆α−1 − �  Π∈T Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (i), we  already have  w(ui) = −w(u−1  i ) = 1  3(1 + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  22  We define w(P) (the definition of w(Q) is similar):  (7-3)  w(P) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  if P has type I or II  1  3(1 − 2ν)  if P has type III and Q has type I  1  6(1 − 2ν)  if P has type III and Q has type II or III  (iii) Let D be a cell of rank α of ∆ and S be a tiling side occurring in δD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The value of w(S)  is already defined by (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have:  If S−1 is the contiguity arc of a contiguity subdiagram Π ∈ T of D to a side of ∆α−1  of type I or II then w(S) = −1  3(1 − 2ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If S−1 is the contiguity arc of a contiguity subdiagram Π ∈ T of D to a side of ∆α−1  of type III then w(S) = −1  6(1 − 2ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If S−1 occurs in the boundary of a connected component of ∆α−1 − �  Π∈T Π then  w(S) = −ζθ|S|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We provide an upper bound for the curvature of any internal tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For contiguity subdia-  grams Π ∈ T we immediately have κ(Π) ≤ 0 by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Θ be a connected component of ∆α−1 − �  Π∈T Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have  κ(Θ) = χ(Θ) − 1  3(1 + ν)b(Θ) + ζθ  �  S  |S|α−1  where the sum is taken over the sides S of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If α = 1 then � |S|α−1 = 0 (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4(iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If α ≥ 2 then by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8α−1,  θ  �  |S|α−1 ≤ 2  3(1 + ν)b(Θ) − χ(Θ)  Using the fact that c(Θ) > 0 it is easy to check that κ(Θ) ≤ 0 in both cases α = 1 and  α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (The critical case is when b(Θ) = 3 and χ(Θ) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' in this case we have κ(Θ) = −ν if  α = 1 and κ(Θ) = 0 if α ≥ 2 by definition (7-1) of ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Finally, let D be a cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We prove that κ(D) ≤ −θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (*), D has at most  one contiguity subdiagram to sides of ∆α−1 of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider first the case when D  has one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let r be the number of contiguity subdiagrams of D to sides of types II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The remaining r + 1 subpaths S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Sr+1 of δD are tiling sides such that S−1  i  belong to  boundary loops of connected components of ∆α−1 − �  Π∈T Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' so we have  κ(D) ≤ 1 − 1  3(1 − 2ν) − r  �1  6(1 − 2ν)  �  − ζθ  r+1  �  i=1  |Si|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By condition (S1) in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 and Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8,  r+1  �  i=1  |Si|α−1 ≥ (1 − ρ − rλ)Ω = (9 − r)λΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Hence  (7-4)  κ(D) ≤ 2  3(1 + ν) − r  �1  6(1 − 2ν)  �  − ζθλΩ max(0, 9 − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  23  If r ≥ 9 then the coefficient before r in the right-hand side of (7-4) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If r ≤ 9 then  the coefficient is  −1  6(1 − 2ν) + ζθλΩ  which is positive since by the second inequality (2-3) we have ζθλΩ ≥ 20ζθ = θ > 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence  the maximal value of the expression in (7-4) is when r = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Substituting r = 9 into the  right-hand side of (7-4) we obtain the expression  2  3(1 + ν) − 9  6(1 − 2ν)  which is equal −θ by (7-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This shows that κ(D) ≤ −θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that D has no contiguity subdiagrams to sides of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let, as above, r be the  number of contiguity subdiagrams of D to sides of types II and III and S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Sr be the  remaining r tiling sides occurring in δD such that S−1  i  belong to boundary loops of connected  components of ∆α−1 − �  Π∈T Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Instead of (7-4) we have  (7-5)  κ(D) ≤ 1 − r  �1  6(1 − 2ν)  �  − ζθN max(0, 1 − rλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If we allow r to be a non-negative real then the maximal value of the right-hand side is when  1 − rλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Substituting r = 1  λ into the left-hand side of (7-5) we obtain the expression  1 − 1 − 2ν  6λ  which is less then −θ since λ ≤  1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Finally, we compute an upper bound for κ(∂∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For a tiling side S occurring in the  boundary loop of an external cell of ∆ (the loop has the form L−1 where L is a boundary  loop of ∆) we have three possibilities: either S−1 is a contiguity arc of a subdiagram Π ∈ T,  S−1 is a side of a component of ∆α−1 − �  Π∈T Π, or S−1 is a bridge of ∆α−1 In the first two  cases we have w(S) ≤ 0 according to (ii) or (i) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S−1 is a bridge of ∆α−1 then  by (**), S−1 is also a bridge of some component of ∆α−1 − �  Π∈T Π and by (i),  w(S) = 1  3(1 + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that each bridge of ∆ produces at most two bridges of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence b(∆α−1) ≤ 2b(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We obtain  (7-6)  κ(∂∆) ≤ 1  3(1 + ν)b(∆α−1) ≤ 2  3(1 + ν)b(∆)  Application of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 gives  2  3(1 + ν)b(∆) − θM ≥ χ(∆)  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced disk diagram of rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If ∆ has a single (cyclic or  non-cyclic) side then ∆ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  24  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced disk diagram of rank α with a single side, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ∆ is of monogon or  nullgon type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that ∆ has a cell of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We choose such ∆ with minimal possible  non-zero number M of cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We then have χ(∆α−1) ≤ 0 and hence c(∆α−1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We can assume that ∆ is given a tight set T of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If each cell of rank α  of ∆ has at most one contiguity subdiagram Π ∈ T to the side of ∆ then application of  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 would give  θM ≤ 2  3(1 + ν) − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Therefore, ∆ has a cell D of rank α having two contiguity subdiagram Π1, Π2 ∈ T to the  side of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The union D ∪ Π1 ∪ Π2 cuts off from ∆ a disk diagram ∆′ of rank α with a single  side and a single bridge (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The assumption that ∆ is reduced implies that ∆′ is  D  ∆′  Π1  Π2  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  reduced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the choice of ∆, ∆′ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then for some component Θ  of ∆α−1 − �  Π∈T Π we have c(Θ) = 0 contrary to the choice of a tight set T of contiguity  subdiagrams of ∆ (Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If a non-empty word X is reduced in Gα then X ̸= 1 in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be reduced in Gα and X = 1 in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consider a reduced disk  diagram ∆ of rank α with one side labeled X and one bridge labeled by the empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 says that ∆ has no cells of rank α and hence we have X = 1 in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since  Rα ⊆ Rα−1, arguing by induction we conclude that X = 1 in the free group G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since X is  freely reduced (definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5) we conclude that X is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced diagram of rank α ≥ 1 and let u be a simple path in ∆  homotopic rel endpoints to a subpath S of a side of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume, moreover, that the label of u  is equal in Gα−1 to a word in Hα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the subdiagram of ∆ with boundary loop Su−1 has  no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆′ be the subdiagram of ∆ with boundary loop Su−1 and let w ∈ Hα−1 be a word  such that label(u) = w in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We attach to ∆′ a diagram Θ over the presentation of Gα−1  with boundary loop uw−1 where label(w) = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider ∆′ ∪ Θ as a diagram of rank α  with one side S and one bridge w−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that any simple path in ∆′ ∪ Θ with endpoints  in ∆′ is homotopic rel endpoints to a simple path in ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Moreover, this holds also if ∆′ ∪ Θ  is refined to a diagram Σ and we take a refinement of ∆′ in Σ instead of ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies  that ∆′ ∪ Θ is a reduced diagram of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, ∆′ ∪ Θ has no cells of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  25  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (bounding sides of a small diagram, raw form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a small diagram  of rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that ∆ is not of bigon type and c(∆α−1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  (7-7)  θ  �  S  |S|α ≤ 2  3(1 + ν)b(∆) − χ(∆)  where the sum is taken over all sides S of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We make ∆ reduced and endow it with a tight set T of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  assign to subpaths of sides of ∆α−1 type I, II and III as in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 and  make several observations about T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim 1: There are no contiguity subdiagrams Π ∈ T between two (not necessarily distinct)  sides of type I of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume Π is such a contiguity subdiagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let δΠ = Pu1Qu2 where P and Q are the  contiguity arcs of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 at least one of ui’s, say u1, is a bond  in ∆α−1 (otherwise Π = ∆α−1 contrary to the assumption c(∆α−1) > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Checking with  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 we see that u1 is also a bond in ∆ (condition (iii) of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 holds due  to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This contradicts the assumption that ∆ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim 2: Up to inessential change of ∆ we may assume that condition (*) of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4  is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' each cell of rank α of ∆ has at most one contiguity subdiagram Π ∈ T to  sides of type I of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that a cell D of rank α has two contiguity subdiagrams Πi ∈ T (i = 1, 2) to  sides Si of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Pi be the contiguity arc of Πi that occurs in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The boundary loop  of D ∪ Π1 ∪ Π2 has the form P1u1P2u2 where labels of ui are in Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since ∆ is small, at  least one of the conditions (iii) or (iv) of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 should be violated for each of the  paths ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S1 = S2 and some ui (and hence both u1 and u2) are homotopic rel endpoints  to a subpath of S1 then D ∪ Π1 ∪ Π2 cuts off a reduced disk subdiagram ∆′ of ∆ with one  bridge u−1  1  or u−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, ∆′ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then either ∆′ is a component  of ∆α−1 −�  Π∈T Π or ∆′ contains a component Θ of ∆α−1 −�  Π∈T Π with c(Θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We come  to a contradiction with the choice of a tight set T of contiguity subdiagrams of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that condition (iv) of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 fails for both u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then, up to renumer-  ation of Π1 and Π2, D ∪ Π1 ∪ Π2 cuts off a simply connected subdiagram ∆′ with boundary  loop u−1  1 T1vT2 where P1T1 is an ending subpath of S1, v is a bridge of ∆, T2P2 is a starting  subpath of S2 and labels of P1T1 and T2P2 are empty, see Figure 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case, we cut  off the subdiagram D ∪ Π1 ∪ Π2 ∪ ∆′ from ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The operation does not change the values  of � |S|α, b(∆) and χ(∆) in (7-7) and preserves the assumption that ∆ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have  also c(∆α−1) > 0 for the modified ∆ (otherwise ∆ would be a monogon type contradicting  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim 3: Up to inessential change of ∆ we may assume that there are no contiguity subdia-  grams Π ∈ T between sides of type I and II of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that Π ∈ T is a contiguity subdiagram between sides of type I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let δΠ =  Pu1Qu2 where P occurs in a side S of ∆ and Q occurs in the central arc R of a bridge  v = v1Rv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe that any of the endpoints of P can be joined with any of the endpoints  of v by a path labeled with a word in Hα in a graph composed from paths u1, u2 and v, see  26  P1  T1  v  D u1  P2  T2  u2  S1  S2  ∆′  Π1  Π2  v1  S1  P  S2  u1  Q  R1  u2  Π  v2  R2  a  b  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Figure 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since ∆ is small, this easily implies that v and S are adjacent in the boundary  of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to symmetry, assume that vS occurs in a boundary loop of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' so R = R1QR2  and S = S1PS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that label(S1P) is empty (otherwise v1R1u−1  1  would give a bond in ∆  after refinement) and label(QR2) is nonempty (because u1 is a bond in ∆α−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We cut off  the subdiagram of ∆ bounded by QR2v2S1Pu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As in the proof of the previous claim, the  operation does not change the values of terms in (7-7), the value of c(∆α−1) and keeps the  assumption that ∆ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' On the other hand, we decrease the total length of labels of  sides ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We now define a discrete connection w∗ on (∆, T) by changing the function w defined in  the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The new function w∗ differs from w only on contiguity arcs of  contiguity subdiagrams Π ∈ T as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let δΠ = Pu1Qu2 where P and Q are the contiguity  arcs of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Claims 1 and 3, if P has type I then Q has necessarily type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Instead of  (7-3) we define  w∗(P) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  θ  if P has type I  1  3(1 − 2ν) − θ  if P has type III and Q has type I  1  6(1 − 2ν)  in all other cases  For contiguity subdiagrams Π ∈ T we immediately have κ∗(Π) ≤ 0 where κ∗ denotes the  curvature function defined from w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If Θ is a connected component of ∆α−1 − �  Π∈T Π then  κ∗(Θ) = κ(Θ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let D be a cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In view of Claim 2  κ∗(D) ≤ κ(D) + θ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We provide a bound for κ∗(∂∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let t be the number of all contiguity subdiagrams Π ∈ T  between sides of type I and sides of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  κ∗(∂∆) ≤ 1  3(1 + ν)b(∆α−1) − θt − ζθ  �  S∈sides(Θ)  |S|α−1  ≤ 2  3(1 + ν)b(∆) − θ  �  S∈sides(∆)  |S|α  27  where Θ runs over all connected components of ∆α−1 − �  Π∈T Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3  we obtain  2  3(1 + ν)b(∆) − θ  �  S∈sides(∆)  |S|α ≥ χ(∆)  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  Below we will often use Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 in a slightly simplified form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We introduce yet  another numerical parameter  η = 1 + 2ν  θ  = 30  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (bounding sides of a small diagram, simplified form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If ∆ is a small  diagram of rank α of positive complexity then  (7-8)  �  S∈ sides(∆)  |S|α ≤ η c(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4(iii) we may assume that α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  It remains to notice that if  c(∆) ≥ 1 then  1  θ  �2  3(1 + ν)b(∆) − χ(∆)  �  ≤ η c(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (The critical case is when b(∆) = 3 and χ(∆) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case we have the equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced diagram of rank α ≥ 1 and let T be a tight set of contiguity  subdiagrams of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let D be a cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let Π1 and Π2 be two contiguity subdiagrams of D to a side S of ∆α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then a  subdiagram Θ of ∆ bounded by δD, Π1, Π2 and S (there are two of them if S is a  cyclic side) is not simply connected (see Figure 11a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let Π be a contiguity subdiagram of D to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the subdiagram Θ′ of ∆ bounded  by δD and Π (see Figure 11b) is not simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If ∆ is simply connected then any cell of rank α has at most one contiguity subdia-  gram to each side of ∆α−1 and has no contiguity subdiagrams to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  S  Π1  Π2  Θ  R  u  Θ′  Π  D  D  a  b  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  28  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) Assume that Θ is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider Θ as a diagram of rank α with  a single side that is a subpath of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The assumption that ∆ is reduced implies that Θ is  reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 Θ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we obtain a contradiction with the  choice of a tight set T of contiguity subdiagrams of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Assume that Θ′ is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∂Θ′ = Ru where R−1 occurs in the boundary  loop of D and u−1 is the bond in ∆α−1 that occurs in ∂Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider Θ′ as a a diagram  of rank α with one side S labeled by the empty word and one bridge Ru (formally, to fit  the definition in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 we have to take a copy of Θ′ and perform a refinement to make S a  non-empty path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 Θ′ has no cells of rank α and we come to a contradiction  since in this case u−1 cannot be a bond in ∆α−1 due to condition (iii) of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) follows from (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (diagrams of small complexity are single layered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced  diagram of rank α ≥ 1 and let T be a tight set of contiguity subdiagrams of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) If ∆ is a disk diagram of bigon type then every cell of rank α of ∆ has a contiguity  subdiagram Π ∈ T to each of the two sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) If ∆ is a disk diagram of trigon or tetragon type then every cell of rank α of ∆ has  contiguity subdiagrams Π ∈ T to at least two sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If ∆ is an annular diagram with two cyclic sides then every cell of rank α of ∆ has  a contiguity subdiagram Π ∈ T to each of the sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iv) If ∆ is an annular diagram with one cyclic side and one non-cyclic side then every  cell D of rank α of ∆ has at least two contiguity subdiagrams Π, Π′ ∈ T to sides  of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Here we admit the possibility that both Π and Π′ are contiguity subdiagrams  between D and the non-cyclic side of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced diagram of rank α of a type listed in (i)–(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We call a cell D of  rank α of ∆ regular if it satisfies the conclusion of the corresponding statement (i)–(iv) and  exceptional otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We need to prove that ∆ has no exceptional cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe that by  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, an exceptional cell has at most one contiguity subdiagram to sides of ∆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  such a cell satisfies condition (*) of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We use induction on the number M of  cells of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let ∆ be of bigon type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' a disk diagram with two sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If ∆ has no regular cells  of rank α but has at least one exceptional cell then application of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 gives a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that D is a regular cell of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Πi (i = 1, 2) be the contiguity subdiagram  of D to Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The complement of D ∪ Π1 ∪ Π2 in ∆ consists of two components ∆1 and ∆2  of bigon type with the induced boundary marking of rank α (see Figure 12a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The set of  subdiagrams Π ∈ T contained in ∆i is a tight set of contiguity subdiagrams of ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Each  of the subdiagrams ∆i has a smaller number of cells of rank α, so the statement follows by  induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let ∆ be of trigon or tetragon type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that ∆ has a regular cell D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Πi (i =  1, 2) be contiguity subdiagrams of D to sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The complement of ∆−D∪Π1∪Π2 consists  of two components ∆1 and ∆2 with the induced boundary marking of rank α (Figure 12b)  making them diagrams of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If ∆ is of trigon type then ∆1 and ∆2 are of trigon and  bigon types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If ∆ is of tetragon type then either ∆1 and ∆2 are of tetragon and bigon types,  or both ∆i are of trigon type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we can refer to (i) and the inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  29  D  D  ∆1  ∆2  ∆1  ∆2  a  b  Π2  Π1  Π1  Π2  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that all cells of rank α of ∆ are exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4  (7-9)  θM ≤ 8  3(1 + ν) − 1  which implies M ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Following the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 we compute a better bound  for M and conclude that M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that M ≥ 1 and let D be a cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consider the discrete connection w  on (∆, T) defined in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An upper bound for κ(D) is given by (7-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The right-hand side of (7-4) is a linear expression on r and, as we have seen in the proof of  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, in the case r ≤ 9 the coefficient before r is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To get a value for the  upper bound, we compute the maximal possible value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe that by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, D  has no contiguity subdiagrams to itself, has at most one contiguity subdiagram to another  cell of rank α of ∆ (if that cell exists) and the number of contiguity subdiagrams of D to  sides of type II is at most 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' so r ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the maximal value of the right-hand side of  (7-4) is achieved when r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Substituting r = 5 into (7-4) and using (2-3) we obtain  κ(D) ≤ 2  3(1 + ν) − 5  6(1 − 2ν) − 4ζθλΩ  ≤ −1  6 + 7  3ν − 4θ = −138  54 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By (7-6)  κ(∂∆) ≤ 8  3(1 + ν) = 152  54 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 gives  1 = κ(∆) + κ(∂∆) ≤ 14  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The contradiction shows that the assumption M ≥ 1 is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii): Similarly to the proof of (ii), assume first that ∆ has a regular cell D of rank α with  two contiguity subdiagrams Π1 and Π2 to sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(i) these are contiguity  subdiagrams to distinct sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the complement ∆ − (D ∪ Π1 ∪ Π2) is a diagram  of bigon type and the statement follows directly from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If all cells of rank α of ∆ are exceptional and there is at least one cell of rank α then  application of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 gives an immediate contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iv): Assume that ∆ has a regular cell D of rank α with two contiguity subdiagrams Πi  (i = 1, 2) to sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  There are two cases depending on whether or not Π1 and Π2  are contiguity subdiagrams to distinct sides of ∆ (see Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In the first case, the  30  D  D  Π1  Π2  Π1  Π2  ∆1  ∆2  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  complement ∆ − (D ∪ Π1 ∪ Π2) is a diagram of trigon type and the statement follows from  the already proved part (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the second case, ∆ − (D ∪ Π1 ∪ Π2) consists of a simply  connected component ∆1 and and an annular component ∆2 with one non-cyclic side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For  cells of rank α in ∆1 the statement follows by (i) and for cells of rank α in ∆2 we can apply  induction since ∆2 has a strictly smaller number of cells of rank α than ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If all cells of rank α of ∆ are exceptional then application of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 gives M =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (small diagrams of trigon or tetragon type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a small diagram  of rank α of trigon or tetragon type with sides Si (1 ≤ i ≤ k, k = 3 or k = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  3  �  i=1  |Si|α ≤ 4ζη  or  4  �  i=1  |Si|α ≤ 6ζη  in the trigon and tetragon cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4(iii) we may assume that α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We assume that ∆ is reduced and is given a tight set T of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fol-  lowing arguments from the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 we can assume that Claims 1–3 from  that proof hold in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Claim 2 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11(ii), ∆ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Claims 1 and 3, T has only contiguity subdiagrams between sides of ∆α−1 of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Hence any side of ∆ occurs entirely in a boundary loop of a connected component Θ of  ∆α−1 − �  Π∈T Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, �  Θ c(Θ) = c(∆α−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9α−1 to  components Θ of ∆α−1 − �  Π∈T Π we obtain  �  i  |Si|α−1 ≤ ηc(∆α−1) ≤ (b(∆α−1) − 2)η  which gives the required inequality by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (cell in a diagram of small complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a reduced diagram of  rank α ≥ 1 of one of the types listed in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let T be a tight set of contiguity  subdiagrams on ∆ and let D be a cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Pi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , r be the contiguity  arcs of contiguity subdiagrams of D to sides of ∆ that occur in δD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then:  (i) If ∆ has bigon type or is an annular diagram with two cyclic sides then r = 2 and  µ(P1) + µ(P2) ≥ 1 − 2λ − 16ζηω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  31  (ii) If ∆ has trigon type then 2 ≤ k ≤ 3 and  k  �  i=1  µ(Pi) ≥ 1 − 3λ − 24ζηω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If ∆ is an annular diagram with one cyclic side and one non-cyclic side then 2 ≤  k ≤ 3 and  k  �  i=1  µ(Pi) ≥ 1 − 4λ − 24ζηω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that C is another cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11, C has at least  two contiguity subdiagrams Π1, Π2 to sides of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆′ be the connected component of  ∆−C−Π1 −Π2 containing D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then ∆′ inherits from ∆ the boundary marking of rank α and  the tight set of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe also that ∆′ is also a diagram of rank α of  one of the types in cases (i)–(iii);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' moreover, it is of the same type (i)—(iii) or has a smaller  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case the statement is reduced by induction to the case of a diagram with  a smaller number of cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  It remains to consider the case when D is a single cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The equality r = 2  in (i) and the bound 2 ≤ r ≤ 3 in (ii) and (iii) follow from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' With bounds from  Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8, Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 for α := α − 1 and inequality (4-2), an easy analysis  shows that the worst cases for the lower bound on �  i µ(Pi) are as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  then get the corresponding inequality in (i)–(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  32  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fragments  In this section we establish several properties of fragments of rank α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Most of them  are proved using facts about relations in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Starting from this point we use extensively  statements from subsequent Sections 9–13 for values of rank β < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We also switch our main  action scene to Cayley graphs Γα−1 and Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  All statements in this section are formulated and proved under assumption α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following observation is a consequence of the assumption that the graded presentation  of Gα is normalized, condition (S3) and the fact that centralizers of non-torsion elements  of Gα−1 are cyclic (Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8α−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Recall that two periodic lines L1 and L2 in Γα−1  are called parallel if sP1,L1 = sP2,L2 where Pi is the period of Li (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If L1 and L2 are two parallel periodic lines in Γα−1 whose periods are relators  of rank α then L1 = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Li (i = 1, 2) be two parallel periodic lines in Γα−1 whose periods Ri are relators  of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to cyclic shift of Ri we can assume that Ri ∈ X±1  α  where Xα is the set of  defining relators of rank α in the presentation (2-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let vi be a vertex on Li such that the  label of Li starts at vi with Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let g = v−1  1 v2 ∈ Gα (recall that we identify vertices of Γα  with elements of Gα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since L1 and L2 are parallel we have gR2g−1 = R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (S3) we have  either R1, R2 ∈ Xα or R−1  1 , R−1  2  ∈ Xα, so according to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, we get R1 ≖ R2 and  R1 ≖ Rt  0 where R0 it the root of R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since the centralizer of R1 is cyclic, we have g = Rk  0  for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies L1 = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary (Small cancellation in the Cayley graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let L1 and L2 be periodic lines  in Γα−1 with periods R1 and R2, respectively, where both Ri are relators of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  that L1 and L2 have close subpaths S1 and S2 such that |S1|α−1 ≥ λ|R1|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then L1 = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If |Si| ≤ |Ri| for i = 1, 2 then the statement follows directly from condition (S2-  Cayley) in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let |S1| > |R1| or |S2| > |R2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 and condition (S1)  we find close subpaths S′  1 and S′  2 of S1 and S2 with |Si| ≤ |Ri|, i = 1, 2 and |Sj|α−1 ≥ λ|Rj|α−1  for j = 1 or j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This reduces the statement to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A relator of rank α is strongly cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let R be a relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that some power Rt is not reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, for some 1 ≤ β ≤ α − 1 there exists a subword S of Rt which is  close in Gβ−1 to a piece P of rank β with µ(P) > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since R is cyclically reduced in Gα−1  we have |S| > |R|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then according to the definition in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6 we have |R◦|β ≤ 1 and hence  |R◦|α−1 ≤ ζα−β−1|R◦|β ≤ 1  contradicting (S1) and (2-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A fragment path of rank α in Γα−1 is a path F labeled by a fragment of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  assume that F has an associated R-periodic segment P with R ∈ Xα which is close to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  call P the base for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that this agrees with the definition in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F is a fragment of rank α with asso-  ciated triple (P, u, v) and F is a path in Γα−1 with label(F) ≖ F then the loop F−1uPv with  label(uPv) ≖ uPv gives a base P for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Conversely, if F is a fragment of rank α in Γα−1  with base P then choosing a loop F−1uPv with label(u), label(v) ∈ Hα−1 and denoting F, P,  33  u and v the corresponding labels we obtain a fragment F of rank α with associated triple  (P, u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If β ≥ α and paths F and P in Γβ are obtained by mapping a fragment ¯F of rank α with  base ¯P in Γα−1 then, by definition, we consider F as a fragment of rank α with base P in Γβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Abusing the language we will use the term ‘fragment’ for both fragment words and frag-  ment paths in Γβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Recall that by a convention in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, a base P for a fragment F of rank α in Γβ has an  associated relator R of rank α and the unique infinite R-periodic extension L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β = α − 1  then L is a bi-infinite path (which is simple by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3) that we call the base axis  for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β > α then L is winding over a relator loop labeled R that we call the base relator  loop for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We describe a way to measure fragments of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If P is a subword of a word Rk  where R is a relator of rank α then we define  (8-1)  µ(P) = |P|α−1  |R◦|α−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that this agrees with the definition in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 of the function µ(S) on the set of pieces S  of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F is a fragment of rank α ≥ 1 then the size µf(F) of F is defined to be equal  to µ(P) where P is the associated subword of Rk and R is the associated relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Thus, for example, µf(F) = 1  2 means approximately that F is close in rank α − 1 to a “half”  of its associated relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If F is a fragment of rank α in Γβ then we set µf(F) = µf(label(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This means that µf(F)  is given by the formula  µf(F) = |P|α−1  |R◦|α−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  where P is the base for F and R is the relator associated with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Using Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21<α we can easily reformulate the definition of a reduced in Gα word  in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 in the following way: a word X is reduced in Gα if and only if X is freely reduced and  contains no fragments F of rank 1 ≤ β ≤ α with µf(F) > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Two fragments F and G of rank α in Γα−1 are compatible if their base axes  are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1, the base axes of fragments of rank α are parallel if  and only if they coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In the case β ≥ α, two fragments F and G of rank α in Γβ are defined to be compatible if  they have compatible lifts in Γα−1, or, equivalently, F and G have the same base relator loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  It will be convenient to extend compatibility relation to fragments of rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Recall that  according to the definition in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6 fragments of rank 0 are letters in A±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, fragments  of rank 0 in Γβ are paths of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By definition, fragments F and G of rank 0 in Γβ are  compatible if and only if F = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We write compatibility of fragments as F ∼ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that we have in fact a family of  relations with two parameters α ≥ 0 and β ≥ max(0, α − 1): compatibility of fragments of  rank α in Γβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The values of β and α will be always clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Below we will use  also “compatibility up to invertion” relation on the set of fragments of rank α in Γβ, denoted  F ∼ G±1 and meaning that F ∼ G or F ∼ G−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Both are obviously equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  34  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (fragment stability in bigon of the previous rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y  be reduced close paths in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K be a fragment of rank α in X with µf(K) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  there exists a fragment M of rank α in Y such that M ∼ K and  µf(M) > µf(K) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P be the base for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (4-2) and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16α−1 we have P = z1P′z2 where  P′ is close to a subpath M of Y and |zi|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then M is a fragment of rank α  with base P′, so µf(M) = µ(P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (4-2)  µ(z1) + µ(z2) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω  and hence  µ(P′) > µ(P) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω = µf(K) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (fragment stability in trigon of the previous rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X−1∗Y1∗Y2∗ be a  coarse trigon in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K be a fragment of rank α in X such that µf(K) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then at  least one of the following statements holds:  For i = 1 or i = 2 there is a fragment Mi of rank α in Yi such that Mi ∼ K and  µf(Mi) > µf(K) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For each i = 1, 2 there is a fragments Mi of rank α in Yi such that Mi ∼ K and  µf(M1) + µf(M2) > µf(K) − 3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This follows from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 in a similar way as in the proof of Proposi-  tion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (fragment stability in conjugacy relations of the previous rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  X be a word cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Y be a word reduced in Gα−1, u ∈ Hα−1  and Y u = z−1Xz in Gα−1 for some z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We represent the conjugacy relation by two lines  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y−1u−1Y0u0Y1u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' and ¯X = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' X−1X0X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' in Γα−1 where label(Xi) ≖ X, label(Yi) ≖ Y  and label(ui) ≖ u (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let K be a fragment of rank α in ¯X with |K| ≤ |X| and  µf(K) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then at least one of the following statements is true:  For some i, there is a fragment M of rank α in Yi such that M ∼ K and  µf(M) > µf(K) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For some i, there are fragments M1 and M2 of rank α in Yi and Yi+1 respectively  such that Mi ∼ K (i = 1, 2) and  µf(M1) + µf(M2) > µf(K) − 3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Follows from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (inclusion implies compatibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K and M be fragments of rank α  in Γβ, β ≥ α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K is contained in M and µf(K) ≥ λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K ∼ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' First consider the case β = α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P and Q be bases for K and M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16α−1, there are close subpaths P′ of P and Q′ of Q such that µ(P′) ≥ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then by Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 P and Q have the same infinite periodic extension and we conclude  that K and M are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  35  If β ≥ α then we consider lifts ˜K and ˜M of K and M in Γα−1 such that ˜K is contained in  ˜M and apply the already proved part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (dividing a fragment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K be a fragment of rank α in Γβ, β ≥ α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If K = K1K2 then either K1 or K2 contains a fragment F of rank α with F ∼ K and µf(F) >  µf(K) − ζω, or K can be represented as K = F1uF2 where Fi are fragments of rank α, F1 is a  start of K1, F2 is an end of K2, F1 ∼ F2 ∼ K and  µf(F1) + µf(F2) > µf(K) − ζω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If α = 1 then u can be taken empty and the statement is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β = α − 1 ≥ 1  then the statement follows from Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The case β > α − 1 follows from the  case β = α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  As an immediate consequence of Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 we get:  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (overlapping fragments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a reduced path in Γβ, β ≥ α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K  and M be non-compatible fragments of rank α in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K ≤ M and µf(K), µf(M) ≥  λ+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there are a start K1 of K disjoint from M and an end M1 of M disjoint from K  such that K1 and M1 are fragments of rank α, K1 ∼ K, M1 ∼ M, µf(K) − µf(K1) < λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω  and µf(M) − µf(M1) < λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (union of fragments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a reduced path in Γα−1 and let Ki (i = 1, 2)  be compatible fragments of rank α in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that µf(Ki) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω for i = 1 or i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  the union of K1 and K2 is a fragment of rank α with the same base axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Moreover, if K1  and K2 are disjoint then µf(K1 ∪ K2) ≥ µf(K1) + µf(K2) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1, K1 and K2 have a common base axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If some of the Ki’s is contained  in the other then there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise the statement easily follows from  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary (compatibility preserves order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a reduced path in Γα−1, let Ki, Mi  (i = 1, 2) be fragments of rank α in X and let µf(Ki), µf(Mi) ≥ λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that  K1 ∼ K2, M1 ∼ M2 and K1 ̸∼ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K1 < M1 if and only if K2 < M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, for each i = 1, 2 neither of Ki or Mi can be contained in the  other, so we have either Ki < Mi or Mi < Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is enough to prove the statement in the case  K1 = K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume, for example, that M1 < K1 < M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 M1 ∪ M2 is a  fragment of rank α with M1 ∪M2 ̸∼ K1 and we get a contradiction with Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (no inverse compatibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K and M be fragments of rank α in a  reduced path X in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let µf(K), µf(M) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K ̸∼ M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Follows from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K be a fragment of rank β in Γα where 1 ≤ β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let R be the base loop for K labeled by a relator R of rank β and let R0 be the root  of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the subgroup {g ∈ Gα | gK ∼ K} is finite cyclic and conjugate to ⟨R0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let X be a word representing an element of Gα which is not conjugate to a power  of R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯X be an X-periodic line in Γα labeled X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then sX, ¯  XK ̸∼ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) Under hypothesis of (ii), if K is a subpath of ¯X and µf(K) ≥ 2λ+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ω then |K| < 2|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  36  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) It follows from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1β that gK ∼ K if and only if gR = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since label(R) ≖ Rt  0  and R0 is a non-power, the stabilizer of K in Gα is a subgroup conjugate to ⟨R0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) follows immediately from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If K is a subpath of ¯X, µf(K) ≥ 2λ+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ω and |K| ≥ 2|X| then using Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11β  and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10β we conclude that either s−1  X,¯XK ∼ K or sX,¯XK ∼ K, a contradiction with (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consequences of diagram analysis  Following the terminology introduced in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16, a coarse r-gon in Γα is a loop of the form  P = X1u1X2u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Xrur  where paths Xi are reduced and ui are bridges of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let us assume that each bridge ui of P is given an associate bridge partition of rank α (see  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13) and consider a filling φ : ∆(1) → Γα of P by a disk diagram ∆ over the presentation  of Gα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ∆ has boundary loop ˜X1˜u1˜X2˜u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , ˜Xr˜ur where φ(˜Xi) ≖ Xi and φ(˜ui) ≖ ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  can assume that ∆ has a boundary marking of rank α with sides ˜Xi and bridges ˜ui (see 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1)  and that each ˜ui has an induced bridge partition of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying to ∆ the reduction  process described in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 we get a reduced diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that during the process, bridges ˜ui  of ∆ can be changed by switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To keep the equality φ(˜ui) ≖ ui we have to perform  appropriated switching of bridges ui (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As a consequence we obtain:  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (filling coarse polygons by diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1 and P = X1u1X2u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Xrur  be a coarse r-gon in Γα with fixed bridge partitions of all bridges ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then, after possible  switching of bridges ui, there exists a reduced disk diagram ∆ of rank α which fills P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The α-area of P, denoted Areaα(P), is the number of cells of rank α of a  filling diagram ∆ as in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To avoid correctness issues, we assume formally that  Areaα(P) is defined with respect to a particular choice of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The image φ(δD) in Γα of the boundary loop of a cell of rank α of ∆ is an active relator  loop for P for a particular choice ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus Areaα(P) is the number of active relator loops  for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Abusing the language, we call the inverse loop φ(δD)−1 an active relator loop for P  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Equality Areaα(X1u1X2u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Xrur) = 0 is equivalent to the assertion that  X1u1X2u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Xrur lifts to Γα−1 after possible switching of bridges ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As a special case of a coarse polygon, consider a coarse bigon X−1uYv in Γα, α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up  to switching of bridges u and v we can assume that there is a reduced diagram ∆ of rank α  which fills X−1uYv via a map φ : ∆(1) → Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume also that ∆ is given a tight set T  of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The boundary loop of ∆ has the form ˜X−1˜u˜Y˜v with sides ˜X−1  and ˜Y which are mapped onto X−1 and Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11(i) each cell of  rank α of ∆ has a contiguity subdiagram to each of the sides ˜X−1 and ˜Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The boundary  loops of cells of rank α and the bridges of these contiguity subdiagrams form a graph mapped  in Γα as in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Ri be images in Γα of boundary loops of cells of rank α of ∆  and let Ki, Mi, Qi and Si be subpaths of X, Y and Ri, respectively, that are images of the  corresponding contiguity arcs of contiguity subdiagrams of cells of rank α to ˜X−1 and ˜Y, as  shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' According to the definition in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, Ki and Mi are fragments of rank α  37  u  Y  M1  M2  M3  v  R1  R2  R3  X  K1  K2  K3  S1  S2  S3  Q1  Q2  Q3  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  with bases Q−1  i  and Si and base relator loops R−1  i  and Ri respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We call Ki and Mi  active fragments of rank α of the coarse bigon X−1uYv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Thus, if Areaα(X−1uYv) = t then there are precisely t disjoint active fragments of rank α  in each of the paths X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note again that the set of active relator loops and the set of  active fragments formally depend on the choice of particular ∆ and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let, as above, P = X−1uYv be a coarse bigon in Γα and ∆ a reduced diagram of  rank α with δ∆ = ˜X−1˜u˜Y˜v filling P via a map φ : ∆(1) → Γα (we assume that the switching  operation is already applied to u and v if needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that ∆ has a tight set T of  contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let R = φ(δD) be an active relator loop of P and let Q−1w1K−1w2  and S−1w3Mw4 be images of boundary loop of contiguity subdiagrams in T of the cell D to  sides ˜X−1 and ˜Y respectively as in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then two loops P1 and P2 as shown in the figure  can be considered as coarse bigons in Γα with sides that are subpaths of X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' They are  X  Y  u  v  K  Q  S  w1  w2  w3  M  w4  P1  R  P2  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  filled by reduced subdiagrams of ∆, so we have Areaα(P1) + Areaα(P2) = Areaα(P) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  will use this simple observation in inductive arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  38  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In a similar way, let P = X1u1X2u2X3u3 be a coarse trigon in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' After possible switching  of bridges ui, we can find a reduced diagram ∆ of rank α with boundary loop ˜X1˜u1˜X2˜u2˜X3˜u3  which fills P via a map φ : ∆(1) → Γα of P where φ(˜Xi) = Xi and φ(˜ui) = ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can also  assume that ∆ has a tight set T of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11(ii) each cell  of rank α of ∆ has contiguity subdiagrams in T to at least two sides ˜Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies that  for any active relator loop R of P there are two or three fragments Ki (i = 1, 2 or i = 1, 2, 3)  of rank α with base loop R that occur in distinct paths Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly to the bigon case, we  call them active fragments of rank α of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As in the bigon case, for any active relator loop R of P we can consider a coarse bigon P1  and a coarse trigon P2 respectively, as shown in Figure 17, with Areaα(P1) + Areaα(P2) =  Areaα(P) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  X1  K1  K2  X2  X3  R  P1  P2  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (active fragments in bigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P = X−1uYv be a coarse bigon in Γα,  α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let K and M be active fragments of rank α of P in X and Y, respectively, with  mutually inverse base active relator loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K ∼ M−1,  µf(K) + µf(M) > 1 − 2λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω  and  µf(K), µf(M) > 7λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let K and K′ be two distinct active fragments of rank α in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K ̸∼ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i): It follows directly from the construction that K ∼ M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The first inequality follows  from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since X and Y are reduced we have µf(K) ≤ ρ and µf(M) ≤ ρ which  implies the lower bound on µf(K) and µf(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii): Assume that K ∼ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let M and M′ be the corresponding active fragments of rank α  in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (i), we have M ∼ M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 and the first inequality of (i),  µf(K ∪ K′) + µf(M ∪ M′) ≥ 2 − 4λ − 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω > 2ρ  which contradicts the hypothesis that X and Y are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  We introduce the notation for the lower bound on the size of active fragments in (i):  ξ0 = 7λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  39  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that paths X and Y in Γα are close in rank β ≤ α if there exist  bridges u and v of rank β such that X−1uYv is a loop that can be lifted to Γβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (So ‘being  close’ for paths in Γα means the same as ‘being close in rank α’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X and Y are labeled with freely reduced words then X and Y are close in  rank 0 if and only if X = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (lifting bigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let 0 ≤ β < α and X−1uYv be a coarse bigon in Γα where  u and v are bridges of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that for all γ in the interval β + 1 ≤ γ ≤ α either X  or Y has no fragments K of rank γ with µf(K) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X−1uYv can be lifted to Γβ and,  consequently, X and Y are close in rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This is a consequence of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 and Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (no active relators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1, X−1uYv be a coarse bigon in Γα and  Areaα(X−1uYv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that |X|α > 2 + 6ζ2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X and Y can be represented as  X = w1X1w2 and Y = z1Y1z2 where X1 and Y1 are close in rank α−1 and |wi|α, |zi|α ≤ 1+4ζ2η  (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 we can assume that X−1uYv lifts to Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To simplify notations, we  assume that X−1uYv is already in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let u = u1Pu2 and v = v1Qv2 where ui, vi are  bridges of rank α − 1 and P, Q are paths labeled by pieces of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We apply Proposition  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(ii)α−1 to the coarse tetragon X−1u1Pu2Yv1Qv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe that if a subpath of P or Q is  close (in Γα−1) to a subpath S of X then |S|α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since |X|α > 2 + 6ζ2η we cannot get the  first case of the conclusion of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(ii)α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Therefore, the second case holds: we  have X = X1z1X2z2X3 where X1 is close to a start of P, X2 is close to a subpath of Y, X3 is  close to an end of Q and |zi|α−1 ≤ 4ζη (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then |X1z1|α ≤ 1+4ζ2η, |z2X3|α ≤ 1+4ζ2η  and we get the required bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary (no active fragments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be close reduced paths in Γα, α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that either X or Y has no fragments K of rank α with µf(K) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume also that  |X|α > 2 + 6ζ2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X and Y can be represented as X = w1X1w2 and Y = z1Y1z2 where X1  and Y1 are close in rank α − 1 and |wi|α, |zi|α ≤ 1 + 4ζ2η (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary (no active fragments, iterated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be close reduced paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let 0 ≤ β < α and assume that for all γ in the interval β + 1 ≤ γ ≤ α either X or Y has no  fragments K of rank γ with µf(K) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let |X|α ≥ 2 + 3ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X and Y can be represented  as X = w1X1w2 and Y = z1Y1z2 where X1 and Y1 are close in rank β and |wi|α < 1 + 5ζ2η  (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a nonempty freely reduced word equal 1 in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X has a  subword P which is a piece of rank β where 1 ≤ β ≤ α and µ(P) > 136ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, X is not reduced in Gα and therefore contains a fragment K of  rank β where 1 ≤ β ≤ α and µf(K) ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β ≥ 1 be the minimal rank such that X contains  a fragment K of rank β with µf(K) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β = 1 then K is already a piece of rank 1 with  µ(K) ≥ ξ0 > 138ω by (4-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K be a fragment in Γβ−1 with label(K) ≖ K and  S a base for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13β−1 we have S = w1Pw2 where |wi|β−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='03 (i = 1, 2) and  P = label(P) occurs in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (4-1), µ(P) ≥ ξ0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='06ω = 7λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='56ω > 136ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  40  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (active fragments in trigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P = X1u1X2u2X3u3 be a coarse trigon  in Γα, let R be an active relator loop for P and let Ki (i = 1, 2 or i = 1, 2, 3) be active  fragments of rank α with base loop R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then Ki ∼ Kj for all i, j,  �  i  µf(Ki) > 1 − 3λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω  and  µf(Ki) > 3λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω  for at least two indices i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have Ki ∼ Kj by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The first inequality follows from Proposition  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since Xi is reduced in Gα we have µ(Ki) ≤ ρ = 1 − 9λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies the second  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (no active fragments in conjugacy relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be words  cyclically reduced in Gα, α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X = Z−1Y Z in Gα for some Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that no cyclic  shift of X contains a fragment K of rank α with µf(K) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists a word Z1  such that Z1 = Z in Gα and X = Z−1  1 Y Z1 in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆0 be a disk diagram of rank α with boundary label X−1Z−1Y Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We produce  an annular diagram ∆1 by gluing two boundary segments of ∆0 labeled Z−1 and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The  diagram ∆1 can be assigned a boundary marking of rank α with two cyclic sides X−1 and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We denote Z the path in ∆ with label(Z) ≖ Z that joins starting vertices of Y and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  ∆2 be a reduced diagram of rank α obtained from ∆1 by reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' According to  the remark in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7, ∆1 and ∆2 have the same frame type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 that  there exists a path Z1 in ∆2 joining starting vertices of boundary loops Y1 and X−1  1  such  that label(X1) ≖ X, label(Y1) ≖ Y and Z1 = Z in Gα where Z1 ≖ label(Z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i), ∆2 has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X = Z−1  1 Y Z1 in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (no active fragments in conjugacy relations, iterated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y  be cyclically reduced in Gα words which represent conjugate elements of Gα, α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that at least one of the words X or Y has the property that no its cyclic shift  contains a fragment K of rank γ with µf(K) ≥ ξ0 and β < γ ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯X = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' X−1X0X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  and ¯Y = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y−1Y0Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' be parallel periodic lines in Γα with label(Xi) ≖ X and label(Yi) ≖ Y  representing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then some vertices on ¯X and ¯Y are joined by a bridge  of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Moreover, for any subpath Z of ¯X there exists a loop S−1uTv which can lifted to Γβ such  that S and T are subpaths of ¯X and ¯Y respectively, u and v are bridges of rank β and Z is  contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since ¯X and ¯Y are parallel, if vertices a on ¯X and b on ¯Y are joined by a path labeled Z  then the same is true for all their translates sk  X,¯Xa and sk  Y,¯Yb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the second statement  follows from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ∆ be an annular diagram of rank α with boundary loops ˆX−1 and ˆY and φ : ˜∆(1) → Γα  a combinatorially continuous map of the 1-skeleton of the universal cover ˜∆ of ∆ to Γα  sending lifts ˜X of ˆX and ˜Y of ˆY to ¯X and ¯Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that ∆ is reduced  and has a tight set of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β = α and ∆ has a cell of rank α then the  statement follows from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If ∆ has no cells of rank α then we can lift ¯X  and ¯Y to Γα−1 and use induction on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β < α and at least one of the words X or Y has no  41  cyclic shift containing a fragment K of rank α with µf(K) > ξ0 then by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i),  ∆ has no cells of rank α and, again, the statement follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (small coarse polygons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P = X1∗X2∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xr∗ be a coarse r-gon in Γα  where r ≥ 3 and Xi are sides of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that there are no pairs of close vertices lying on  distinct paths Xi and Xj except pairs {τ(Xi), ι(Xi+1)} and {τ(Xr), ι(X1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  �  i  |Xi|α ≤ (r − 2)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If r = 3 or r = 4 then we have a stronger bound  �  i  |Xi|α ≤ 2(r − 1)ζη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consider a filling φ : ∆(1) → Γα of P by a reduced disk diagram ∆ of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  δ∆ = ¯X1u1¯X2u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ¯Xrur where ui are bridges and Xi are sides of ∆ with φ(¯Xi) = Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The  hypothesis of the proposition implies that ∆ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then the statement follows from  Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (trigons and tetragons are thin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let X−1∗Y1∗Y2∗ be a coarse trigon in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X can be represented as X = X1zX2  where X1 is close to a start of Y1, X2 is close to an end of Y2 and |z|α ≤ 4ζη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let X−1∗Y1∗Y2∗Y3∗ be a coarse tetragon in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then at least one of the following  possibilities holds:  X can be represented as X = X1zX2 where X1 is close to a start of Y1, X2 is close  to an end of Y3 and |z|α ≤ 6ζη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' or  X can be represented as X = X1z1X2z2X3 where X1 is close to a start of Y1, X2 is  close to a subpath of Y2, X3 is close to an end of Y3 and |zi|α ≤ 4ζη (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) We can represent X1 = X1zX2, Yi = Yi1wiYi2 (i = 1, 2) with close pairs (X1, Y11),  (Y12, Y−1  21 ) and (Y22, X2) where no vertices lying on distinct paths z, w1 and w2 are close  except appropriate endpoints (Figure 18a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement follows by application of  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18 to z−1∗w1∗w2∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  X1  z  X2  Y11  w1  Y12  Y21  w2  Y22  X1  X2  Y1  Y21  Y22  Y3  a  b  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) If there is a pair of close vertices on Y1 and Y3 then the statement follows from (i) giving  the first alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If there is a pair of close vertices on X and on Y2 then we represent  X and Y2 as X = X1X2, Y2 = Y21Y22 where τ(X1) and τ(Y21) are close, and apply (i) to  42  X−1  1 ∗Y1∗Y21∗ and X−1  2 ∗Y22∗Y3∗ (Figure 18b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We then come to the second alternative to  the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise we use an argument similar to the proof of (i) coming to the first  alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (small cyclic monogon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a word cyclically reduced in Gα and let  X be conjugate in Gα to a word Y u where Y is reduced in Gα and u is a bridge of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ¯X = �  i∈Z Xi and �  i∈Z Yiui be lines in Γα representing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  that no vertex on X0 is close to a vertex on Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then |X|α ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be an annular diagram of rank α with boundary loops ˆX and ˆY−1ˆu−1 represent-  ing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider ∆ as having a cyclic side ˆX, a non-cyclic side ˆY−1  and a bridge ˆu−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to switching of ˆu−1 we can assume that ∆ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The hypothesis  implies that ∆ cannot have a bond between ˆX and ˆY−1 after any refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that  ∆ has a bond v (possibly after refinement) joining two vertices on the same side ˆY−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  v cuts off from ∆ a simply connected subdiagram Σ with boundary loop Z1ˆu−1Z2v±1 where  ˆY−1 = Z2WZ1 for some W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' According to Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1, at least one of the words label(Zi)  (i = 1, 2) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Removing Σ from ∆ we obtain a diagram ∆′ with a shorter total label  of its two sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence, by induction, we can assume that ∆′ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then |X|α = |ˆX|α ≤ η  by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (closeness fellow traveling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be close reduced paths in Γα,  α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X and Y can be represented as X = U1U2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Uk and Y = V1V2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Vk (Ui and Vi  can be empty) where the starting vertex of each Ui is close to the starting vertex of Vi and  |Ui|α, |Vi|α ≤ ζ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe that the statement of the lemma holds in the case α = 0 with |Ui|0, |Vi|0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Thus we may refer to the statement of the lemma in rank α−1 with bounds |Ui|α−1, |Vi|α−1 ≤  1 which imply |Ui|α, |Vi|α ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe also that if X = X1X2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xr and Y = Y1Y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yr  where for each i, Xi and Yi are close then the statement of the lemma for each pair (Xi, Yi)  implies the statement of the lemma for X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 we represent X and Y as X =  X1X2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xr and Y = Y1Y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yr where pairs (Xi, Yi) satisfy the following conditions (1)  or (2) in the alternate way: (1) for some bridges ui and vi of rank α the loop X−1  i uiYivi lifts  to Γα−1 or (2) there are loops X−1  i wi1Riwi2 and Yiwi3Siwi4 which can be lifted to Γα−1 such  that Si and Ri occur in one relation loop of rank α and wij are bridges of rank α − 1 (see  Figure 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that pairs (X1, Y1) and (Xr, Yr) satisfy (2) and that in the case  Xi  wi1  wi2  wi3  wi4  Ri  Si  Yi  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  43  of (2), subpaths Xi, Yi of X, Y and Si, Ri of the appropriate relation loop cannot be extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We prove the statement for each of the pair (Xi, Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case of (1): Omitting the index i for Xi and Yi, assume that a loop X−1w1Pw2Yw3Qw4  lifts to Γα−1 where wi are bridges of rank α − 1 and P and Q are labeled by pieces of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without changing notations, we assume that X−1w1Pw2Yw3Qw4 is already in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By the maximal choice of Xi, Yi, Si and Ri in the case of (2), there are no close vertices  on pairs (X, P), (X, Q), (Y, P) and (Y, Q) except appropriate endpoints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' except ι(X) and  ι(P) for (X, P) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Depending on existence of close vertices on pairs (P, Q) and (X, Y) we  consider three cases (a)–(c) as in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In case (a) we have |X|α, |Y|α ≤ 6ζ2η < ζ by  X  Y  w1  P  w2  w3  Q  w4  X1  X2  X3  Y1  Y2  Y3  (a)  (b)  (c)  Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In case (b) taking the maximal pair of close subpaths of P and Q we  get |X|α, |Y|α ≤ 4ζ2η < ζ again by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In case (c) we have X = X1X2X3  and Y = Y1Y2Y3 where X2 and Y2 are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Taking X2 and Y2 maximal possible we get  |Xi|α, |Yi|α ≤ 4ζ2η for i = 1, 3 by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For X2 and Y2 we can apply the  statement for α := α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case of (2): In the second case by the statement of the lemma for α := α − 1 we have  X = U1U2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Uk and Y = W1W2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Wl where |Ui|α, |Wi|α ≤ ζ, the starting vertex of each Ui  can be joined by a bridge of rank α − 1 with a vertex on R and the starting vertex of each  Wi can be joined by a bridge of rank α − 1 with a vertex on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then each ι(Ui) is close to  ι(Y) and each ι(Wi) is close to τ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We take X = U1U2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Uk+l and Y = V1V2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Vk+l where  Uk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Uk+l, V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Vk are empty and Vj = Wj−k for k + 1 ≤ j ≤ k + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a reduced path and R a relation loop of rank α in Γα, α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ui  (i = 1, 2) be a path labeled by a word in Hα−1 and joining vertices ai on X and bi on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  Y be the subpath of X±1 that starts at a1 and ends at a2, and let R = R1R2 where Ri starts  at bi (Figure 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then one of the two loops Yu2R−1  1 u−1  1  or Yu2R2u−1  1  lifts to Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We fill the loop Yu2R−1  1 u−1  1  by a disk diagram ∆ of rank α with boundary loop ¯Y¯u2S¯u−1  1  where label(S) ≖ label(R−1  1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We take ¯Y as a side and ¯u2S¯u−1  1  as a bridge of ∆ with bridge  partition ¯u2·S·¯u−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we apply the reduction process making ∆ reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' After reduction,  we get either label(S) ≖ label(R−1  1 ) or label(S) ≖ label(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, ∆ has no cells  of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Depending on the case, this implies that either Yu2R−1  1 u−1  1  or Yu2R2u−1  1  lifts  to Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  44  X  a1  Y  a2  u1  u2  R1  R2  b1  b2  Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (compatibility lifting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let 1 ≤ β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K and M be fragments of  rank β which occur in a reduced path X in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ˆX be a lift of X in Γβ−1 and ˆK and ˆM  be the subpaths of ˆX which are projected onto K and M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K ∼ M implies  ˆK ∼ ˆM and K ∼ M−1 implies ˆK ∼ ˆM−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K ∼ Mε where ε = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let R be the common base loop for K and Mε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='22 implies that R can be lifted to a line ˆR which is the common base axis for both  ˆK and ˆMε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies ˆK ∼ ˆMε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let 1 ≤ β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then statements of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 and  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 hold for fragments of rank β in a reduced path X in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  More precisely, let X be a reduced path in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let Ki (i = 1, 2) be fragments of rank β in X, K1 ∼ K2 and µf(Ki) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω for i = 1  or i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K1 ∪ K2 is a fragment of rank β with K1 ∪ K2 ∼ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K1 and K2 are  disjoint then µf(K1 ∪ K2) ≥ µf(K1) + µf(K2) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let Ki, Mi (i = 1, 2) be fragments of rank β in X with µf(Ki), µf(Mi) ≥ γ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that K1 ∼ K2, M1 ∼ M2 and K1 ̸∼ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K1 < M1 if and only if  K2 < M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If K and M are fragments of rank β in X and µf(K), µf(M) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω then K ̸∼ M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Stability  Let FA be a free group with basis A and let X−1Y1Y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yk+1 = 1 be a relation in FA where  X, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Yk are freely reduced words in the generators A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then for any occurrence of a  letter aε ∈ A±1 in X there is a unique occurrence of the same letter aε in some Yi which cancels  with a−ε in X−1Y1Y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The main goal of this section is to establish an analog of this  statement for relations in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The role of letters aε will be played by fragments of rank α and  instead of relation X−1Y1Y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yk+1 = 1 we consider coarse polygons X−1∗Y1∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yk∗ in Γα  (for our considerations, it is enough to consider cases k = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The role of correspondence  of canceled letters will be played by equivalence relation ‘K ∼ L±1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  There are two essential differences of the case of groups Gα from the case of a free group FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  One is a “fading effect”: a fragment in Yi can be of a “smaller size” than an initial fragment  in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Another difference is that bridges of the coarse polygon can produce exceptions for  stability (to describe such situations we introduce a special relation between fragments and  bridges of the same rank β, see Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We start with a statement which shows how closeness is propagated in coarse tetragons  in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This is essentially a consequence of inductive hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  45  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition (uniformly close).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For α ≥ 1, we say that vertices a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , ar of Γα are  uniformly close if at least one of the following is true:  they are pairwise close in rank α − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' or  there exists a relator loop R of rank α such that each ai is close in rank α − 1 to a  vertex on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We cover also the case α = 0: vertices a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , ar of Γ0 are said to be uniformly close if  a1 = a2 = · · · = ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that uniformly close vertices are pairwise close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If r = 2 then being uniformly close  and being close is equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1, X and Y be close reduced paths in Γα−1, and let S−1∗T1∗T2∗T3∗  be a coarse tetragon in Γα−1 such that Y is a subpath of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that |X|α−1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  X can be represented as z0X1z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xrzr (1 ≤ r ≤ 3) where Xi is close to a subpath Wi of  some Tji, j1 < · · · < jr and  (10-1)  �  i  |Xi|α−1 > |X|α−1 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Moreover:  (i) if r = 3 then we have a stronger bound  �  i  |Xi|α−1 > |X|α−1 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) There is a subpath Y1 of Y such that the starting vertices ι(X1), ι(Y1) and ι(W1) are  uniformly close and the same is true for the ending vertices ι(Xr), ι(Y1) and ι(Wr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If α = 1 the statement is obvious (see Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Z be  a reduced path joining ι(S) and τ(T2) which exists by Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1 (see Figure 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We apply Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 first to the coarse trigon S−1∗Z∗T3∗ and then, possibly, to  X  Y  S  Z  T1  T2  T3  Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  the coarse trigon Z−1∗T1∗T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since |X|α−1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, after the first application of Proposition  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1, we find either a subpath X3 of X that is close to a subpath of T3 or a subpath X′  of X that is close to a subpath of Z with |X′|α−1 > |X|α−1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='75 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the latter  case, the second application of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 gives the remaining X1 and/or X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If r < 3 then  for the bound (10-1), the worst cases are when we get two Xi’s after double application of  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In those cases we have once case (iii) of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 and another time case (i) or (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  46  Hence �  i |Xi|α−1 > |X|α−1 − 3 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Statement (ii) follows from the appropriate part of  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that r = 3 and therefore X = z0X1z1X2z2X3z3 where each Xi is close to a subpath  of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' From application of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 we have |z0|α−1, |z3|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then using  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(i)α−1 we extend all Xi to get |z1|α−1, |z2|α−1 ≤ 4ζη < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If α = 1 then hypotheses of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 say that X = Y and S−1T1T2T3 is a  loop in the Cayley graph Γ0 of the free group G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement of the lemma holds  without the assumption |X|α−1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Furthermore, in the conclusion we have �  i |Xi|α−1 =  |X|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition (independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let 1 ≤ β ≤ α, K be a fragment of rank β in Γα and u  be a bridge of rank β in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Recall that K is considered with the associated base loop R of  rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that K is independent of u if either label(u) ∈ Hβ−1 or u possesses a bridge  partition u = v·S·w of rank β where S occurs in a relator loop L of rank β such that L ̸= R±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  It follows from the definition that if K is independent of u and M ∼ K±1 then M is also  independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (non-active fragment in bigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1, X−1uYv be a coarse bigon  in Γα and let X = F0K1F1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' KrFr where Ki are the associated active fragments of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let K be a fragment of rank α in X with µf(K) ≥ 2λ + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K ̸∼ Ki for all i  and that K is independent of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists a fragment of rank α in Y such that  M ∼ K and  µf(M) ≥ µf(K) − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 K is a subpath of one of the paths F0K1, K1F1K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , KrFr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  consider the case when K is a subpath of some KiFiKi+1 (the cases when K is a subpath of F0K1  or KrFr are similar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' see also the remark in the end of the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Y = H0M0H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' MrHr  where Mi are the corresponding active fragments of rank α in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As we can see from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, there is a loop T = (KiFiKi+1)−1w1S1w2Hiw3S2w4 which can be  lifted to Γα−1 and where wj are bridges of rank α − 1 and S1 and S2 occur in base loops  for Ki and Ki+1 respectively (see Figure 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Abusing notation we assume that T is already  S  L  Ki  Fi  Ki+1  X  K  w1  S1  w2  Hi  w3  S2  w4  Y  Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  47  in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then, instead of base loops, S1 and S2 occur in base axes L1 and L2 for Ki and Ki+1  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let L be the base axis for K and S the base for K (which is contained in L by definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assumptions K ̸∼ Ki and K ̸∼ Ki+1 imply L ̸= Li (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, if a subpath P  of S is close to a subpath of Si then µ(P) < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 we find a subpath Q  of S which is close to a subpath M of Hi and satisfies  µ(Q) > µ(S) − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then M is a fragment of rank α with base Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Clearly, M satisfies the conclusion of the  proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If K is a subpath of F0K1 or KrFr, a similar argument applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For example, assume that  K is a subpath of F0K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As above, we assume that all paths are in Γα−1 not changing their  notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let L be a base axis for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By hypothesis, either label(u) ∈ Hα−1 or u = u1Vu2  where V occurs in a line L1 labeled by the infinite power R∞ of a relator R of rank α and  L1 is distinct from L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case label(u) ∈ Hα−1 we apply Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise  the argument is the same as in the case when K is a subpath of KiFiKi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The case when K  is a subpath of KrFr is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Finally, there is a “degenerate” case when Areaα(X−1uYv) = 0 and both u and v are bridges  of rank α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case, the statement follows directly from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (fragment stability in bigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1, X−1uYv be a coarse bigon in Γα  and let K be a fragment of rank α in X with µf(K) ≥ 2λ+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K is independent  of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists a fragment M of rank α in Y such that M ∼ K±1 and  µf(M) ≥ min{µf(K) − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω, ξ0}  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X = F0K1F1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' KrFr and Y = H0M0H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' MrHr where Ki and Mi are the associ-  ated active fragments of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K ∼ Ki for some i then we can take M = Mi due to  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise we apply Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (fragment stability in trigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1, X−1u1Y1u2Y2u3 be a coarse  trigon in Γα and let K be a fragment of rank α in X with µf(K) ≥ 3λ + 10ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that  K is independent of any of ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there is a fragment M of rank α in Y1 or Y2 such that  M ∼ K±1 and  µf(M) > min  �  3λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω, 1  2(µf(K) − 3λ − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The idea of the proof is the same as in the proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  To avoid  complicated notations, we proceed by induction on the α-area of P = X−1u1Y1u2Y2u3 as  described in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that R is an active relator loop of rank α of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As observed  in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, there are two or three fragments Ni (i = 1, 2 or i = 1, 2, 3) of rank α with base  loop R that occur in distinct paths X−1, Y1 or Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 we can assume that  µf(Ni) ≥ 3λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K ∼ N±1  1  then we for the required M we take that Ni which  occurs in Y1 or Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K ̸∼ N±1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If N1 and N2 occur in Y1 and Y2 then we can replace P by a coarse trigon with smaller  α-area and use induction (see Figure 24a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (In this case u2 is replaced by a new bridge u′  2  and the assumption K ̸∼ N±1  1  implies that K is independent of u′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=') Otherwise, assume that  N1 occurs in X−1 and N2 occurs in Y1 (the case when N2 occurs in Y2 is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  48  K  X  Y1  Y2  N1  N2  N1  N2  K  a  b  Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since K ̸∼ N−1  1  we have either K < N−1  1  or K > N−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the first case, we reduce the  statement to the case of a coarse bigon as in Figure 24b and apply Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the  second case, the statement follows by inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  It remains to consider the case Areaα(P) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the loop P can be lifted to Γα−1 and  we assume that P is already in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let L be the base axis for K and S the base for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since  K is independent of ui (when viewed in Γα), we have either label(ui) ∈ Hα−1 or ui = viQiwi  where label(vi), label(wi) ∈ Hα−1 and Qi occurs in a line Li labeled by the infinite power R∞  i  of a relator Ri of rank α such that Li ̸= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We obtain a coarse r-gon with sides X−1, Y1, Y2  and Qi where 3 ≤ r ≤ 6 (see Figure 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider the “worst” case r = 6 (the other cases  are similar, with application of Propositions 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 or 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7α−1 where needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Z be a  K  X  v1  Q1  w1  Y1  v2  Q2  w2  Y2  v3  Q3  w3  Z  Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  reduced path joining τ(u1) and ι(u3) existing by Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, if a  subpath P of S is close to a subpath of Qi then µ(P) < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement easily follows by  applying Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 twice to coarse tetragons X−1v1Q1w1Zv3Q3w3 and Z−1Y1v2Q2w2Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1, X be a piece of rank 1 ≤ β < α or a fragment of rank β < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then X contains no fragment K of rank α with µf(K) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In particular, any fragment K of rank α with µf(K) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω is a nonempty word (since  otherwise it would occur in a fragment of rank 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider the case when X is a fragment of rank β < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We represent X by  a path X in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that X contains a fragment K of rank α with µf(K) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  49  Let S be a base for K with |S|α−1 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8≤α−1 and Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 we  have S = w1S1w2 and K = z1K1z2 where S1 and K1 are close in rank max(0, β − 1) and  |S1|α−1 > |S|α−1 −2−10ζ2η > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β = 0 we already get a contradiction since in this case  |K1| ≤ 1 but |S1| ≥ |S1|α−1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to change of notation, we assume that X,  K1 and S1 are lifted to Γβ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let T be a base for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16β−1 a subpath T1  of T is close to a subpath S2 of S with |S2|α−1 > |S1|α−1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ζ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then S2 is a fragment  of rank β with base T1 and we should have |S2|α−1 ≤ 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In the case when X is a piece of rank α a similar argument works with skipping application  of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16β−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1 and X be a word cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that a cyclic  shift of X contains a fragment K of rank α with µf(K) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X is strongly cyclically  reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let F be a fragment of rank 1 ≤ β ≤ α − 1 in a word Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that |F| > |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 represent K as K ≖ K1uK2 where µf(K1), µf(K2) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since  |K| ≤ |X|, F should contain a translate of K1 or K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' But this is impossible by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Hence |F| ≤ |X| and then µf(F) ≤ ρ since X is cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This shows that  any power Xt is reduced in Gα−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' X is strongly cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (fragment stability in conjugacy relations with cyclic sides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1  and X and Y be words which are cyclically reduced in Gα and represent conjugate elements  of Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯X = �  i∈Z Xi and ¯Y = �  i∈Z Yi be parallel lines in Γα representing the conjugacy  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K be a fragment of rank α in ¯X with µf(K) ≥ 2λ + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8ω and |K| ≤ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  there is a fragment M of rank α in ¯Y such that M ∼ K±1 and  µf(M) ≥ min{µf(K) − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω, ξ0}  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9 X is strongly cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We claim that a cyclic shift  of Y also contains a fragment F of rank α with µf(F) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 and thus Y is strongly cyclically  reduced in Gα−1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Indeed, by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17 with β := α − 1 we may assume for  some cyclic shifts X′ and Y ′ of X and Y we have Y ′ = w−1X′w in Gα−1 where w ∈ Hα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then existence of F easily follows by Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Consider a reduced annular diagram ∆ of rank α with boundary loops ˆX and ˆY−1 repre-  senting the conjugacy relation given in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ˜∆ be the universal cover of ∆  and let φ : ˜∆(1) → Γα be a combinatorially continuous map which sends lifts of ˆX and ˆY to  ¯X and ¯Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that ∆ has a cell of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let D be some lift of this cell in ˜∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i), φ(δD) and φ(δD)−1 are base loops for fragments Ni (i = 1, 2) of rank α in ¯X and ¯Y  respectively, such that µf(N1)+µf(N2) ≥ 1−2λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since X and Y are cyclically reduced  in Gα we have µf(Ni) ≤ ρ and hence µf(Ni) ≥ 1−ρ−2λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω = ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By construction, we have  N1 ∼ N−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since ¯X and ¯Y are parallel, we have sk  X,¯XN1 ∼ sk  Y,¯YN−1  2  for any k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K ∼ sk  X,¯XN1  for some k then we can take sk  Y,¯YN2 for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise we have sk  X,¯X,N1 < K < sk+1  X,¯XN1 for  some k and the rest of the argument is the same as in the proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Now assume that ∆ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that ∆ is a reduced diagram  of rank β for some β ≤ α − 1 and in case β ≥ 1, ∆ has at least one cell of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β = 0  then ¯X = ¯Y and there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to change of notations, we assume  that K, ¯X and ¯Y are lifted to Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i)β implies that some vertices a on ¯X  50  and b on ¯Y are joined by a bridge of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This is true also for any translates si  X,¯Xa and  si  Y,¯Yb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement follows by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 (here we use that X and Y are strongly  cyclically reduced in Gα−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1 and S be a word cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that S is  conjugate in Gα−1 to a word T1v1T2v2 where Ti are reduced in Gα−1 and vi are bridges of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯S = �  i∈Z Si and �  i∈Z T(i)  1 v(i)  1 T(i)  2 v(i)  2  be parallel lines in Γα−1 representing the  conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote U2i = T(i)  1  and U2i+1 = T(i)  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that a reduced path X in Γα−1 is close to a subpath Y of ¯S with |Y| ≤ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  |X|α−1 ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X can be represented as z0X1z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Xrzr (1 ≤ r ≤ 4) where each Xi is close  to a subpath of some Uji, j1 < · · · < jr, jr − j1 ≤ 3 and  �  i  |Xi|α−1 ≥ |X|α−1 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Z be a word reduced in Gα−1 such that Z = T1v1T2 in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We join ι(T(i)  1 ) and  τ(T(i)  2 ) with the path Zi labeled Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since |X|α−1 ≥ 8, application of Propositions 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19α−1  gives X = w1X′w2 or X = w1X′w2X′′w3 where, respectively, X′ is close to a subpath of some Zi  and |X′|α−1 ≥ |X|α−1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9 or for some i, X′ is close to a subpath of Zi, X′′ is close to a  subpath of Zi+1 and |X′|α−1 + |X′′|α−1 ≥ |X|α−1 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then a single or double application of  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 gives the required Xi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (fragment stability in conjugacy relations with non-cyclic side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  α ≥ 1 and X be a word cyclically reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that X is conjugate in Gα to  a word Y u where Y is reduced in Gα and u is a bridge of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯X = �  i∈Z Xi and  �  i∈Z Yiui be parallel lines in Γα representing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K be a fragment of  rank α in ¯X with µf(K) ≥ 3λ + 9ω and |K| ≤ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K is independent of any of  the bridges ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there is a fragment M of rank α in some Yk such that M ∼ K±1 and  µf(M) > min  �5  2λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω, 1  2(µf(K) − 3λ − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be an annular diagram of rank α with boundary loops ˆX−1 and ˆYˆu representing  the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ˜∆ be the universal cover of ∆ and φ : ˜∆(1) → Γα a combinatorially  continuous map sending lifts ˜Xi, ˜Yi and ˜ui of ˆX, ˆY and ˆu to Xi, Yi and ui respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to  switching of ˆu, we assume that ∆ is reduced and has a tight set T of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1: ∆ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then parallel lines ¯X = �  i∈Z Xi and �  i∈Z Yiui can be  lifted to Γα−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we assume that they and the subpath K of ¯X are already lifted to Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If  u ∈ Hα−1 then the statement follows by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19α−1, so we assume that u /∈ Hα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let L be the base axis for K and S the base for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since K is independent of ui (when viewed  in Γα) we have ui = w(i)  1 Qiw(i)  2  where label(w(i)  j ) ∈ Hα−1 and Qi occurs in a line Li labeled  by the infinite power R∞  i  of a relator Ri of rank α such that Li ̸= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, if  a subpath P of S is close to a subpath of Qi then µ(P) < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 we  conclude that either there exists a fragment M of rank α in some Yk such that M ∼ ¯K and  µf(M) > µf(K) −2λ −9ω or there exist fragments M1 and M2 of rank α in some Yk and Yk+1  respectively such that M1 ∼ M2 ∼ K and  µf(M1) + µf(M2) > µf(K) − 2λ − 9ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  51  In the latter case, for at least one Mi we have µf(Mi) > 1  2(µf(K) − 2λ − 9ω) and we can take  its image in Γα for the required M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: ∆ has at least one cell of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let D be such a cell and let ˜D be a lift of D  in ˜∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11(iv) and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(i), D has two or three contiguity subdiagrams  Πi ∈ T to sides of ∆, at most two to ˆY and at most one to ˆX−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(iii),  φ(δ˜D) is the base loop for two or three fragments Ni (i = 1, 2 or i = 1, 2, 3) of rank α in two  or three of the paths ¯X−1, Yj and Yj+1 for some j, respectively, with  (10-2)  �  i  µf(Ni) > 1 − 4λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since µf(Ni) ≤ ρ for each i, for at least two indices i we have  µf(Ni) > 1  2(1 − 4λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω − ρ) = 5  2λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that all Ni are pairwise compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K ∼ N±1  1  then for the required M we can take  that Ni which occurs in Yi or in Yj+1 and has a larger µf(Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence we can assume that  K ̸∼ N±1  i  for all Ni produced by all lifts ˜D of all cells D of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that D has two contiguity subdiagrams Πi ∈ T (i = 1, 2) to ˆY, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' the corre-  sponding fragments N1 and N2 of rank α occur in Yk and Yk+1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we cut  off from ∆ the subdiagram ∆ ∪ Π1 ∪ Π2 and the remaining simply connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  This replaces ∆ with a new diagram ∆′ with a smaller number of cells of rank α, Yi with a  subpath of Yi, bridges ui with another bridges u′  i and the assumption that K ̸∼ N±1  i  for Ni  produced by all lifts ˜D of D implies that K is independent of all new bridges u′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case  we can apply induction on the number of the cells of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We may assume now that each cell D of rank α of ∆ has precisely two contiguity subdi-  agrams Πi ∈ T to sides of ∆, one to ˆX−1 and another one to ˆY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies that each lift  of D produces two fragments Ni, one in ¯X−1 and one in some Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Dk} be  the set of all cells of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For each lift ˜D(j)  i  (t ∈ Z) of Di, denote N(j)  i,1 and N(j)  i,2 the  corresponding fragments of rank α that occurs in ¯X−1 and Yj respectively (the requirement  that N(j)  i,2 occurs in Yj determines uniquely the lift ˜D(j)  i  and the fragment N(j)  i,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that  (10-2) implies  µf(N(j)  i,k) > 1 − 4λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω − ρ = 5λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We order cells Di to get N(j)  i,2 ordered in Yj as N(j)  1,2 ≪ · · · ≪ N(j)  k,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consequently, in ¯X we  have · · ·N(j)  1,1  −1 ≪ · · · ≪ N(j)  k,1  −1 ≪ N(j+1)  1,1  −1 ≪ · · · ≪ N(j+1)  k,1  −1 · · · (Figure 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By the  N(i)  11  N(i)  12  N(i)  13  N(i)  21  N(i)  22  N(i)  23  Yi  N(i+1)  11  N(i+1)  12  N(i+1)  13  N(i+1)  21  N(i+1)  22  N(i+1)  23  Yi+1  ¯X  ui  ui−1  Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  assumption above, we have K ̸∼ N(j)  i,1  −1 for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 we have either  52  N(j)  i,1  −1 < K < N(j)  i+1,1  −1 for some i, j or N(j)  k,1  −1 < K < N(j+1)  1,1  −1 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In each of these  cases, we find the required M by applying an appropriate part of the proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5  or Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  We will use the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let K be a fragment of rank 1 ≤ β ≤ α in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let M be either  another fragment of rank β in Γα such that K ∼ M±1 or a bridge of rank β such that  K is not independent of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then any of the endpoints of K can be joined with any  of the endpoints of M by a bridge w of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Moreover, w can be chosen with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If N is any other fragment  of rank β such that N ̸∼ M±1 then N is independent of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let K1, K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Kr be fragments of rank β ≤ α in Γα such that K1 ∼ K±1  i  for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then all endpoints of all Ki are uniformly close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Follows from definitions in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 and Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let (Xi, Yi) (i = 1, 2) be two pairs of close reduced paths in Γα where X1  and X2 are subpaths of a reduced path ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that for the common subpath Z of X1 and X2  we have |Z|α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists a triple ai (i = 1, 2, 3) of uniformly close vertices on  Z, Y1 and Y2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If α = 0 there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X−1  i uiYivi (i = 1, 2) be a coarse  bigon where ui and vi are bridges of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1: Areaα(X−1  i uiYivi) = 0 for both i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We apply Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 and find  loops X′−1  i  u′  iY′  iv′  i that can be lifted to Γα−1 where X′  i and Y′  i are subpaths of Xi and Yi  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For the common part Z′ of X′  1 and Z′  2 we have |Z′|α ≥ |Z|α − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='04 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16 and  hence |Z′|α−1 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: Areaα(X−1  i uiYivi) > 0 for i = 1 or i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without loss of generality, assume that  K and M are active fragments of rank α in X1 and in Y1, respectively, such that K ∼ M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let X1 = S1KS2 and Y1 = T1MT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S1K contains Z then we shorten X1 and Y1 replacing  them with S1K and T1 thereby decreasing Areaα(X−1  1 u1Y1v1) as described in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly,  if KS2 contains Z then we can replace X1 and Y1 with KS2 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Therefore, we can assume  that K is contained in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We take a1 = ι(K) and a2 = ι(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K is not independent of u2 or  from v2 then for a3 we can take ι(Y2) or τ(Y2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6  there exists a fragment N of rank α in Y2 such that N ∼ K±1 and we can take a3 = ι(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let (S, T) and (X, Y) be pairs of close reduced paths in Γα where Y is an end  of S and the ending vertices τ(X), τ(Y) = τ(S) and τ(T) are uniformly close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there  exists a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T respectively, such that  a1 cuts off a start X1 of X with |X1|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and a2 cuts off a start Y1 of Y with |Y1|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We use induction on |X|+|Y|+|T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If |X|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and |Y|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2  there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that |X|α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 or |Y|α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is enough to find  a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T respectively, such that at  least one ai cuts off a proper start of appropriate path X, Y or T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let X−1u1Yu2 and S−1v1Tv2 be coarse bigons in Γα where ui and vi are bridges of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1: Areaα(X−1u1Yu2) = Areaα(S−1v1Tv2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that u2 and v2 are defined  from the condition that τ(X), τ(Y) and τ(T) are uniformly close;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' that is, either u2 and v2 are  53  bridges of rank α − 1 or have the form u2 = w1P1w2 and v2 = w3P2w4 where wi are bridges  of rank α − 1 and P±1  i  are subpaths of a relator loop R of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider the second  case (the case when u2 and v2 are bridges of rank α − 1 is treated in a similar manner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Without changing notations, we assume that loops X−1u1Yu2 and S−1v1Tv2 are lifted  to Γα−1 and, consequently, all paths introduced are in Γα−1 (the only change is that P±1  i  become subpaths of an R-periodic line ˜R where R is a relator of rank α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' After choosing ai  (i = 1, 2, 3) in Γα−1 we pass on to their images in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1a: |X|α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If a vertex b1 ̸= τ(X) on X is close in rank α − 1 to a vertex b2 on P1  then we can take a1 := b1, a2 := τ(Y) and a3 := τ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that no such b1 and b2  exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then application of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(ii)α−1 shows that X = z1X′z2 where X′ is close to  a subpath Y′ of Y, |z1|α ≤ 1 + 4ζ2η, |z2|α ≤ 4ζ2η and hence |X′|α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 − 8ζ2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume first that α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then shortening X′ from the end by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 we can  assume that z1X′ is a proper start of X (and that X′ is still close to a subpath Y′ of Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For  the shortened X′, we have |X′|α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3−8ζ2η−ζ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='26 which implies |X′|α−1 ≥ 1  ζ|X′|α > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let v1 = w5Qw6 where w5, w6 are bridges of rank α −1 and Q is labeled by a piece of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Application of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 gives a triple of uniformly close vertices ai (i = 1, 2, 3) where  a1 lies on X′, a2 lies on Y′ and a3 lies either on Q or T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If a3 lies on Q then we replace  it with ι(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case α = 1 we shorten X′ by one edge and for the new X′ we have  |X′|α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 − 8ζ2η − ζ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can still apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 due to Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3, so the  argument remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1b: |Y|α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly to Case 1, we can assume that there is no vertex b ̸= τ(Y)  on Y (and hence on S since |Y|α−1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15  ζ  = 23) close in rank α − 1 to a vertex on P1 or  on P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(ii)α−1 we represent Y and S as Y = z1Y′z2, S = z3S′z4  where Y′ is close (in rank α − 1) to a subpath X′ of X, S′ is close to a subpath T′ of  T and |z1|α, |z3|α < 1 + 4ζ2η, |z2|α, |z4|α < 4ζ2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case α = 1 there is a common  subpath Z of X′, Y′, S′ and T′ of size |Z|α ≥ |Y|α − 1 − 8ζ2η > 0 and we can take ι(Z) for  all ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case α ≥ 2, shortening Y′ from the end by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 we can assume  that z1Y′ is a proper start of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Z be the common subpath of Y′ and S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have  |Z|α > |Y|α − 1 − 8ζ2η − ζ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 and hence |Z|α−1 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement follows by  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: Areaα(S−1v1Tv2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K and M be active fragments of rank α in S and in T,  respectively, such that K ∼ M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S = G1KG2 and T = H1MH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that |K|, |M| > 0 by  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K is not contained in Y then we replace S and T with KG2 and H2 respectively  and use induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K is contained in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We first take a2 := ι(K), a3 := ι(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If M is not independent on u1 or from u2 then we take a1 := ι(X) or a1 := τ(X) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Otherwise by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6 there exits a fragment N of rank α in X such that N ∼ M±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In this case we take a1 := ι(N) by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 3: Areaα(X−1u1Yu2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K and M be active fragments of rank α in X and Y  respectively such that K ∼ M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then take a1 := ι(K), a2 := ι(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Depending on whether  M is not independent of v1 or v2 we find a3 similarly to the case 2 using Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6  and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (closeness transition in bigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let (X, Y) and (S, T) be pairs of close  reduced paths in Γα where Y is a subpath of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that |X|α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X = z1X′z2  where X′ is close to a subpath W of T and |zi|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  54  Moreover, we have Y = t1Y′t2 where |t1|α, |t2|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 and triples (ι(X′), ι(Y′), ι(W)) and  (τ(X′), τ(Y′), τ(W)) are uniformly close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X−1u1Yu2 and S−1v1Tv2 be coarse bigons in Γα where  ui and vi are bridges of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 it is enough to find a triple ai (i = 1, 2, 3)  of uniformly close vertices on X, Y and T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An easy analysis involving Proposi-  tion 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6 shows how to do this in the case when Areaα(X−1u1Yu2) > 0 or Areaα(S−1v1Tv2) >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It remains to consider the case when Areaα(X−1u1Yu2) = Areaα(S−1v1Tv2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  vi = vi1Rivi2 (i = 1, 2) where vij is a bridge of rank α − 1 and Ri is labeled by a piece of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 we have X = w1X1w2 where endpoints of X1 and a subpath Y1  of Y can be joined by bridges u′  1 and u′  2 of rank α − 1, so that the loop X−1  1 u′  1Y1u′  2 can be  lifted to Γα−1 and |wi|α ≤ 1 + 4ζ2η (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without changing notations, we assume that  loops X−1  1 u′  1Y1u′  2 and S−1v1Tv2 are already lifted to Γα−1 (and Y1 is still a subpath of S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  have  |X1|α ≥ |X|α − |w1|α − |w2|α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 − 8ζ2η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='26  and, consequently, |X1|α−1 > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 there is a triple of uniformly close ver-  tices b1 on X, b2 on Y and b3 on one of the paths R1, T or R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For a1 and a2 we take images  of b1 and b2 in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Depending on the location of b3 we take for a3 the image of either ι(T),  b3 or τ(T) as shown in Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  a1 X1  X  a2 Y1  S  u′  1  u′  2  v11  v12  v21  v22  T  R1  R2  b  a3  a3 = b  b  a3  Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let (X, Y) be a pair of close reduced paths in Γα, and let S−1∗T1∗T2∗ be a  coarse trigon in Γα where Y is an end of S and ending vertices τ(X), τ(Y) and τ(T2) are  uniformly close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then either  (i) there exists a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T1  respectively, such that a1 cuts off a start X1 of X with |X1|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) there exists a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and T2  respectively, such that a1 cuts off a start X1 of X with |X1|α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We use the same strategy as in the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 and  proceed by induction on |X| + |Y| + |T2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In view of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15, it is enough to prove that  if |X| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45 then there exists a triple ai of uniformly close vertices on X, Y and some Ti  respectively such that a1 or a2 cuts off a proper start of the appropriate path X or Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ui (i = 1, 2) and vj (j = 1, 2, 3) be bridges of rank α in Γα such that u1Xu2Y−1 is a  coarse bigon and S−1v1T1v2T2v3 is a coarse trigon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  55  Case 1: Areaα(X−1u1Yu2) = Areaα(S−1v1T1v2T2v3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that u2 and v3 are  defined from the condition that τ(X), τ(Y) and τ(T2) are uniformly close;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' that is, either u2  and v3 are bridges of rank α − 1 or have the form u2 = u21Qu22 and v3 = v31P3v32 where  u2i, v3i are bridges of rank α − 1 and Q±1, P±1  3  are subpaths of a relator loop R of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We consider the second case (in the first case the argument is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let vi = vi1Pivi2  (i = 1, 2) where vij is a bridge of rank α − 1 and label(Pi) is a piece of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We can assume that there is no vertex on X other than τ(X) which is close in rank α−1 to  a vertex on R (otherwise we can take those for a1 and a2 as in the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3, we can assume that loops X−1u1Yu2 and S−1v1T1v2T2v3 can be lifted to Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Abusing notations, we assume that they are already in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Application of Proposition  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(ii)α−1 shows that X = w1X′w2 where X′ is close to a subpath Y′ of Y, |w1|α ≤ 1 + 4ηζ2,  |w2|α ≤ 4ηζ2 and hence |X′|α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45 − 8ηζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As in the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 the proof slightly differs in cases α ≥ 2 and α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In  the case α ≥ 2, shortening X′ from the end by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 we can assume that w1X′  is a proper start of X, with a new bound |X′|α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45 − 8ηζ2 − ζ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='41 which implies  |X′|α−1 > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If there is a triple of uniformly close vertices on X′, Y′ and some Pi then we  are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that no such triple exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S1 be a reduced path joining ι(T1) and  τ(T2) (see Figure 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 we have X′ = z1X′′z2 where X′′ is close to a subpath  X′  w1  w2  S  Y′  u22  R  v11  P1  v12  T1  v21  P2  v22  T2  v31  P3  v32  S1  Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Moreover, the lemma says that there exists a triple of uniformly close vertices on X′,  Y′ and S1 and then applying Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17α−1 we may assume that |zi|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  |X′′|α−1 ≥ |X′|α−1 − |z1|α−1 − |z2|α−1 > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Another application of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 gives a triple of uniformly close vertices bi (i = 1, 2, 3)  where b1 lies on X′, b2 lies on Y′ and b3 lies either on T1 or on T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For ai we take the images  of bi in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In the case α = 1 the argument is similar (see Case 1a in the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15) with  no need for a lower bound on |X′′|α−1 for application of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: r = Areaα(S−1v1T1v2T2v3) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let L be an active relator loop for S−1v1T1v2T2v3  and Ki (i = 1, 2 or i = 1, 2, 3) be the associated active fragments of rank α occurring in  S, T1 or T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If some Ki occurs in T1 and some Kj occur in T2 then we can shorten T1  56  and T2 decreasing r as described in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A similar inductive argument works in the case  when some Ki occurs in S and is not contained in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus we may assume that there are  only K1 and K2„ K1 is contained in Y and K2 occurs in T1 or T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15,  µf(Ki) ≥ 3λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The rest of the argument is the same as in the Case 2 of the proof of  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 3: Areaα(X−1u1Yu2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K and M be active fragments of rank α in X and in Y  respectively such that K ∼ M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We take a1 := ι(K), a2 := ι(M) and define a3 according to  the following cases:  If M is not independent of v1 then a3 := ι(T1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If M is not independent of v2 then a3 := τ(T1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If M is not independent of v3 then a3 := τ(T2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Otherwise by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 applied to M there exists a fragment N or rank α in  T1 or T2 such that M ∼ N±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then a3 := ι(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (closeness transition in trigon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let (X, Y) be a pair of close reduced  paths in Γα, and let S−1∗T1∗T2∗ be a coarse trigon in Γα where Y is a subpath of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  that |X|α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X can be represented as in one of the following three cases:  (i) X = z1X1z2 where X1 is close to a subpath W1 of T1 and |z1|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3, |z2|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) X = z1X2z2 where X2 is close to a subpath W2 of T2 and |z1|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45, |z2|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) X = z1X1z3X2z2 where Xi is close to a subpath Wi of Ti (i = 1, 2), |z1|α, |z2|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3  and |z3|α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Moreover, we can assume that there exists a subpath Y′ of Y such that triples (ι(Xp), ι(Y′), ι(Wp))  and (τ(Xq), τ(Y′), τ(Wq)) are uniformly close where p and q are the smallest and the greatest  indices of Xi in (i)–(iii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' p = q = 1 in (i), p = q = 2 in (ii) and p = 1, q = 2 in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ui (i = 1, 2) and vj (j = 1, 2, 3) be bridges of rank α such that u1Xu2Y−1 is a  coarse bigon and S−1v1T1v2T2v3 is a coarse trigon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In view of Lemmas 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17,  finding a triple ai (i = 1, 2, 3) of uniformly close vertices on X, Y and some Ti implies the  conclusion of the proposition except the bound |z3|α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The latter follows from  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An easy analysis as in Cases 2 and 3 of the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17 shows  how to find the vertices ai in the case when Areaα(X−1u1Yu2) > 0 or Areaα(S−1v1Tv2T2v3) >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It remains to consider the case when Areaα(X−1u1Yu2) = Areaα(S−1v1Tv2T2v3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  vi = wi1Riwi2 (i = 1, 2, 3) where label(wij) ∈ Hα−1 and the label of Ri is a piece of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 we have X = w1X1w2 where endpoints of X1 and a subpath Y1 of Y  can be joined by bridges u′  1 and u′  2 of rank α − 1 and the loop X1u′  1Y−1  1 u′−1  2  can be lifted  to Γα−1 and |wi|α ≤ 1 + 4ζ2η (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without changing notations, we assume that loops  X−1  1 u′  1Y1u′  2 and S−1v1Tv2 are already in Γα−1 (and Y1 is still a subpath of S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have  |X1|α ≥ |X|α − |w1|α − |w2|α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='41  and, consequently, |X1|α−1 > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we find ai applying Lemmas 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17α−1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 as in  the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (closeness transition in conjugacy relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S be a word cyclically  reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that S is conjugate in Gα to a word Tv where T ∈ Rα and v ∈ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ¯S = �  i∈Z Si and �  i∈Z Tivi be lines in Γα representing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  57  Assume that a reduced path X in Γα is close to a subpath Y of ¯S with |Y| ≤ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  |X|α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then either:  (i) X can be represented as X = z1X1z2 where X1 is close to a subpath W1 of Ti for  some i and |z1|α, |z2|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) X can be represented as X = z1X1z3X2z2 where for some i, X1 is close to a subpath  W1 of Ti, X2 is close to a subpath W2 of Ti+1, |z1|α, |z2|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and |z3|α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Moreover, we can assume that there exists a subpath Y′ of Y such that triples (ι(X1), ι(Y′), ι(W1))  and (τ(Xq), τ(Y′), τ(Wq)) are uniformly close where q = 1 in (i) and q = 2 in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is enough to find a uniformly close triple of vertices ai (i = 1, 2, 3) on X, Y and  some Ti and then use Lemmas 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17 or 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X−1u1Yu2 be a coarse bigon where u1 and u2  are bridges of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If Areaα(X−1u1Yu2) > 0 then we reach the goal using Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12  and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that Areaα(X−1u1Yu2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ∆ be an annular diagram of rank α with boundary loops ˆS−1 and ˆTˆv representing the  conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ˜∆ be the universal cover of ∆ and φ : ˜∆(1) → Γα the combinatorially  continuous map sending lifts ˜Si, ˜Ti and ˜vi to Si, Ti and vi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that ∆  is reduced and has a tight set of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let r be the number of cells of  rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that r > 0 and let D be a cell of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11(iv) and Lemma  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(i), D has two or three contiguity subdiagrams Πi ∈ T to sides of ∆, at most two to ˆT  and at most one to ˆS−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If there are two contiguity subdiagrams Πi (i = 1, 2) of D to ˆT then  we consider a new annular diagram ∆′ obtained by cutting off D∪Π1 ∪Π2 and the remaining  simply connected component from ∆, and new words T ′ and v′ where T ′ is a subword of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In this case, the statement follows by induction on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We can assume now that D has one contiguity subdiagram to ˆS−1 and one to ˆT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ˜Di  (i ∈ Z) be the lifts of D in ˜∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' With an appropriate numeration of ˜Di’s, each relation loop  φ(δ˜Di) is a base loop for a fragment Ki in ¯S−1 and a fragment Mi in Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(iii),  µf(K−1  i ) + µf(Mi) > 1 − 4λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since T is reduced in Gα, we have µf(Mi) ≤ ρ and hence  µf(K−1  i ) > 5λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If none of K−1  i ’s is contained in Y then we can apply Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise we use  an argument similar to one in Case 2 of the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Now assume that ∆ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without changing notations, we assume that  parallel lines ¯S = �  i∈Z Si, �  i∈Z Tivi and paths X and Y are lifted to Γα−1 so that Y is still a  subpath of ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let v ≖ w1Rw2 where wi ∈ Hα−1 and R is a piece of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We represent vi  accordingly as vi = w(i)  1 Riw(i)  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Z be a word reduced in Gα−1 such that Z = Tw1R and  let Zi (i ∈ Z) be appropriate paths in Γα−1 with label(Zi) ≖ Z (Figure 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since |X|α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='45  we have |X|α−1 ≥ 1  ζ |X|α ≥ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19α−1, a subpath X′ of X with |X′|α−1 > 23  is close to a subpath of some Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then using Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 we find a triple bi of  uniformly close vertices on X′, Y and Ti or Ri respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If b3 lies on Ti then for the  desired ai we take images of bi in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If b3 lies on Ri then for ai (i = 1, 2, 3) we take images  of b1, b2 and τ(Ti), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  58  X  u1  Y  u2  Zi  Zi+1  Ti  Ti+1  w(i)  1  w(i)  2  w(i+1)  1  w(i+1)  2  Ri  Ri+1  ¯S  Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let 1 ≤ β ≤ α and X be a reduced path in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K1 and K2 be fragments  of rank β in X such that µf(Ki) ≥ λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω (i = 1, 2), K1 < K2 and K1 ̸∼ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If a bridge of  rank β starts or ends at ι(X) then K2 is independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, if a bridge of rank β  starts or ends at τ(X) then K1 is independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider the case when ι(u) = ι(X) (all other cases are similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that K2  is not independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, u = vSw where S occurs in a relation loop R of  rank β, v and w are bridges of rank β − 1 and R±1 is the base relation loop for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ˜R  and ˜X be lifts of R and X in Γβ−1 so that ˜R±1 is the base axis for ˜K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='22 implies  that the starting vertex of ˜X is close to a vertex on ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then using Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 we  conclude that the starting segment ˜X1˜K2 of ˜X is a fragment of rank α with base axis ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since  K1 is contained in ˜X1˜K2, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 gives K1 ∼ K2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (closeness preserves order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X1X2 and Y1Y2 be reduced paths in Γα  such that endpoints of Xi and Yi are close in the order as in Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then |X1|α, |Y2|α <  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  X1  X2  Y1  Y2  u1  u2  u3  Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Due to symmetry, it is enough to show that |X1|α < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Denote ui (i = 1, 2, 3) bridges of rank α joining endpoints of Xi and Yi as shown in Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim 1: Areaα(X−1  1 u1Y2u−1  2 ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that Areaα(X−1  1 u1Y2u−1  2 ) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Ki and Mi (i = 1, 2) be active  fragments of rank α in X1 and Y2, respectively, such that K1 < K2 and Ki ∼ M−1  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7(ii) and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='20, K2 is independent of u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, M2 and hence K2,  are independent of u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 applied to (X1X2)−1u1Y−1  1 u−1  3 , there is  59  a fragment N of rank α in Y1 such that N ∼ K±1  2  and µf(N) ≥ 5λ − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We obtain a  contradiction with Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(ii),(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  Claim 2: If Areaα(X−1  1 u1Y2u−1  2 ) = 0 and label(u1), label(u2) ∈ Hα−1 then |X1|α < 1 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If r = Areaα(X2u3Y1Y2u−1  2 ) > 0 then we can reduce the statement to the  case of a smaller r as explained in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' So we can assume that Areaα(X2u3Y1Y2u−1  2 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  loops X−1  1 u1Y2u−1  2  and X2u3Y1Y2u−1  2  can be lifted to Γα−1 (up to possible switching of u3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  To simplify notations, we assume that these loops are already in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let u3 = v1Qv2  where label(vi) ∈ Hα−1 and label(Q) is a piece of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We obtain a coarse trigon in Γα−1  with sides X1X2, Q and Y1, see Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(i)α−1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 we  obtain  |X1X2|α < 1 + 4ζ2η + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ζ < 1 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  u1  u2  X1  X2  Y1  Y2  v1  Q  v2  Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The rest of the proof: If Areaα(X−1  1 u1Y2u−1  2 ) = 0 then the statement follows from Claim 2  and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Claim 1, it remains to consider the case Areaα(X−1  1 u1Y2u−1  2 ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then X1 can be represented as R1S1R2S2R3 (see Figure 32) where each Ri is a fragment of  rank α and by Claim 2 and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(ii)α−1 each Si satisfies |Si|α < 1 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ζ + 8ζ2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We obtain  |X1|α < 3 + 2(1 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ζ + 8ζ2η) < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  R1  S1  R2  S2  R3  X2  Y1  Y2  u1  u3  u2  Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In the end of the section we formulate several statements about stability of fragments in  a more general setup when fragments have arbitrary rank β in the interval 0 ≤ β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  60  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S and T be close reduced paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let 0 ≤ β < α and let X  and Y be close in rank β reduced paths in Γα such that Y is a subpath of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that  |X|α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and Y contains no fragments K of rank γ with β < γ ≤ α and µf(K) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then X can be represented as X = w1X′w2 where X′ is close in rank β to a subpath of T and  |wi|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S−1u1Tu2 and X−1v1Yv2 be corresponding coarse bigons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If Areaα(S−1u1Tu2) >  0 then by the argument from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 we reduce the statement to a new pair (S, T) and a  coarse bigon S−1u1Tu2 with a smaller value of Areaα(S−1u1Tu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence we can assume that  Areaα(S−1u1Tu2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without changing notations, we assume that both loops S−1u1Tu2  and X−1v1Yv2 are in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ui = ui1Piui2 where label(uij) ∈ Hα−1 and label(Pi) is a piece  of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Observe that if a subpath X′ is close to a subpath of P1 or P2 then |X′|α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since |X|α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 applying Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 we find a subpath of X close to a subpath of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  consider the case when X = z0X1z1X2z2X3z3 where Xi (i = 1, 2, 3) are close to subpaths of P1,  T and P2 respectively (the other cases from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 give a better lower bound on |X2|α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 we can assume that |z0|α−1, |z3|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(i)α−1 we  can assume that |z1|α−1, |z2|α−1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have |X1|α, |X3|α ≤ 1, so |X2|α > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3−2−3ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15  and hence |X2|α−1 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13α−1 we have X2 = t1X′t2 where X′ is close  in rank β to a subpath of T and |ti|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have X = z1X1z2t1X′t2z3X3z4 where  |z1X1z2t1|α < 1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='73ζ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 and a similar bound holds for |t2z3X3z4|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be reduced paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let 1 ≤ β ≤ α and assume that  either X or Y contains no fragments N of rank γ with β < γ ≤ α and µf(N) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Ki (i = 1, 2) be fragments of rank β in X such that K1 ̸∼ K2 and K1 < K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  that at least one of the following conditions holds:  (*) there exist fragments Mi (i = 1, 2) of rank β in Y such that µf(Mi) ≥ λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω,  Ki ∼ M±1  i  and M1 < M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' or  (**) X and Y are close in rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then the following is true:  (i) Let N be a fragment of rank β in X with µf(N) ≥ 2λ + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω such that K1 < N < K2  and N ̸∼ Ki for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists a fragment N′ of rank β in Y such that  N′ ∼ N±1, M1 < N′ < M2 in case (*) and  (10-3)  µf(N′) ≥ min{µf(Ni) − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω, ξ0}  In case (*), if M1 and M2 are disjoint then we can assume that M1 ≪ N′ ≪ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  This is the case (that is, M1 and M2 are necessarily disjoint) if µf(N) ≥ 4λ + 9ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Assume that µf(Ki) ≥ 2λ + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω and in case (*), µf(Mi) ≥ 2λ + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let K′  i  (i = 1, 2) be a pair of another fragments of rank β in X and M′  i (i = 1, 2) a pair of  another fragments of rank β in Y such that µf(K′  i), µf(M′  i) ≥ 2λ + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω, K′  i ∼ M′±1  i  (i = 1, 2) and K′  1 ̸∼ K′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K′  1 < K′  2 if and only if M′  1 < M′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Furthermore, the statement of the proposition is true also in the case β = 0 if we drop all  conditions of the form µf(·) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for fragments of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β = 0 then by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 we have Mi = Ki (i = 1, 2), M1 ∪ M2 = K1 ∪ K2 in  case (*) and X = Y in case (**).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i): Assume that (*) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' First assume that M1 and M2 are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X1 = K1 ∪ K2  and Y1 be the subpath of Y between M1 and M2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y = ∗M1Y1M2∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i)  61  and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 we have a loop X−1  1 uY1v that can be lifted to Γβ where u and v are  bridges of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to change of notation, we assume that X−1  1 uY1v is already in Γβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Again  by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i)β, N is independent of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6β, there exists N′ in  Y1 satisfying (10-3) such that N′ ∼ N±1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we have M1 ≪ N′ ≪ M2 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that M1 and M2 have a nonempty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12β there exist  fragments M′  1 and M′  2 of rank β such that M′  i ∼ Mi, M′  1 is a start of M1 disjoint from M2  and M′  2 is an end of M2 disjoint from M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Y2 = M1 ∪ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using the argument above  with Y2 instead of Y1 and M′  1 instead of M1 we find N1 in Y2 disjoint from M2 such that  µf(N1) > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω and N1 ∼ N±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, using Y2 instead of Y1 and M′  2 instead of M2 we  find N2 in Y2 disjoint from M1 such that µf(N2) > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω and N2 ∼ N±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we can take  N′ = N1 ∪ N2 by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(i), (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If µf(N) ≥ 4λ + 9ω then µf(N′) > 2λ + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω and using Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11β and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10β we  conclude that M1 and M2 cannot cover N′ together, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' M1 ≪ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In case (**) we already have a loop X−1uYv with bridges u and v of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We lift it  to Γβ and then apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='20β to see that the lift of N is independent of the lifts of u  and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then application of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6β gives the required N′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii): An easy analysis with a help of Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(ii) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10β shows that it is enough  to prove the following: Let X and Y be reduced paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Ki (i = 1, 2, 3) be fragments of  rank β in X, Mi (i = 1, 2, 3) be fragments of rank β in Y, µf(Ki), µf(Mi) ≥ λ+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω, Ki ∼ M±1  i  for all i and Ki ̸∼ Kj for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K1 < K2 < K3 and M1 < M3 then M1 < M2 < M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that this is not the case, that is, we have K1 < K2 < K3, M1 < M3 and either  M2 < M1 or M3 < M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (i), there exists a fragment N of rank α in Y such that K2 ∼ N±1  and M1 < N < M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(i) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10β we obtain M1 ∼ N or M3 ∼ N, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be words strongly cyclically reduced in Gα, representing  conjugate elements of Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯X and ¯Y be lines in Γα representing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let 1 ≤ β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that at least one of the words X or Y has the property that no  its cyclic shift contains a fragment K of rank γ with µf(K) > ξ0 and β < γ ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  ¯X = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' X−1X0X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' and ¯Y = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y−1Y0Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' be lines in Γα representing the conjugacy  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Then for any fragment K of rank β in ¯X with µf(K) ≥ 2λ + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω there exists a  fragment M of rank β in ¯Y such that M ∼ K±1 and  µf(M) ≥ min{µf(K) − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω, ξ0}  (ii) If X and Y are strongly cyclically reduced in Gα then the correspondence between  fragments of rank β in ¯X and in ¯Y preserves the ordering in the following sense: if  Ki (i = 1, 2) are fragments of rank β in ¯X, Mi (i = 1, 2) are fragments of rank β  in ¯Y, µf(Ki), µf(Mi) ≥ 2λ + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω, Ki ∼ M±1  i  (i = 1, 2) and K1 ̸∼ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then K1 < K2  if and only if M1 < M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Furthermore, the statement of the proposition is true also in the case β = 0 if we drop all  conditions of the form µf(·) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for fragments of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17 every subpath of ¯X can be extended to be close in rank β to  a subpath of ¯Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then (i) follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(ii) and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(i) with  K1 = s−1  X,¯XK and K2 = sX,¯XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Statement (ii) follows by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case  β = 0 the statement becomes trivial after application of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  62  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Reduced representatives  The main goal of this section is to prove that any element of Gα can be represented by a  reduced word and to prove a cyclic analog of this statement (Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (reduced representative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Every element of Gα can be represented by  a reduced in Gα word which contains no fragments F of rank 1 ≤ β ≤ α with µf(F) ≥  1  2 + 2λ + 15ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let m ≥ 3 and X−1∗Y1∗Y2∗ · · ·∗Ym∗ be a coarse (m+1)-gon in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  that there are indices 1 ≤ t1 < t2 < · · · < tk ≤ m (k ≥ 1) such that  t1 ≤ 3,  tk ≥ m − 2,  tj − tj−1 ≤ 2 for all j  and  |Ytj|α−1 > 4η  for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume further that there are no close vertices in each of the pairs (Yi, Yi+1), (Y1, Yt1),  (Ytj, Ytj+1), (Ytk, Ym) except appropriate endpoints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' except τ(Yi) and ι(Yi+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  each of the paths Ytj has a vertex close to a vertex aj on X and these vertices aj are in X in  the (non-strict) order from start to end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We first claim that there are no close vertices in pairs (Yi, Yj) for j − i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  there are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We choose such a pair with minimal possible j − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then an ending segment Y′  i  of Yi, paths Yi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Yj−1 and a starting segment Y′  j of Yj form a coarse r-gon with  r = j − i + 1 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 we get  j−1  �  k=i+1  |Yi|α−1 ≤ (r − 2)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  On the other hand, it follows from the hypothesis of the lemma that there are at least  min(1, 1  2(r − 3)) paths Ytk among Yi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Yj−1 and hence  j−1  �  k=i+1  |Yi|α−1 > 4η min  �  1, 1  2(r − 3)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We get a contradiction since the right hand side of the inequality is at least (r − 2)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This  proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Shortening if necessary Y1 and X we can assume that there is no pair of close vertices  on Y1 and X other that (ι(Y1), ι(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Similarly, we can assume that there is no pair of  close vertices on Ym and X other than (τ(Ym), τ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Now we claim that there is a pair  of close vertices on Yi and X for some 2 ≤ i ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Indeed, otherwise we can apply  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='18α−1 to the whole coarse (m + 1)-gon X−1∗Y1∗Y2∗ · · ·∗Ym∗ and obtain a  contradiction since 4kη ≥ (m − 1)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let (b, c) be a pair of close vertices on X and Yi0 where 2 ≤ i0 ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let b divide X as  X1X2 and c divide Yi0 as Z1Z2 If there is at least one index tj in the interval 2 ≤ tj ≤ i0 − 1  then the conditions of the lemma are satisfied for the coarse (i0+1)-gon X−1  1 ∗Y1∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yi0−1∗Z1∗  and we conclude by induction that every Ytj with tj < i0 has a vertex close to a vertex aj on  X and the vertices aj occur in X in the appropriate order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, we conclude the same  for every path Ytj with tj > i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies the statement for all Ytj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  63  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a word reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that for any fragment K of rank α  in X we have  µf(K) ≤ 1 − 3λ − 5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then there exists a word Y equal to X in Gα which is reduced in Gα−1 and such that for any  fragment M of rank α in Y we have  µf(M) < 1  2 + 2λ + 15ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In particular, Y is reduced in Gα (note that 1  2 + 2λ + 15ω < ρ = 1 − 9λ by (2-3) and (4-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We represent X by a reduced path X in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote  t = 1  2 + 11ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Kr be a maximal set of pairwise non-compatible fragments of rank α in X with  µf(Ki) ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that each Ki has maximal size µf(Ki) in its equivalence class of  compatible fragments of rank α occurring in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 we shorten each Ki  from the start obtaining a fragment ¯Ki of rank α so that ¯Ki do not intersect pairwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we  have µf(¯Ki) > µf(Ki) − λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  X = S0¯K1S1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ¯KrSr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Pi be a base for ¯Ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for each i, we have a coarse bigon ¯K−1  i uiPivi with bridges ui and vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Pi ≖ label(Pi) and PiQ−1  i  be the associated relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider a path in Γα−1  Z = S∗  0u∗  1Q1v∗  1S∗  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' u∗  rQrv∗  rS∗  r  where labels of S∗  i , u∗  i and v∗  i are equal to corresponding labels of Si, ui and vi and label(Qi) ≖  Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that label(Z) = X in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We perform the following procedure:  (i) if a pair of vertices on Qi and S∗  i are close and is distinct from (τ(Qi), ι(S∗  i )) then  we choose a bridge w of rank α − 1 joining these vertices, replace v∗  i with w and  shorten Qi from the end and S∗  i from the start;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' similarly, if a pair of vertices on Qi  and S∗  i−1 are close and is distinct from (ι(Qi), τ(S∗  i−1)) then we choose a bridge w of  rank α−1 joining them and replace u∗  i with w shortening Qi from the start and S∗  i−1  from the end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we apply recursively the operation until possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) if a vertex on Qi is close to a vertex on Q∗  i+1 then we choose a bridge w of rank α −1  joining these vertices, shorten Qi from the end and Qi+1 from the end and join then  by w (so S∗  i is eliminated and v∗  i S∗  i u∗  i is replaced with a bridge w of rank α − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we  apply recursively the operation until possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  After the procedure, we obtain a path  Z1 = T0U0R1U1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' RrUrTr  where for each i, Ri is a subpath of Qi and Ui either is a bridge of rank α − 1 or has the  form wiTizi where Ti is a subpath of S∗  i and wi and zi are bridges of rank α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Y be  a reduced path with the same endpoints as Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Our goal is to prove that the label Y of Y  satisfies the requirement of the lemma, that is, for any fragment N of rank α in Y we have  µf(N) < 1  2 + 2λ + 15ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We compute a lower bound for µ(Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fix i and let Qi = Q′RiQ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' At step (i) of the  procedure, we do not shorten Qi more than this would give a fragment of rank α in X with  a base that is a proper extension of Pi, so we get µ(Qi) ≥ 1 − µf(Ki) ≥ 3λ + 5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' At step (ii)  64  we shorten Qi from each side by less than λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω (this follows from Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(i)α−1,  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 and Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies µ(Ri) > λ+4ω and, in particular, |Ri|α−1 >  4η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We apply Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 with X := Y where Ri and Ti play the role of Yi’s and Ri are taken  as Yti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The lemma says that each path Ri has a vertex close to a vertex on Y and these  vertices on Y are appropriately ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can write  Y = V0M1V1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' MrVr  where each Mi is close to a subpath of Qi (at the moment each Mi is empty because it is  represented by a vertex on Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Extending Mi’s we make them maximal so that no vertex  on Wi except ι(Vi) is close to a vertex on Qi and no vertex on Vi except τ(Vi) is close to a  vertex on Qi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to location of Z in Γα−1 we can assume that it starts at ι(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Combining  the two graphs shown in Figure 33a and mapping them to Γα we obtain a graph as shown  in Figure 33b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This graph is similar to one obtained from a single-layer diagram (as in Fig-  u1  v1  u2  v2  ur  vr  S0  S1  S2  S∗  r−1  Sr  u∗  1  v∗  1  u∗  2  v∗  2  u∗  r  v∗  r  S∗  0  S∗  1  S∗  2  S∗  r−1  S∗  r  Q1  Q2  Y  T0  U0  R1  U1  R2  U2  Ur−1  Rr  Ur  Tr  a  b  Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  ure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An easy analysis with use of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19α−1, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 and Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2  shows that Mi and some extension ˜Ki of ¯Ki satisfy the bound as in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  µf(Mi) + µf(˜Ki) > 1 − 2λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since µf(˜Ki) ≤ µf(Ki) ≤ 1 − 3λ − 5ω we obtain that for all i,  µf(Mi) > λ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  65  Let N be a fragment of rank α in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, we have either N ∼ Mi or  N ⊆ Mi ∪ Mi+1 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case when N ⊆ Mi ∪ Mi+1, N ̸∼ Mi and N ̸∼ Mi+1 we can  apply the argument from the proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 and find a fragment N′ in X such that  µf(N′) > µf(N) − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We have also N′ ̸∼ Ki, Ki+1 and hence N′ ̸∼ Kj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the choice of the Ki’s, we have  µf(K′) < t and hence  µf(N) < t + 2λ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω < 1  2 + 2λ + 15ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that N ∼ Mi for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯Q and ¯P be bases for N and Ki respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Images  of ¯Q−1 and ¯P in Γα are subpaths of a relator loop and have at most two overlapping parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We give an upper bound for µ(¯Q) + µ(¯P) by finding an upper bound for the size of each  overlapping part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume, for example, that an end of the image of ¯P in Γα overlaps with a  start of the image of ¯Q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Changing the location of Z in Γα−1 we can assume that ¯P and ¯Q−1  overlap on a subpath W of the same size already in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We consider the case i < r (see Figure 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' the case i = r is similar with a better upper  bound on µ(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We apply Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(ii)α−1 to a coarse tetragon with one side W and  Ki  ¯Ki+1  S  X  ¯P  ¯Q  W  V  N  L  Y  Pi+1  Mi+1  Figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  other sides which are an end S of Si¯Ki+1, a start V of M−1  i+1V−1  i  and a subpath of a common  base axis L for K−1  i+1 and Ni+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the worst case we have W = W1z1W2z2W3 where W1 is  close to a subpath of V−1, W2 is close to a subpath of L−1, W3 is close to a subpath of S−1  and |zi|α−1 ≤ 4ηζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 implies |W1|α−1 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 and |W3|α−1 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since  Ki ̸∼ Ki+1 we obtain µ(W2) < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence  µ(W) < λ + 2ω(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 + 4ηζ) < λ + 13ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We obtain  µf(N) + µf(Ki) < 1 + 2λ + 26ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since µf(Ki) ≥ t this implies the required bound µf(N) < 1  2 + 2λ + 15ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1 and X be a word reduced in Gα and a ∈ A±1 a letter in the  generators of Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Y be a word reduced in Gα−1 such that Y = Xa in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then Y has  no fragments K of rank α with µf(K) ≥ ρ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Follows from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  66  Proof of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is trivial if α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case α ≥ 1 Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 follows  by induction from Lemmas 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 since ρ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω < 1 − 3λ − 5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  We turn to the cyclic analogue of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1:  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (cyclically reduced representative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Every element of Gα of finite order  is conjugate to a cyclically reduced word of the form Rk  0 where R0 is the root of a relator of  rank β, 1 ≤ β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Every element of Gα of infinite order is conjugate to a strongly cyclically reduced word  in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma (a cyclic version of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a word cyclically reduced in Gα−1  representing an element of Gα−1 of infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let m ≥ 2, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Ym be words reduced in  Gα−1, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , um be bridges of rank α−1 and let X be conjugate to Y1u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ymum in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  �  i∈Z Y(i)  1 u(i)  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y(i)  m u(i)  m and ¯X = �  i∈Z X(i) be lines in Γα−1 labeled (Y1u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ymum)∞ and X∞  respectively representing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that there are indices 1 ≤ t1 < t2 < · · · < tk ≤ m (k ≥ 1) such that  m + t1 − tm ≤ 2,  tj − tj−1 ≤ 2  for all j,  and  |Ytj|α−1 > 4η  for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that there are no close vertices in each of the pairs (Y(0)  i , Y(0)  i+1), (Y(0)  m , Y(1)  1 ), (Y(0)  tj , Y(0)  tj+1),  (Y(0)  tk , Y(1)  t1 ) except appropriate endpoints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' except pairs (τ(Y(0)  i ), ι(Y(0)  i+1)) and (τ(Y(0)  m ), ι(Y(1)  1 ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then each of the paths Y(0)  tj , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , k has a vertex close to a vertex aj on ¯X and these  vertices aj are in the (non-strict) order corresponding to the order of the Y(0)  j ’s (and ak is  located non-strictly before sX,¯Xa0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The proof follows the proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 with appropriate changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim 1: There are no close vertices in pairs (Y(0)  i , Y(0)  j ) with j − i > 1 and (Y(0)  i , Y(1)  j ) with  j + m − i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The proof repeats the argument from the proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim 2: For some i, there are close vertices in the pair (Y(0)  i , ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consider an annular diagram ∆ of rank α − 1 with boundary  loops ˆX−1 and ˆY1ˆu1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ˆYmˆum and a combinatorially continuous map φ : ˜∆ → Γα−1 such that  φ maps the boundary of ˜∆ to ¯X−1 and �  i Y(i)  1 u(i)  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y(i)  m u(i)  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The assumption, Claim 1 and  the hypothesis of the lemma imply that ∆ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Application of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9α−1 gives  �  i  |Yi|α−1 ≤ ηm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  On the other hand, from the hypothesis of the lemma we have �  i |Yi|α−1 ≥ 4kη > ηm, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Claim 2, assume without loss of generality that there is a vertex b on Y(0)  1  which is  close to a vertex c on ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let b divide Y(0)  1  as Y(0)  1  = Z1Z2 and up to cyclic shift of X, assume  67  that X(0) starts at c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Now we can directly apply Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 to the coarse (m + 2)-gon  (X(0))−1∗Z2u(0)  1 Y(0)  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' u(0)  m−1Y(0)  m u(0)  m Z1∗  and get the required conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma (a cyclic version of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a word strongly cyclically reduced  in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that X is not conjugate in Gα to a power of the root of a relator of rank  β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Next, assume that for any fragment K of rank α in a cyclic shift of X we have  µf(K) ≤ 1 − 4λ − 8ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then there exists a word Z conjugate to X in Gα which is strongly cyclically reduced in Gα−1  and such that no power Zk contains a fragment L of rank α with  µf(L) < 1  2 + 2λ + 15ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In particular, Z is strongly cyclically reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The general scheme is the same as in the proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯X = �  i∈Z Xi be  a line in Γα−1 labeled X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' First we note that for any fragment K of rank α in ¯X we have  sX,¯XK ̸∼ K by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 there exists a starting  segment K′ of K that is a fragment of rank α with µf(K′) > µf(K) − λ − 3ω and |K′| ≤ |X|,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' label(K′) occurs in a cyclic shift of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the hypothesis of the lemma implies that ¯X  contains no fragments K of rank α with µf(K) ≥ 1 − 3λ − 5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Denote t =  1  2 + 11ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that there is at least one fragment K of rank α  in ¯X with µf(K) ≥ t (otherwise we can take Z := X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We choose a maximal set K1,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Kr of pairwise non-compatible fragments of rank α in ¯X with µf(Ki) ≥ t such that  K1 < · · · < Kr < sX,¯XK1 and Kr ̸∼ sX,¯XK1 (after choosing K1 we use Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(ii) to  get sX,¯XK1 ̸∼ K1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that each Ki has maximal size µf(Ki) in its class of compatible  fragments of rank α in ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 we shorten each Ki from its start obtaining  a fragment ¯Ki of rank α so that all ¯Ki do not intersect pairwise and |K1 ∪Kr| ≤ |X|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we have  µf(¯Ki) > µf(Ki) − λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Passing to a cyclic shift of X (and changing all Xi accordingly)  we may assume also that  X0 = ¯K1S1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ¯KrSr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Pi be the base for ¯Ki and ¯K−1  i uiPivi a loop in Γα−1 with bridges ui and vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote  Si ≖ label(Si), Pi ≖ label(Pi), ui ≖ label(ui), vi ≖ label(vi) and let PiQ−1  i  be the associated  relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  Z = u1Q1v1S1u2Q1v2S2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' urQrvrSr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let Y be a word strongly cyclically reduced in Gα−1 that is conjugate to Z in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  prove that Y satisfies the requirements of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that Y and hence Z are conjugate  to X in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We transform Z using a procedure analogous to the procedure described in the proof of  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' At any moment, we will have a word Z1 of the form  Z1 = R1U1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' RrUr,  conjugate to Z in Gα−1 where each Ri is a subword of Qi and each Ui either is a bridge of  rank α − 1 or has the form wiTizi where wi, zi are bridges of rank α − 1 and Ti is a subword  of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' At the start, we have Ri = Qi and Ui = viSiui+1 (here and below i + 1 is taken  68  modulo r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The transformation procedure consists of the following steps applied recursively  until possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Suppose that Ui has the form wiTizi above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If Ri = R′R′′, Ti = T ′T ′′ where |R′′| +  |T ′| > 0 and R′′wiT ′ is equal in Gα−1 to a bridge w of rank α −1 then replace Ri, wi  and Ti with R′, w and T ′′ respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' similarly, if Ti = T ′T ′′, Ri+1 = R′R′′ where  |T ′′| + |R′| > 0 and T ′′ziR′ is equal in Gα−1 to a bridge w of rank α − 1 then replace  Ti, zi and Ri+1 with T ′, w and R′′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) If Ri = R′R′′ and Ri+1 = R∗R∗∗ where |R′′| + |R∗| > 0 and R′′UiR∗ is equal in  Gα−1 to a bridge w of rank α − 1 then replace Ri, Ui and Ri+1 with R′, w and R∗∗  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Similar to the proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3, after performing the procedure we obtain |Ri|α−1 > 4η  for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ¯Z = �  i∈Z Z(i) be a line in Gα−1 labeled Z∞ and let Q(i)  j denote the appropriate subpath  of Z(i) labeled Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can implement the procedure above on the line ¯Z instead of a word Z  by changing appropriate paths instead of words (to each change of words in (i) or (ii) there  corresponds infinitely many changes of paths translated by sX,¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As a result, we get a line  �  i∈Z Z(i)  1  so that the corresponding subpath R(i)  j  of Z(i)  1  is also a subpath of Q(i)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote  also T(i)  j  the appropriate subpath of Z(i)  1  labeled Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ¯Y = �  i∈Z Y(i) be the line in Gα−1  such that ¯Z and ¯Y are associated with conjugate words Z and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We apply Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6  with ¯X := ¯Y where R(i)  j  and T(i)  j  play the role of Y(i)  j ’s and R(i)  j  are taken as Y(i)  tj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' According  to the lemma, each path R(0)  j  has a vertex close to a vertex on ¯Y, these vertices on ¯Y are  ordered along ¯Y in the increasing order of the index j, and the length of the segment of ¯Y  between the first and the last one is not more that |Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to cyclic shift of Y , we can write  Y(0) = W0M1W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' MrWr  where each Mj is close to a subpath of Q(0)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Taking Mj maximal with these properties we  obtain, as in the proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3,  µf(Mi) > λ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5ω  for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The rest of the proof is similar to the proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X is a reduced path in Γα and the endpoints of X are close then |X|α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For α ≥ 1 this follows from Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If P is a piece of rank α then for any fragment K of rank α in P we have  µf(K) ≤ max{λ, µ(P) + 2ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P be a path in Γα−1 with label(P) ≖ P, let R be the associated relator of rank α  and let L be the line labeled R∞ extending P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that K is a fragment of rank α  contained in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If the base axis for K is distinct from L then µf(K) < λ by Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Otherwise the base Q for K is contained in L and Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8α−1 implies  µf(K) = µ(Q) ≤ µ(K) + 2ω ≤ µ(P) + 2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  69  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P be a piece of rank 1 ≤ β ≤ α with µ(P) ≤ ρ − 2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then P  is reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If R ≖ QS where R is a relator of rank β then either Q or S is reduced  in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The first statement follows from Lemmas 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If R is a relator of rank β  and R ≖ QS then by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14(ii), we have either µ(Q) ≤ 1  2 + ω or µ(S) ≤ 1  2 + ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It remains to  note that 1  2 + ω < ρ − 2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  Proof of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a word representing an element of Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We may assume  that X is reduced in Gα as a non-cyclic word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We perform a “coarse cyclic cancellation”  in X: represent X as UX1V where V U is equal in Gα to a bridge u of rank α and X1 has  the minimal possible length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let u ≖ v1Pv2 where P is a piece of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume  that µ(P) ≤  1  2 + ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Y be a word cyclically reduced in Gα−1 and conjugate to X1u  in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that X1u and hence Y are conjugate to X in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We show that either Y  is conjugate in Gα−1 to a power Rt  0 of the root R0 of a relator of rank β ≤ α or no cyclic  shift of Y contains a fragment K of rank α with µf(K) ≥ ρ + 2λ + 16ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the first case, by  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 we can assume that Rk  0 is cyclically reduced in Gα and we come to the first  alternative of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise, according to Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5α−1 we can assume  that Y is strongly cyclically reduced in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we apply Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 to find a strongly  cyclically reduced in Gα word Z conjugate to Y in Gα (note that ρ+2λ+16ω < 1−4λ−8ω),  coming to the second alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let ¯Y = �  i∈Z Yi and �  i∈Z X(i)  1 v(i)  1 Piv(i)  2 be lines in Γα−1 representing the conjugacy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We observe that  (i) The base axis of any fragment N of rank α in Pi with µf(N) ≥ λ is the infinite  periodic extension of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, If N1 and N2 are fragments of rank α in Pi  with µf(Nj) ≥ λ then N1 ∼ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (This follows from Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  Now formulate some consequences of the choice of X1 of minimal possible length:  (ii) There exist no fragments N1 and N2 of rank α in X(i)  1 and in X(i+1)  1  , respectively, such  that N1 ∼ N2 and µf(Ni) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Indeed, assume that such N1 and N2 do exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that both N1 and N2 are nonempty  by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i), any two of the endpoints of the images of N1 and N2  in Γα are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we can shorten X1 to its subword X2 so that X2u′ is conjugate to X  in Gα for some u′ ∈ Hα contrary to the choice of X1 (see Figure 35a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' in the figure we have  N2 ≪ sY,¯YN1 in X(i+1)  1  but in all other cases we can easily find an appropriate path X2 with  |X2| < X1 and take X2 := label(X2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) There exist no fragments N1 and N2 of rank α in X(i)  1 and in Pi or Pi−1, respectively,  such that N1 ∼ N2, µf(N1) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω and µf(N2) ≥ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (Otherwise using (i) we can  shorten X1 to X2 := label(X2) as shown in Figure 35b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  Let Q be a word reduced in Gα−1 which is equal to X1v1P in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We denote Qi the corre-  sponding path in Γα−1 joining ι(X(i)  1 ) with τ(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using (iii), Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 and Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9  we conclude that  (iv) There are no fragments M of rank α in Qi with µf(M) ≥ ρ + λ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that K is a fragment of rank α in ¯Y with µf(K) ≥ ρ + 2λ + 16ω and |K| ≤ |Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  (iv) and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9, for some i there are fragments M1 and M2 of rank α in Qi and Qi+1  70  X(i+1)  1  N1  N2  sY,¯YN1  X2  X(i)  1  X(i)  1  X(i+1)  1  N1  N2  Pi  Pi  v(i)  1  v(i)  2  v(i)  1  v(i)  2  a  b  Figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  respectively such that Mj ∼ K (i = 1, 2) and µf(Mj) > λ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 there is  a fragment N1 of rank α such that M1 ∼ N1 and either N1 occurs in X(i)  1 and µf(N1) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω or  N1 occurs in Pi and µf(N1) > λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, there is a fragment N2 of rank α such that M2 ∼ N2  and either N2 occurs in X(i+1)  1  and µf(N2) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω or N2 occurs in Pi+1 and µf(N2) > λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If  N1 occurs in X(i)  1  and N2 occurs in X(i+1)  1  we get a contradiction with (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If N1 occurs in  Pi and N2 occurs in X(i+1)  1  or N1 occurs in X(i)  1  and N2 occurs in Pi+1 we get a contradiction  with (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Finally, if N1 occurs in Pi and N2 occurs in Pi+1 then by (i), we have sY,¯YN1 ∼ N2  and hence K ∼ sY,¯YK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(i)α−1 this implies that Y is conjugate in Gα−1 to  a power of the root of a relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let R be a relator of rank β ≤ α and let R ≖ Rn  0 where R0 is the root  of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then R0 has order n in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let k be a proper divisor of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8, Rk  0 contains no fragments K of rank γ  with µf(K) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω, for all γ = β + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10β, Rk  0 is cyclically reduced  in Gβ and hence also in rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence Rk  0 ̸= 1 in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (conjugate powers of relator roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let R be a relator of rank 1 ≤ β ≤ α  and let R ≖ Rn  0 where R0 is the root of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If Rk  0 = g−1Rl  0g in Gα for some k, l ̸≡ 0 (mod n)  then g ∈ ⟨R0⟩ and k ≡ l (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11, if Rk  0 = g−1Rl  0g in Gα and g ∈ ⟨R0⟩ then k ≡ l (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It  remains to prove that equality Rk  0 = g−1Rl  0g for k, l ̸≡ 0 (mod n) implies g ∈ ⟨R0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 we can assume that Rk  0 and Rl  0 are cyclically reduced in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  represent g by a word Z and consider an annular diagram ∆ of rank α with two cyclic  sides X1 and X2 labeled R−k  0  and Rl  0 which is obtained from a disk diagram with boundary  label R−k  0 Z−1Rl  0Z by gluing two boundary segments labeled Z−1 and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Z be the path  in ∆ with label(Z) ≖ Z that joins starting vertices of X2 and X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We apply to ∆ the reduction process 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8, we can replace Z by a new  path Z1 with the same endpoints such that label(Z1) = Z in Gα (so label(Z1) represents g  in Gα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume also that ∆ has a tight set T of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1: ∆ has a cell D of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i), D has a contiguity subdiagram  Πi ∈ T to each of the sides Xi of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Moreover, if δΠi = SiuiQivi where S−1  i  is a contiguity  arc occurring in δD then µ(Si) > λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 this implies β = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let label(δ∆) ≖ R′  71  where R′ is a relator of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consider lines ¯X1, ¯X2 and ¯R in Γα−1 labeled R±∞, R±∞  and R′∞ which are obtained by mapping the universal cover of the subgraph of ∆ shown  in Figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 we get ¯X1 = ¯X2 = ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies that label(Z1) is equal  X1  X2  Z1  S1  Q1  u1  v1  S2  u2  Q2  v2  R  ¯X1  ¯X2  D  Figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  in Gα−1 to a power of R0, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: ∆ has no cells of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we have equality Rk  0 = Z−1  1 Rl  0Z1 in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If  β < α then the statement follows from Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If kl > 0 then  the statement follows from Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If kl < 0 then by Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(i)α−1 we  obtain R0 = g−1R−1  0 g which contradicts our condition (S3) on the presentation of Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Every element of Gα of infinite order has the form hm where h is a  non-power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We need to prove this only in the case α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let g ∈ Gα be an element of infinite  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is enough to find an upper bound on |m| in the equality of the form g = hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up  to conjugation, we represent g and h by a strongly cyclically reduced in Gα words X and Y  by Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β be the maximal rank with 1 ≤ β ≤ α such that a cyclic shift  of X contains a fragment K of rank β with µf(K) ≥ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (It there is no such K then by  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16 X in conjugate to Y m in the free group G0 and then |m| ≤ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=') Using  Propositions 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(i) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(ii) we find m pairwise non-compatible fragments M of rank β  with µf(M) ≥ ξ0 − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω in a cyclic shift of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This again implies |m| ≤ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Coarsely periodic words and segments over Gα  In this section we analyze words which are “geometrically close” in Gα to periodic words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In Sections 12 and 13 we use the following notation for numeric parameters:  ξ1 = ξ0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω,  ξ2 = ξ1 − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A simple period over Gα is a strongly cyclically reduced word representing  a non-power element of Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, if A is a simple period over Gα then any word An is reduced over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6 implies that A has infinite order in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  72  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a simple period over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The activity rank of A is the maximal  rank β such that an A-periodic word contains a fragment K of rank β ≥ 1 with µf(K) ≥ ξ1  or it is 0 if no such fragments exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Case of activity rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The arguments below differ depending on whether the activity  rank β of a simple period over Gα is positive or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' However, the difference is only that in  the case β ≥ 1 we use various conditions on the size µf(F) of fragments F of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' All  definitions, statements and proofs in Sections 12 and 13 apply in cases when the activity  rank β of a simple period over Gα is 0 simply ignoring conditions of the form µf(·) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' for  fragments of rank β (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' assuming that these conditions are all formally true in case β = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Below we do not distinguish this special case β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If K and M are fragments of the same rank 0 ≤ β ≤ α  occurring in a reduced path X in Γγ then K ≲ M means K < M or K ∼ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' similarly, K � M  means K < M and K ̸∼ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(ii), for fragments K, M of rank  β ≥ 1 with µf(K), µf(L) ≥ γ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω the relation ‘K ≲ M’ depends only on their equivalence  classes with respect to compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, for fixed X and β it induces the linear order on  the set of equivalence classes of ‘∼’ of fragments N of rank β in X with µf(N) ≥ γ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (In  case β = 0 relation K ≲ M is defined on subpaths on length 1 and means K ≪ M or K = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a simple period over Gα and β the activity rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A reduced  path S in Γα is a coarsely periodic segment with period A (or a coarsely A-periodic segment  for short) if there exists a path P labeled by an A-periodic word, fragments K0, K1 of rank β  in P and fragments M0, M1 of rank β in S such that:  P starts with K0 and ends with K1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' S starts with M0 and ends with M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  K0 ∼ M±1  0 , K1 ∼ M±1  1  and K0 ̸∼ K1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  µf(Ki) ≥ ξ1, µf(Mi) ≥ ξ2 (i = 0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  sA,PK0 ≲ K1 (informally, P “contains at least one period A”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The path P is a periodic base for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The infinite A-periodic extension of P is an axis for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that the starting fragment M0 and the ending fragment M1 of S are defined up to  compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note also that by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i) and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, P and S are close in rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In  particular, if β = 0 then P = Q and thus P is an A-periodic segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will be assuming that a coarsely A-periodic segment is always considered with a fixed  associated axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (In fact, we prove later that the axis of a coarsely A-periodic segment is  defined in a unique way, see Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that under this assumption, the periodic  base P for S is defined up to changing the starting and the ending fragments K0 and K1 of  rank β with compatible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The label of a coarsely A-periodic segment in Γα is a coarsely A-periodic word over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that a simple period A over G0 is any cyclically freely reduced word that is not a  proper power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A coarsely A-periodic word over G0 is simply any A-periodic word P with  |P| > |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We measure the size of a coarsely A-periodic segment S, which roughly  corresponds to the number of periods A, in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P be the periodic base  for S and K0, K1 as in Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we write ℓA(S) = t where t is the maximal  integer such that st  A,PK0 ≲ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, we always have ℓA(S) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  73  Since we consider a fixed associated axis for S, the number ℓA(S) does not depend on the  choice of a periodic base P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If S is a coarsely A-periodic word over Gα then we formally define ℓA(S) to be the maximal  possible value of ℓA(S) where S is a coarsely A-periodic segment labeled S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) It immediately follows from the definition that t is also the maximal integer  such that K0 ≲ s−t  A,PK1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, ℓA(S) = ℓA−1(S−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) To compute ℓA(S) we have to take a path S in Γα with label(S) ≖ S and then choose a  periodic base P for S so that ℓA(S) is maximal possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' it will follow from Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7  that any choice of P gives in fact the same value ℓA(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to changing the periodic base P, we can always assume in Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5  that both K0 and its translation st  A,PK0 occur in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case we have |P| ≥ ℓA(S)|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S1 and S2 be coarsely A-periodic segments in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We say that S1 and S2 are compatible if they have the same axis and strongly compatible  if they share a common periodic base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We use notations S1 ∼ S2 and S1 ≈ S2 for compatibility and strong compatibility respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that in the case S1 ≈ S2 any periodic base for S1 is a periodic base for S2 and vice  versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This easily follows from Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If S1 and S2 are coarsely A-periodic segments in Γ0 then S1 ∼ S2 if and only if they have  a common periodic extension and S1 ≈ S2 if and only if S1 = S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S1 and S2 be coarsely A-periodic segments in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) If S1 ≈ S2 then ℓA(S1) = ℓA(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Assume that S1 and S2 occur in a reduced path X in Γα and S1 ∼ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the union  of S1 and S2 in X is an A-coarsely periodic segment where a periodic base for S1 ∪S2  is the union of periodic bases f or S1 and S2 in their common infinite A-periodic  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) is immediate consequence of Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) follows from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We describe a procedure of shortening a coarsely A-periodic segment S by a “given  number k of periods”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let k ≥ 1 and ℓA(S) ≥ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β be the activity rank of S, let  P a periodic base for S and let Ki and Mi (i = 0, 1) be starting and ending fragments of  rank β of P and S respectively as in Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have K0 < sk  A,PK0 ≲ s−1  A,PK1 < K1  and it follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(ii) that sk  A,PK0 ̸∼ K0 and sk  A,PK0 ̸∼ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(i) there exists a fragment N of rank β in S with µf(N′) ≥ ξ2 such that sk  A,PK0 ∼ N±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then S1 = N ∪ M1 is an end of S which is a coarsely A-periodic segment with periodic base  P1 = sk  A,PK0 ∪ K1 and ℓA(S1) = ℓA(S) − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We note that:  (i) The result of the operation is defined up to the strict compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) We have P = XP1 where |X| = k|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If k ≥ 2 then by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(i) we can find also a fragment N′ of rank β in S  with µf(N′) ≥ ξ2 such that sk−1  A,P K0 ∼ N′±1 and N′ and N are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then S = S0uS1  where S0 = M0 ∪ N′ is a coarsely A-periodic segment with periodic base K0 ∪ sk−1  A,PK0  and ℓA(S0) = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  74  (iv) The starting position of S1 depends only on the starting position of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' more precisely,  if S′ is a start of S and S1 and S′  1 are obtained from S and S′ as above then S′  1 is a  start of S1 up to strict compatibility of S′  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' if S ≈ S′ then S1 ≈ S′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S1 is obtained from S by the procedure in 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 then we say that S1  is obtained by shortening of S by t periods from the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the symmetric way, we define  shortening of S by t periods from the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If ℓA(S) ≥ 2t+1 and S′ is obtained from S by applying the operation from both sides then  S′ is the result of truncation of S by t periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We define two numeric parameters associated with a simple period A  over Gα: the stable size [A]α of A in rank α,  [A]α = inf  m≥1  |(Am)◦|α  m  and the stability decrement hα(A):  hα(A) =  � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2  [A]α  �  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If ℓA(S) ≥ 2hα(A) + 1 then the result of truncation of S by hα(A) periods is the stable  part of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(iv) and its symmetric version, the function ‘S → stable part of S’  respects strict compatibility: if S1 ≈ S2 and S∗  i is the stable part of Si then S∗  1 ≈ S∗  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The basic fact about [A]α and hα(A) is the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X is an A-periodic word and |X| ≥ m|A| then |X|α ≥ m[A]α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In partic-  ular, if |X| ≥ (hα(A) − 1)|A| then |X|α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have  |X|α ≥ |Am  1 |α ≥ |(Am)◦|α ≥ m[A]α  where A1 is the cyclic shift of A at which X starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The second statement follows from the  first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  The principal role of the stable part is described by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (stability of coarsely periodic words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S be a coarsely A-periodic  segment in Γα with ℓA(S) ≥ 2hα(A) + 1 and let S∗ be the stable part of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If X and Y are  close reduced paths in Γα and S is a subpath of X then Y contains a coarsely A-periodic  segment T such that T ≈ S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P and P∗ be periodic bases for S and S∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β be the activity rank  of A and let Ki and Mi (i = 0, 1) be fragments of rank β in P and in S, respectively, from  Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 applied to P and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote t = hα(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let X and Y be as in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If α = 0 then X = Y and there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We claim that P = z1P′z2 where P′ is close in rank β to a subpath of Y and  |zi|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Indeed, if β = α then it easily follows from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6 and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i)  that P is already close to a subpath of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If β < α then we observe that S contains no  fragments K of rank γ with β < γ ≤ α and µf(K) ≥ ξ0 due to the definition of the activity  rank and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7≤α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the claim follows by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  75  By Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 we have |zi| < (t−1)|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies that st−1  A,PK0 ∪s−t+1  A,P K1 is contained  in P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that P∗ = st  A,PK0 ∪ s−t  A,PK1 where µf(K0), µf(K1) ≥ ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Proposition  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(i) we find a subpath T which is a coarsely A-periodic segment with periodic base P∗  and, consequently, we have T ≈ S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  We use parameter hα(A) also in several other situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P be a periodic segment in Γα with a simple period A over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that |P| ≥ m|A| where m ≥ 2hα(A) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a reduced path in Γα such that P  and X are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exist a subpath P1 of P and a subpath X1 of X such that X1 is a  coarsely A-periodic segment with periodic base P1 and ℓA(X1) = m − 2hα(A) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β be the activity rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 and Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 we find close  in rank β subpaths P2 of P and X2 of X with |P2| ≥ m−2hα(A)+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(iii)  any fragment K of rank β in P with µf(K) ≥ 2λ + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ω satisfies |K| < 2|A|, so according  to Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 there exists a fragment K of rank β in P with µf(K) ≥ ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Shortening K  from the end by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 if β ≥ 1 and using again Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(ii) we find a  fragment K1 of rank β with µf(K1) > ξ1 − λ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7ω that is a start of K disjoint from sA,PK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  hence |K1| ≤ |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that K occurs in P2 and is closest to the start of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  P2 contains m−2hα(A) translates si  A,PK of K for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , m−2hα(A)−1 and contains also  sm−2hα(A)  A,P  K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(i) we find fragments Mi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , m−2hα(A)−1)  of rank β in X2 with µf(Mi) ≥ ξ2 such that si  A,PK ∼ M±1  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then X1 = M1 ∪ Mm−2hα(A)−1 is a  coarsely A-periodic segment with periodic base sA,PK ∪ sm−2hα(A)−1  A,P  K and we have ℓA(X1) =  m − 2hα(A) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S be a coarsely A-periodic word over Gα and B a simple period  over Gα conjugate to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ℓA(S) ≥ 2hα(A) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then a subword T of S is a coarsely  B-periodic word over Gα with ℓB(T) ≥ ℓA(S) − 2hα(A) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We represent S by a coarsely A-periodic segment S in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P a periodic base for S,  let L1 be the axis of S and let L2 be the B-periodic line parallel to L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote β1 and β2  activity ranks of A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, either L1 or L2 contains no fragments K of rank γ with  β1 < γ ≤ α and µf(K) ≥ ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K0 and K1 be fragments of rank β1 with µf(Ki) ≥ ξ1 that are  a start and an end of P respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have sℓA(S)  A,L1 K0 ≲ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(i), there  exist fragments M0 and M1 of rank β1 in L2 with µf(Mi) ≥ ξ2 such that Ki ∼ M±1  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since L1  and L2 are parallel, we have sA,L1 = sB,L2 and hence sℓA(S)  B,L2 M0 ≲ M1 by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then Q = M0 ∪ sℓA(S)  B,L2 M0 ∪ M1 is close in rank β1 to P, |Q| ≥ ℓA(S) and the statement follows  by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Overlapped coarse periodicity  The main result of this section is Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 which can be thought as an analog  of a well known property of periodic words: if two periodic words have a sufficiently large  overlapping then they have a common period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We need such an analog in a more general  context where closeness plays the role of overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As a main technical tool, instead of  coincidence of letters in the overlapping case we use correspondence of fragments of rank β ≤  76  α in strictly close in rank β segments in Γα given by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A difficulty is caused  by the “fading effect” of this correspondence: a fragment size can decrease when passing  from one segment to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To overcome this difficulty, we use a special combinatorial  argument [9, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma (penetration lemma, [9, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S0, S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Sk be a finite collection  of disjoint sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that the following assertions hold:  (i) Each Si is pre-ordered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' endowed with a transitive relation ‘<i’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) There is an equivalence relation a ∼ b on the union �  i Si such that for any a, b in  the same set Si we have either a <i b, b <i a or a ∼ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' in other words, we have an  induced linear ordering on the set of equivalence classes on each Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) We assume that the equivalence preserves the pre-ordering in neighboring sets: if  a, b ∈ Si, a′, b′ ∈ Si+1, a ∼ a′ and b ∼ b′ then a <i b ⇔ a′ <i+1 b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If c ∈ Si, a, b ∈ Sj and a ≲j b (where a ≲j b denotes ‘a <j b or a ∼ b’) then we  say that c penetrates between a and b if there exists c′ ∼ c such that a ≲j c′ ≲j b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iv) There is a subset of �  i Si of stable elements that have the following property: if  c ∈ Si is stable, a ≲i c ≲i b, a′, b′ ∈ Sj, a′ ≲j b′, a ∼ a′ and b ∼ b′ then c penetrates  between a′ and b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (v) For each i ≤ k − 1, there are stable elements ai, bi ∈ Si and a′  i, b′  i ∈ Si+1 such that  ai ∼ a′  i, bi ∼ b′  i and ai <i bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Finally, let c0 ∈ S0 be stable and a0 ≲0 c0 ≲0 b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that c0 penetrates between ai  and bi for each i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then c0 penetrates between ak and bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following observation is a special case of [9, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Suppose a group G acts on set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let g, h ∈ G, x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , xt ∈ X and for  some r, s ≥ 0 with gcd(r, s) = 1 and r + s ≤ t,  gxi = xi+r (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , t − r),  hxi = xi+s (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , t − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that the stabilizer H of x0 is malnormal in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then either g, h ∈ H (and hence  x0 = x1 = · · · = xt) or there exists d ∈ G such that g = dr and h = ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Induction on r +s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that r ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If r > 0 then we have g−1hxi = xi+s−r  for 0 ≤ i ≤ t − s and the statement follows from the inductive hypothesis with h := g−1h,  s := s−r and t := t−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise we have r = 0 and s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then h−1ghx0 = gx0 = x0 and  by malnormality of H, we have either g, h ∈ H or g = 1 (and then g = h0 and h = h1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be reduced paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that X and Y are strictly  close in rank β ≤ α if there are fragments K0, K1 of rank β in X and fragments M0, M1 of  rank β in Y such that:  µf(Ki), µf(Mi) ≥ ξ2 (i = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  X starts with K0 and ends with K1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Y starts with M0 and ends with M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  K0 ∼ M±1  0 , K1 ∼ M±1  1  and K0 ̸∼ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i), paths which are strictly close in rank β are also close in rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' One  of the advantages of strict closeness is that this relation is transitive (this follows immediately  from Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that a coarsely periodic segment P in Γα and its periodic base S  are strictly close according to Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 (and the condition in Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 is slightly  77  stronger because of the lower bound on the size of the starting and the ending fragments  of S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a simple period over Gα, β the activity rank of A and Pi  (i = 0, 1) be two A-periodic segments in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Si (i = 0, 1) be a reduced path in Γα which  is strictly close to Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that S0 is contained in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume also that P0 contains at  least one period A in the sense that there exist fragments K and K′ of rank β in P0 such that  µf(K), µf(K′) ≥ ξ2 and K′ ∼ sA,P0K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then P0 and P1 have a common periodic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote  ξ3 = ξ2 − 2λ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ω = 3λ − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Throughout the proof, “fragment M” means “fragment M of rank β with µf(M) ≥ ζ3” (or  simply “fragment M of rank 0” if β = 0, see 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let a line Li be the infinite periodic extension of Pi and let g be an element of Gα such  that L1 = gL0, so sA,P1 = gsA,P0g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Our argument relies on establishing a correspondence  between fragments of rank β in Pi and Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It will be convenient to consider fragments of  rank β in four paths Pi and Si as four disjoint sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we will formally consider pairs (M, X)  where X ∈ {P0, P1, S0, S1} and M is a fragment occurring in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We will refer to M as a  “fragment belonging to X” or simply as a “fragment in X”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We introduce two operations on fragments in Pi and Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let M and N be fragments each  belonging to some Pi or Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) If M belongs to Pi, N belongs to Si and M ∼ N±1 then either of M and N jumps to  the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) M translates to N in the following cases (a)–(d):  (a) M and N belong to the same Pi and N ∼ sk  A,PiM for some k ∈ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' or  (b) M belongs to P0, N belongs to P1 and N ∼ gsk  A,P0M for some k ∈ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' or  (c) M belongs to P1, N belongs to P0 and N ∼ g−1sk  A,P1M for some k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (In other words, M translates to N in cases (a)–(c) if they have the same position in  their corresponding periodic lines Li with respect to the period A up to compatibil-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  (d) An “identical” case: M ∼ N and they belong to some Si and Sj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that the two operations are reversible and are defined up to compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let K and K′ be fragments in P0 such that µf(K), µf(K′) ≥ ξ1 and K′ ∼ sA,P0K, as assumed  in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let M be a maximal set of pairwise non-compatible fragments which can  be obtained by operations (i) and (ii) starting from K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10, neither of any  two fragments in M is contained in the other, so M is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The following assertion is the principal step of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim: The jump operation is always possible inside M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' that is, for any M ∈ M in Pi or  in Si, i ∈ {0, 1}, there exists a fragment N of rank α in Si or, respectively, in Pi such that  M ∼ N±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that some M ∈ M is given and prove existence of the  required N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The proof will consist of application of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We do a necessary prepa-  ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to the definition of M, there is a sequence T0 = K, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Tl = M of fragments  Tj ∈ M such that Tj+1 is obtained from Tj by one of the operations (i) or (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can  78  assume that the sequence has no two translations in a row (otherwise we can replace them  by a single translation) and has no two jumps in a row (otherwise they eliminate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  also for convenience that T0 → T1 is a translation (by inserting a trivial translation if  needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus for each i, T2j translates to T2j+1 and T2j+1 jumps to T2j+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume  that the last step Tl−1 → Tl is a translation, so l = 2k − 1 for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Now roughly speaking, we move all fragments Tj along with the corresponding paths Pi  or Si belonging them, to the same location up to compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We define a sequence Y0,  Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Yk of paths in Γα and a sequence Wj of fragments in Yj for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For  each j we will have Wj = fjT2j+1 for some fj ∈ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The definition of Yj and fj goes as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Denote (X1, X2, X3, X4) = (P0, S0, P1, S1) and let J(i) denote the index such that a fragment  in Xi jumps to a fragment in XJ(i) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (J(1), J(2), J(3), J(4)) = (2, 1, 4, 3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote also I(j)  the index such that T2j−1 belongs to XI(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, T2j belongs to XJ(I(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We start with Y0 = XI(0) and W0 = T1, so f0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that j < k−1 and Yj and fj are  already defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If T2j → T2j+1 is a translation by (a)–(c) then there exists fj+1 ∈ Gα such  that fj+1XI(j+1) and fjXJ(I(j)) belong to the same A-periodic line and fj+1T2j+1 ∼ fjT2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We take Yj+1 = fj+1XI(j+1) ∪ fjXJ(I(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Otherwise T2j → T2j+1 is a translation by (d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  XJ(I(j)) is either S0 or S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case we take fj+1 = fj and Yj+1 = fjS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Finally, define  Yk = fkXJ(I(k−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have fj+1T±1  2j+2 ∼ fj+1T2j+1 ∼ fjT2j for all j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , k − 2 and  hence W0 ∼ W±1  1  ∼ · · · ∼ W±1  k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Figure 37 illustrates the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  W0  W1  W2  W3  Figure 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By strict closeness of pairs (P0, S0) and (P1, S1), each Xi starts with a fragment Ui and  ends with a fragment Vi such that µf(Ui), µf(Vi) ≥ ξ2, Ui ̸∼ Vi and we have Ui ∼ U±1  J(i) and  Vi ∼ V±1  J(i)  We now apply Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 where:  Sj is the set of all fragments N in Sj with µf(N) ≥ ξ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  N <i N′ is defined as ‘N ̸∼ N′ and N < N′ in Sj’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Equivalence of N, N′ ∈ �  j Sj is defined as N ∼ N′±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  N ∈ �  j Sj is defined to be stable iff µf(N) ≥ ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For aj, bj, a′  j and b′  j we take appropriate translates of Ui and Vi, namely, fjUI(j),  fjVI(j), fjUJ(I(j)) and fjVJ(I(j)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We have conditions (i)–(v) of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 satisfied: condition (i) holds in case β ≥ 1 by  Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(ii), condition (ii) holds by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10β, conditions (iii) and (iv) hold by  79  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23 in view of the inequality ξ3 ≥ 2λ + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1ω and, finally, condition (v) holds  immediately by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For c0 in Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 we take T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that up to compatibility, we can assume that  µf(T1) ≥ ξ2, so T1 is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (By construction, T1 is obtained from T0 = K by translation  to XI(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' if T1 is compatible with the starting or the ending fragment of XI(0) then we can  assume µf(T1) ≥ ξ2 due to Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' otherwise we can assume that T1 is a literal  translation of K and then µf(T1) = µf(K) ≥ ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=') Since T1 = W0 ∼ W±1  1  ∼ · · · ∼ W±1  k−1  and each Wj occurs in fjXI(j), T1 penetrates between each pair fjUI(j) and fjVI(j) for j =  0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' All the hypotheses of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 are satisfied and applying it we find a  fragment Wk in fk−1XJ(I(k−1)) such that W±1  k  ∼ Wk−1 = fk−1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then M → f −1  k−1Wk is the  required jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This finishes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We finish the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let K0 = K, K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Km ∼ sA,P0K be all fragments  in M between K and sA,P0K in their natural order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we have K0 < K1 < · · · < Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  M0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Mm ∈ M be fragments in P1 such that Mi ∼ K±1  i  for all i (each Mi is obtained  from Ki by two jumps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that M0 < M1 < · · · < Mm by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since M  is closed under translations, the number of fragments in M between M0 and sA,P1M0 is the  same as the number of fragments in M between K and sA,P0K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we have Mm ∼ sA,P1M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  This implies that K0 translates to some Mq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Mq ∼ gst  A,P0K±1  0  for some t and hence  Mi+q ∼ gst  A,P0K±1  i  for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , m − q,  Mi+q−m ∼ gst−1  A,P0K±1  i  for i = m − q + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that gcd(m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='q) = 1 since M is generated by a single fragment K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(i),  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 and Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24(iii), the subgroup {g ∈ Gα | gM0 ∼ M±1  0 } is malnormal in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  now apply Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 where for xi we take the equivalence class of Mi in the set of fragments  of rank β in Γα under compatibility up to invertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the lemma, ⟨g, sA,P0⟩ is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since  A is a non-power, we get g ∈ ⟨sA,P0⟩ which means that L1 = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  As an immediate consequence of Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 we get:  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary (overlapping coarse periodicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S0 and S1 be coarsely periodic segments  in Γα with the same simple period A over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S0 is contained in S1 then S0 ∼ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S and T be non-compatible coarse periodic segments in Γα with the  same simple period A which occur in a reduced path X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ℓA(S) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that S1 is  obtained from S by shortening by 2 periods from the end if S < T or by shortening by 2  periods from the start if S > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then S1 and T are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without loss of generality, we assume that S < T and S1 is obtained from S by  shortening by 2 periods from the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(iii) we have S = S1uS2 where S2 is a coarsely  A-periodic segment with S2 ∼ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' B y hypothesis we have S2 ̸∼ T and then by Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5,  neither of S2 or T is contained in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies that S1 and T are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition (strictly close periodic paths with one period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a simple period  over Gα and β the activity rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P0 and P1 be strictly close in rank β paths in Γα  labeled by periodic words with period A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that there exist fragments K, K′ of rank β  in P0 such that µf(K), µf(K′) ≥ ξ2 and sA,P0K ∼ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then P0 and P1 have a common periodic  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This is a special case of Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 with S0 = S1 = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  80  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let g ∈ Gα be a non-power of infinite order and let h ∈ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If gk =  h−1glh for some k, l > 0 then h ∈ ⟨g⟩ and k = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, up to conjugation we can assume that g is represented by a  simple period A over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It is enough to prove that h ∈ ⟨A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Consider two periodic lines L0 and L1 in Gα with period A which represent the conjugacy  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have h ∈ ⟨A⟩ if and only if L0 = L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β be the activity rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='24 we find strictly close in rank β subpaths Pi of Li with any desired bound  |P0| ≥ t|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the statement follows from Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  As an immediate consequence we get:  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S0 and S1 be coarsely A-periodic segments in Γα and Li (i = 1, 2) be an  axis for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S0 ∼ S1 then L1 = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let g ∈ Gα be an element of infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) g has the unique root;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' there exists a unique non-power element g0 ∈ Gα such  that g = gt  0 for some t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) If hr ∈ ⟨g⟩ and hr ̸= 1 then h ∈ ⟨g0⟩ where g0 is the root of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) If g is conjugate to g−1 then g is the product of two involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) is direct consequence of Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) follows from (i) and Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 because gt  0 = hr implies gt  0 = h−1gt  0h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) Assume that g = h−1g−1h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  From g = h−2gh2 we conclude that h2 = 1 by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Similarly, we have (hg)2 = 1 and then g = h · hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that each relator R of each rank β ≤ α has the form R = Rn  0  where R0 is the root of R and n is odd (n can vary for different relators R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then Gα has  no involutions and no element of Gα is conjugate to its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, any element of finite order of Gα is conjugate to some power  Rt  0 of the root R0 of a relator R of rank β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11, Rt  0 has an odd  order and cannot be an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The second statement follows from the first by Corollary  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P be an A-periodic segment in Γα with a simple period A over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  S be a coarsely periodic segment in P with another simple period B over Gα and assume that  A and B are not conjugate in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) S ̸∼ st  A,PS for any t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) If ℓB(S) ≥ 3 then |S| < 2|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) Assume that S ∼ st  A,PS for some t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let L1 be the infinite periodic extension  of P, and let L2 be the axis for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9 we have L2 = st  A,PL2, so st  A,P = sr  B,Q for  some r ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since A and B are non-powers, by Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(ii) sε  A,P = sB,Q for ε = ±1  and hence Lε  1 and L2 are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' From the fact that S is a subpath of P we easily deduce by  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23 (taking for β the activity rank of A) that ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We obtain a contradiction  with the assumption that A and B are not conjugate in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) By 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(iii) we represent S as S = S1uS2 where S1 and S2 are coarsely periodic  segments with period B and ℓB(S1) ≥ ℓB(S) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By (i) and Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, s−1  A,PS does not  contain S1 and sA,PS does not contain S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies |S| < 2|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  81  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P and Q be close periodic segments in Γα with the same simple  period A over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If |P| ≥ (2hα(A) + 1)|A| (where hα(A) is defined in 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12) then P and Q  belong to the same A-periodic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Follows from Propositions 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  We finish the section by formulating technical statements which we will need in the con-  struction of relations of Burnside groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We use notation S ⪅ T for ‘S < T or S ≈ T’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S and T be coarsely A-periodic segments occurring in a reduced path X  in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that some periodic bases for S and T have the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If S is contained  in T then S ≈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that S is contained in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Pi (i = 1, 2) be periodic bases for S and T  respectively, with label(P1) ≖ label(P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β be the activity rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4,  P1 and P2 have a common periodic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Ki and Mi (i = 0, 1, 2, 3) be fragments  of rank β with µf(Ki), µf(Mi) ≥ ξ2 such that P1 = K0 ∪ K1, P2 = K2 ∪ K3, S = M0 ∪ M1,  T = M2 ∪M3 and Ki ∼ Mi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have M2 ≲ M0 � M1 ≲ M3 which by Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4  implies K2 ≲ K0 and K1 ≲ K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Now from label(P1) ≖ label(P2) we conclude that K2 ∼ K0  and K1 ∼ K3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' S ≈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be close reduced paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S0, S1 be coarsely A-periodic  segments in X and T0, T1 be coarsely A-periodic segments in Y such that ℓ(Si) ≥ 2hα(A) + 1,  Si ≈ Ti for i = 0, 1 and S0 ̸∼ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then S0 < S1 if and only if T0 < T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, none of S0 and S1 is contained in the other and the same is true  for T0 and T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume, for example, that S0 < S1 and T1 < T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X1 and Y1 be the  starting segments of X and Y ending with S1 and S2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 with  X := X1 and Y := Y1 there exists U in Y1 such that U ≈ S∗  0 where S∗  0 is the stable part of S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then U ∪ T0 is a coarsely A-periodic segment containing T1 and we get a contradiction with  Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X and Y be reduced paths in Γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S0, S1 be coarsely A-periodic seg-  ments in X and T0, T1 be coarsely A-periodic segments in Y such that S0 ⪅ S1, T0 ⪅ T1 and  Si ≈ Ti, i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let U be a coarsely A-periodic segment in X such that S0 ⪅ U ⪅ S1, ℓA(U) ≥ hα(A)+1  and U is the stable part of some other coarsely A-periodic segment in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there  exists a coarsely A-periodic segment V in Y such that T0 ⪅ V ⪅ T1 and U ≈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let Ui (i = 1, 2) be coarsely A-periodic segments in X and Vi (i = 1, 2) be coarsely  A-periodic segments in Y such that ℓA(Ui) ≥ 2hα(A) + 1 (i = 1, 2), S0 ⪅ Ui ⪅ S1,  T0 ⪅ Vi ⪅ T1 and Ui ≈ Vi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that U2 ≈ gU1 for some g ∈ Gα,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' U1 and U2 have periodic bases with the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then U1 ⪅ U2 if and only if  V1 ⪅ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let β be the activity rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i): Let U be the stable part of ¯U and ¯U = Z1UZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1: U ̸∼ Si for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 we have S0 < ¯U < S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since S0 ∪ S1  and T0 ∪ T1 are close, existence of V follows from Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: Exactly one of the relations U ∼ Si (i = 0, 1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without loss of generality,  assume that U ∼ S0 and U ̸∼ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 we have ¯U < S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If U ≈ S0 there is  nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that U ̸≈ S0 and hence UZ2 is contained in S0 ∪ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  82  By the construction of the stable part, UZ2 is a coarsely A-periodic segment with ℓA(UZ2) =  ℓ(U) + hα(A) ≥ 2hα(A) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let W be the stable part of UZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 with  X := S0 ∪S1 and Y := T0 ∪T1 we find a coarsely A-periodic segment W′ in T0 ∪T1 such that  W ≈ W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9(ii),  S0∪U is a coarsely A-periodic segment and since W′ ∼ T0, T0∪W′ is a coarsely A-periodic  segment as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(iv) (more formally, by the symmetric version of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10(iv)) W is  an end of U which implies S0 ∪ U ≈ T0 ∪ W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Now let P be a periodic base for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the construction of the stable part, P starts with  a fragment N of rank β with µf(N) ≥ ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since P is contained in a periodic base for T0 ∪ W′,  by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23 we find a fragment N′ of rank β in T0 ∪ W′ such that µf(N′) ≥ ξ2 and  N′ ∼ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then for the desired V we can take the end of T0 ∪ W′ starting with N′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 3: U ∼ S0 ∼ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then a periodic base P for U is contained in a periodic base for  S0 ∪ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the construction of the stable part, P starts and ends with fragments N0 and N1  of rank β with µf(Ni) ≥ ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then using Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23 we find fragments N′  i (i = 0, 1) of  rank β in T0 ∪ T1 such that µf(N′  i) ≥ ξ2 and N′  i ∼ Ni (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can take V = N′  0 ∪ N′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii): We consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 1: U1 ∼ U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P1 and P2 be periodic bases for U1 and U2 with label(P1) ≖ label(P2)  which have a common periodic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It easily follows from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='23(ii) that  U1 < U2 ⇔ P1 < P2 and U1 ≈ U2 ⇔ P1 = P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since Pi is also a periodic base for Vi, a  similar statement holds for Vi’s which clearly implies the required conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Case 2: U1 ̸∼ U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Without loss of generality, we assume that U1 < U2, V1 > V2 and  come to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume also that X = S0 ∪ S1, Y = T0 ∪ T1 and hence X  and Y are close in rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let U∗  i and V∗  i be stable parts of Ui and Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6,  U1 is disjoint from U∗  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X = X1U1X2U∗  2X3 and Y = Y1V∗  2Y2V1Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14  with X = X1U1X2 and Y := Y1 there exists a coarsely A-periodic segment W in Y1 such  that W ≈ U∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then W ∼ U1 ∼ V1 and by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9(ii) and Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 we get  U1 ∼ W ∼ W ∪ V1 ∼ V2 ∼ U2, the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Comparing α-length of close words  In this section, we prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X, Y ∈ Rα be close in rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  |Y |α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|X|α + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Recall that a fragment word F of rank α is considered with fixed associated words S, u, v  and a relator R of rank α such that F = uSv in Gα−1, u, v ∈ Hα−1 and S is a subword of Rk  for some k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F is a path in Γα−1 labeled F then this uniquely defines the base S for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let F and G be fragments of rank α in a word X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X be a path in Γα−1 labeled X and  F, G the corresponding subpaths of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We write F ∼ G if F ∼ G (so the relation is formally  defined for the occurrences of F and G in X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Recall that the size |X|α of a word X in rank α is the minimal possible value of weightα(F)  of a fragmentation F of rank α of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A fragmentation F of rank α of X is a partition  X ≖ F1 · F2 · · ·Fk where Fi is a nonempty subword of a fragment of rank β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assuming  that each Fi is assigned a unique value of β, the weight in rank α of F is defined by formula  weightα(F) = mα + ζmα−1 + ζ2mα−2 + · · · + ζαm0  83  where mβ is the number of subwords of fragments of rank β in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We call a fragmentation F of X minimal if weightα(F) = |X|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We call a subword F of a fragment of rank β ≥ 1 a truncated fragment of rank β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We  will be assuming that with a truncated fragment F of rank α there is an associated genuine  fragment ¯F of rank β such that F is a subword of ¯F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F is a path in Γα with label(F) ≖ F  then we have the associated fragment ¯F in Γα such that F is a subpath of ¯F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note a truncated  fragment of rank 1 is simply a fragment of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We extend the compatibility relation to truncated fragments of rank β in a word X in the  following natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F and G are truncated fragments of rank β in X and ¯F and ¯G  their associated fragments of rank β in Γα then F ∼ G if and only if ¯F ∼ ¯G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let F = F1 · F2 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' · Fk be a fragmentation of rank α of a word X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Fi be a  truncated fragment of rank β ≥ 1 in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that Fi can be extended in X to a larger  truncated fragment G of rank β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  X ≖ F1F2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' F ′  pF ′′  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' F ′  qF ′′  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fk  where Fp ≖ F ′  pF ′′  p , Fq ≖ F ′  qF ′′  q and G ≖ F ′′  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' F ′  q (here we consider the case 1 < i < k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  cases i = 1 and i = k differ only in notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then we can produce a new fragmentation F′  of rank α, X ≖ F1 · · ·Fp−1 · [F ′  p] · G · [F ′′  q ] · Fq+1 · · · Fk where square brackets mean that F ′  p  and F ′′  q are absent if empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We say that F′ is obtained from F by extending Fi to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note  that if F is minimal then in the case i > 1, we necessarily have p = i − 1 and nonempty F ′  p  and in the case i < k we necesarily have q = i + 1 and nonempty F ′′  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let F = F1 · F2 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' · Fk be a minimal fragmentation of rank α ≥ 1 of a word  X ∈ Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) Let Fi be a truncated fragment of rank α in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then |Fi|α−1 ≥ 1  ζ and Fi = uFv  where F is a fragment of rank α, Fi ∼ F, |u|α−1, |v|α−1 < ζ and the base P for the  corresponding fragment F in Γα−1 satisfies |P|α−1 > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) If K is a fragment of rank α in X and µf(K) ≥ 3λ + 15ω then Fi ∼ K for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) Let X = P0K1P1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' KrPr where Ki are fragments of rank α with µf(Ki) ≥ 3λ + 13ω  for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists another minimal fragmentation F′ of rank α of X such  that each Ki is contained in a compatible truncated fragment of rank α in F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (i) If |Fi|α−1 < 1  ζ then we could replace Fi by its fragmentation of rank α − 1 which  would decrease the weight of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 in the case α ≥ 2 (in the case α = 1  we take u and v empty) we have Fi = uFv where F is a fragment of rank α, Fi ∼ F and  |u|α−1, |v|α−1 < ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F is the corresponding fragment of rank α in Γα−1 and P is the base  for F then by Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1  |P|α−1 > 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3  �1  ζ − 2ζ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2  �  > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Let K be a fragment of rank α in X and µf(K) ≥ 3λ + 15ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that there is  no truncated fragment Fi of rank α such that Fi ∼ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 and the assumption, if H is a common part of K and some Fi of  rank α then H contains no fragment K′ of rank α with µf(K′) ≥ λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8,  if H is a common part of K and some Fi of rank β < α then H contains no fragment K′ of  84  rank α with µf(K′) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, K is not contained in any Fi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  X ≖ F1F2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' F ′  pF ′′  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' F ′  qF ′′  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fk  where  Fp ≖ F ′  pF ′′  p ,  Fq ≖ F ′  qF ′′  q ,  K ≖ F ′′  p Fp+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' F ′  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If some Fi is contained in K and has rank α then by the remark above and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, K is covered  by at most three of the Fj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case, by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 we would have  µf(K) ≤ 3(λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω) + 2ζω < 3λ + 15ω  contrary to the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Therefore, each Fi that contained in K has rank β < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Now  by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11, FpFp+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fq contains a fragment K′ of rank α with  µf(K′) ≥ µf(K) − 2(λ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ω) − 2ζω > 29ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For a base P of K′ we have |P|α−1 > 29 and by Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1, |K′|α−1 > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This  implies that weightα(Fp · Fp+1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' · Fq) > 1 and we get a contradiction with minimality  of F since we can replace FpFp+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Fq in F by a single truncated fragment of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This  finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (iii) By (ii), for each i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , r there exists a truncated fragment Fti of rank α in F  such that Ki ∼ Fti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 easily implies that Fti ∪ Ki is a truncated fragment of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For each i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , r we consequently replace Fti in F by Fti ∪ Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since we do  not increase weightα(F), the resulting fragmentation F′ of X is also minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1 and X, Y ∈ Rα be close in rank α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  |Y |α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|X|α + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let F be a minimal fragmentation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We represent X and Y by close paths X and Y  in Γα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then F induces the partition of X, denoted ¯F, into (path) truncated fragments of  ranks ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let  X = P0H1P1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' HrPr  where H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Hr are all truncated fragments of rank α in ¯F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If r = 0 then |X|α = ζ|X|α−1,  |Y |α ≤ ζ|X|α−1 and the statement simply follows from Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume  r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3(i), for each i we have Hi = uiH′  ivi where H′  i is a fragment of rank α,  H′  i ∼ Hi, |u|α−1, |v|α−1 < ζ, and the base Si for Hi satisfies |Si|α−1 > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Proposition  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16α−1 we find fragments H′′  i and Gi of rank α in X and Y respectively where H′  i = wiH′′  i zi,  |wi|α−1, |zi|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15, Hi ∼ H′′  i ∼ Gi and H′′  i and Gi are close in rank α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Lemma  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i)α−1 after each application of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16α−1 we can assume that Gi are disjoint,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Y = Q0G1Q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' GrQr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1 we have  |Q0|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|P0u1w1|α−1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2,  |Qi|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|ziviPiui+1wi+1|α−1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2  (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , r − 1),  |Qk|α−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|zkvkPk|α−1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We have also  |X|α = r + ζ  r  �  i=1  |Pi|α−1  and  |Y |α ≤ r + ζ  r  �  i=1  |Qi|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  85  Then  |Y |α < r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ζ  r  �  i=1  |Pi|α−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3rζ(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 + 2ζ) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ζ(r + 1)  = (1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ζ(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 + 2ζ))r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ζ  r  �  i=1  |Pi|α−1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ζ  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|X|α + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  Proof of Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let X, Y ∈ Rα be close in rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let F be a minimal fragmen-  tation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider close paths X and Y in Γα labeled X and Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then F  induces the partitions of X into (path) truncated fragments of ranks ≤ α,  X = F1 · F2 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' · Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let X−1uYv be a coarse bigon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We fix some bridge partitions of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let ∆ be a filling  diagram of rank α with boundary loop ˜X−1˜u˜Y˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to switching of u and v we can assume  that ∆ is reduced and has a tight set T of contiguity subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Dr be all  cells of rank α of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the process of forming T we assume that we pick first the contiguity  subdiagrams of Di to ˜X−1 choosing them with maximal possible contiguity segment occurring  in ˜X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  X = P0K1P1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' KrPr  and  Y = Q0M1Q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' MrQr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  where Ki and Mi are the corresponding active fragments of rank α in X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the way  we produce T and by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='21α−1 in the case α ≥ 2 we have the following:  (*) For all i, the fragment Ki cannot be extended in Pi−1KiPi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, if F is a  truncated fragment of rank α contained in Pi−1KiPi and containing Ki then F = w1Kiw2  where |wi|α−1 < ζ (i = 1, 2)  By Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3(iii) we can assume that each Ki is contained in a compatible truncated  fragment Fti of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let  X = P′  0Ft1P′  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' FtrP′  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that  |X|α = r +  �  i  |P′  i|α  and  |Y |α ≤ r +  �  i  |Qi|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By (*),  |P′  i|α ≥ |Pi|α − ζ2 for i = 0, r,  |P′  i|α ≥ |Pi|α − 2ζ2 for 1 ≤ i ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Hence  (14-1)  |X|α ≥ r +  �  i  |Pi|α − 2rζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We give an upper bound on |Qi|α in terms of |Pi|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' First we consider the case 1 ≤ i ≤  r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' There are three possibilities for the subdiagram of ∆ surrounded by Di and Di+1  and contiguity subdiagrams of Di and Di+1 to ˜X−1 and ˜Y, depending on the presence of  contiguity subdiagrams from T (see Figure 38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that according to Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12, all  the components of ∆ − ∪Π∈T are small diagrams of rank α − 1, so we can use bounds from  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In cases (a) and (b) we have |Qi|α ≤ 6ζ2η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6ζ and |Qi|α ≤ 4ζ2η <  86  Di  Di+1  ˜X  ˜Y  (a)  (b)  (c)  Figure 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ζ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that case (c) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then Pi = u1Su2 and Qi = v1Tv2 where S  and T are close in rank α − 1 and |ui|α, |vi|α ≤ 4ζ2η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, we get  |Qi|α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|Pi|α + 3ζ  Note that this inequality holds also in cases (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Now let i = 0 or i = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If r > 0 then the difference of the case i = 0 from the case  1 ≤ i ≤ r − 1 is that we can have an extra contiguity subdiagram between Y and the central  arc of ˜u (see Figure 39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We then have  ˜X  ˜Y  ˜u  D1  Figure 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  |Q0|α < 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|P0|α + 3ζ  and, similarly,  |Qr|α < 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|Pr|α + 3ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If r = 0 we have a single bound instead,  |Q0|α < 2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|P0|α + 3ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Summarizing, with (14-1) we get  |Y |α ≤ r + γ  �  i  |Pi|α + 2 + 3ζ(r + 1)  = (1 + 3ζ)r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3  �  i  |Pi|α + 2 + 3ζ  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3|X|α + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If F is a fragment of rank α and µf(F) ≥ tω then |F|α−1 >  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3(t − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In particular, |F| >  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3ζ1−α(t − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Y = u1X1u2X2u3 in Γα where Xi, Y ∈ Rα and ui ∈ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then  |Y |α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3(|X1|α + |X2|α) + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  87  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Follows from Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='19(i) and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  The following two statements are proved under the assumption that a normalized presen-  tation (2-1) of G satisfies the iterated small cancellation condition (S0)–(S3) for all α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We therefore will be assuming that all statements starting from Section 5 hold for all values  of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let W be a word with |W| ≤ α and let W = X in Gα where X ∈ Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then |X|α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3, X contains no fragments F of rank β > α with µf(F) ≥ 3ω and, in  particular, X ∈ ∩α≥1Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 it is enough to prove that |X|α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We proceed by induction on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If α = 1 then X is the freely reduced form of W and |X|1 ≤ ζ|X| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let W ≖ W1a, a ∈ A±1 and W1 = X1 in Gα−1 where X1 ∈ Rα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, the  inductive hypothesis and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15, equality X = X1a holds already in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6α−1  |X|α ≤ ζ|X|α−1 ≤ ζ(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3) + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Every element of G can be represented by a word X reduced in G such that  for some α ≥ 1, X contains no fragments F of rank β ≥ α with µf(F) ≥ 3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A graded presentation for the Burnside group  In this section we show that for sufficiently large odd n the Burnside group B(m, n) has  a graded presentation which satisfies the iterated small cancellation condition formulated in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We fix an odd number n > 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We are going to construct a graded presentation of the  form  (15-1)  �  A  �� Cn = 1 (C ∈  �  α≥1  Eα)  �  where all relators of all ranks α are n-th powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that values of the parameters λ  and Ω are chosen as in Theorem 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  λ = 80  n ,  Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We will use also the following extra parameters:  p0 = 39,  p1 = p0 + 26 = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  In what follows, we define the set Eα+1 under the assumption that sets Eβ are already  defined for all β ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We fix the value of rank α ≥ 0 and assume that the presentation  (15-1) satisfies small cancellation conditions (S0)–(S3) in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9 and in normalized in the  sense Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10 for all values of the rank up to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We can therefore assume that all statements in Sections 5–13 are true for the current value  of α and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  According to Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 each element of infinite order of Gα is conjugate  to a power of a simple period over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We will define Eα+1 as a certain set of simple periods  over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This will automatically imply condition (S0) with α := α + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since n is odd, by Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11 we obtain also that (S3) holds with α := α + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  88  Before going to the chain of definitions in the next section, we formulate the following two  conditions (P1) and (P2) on Eα+1 (which can be viewed as “periodic” versions of (S1) and  (S2) for the value of rank α := α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (P1) For each A ∈ Eα+1, [A]α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (P2) Let L1 and L2 be periodic lines in Γα with periods A, B ∈ Eα+1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume  that a subpath P of L1 and a subpath of Q of L2 are close and |P| ≥ p1|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then L1  and L2 are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The main goal of the construction of Eα+1 will be to satisfy (P1) and (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that (P1)  immediately implies (S1) for α := α + 1 because of the assumption n > 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Later we  prove that (P2) implies (S2)α+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (The difference between (P2) and (S2)α+1 is that in (P2)  we measure periodic words by the number of periods while in (S2)α+1 we use the length  function | · |α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' An appropriate bound will be given in Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  Our first step is to define a set of simple periods over Gα which potentially violate (P2)  (they will be excluded in the definition of Eα+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A simple period A over Gα is suspended of level 0 if there exist a simple  period B not conjugate in Gα to A and words P ∈ Per(A) and Q ∈ Per(B) such that P  and Q are close in Gα and |Q| ≥ p1|B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  At first sight, we could simply define Eα+1 by excluding periods A as in Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1  from the set of all simple periods over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' However, in this case we cannot guarantee a  necessary lower bound on [A]α for A ∈ Eα+1 in (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Roughly speaking, we need to claim  that a fragment of rank β ≤ α can cover only a “small” part of a periodic word with a period  A ∈ Eα+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' moreover, we need an exponentially decreasing upper bound on the size of this  part when β decreases (compare with the definition of the function | · |α in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' To achieve  this, we enlarge the set of excluded simple periods over Gα+1 by adding potentially “bad”  examples of this sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A simple period A over Gα is suspended of level m ≥ 1 if there exist a  suspended period B of level m − 1 not conjugate to A in Gα, and a reduced in Gα word  of the form XQY such that Q ∈ Per(B), |Q| ≥ 4|B| and XQY is close in Gα to a word  P ∈ Per(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let Pα denote the set of all simple periods over Gα and Sα denote the set  of all suspended simple periods over Gα of all levels m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For Eα+1 we take any set of  representatives of equivalence classes in Pα \\ Sα with respect to the equivalence  A ∼ B ⇔ A is conjugate to B±1 in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The definition implies that any simple period over Gα in Pα \\ Sα has finite order in Gα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since Pα+1 ⊆ Pα, it follows that any simple period over Gα+1 and, in particular, any word  in Eβ for β ≥ α + 1 belongs to Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As a consequence, we prove now that a fragment of rank  α + 1 cannot cover a large periodic word with a simple period A over Gα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (So here is the  trick: the definition of the set of suspended periods over Gα of levels m ≥ 1 serves condition  (P1) for the future rank α + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By construction, we obtain a normalized presentation (15-1) (see Definition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  89  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a simple period over Gα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If an A-periodic word P is a  subword of a fragment of rank α + 1 then |P| < 4|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As observed above, A ∈ Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let UPV be a fragment of rank α+1 where P ∈ Per(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then UPV is close in Gα to a word Q ∈ Per(B) where B ∈ Eα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since A is of infinite order  in Gα+1, it is not conjugate to B in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In this case, Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 says that if |P| ≥ 4|A|  then B ∈ Sα which would contradict Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 with α := α −1 is an important but not sufficient ingredient in the proof  of (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We need also to ensure that if a subword of fragment of rank β < α is a subword  of an A-periodic word with A ∈ Eα+1 then its length compared to |A| is “exponentially  decreasing when β decreases”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We prove a precise form of this statement in the next section  by showing that coarsely periodic words have a certain property of hierarchical containment:  a coarsely A-periodic word S over Gα has t disjoint occurrences of coarsely periodic words  over Gα−1 with sufficiently large number of periods where t is approximately the number of  periods A in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hierarchical containment of coarsely periodic words  Starting from this point, all statements are formulated and proved under assumption that  the group G has a specific presentation (15-1) defined in Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The goal of this section  is to prove the following property of suspended periods over Gα and to finalize the proof of the  fact that the presentation (15-1) satisfies conditions (S0)–(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As in Section 15 we assume  fixed the value of rank α ≥ 0 and assume that the normalized presentation (15-1) satisfies  conditions (S0)–(S3) for ranks less or equal α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' so we can use all statements in Sections 5–15  for any rank up to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a suspended period over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there exists a simple period B  over Gα such that:  (i) A cyclic shift of A contains a coarsely B-periodic word T over Gα with ℓB(T) ≥ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) Moreover, this subword T has the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S be a coarsely A-periodic  segment in Γα with ℓA(S) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then there are an A-periodic base P for S, ℓA(S) − 3  translates T, sA,PT, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , sℓ(S)−4  A,P  T of a coarsely B-periodic segment T in P with  label(T) ≖ T and ℓA(S)−3 disjoint coarsely B-periodic segments Vi (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , ℓ(S)−  4) in S such that Vi ≈ si  A,PT for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We start with showing how Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1 implies (P1) in the case α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a simple period over Gα and let S and Vi (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , ℓA(S) − 4)  be as in Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then for any i, Vi ∪ Vi+4 is not contained in a fragment of  rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As in Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1, let P be an A-periodic base for S in Γα−1 containing t − 3  translates T, sA,PT, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , st−4  A,PT where T is a coarsely periodic segment with another period B  and ℓB(T) ≥ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that a fragment K of rank α in Γα−1 contains Vi and Vi+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let L  be the base axis for K, so L is a C-periodic line with C ∈ Eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denoting V∗  i the stable part  of Vi, by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14α−1 we find W and W′ in L such that W ≈ V∗  i and W′ ≈ V∗  i+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then W ∪ W′ is close to si  A,PT∗ ∪ si+4  A,PT∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since A ∈ Sα−1, according to Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2α−1  this should imply C ∈ Sα−1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  90  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that a (linear or cyclic) word X has r disjoint occurrences  of coarsely A-periodic words Ui (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , r) over Gα−1 with ℓA(Ui) ≥ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then |X|α−1 ≥ 5r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The statement is immediate if α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Consider a fragmentation F of rank α − 1 of X (definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Sk be the  subwords of fragments of rank α−1 in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1 each Ui contains p0−3 = 36  disjoint coarsely B-periodic words Vi,j (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , 36) over Gα−2 with ℓB(Vi,j) ≥ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can  assume that Ui and Vi,j are indexed in their natural order from the start to the end in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, each Si intersects at most 6 consequent subwords Vi,j, Vi,j+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Vi,j+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Excluding Vi,j with 1 ≤ j ≤ 6, we obtain that each Si intersects at most 6 of all the  remaining Vi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By induction, we conclude that  |X|α−1 ≥ k + 5ζ max{0, 30r − 6k}  With fixed r, the minimal value of the right-hand side is achieved when 30r − 6k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This  gives the bound |X|α−1 ≥ 5r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  We prove the following stronger form of (P1):  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For any simple period A over Gα we have [A]α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25 and, consequently,  hα(A) ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If α = 0 then [A]0 ≥ 1 by the definition of [·]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Take any r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Consider  a fragmentation F of rank α of the cyclic word (Ar)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that F consists of words Si,  i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , N where the first k are subwords of fragments of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5α−1  we have |Si| < 4|A| for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This implies that the cyclic word (Ar−4k)◦ can be  partitioned into subwords of words in some subset of the remaining Si, i = k+1, k+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Therefore,  |(Ar)◦|α ≥ k + ζ|(Ar−4k)◦|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1α−1 says that (Ar−4k)◦ has at least r − 4k disjoint occurrences of a coarsely  B-periodic word K over Gα−1 with ℓB(K) ≥ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3,  |(Ar)◦|α ≥ k + 5ζ(r − 4k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  This holds for all r ≥ 1, so by Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 we get [A]α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25 and hence hα(A) ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  The following lemma is a key tool in the proof of Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Very roughly, it  corresponds to the statement “if a word W is periodic with two simple periods A and B at  the same time, and if |W| ≥ 2|A|, |W| ≥ 2|B| then B is a cyclic shift of A”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let L0 and L1 be periodic lines in Γα with simple periods A and B over Gα,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S be a coarsely C-periodic segment in L0 where C is another simple period  over Gα, ℓC(S) ≥ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that there exist coarsely C-periodic segments T0, T1, T2 in L1  such that T0 < T1 < T2 and Ti ≈ si  A,L0S, i = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If T0 ≲ s−1  B,L1T1 or sB,L1T1 ≲ T2 then, if fact, T0 ≈ s−1  B,L1T1, sB,L1T1 ≈ T2, words A and B  represent conjugate elements of Gα and periodic lines L0 and L1 are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote P0 = S ∪ s2  A,L0S and P1 = T0 ∪ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S∗ and T∗  i be stable parts of S and Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The crucial argument is similar to one in the proof of Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote P the  set of all coarsely C-periodic segments U in Γα such that U ≈ gS∗ for some g ∈ Gα (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' U  and S∗ have the same labels of their periodic bases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We introduce translations and jumps  on the set of coarsely C-periodic segments U ∈ P which occur in P0 or P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As in the proof  91  of Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, it will be convenient to consider two disjoint sets of those U ∈ P which  occur in P0 and in P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (So formally we introduce the set Pi (i = 0, 1) of pairs (U, Pi) where  U occurs in Pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' thus si  A,L0S∗ belongs to P0 and T∗  i belongs to P1 for i = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For a coarsely  C-periodic segment U ∈ P, saying ‘U occurs in Pi’ we mean the corresponding element of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=')  Let U, V ∈ P be coarsely C-periodic segments each occurring in some Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (i) If U and V occur in different paths Pi and U ≈ V then U jumps to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  (ii) U translates to V in the following cases:  U and V occur in P0 and U ≈ sk  A,L0V for some k ∈ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' or  U and V occur in P1 and U ≈ sk  B,L1V for some k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let M be a maximal set of pairwise non-(strictly compatible) segments which can be obtained  by these two operations from S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 implies that M is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' As in the proof  of Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 we prove the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Claim: The jump operation is always possible inside M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' that is, for any U ∈ M in Pi,  i ∈ {0, 1}, there exists V ∈ P in P1−i such that V ≈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  To prove the claim, we will apply Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 and do a necessary preparatory work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that U ∈ M belongs to P0 (the other case differs only in notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let V0 = S∗, V1,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Vl = U be a sequence of coarsely C-periodic segments Vi ∈ M such that Vi+1 is obtained  from Vi by one of the operations (i) or (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We can assume that V2j → V2j+1 are translations  and V2j+1 → V2j+2 are jumps, so l = 2k −1 for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Under this assumption, V2j → V2j+1  is a translation inside P0 if j is even and inside P1 if j is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We then define a sequence Y0,  Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Yk of paths in Γα (Yj will be periodic segments with alternating periods A and B)  and a sequence Wj ∈ P of coarsely C-periodic segments in Yj for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , k−1 such that  W0 = V1 and Wi ≈ W0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For each j we will have Wj = fjV2j+1 for some fj ∈ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  The definition of Yj and fj goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We start with Y0 = P0 and W0 = V1, so f0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Assume that j < k − 1 and Yj and fj  are already defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For even j, V2j translates to V2j+1 inside P0, so there exists fj+1 ∈ Gα  of the form fjst  A,P0 such that fj+1V2j+1 ≈ fjV2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Thus, fiP0 and fj+1P0 have a common  A-periodic extension and we take Yj+1 = fiP0 ∪ fj+1P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, for odd j V2j translates  to V2j+1 inside P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We take fj+1 ∈ Gα of the form fjst  B,P1 such that fj+1V2j+1 ≈ fjV2j and  take Yj+1 = fiP0 ∪ fj+1P0 inside a common B-periodic extension of fiP0 and fj+1P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note  that k is odd because V2k+1 = U is assumed to occur in P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We finally set Yk = fk−1P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We now apply Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 where:  Sj is the set of all coarsely C-periodic segments V ∈ P in Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Sj is pre-ordered by ‘≨’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Equivalence is strict compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  A segment V ∈ �  j Sj is defined to be stable if V is the stable part of some coarsely  C-periodic segment in Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For aj, bj, a′  j and b′  j we take appropriate translates of S∗ and T∗  i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' namely, fjS∗,  fjs2  A,L0S∗, fjT∗  0 and fjT∗  2 if j is even or fjT∗  0, fjT∗  2, fjS∗ and fjs2  A,L0S∗ if j is odd,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  c0 is V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note that by Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 we have hα(C) ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence the hypothesis ℓC(S) ≥ 25 implies  ℓC(V) ≥ 13 ≥ 2hα(C)+1 for any V ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Condition (ii) of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 holds by Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Conditions (iii) and (iv) of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 hold by Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the lemma, there exists a  92  coarsely C-periodic segment Vk ∈ P in fk−1P1 such that Vk ≈ fk−1U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This gives the required  jump U → f −1  k−1Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let r be the number of coarsely C-periodic segments V ∈ M such that and K∗ ≨ V ⪅  sA,L0K∗ and let q be the number of coarsely C-periodic segments V ∈ M such that T∗  1 ≨ N ⪅  sB,L1T∗  1 (in other words, r and q are the numbers of coarsely C-periodic segments V ∈ M  in one period A and in one period B, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that gcd(r, q) = 1 because M is  generated by a single segment S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We assume first that either T0 ⪅ s−1  B,L1T1 or sB,L1T1 ⪅ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Since M is closed under  translations modulo equivalence ‘≈’, each of these relations implies q ≤ r and hence implies  the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let U0, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Ut be all coarsely C-periodic segments in M belonging to P0  arranged in their order in P0 (so Ui form a set of representatives of coarsely C-periodic  segments in M modulo ‘≈’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The group Gα acts on the set P/≈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' It follows from Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9  that the action is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For equivalence classes [Ui] of Ui we have  sA,L0[Ui] = [Ui+r], i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , t − r  sB,L1[Ui] = [Ui+q], i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , t − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Note also that t ≥ 2r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Applying Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2 we get sA,L0 = dq and sB,L1 = dr for some  d ∈ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since A and B are non-powers we get q = r = 1 which immediately implies the  conclusion of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  For the proof, it remains to consider cases T0 ∼ s−1  B,L1T1 and sB,L1T1 ∼ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We consider  the case sB,L1T1 ∼ T2 (the case T0 ∼ s−1  B,L1T1 is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By the already proved part, we  can assume that T2 ≨ sB,L1T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We show that the assumption leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We have T0 ≨ s−1  B,L1T2 ≨ T1, so there exists T3 ∈ M such that T3 ≈ s−1  B,L1T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' T3 jumps to  some S3 ∈ M in L0 such that S3 ∼ S and S3 ≨ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then S3 translates to S4 ≈ sA,L0S3 and we  have S4 ∼ S2 and S4 ≨ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then S4 jumps to some T4 in L1 and we can continue the process  infinitely (see Figure 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  T0  T3  T1  T4  T2 sB,L0T1  s−1  A,L0S  S3  S  S4  sA,L0S  L1  L0  Figure 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof of Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A be a suspended period of level m over Gα,  Assume first that m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then by Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 and Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15 an A-periodic  segment R in Gα contains a coarsely B-periodic segment ˆT with ℓB(ˆT) ≥ p1−2hα(B)−2 ≥ 51  where B is not conjugate to A in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 we have ˆT ̸∼ sA,RˆT and |ˆT| < 2|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let T be the stable part of ˆT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since hα(B) ≥ 2 by Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12, we have |T| < |A| by  Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note also that ℓB(T) ≥ ℓB(ˆT) − 2hα(B) ≥ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let T ≖ label(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We show  that T has the required property (ii) formulated in Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1  Let S be a coarsely A-periodic segment in Γα with ℓA(S) ≥ 4 and let P be a periodic base  for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote t = ℓ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 we can assume that |P| ≥ t|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to placing ˆT  93  in Γα we can assume that P contains t−2 translates ˆT, sA,PˆT, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , st−3  A,PˆT of ˆT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Lemma  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(i) (which implies that strictly compatible coarsely periodic segments are close) and  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 we find disjoint Vi (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , t − 3) in S such that Vi ≈ si  A,PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This  proves the proposition in the case m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Let m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The proof consists of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' First we provide a construction of a coarsely  B-periodic segment T satisfying condition (i) of Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='1 and then we prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Construction of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' According to Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='2, there exists a sequence A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ,  Am = A of simple periods over Gα where A0 is suspended of level 0, for each i ≤ m − 1 Ai is  not conjugate to Ai+1 and there are reduced in Gα close words XiQiYi and Pi+1 ∈ Per(Ai+1)  where Qi ∈ Per(Ai) and |Qi| ≥ 4|Ai|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For each i, we consider corresponding close paths  XiQiYi and Pi+1 in Γα and place then in such a way that Qi and Pi have the common infinite  Ai-periodic extension Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We denote also L0 the infinite A0-periodic extension of Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  As we proved above, there is a coarsely B-periodic segment ˆT0 in Q0 with ℓ(ˆT0) ≥ 51 and  the stable part T0 satisfying ℓ(T0) ≥ p0 and |T0| < |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to positioning ˆT0 in L0 we can  assume that Q0 contains translates s−1  A0,L0T0 and sA0,L0T0 of T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In what follows, if Z is a  coarsely B-periodic segment in Γα then Z∗ denotes the stable part of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12,  st  A0,L0T0 ̸∼ T0 for any t ̸= 0 and hence s−1  A0,L0T0 � ˆT0 � sA0,L0T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14  there are T1, U1,1 and W1,1 in P1 such that T1 ≈ T0, U1,1 ≈ s−1  A0,L0T∗  0 and W1,1 ≈ sA0,L0T∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Application of Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 with S := T∗  0 (note that ℓB(T∗  0) ≥ p0 − 12 ≥ 27) gives s−1  A1,L1T1 �  U1,1 and W1,1 � sA1,L1T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, we have |T1| ≤ |A1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In the case m = 1 we take  T := label(T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We continue a procedure of finding coarsely B-periodic segments Ti  in Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to positioning Q1 in L1 we can assume that Q1 contains both s−1  A1,L1T1 and sA1,L1T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 we find U2,2, U2,1, W2,1 and W2,2 in P2 such that U2,2 ≈ s−1  A1,L1T∗  1,  U2,1 ≈ U∗  1,1, W2,1 ≈ W∗  1,1 and W2,2 ≈ sA1,L1T∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='15, U2,2 � U2,1 � W2,1 � W2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We have U2,1 ≈ s−1  A0,L0T∗∗  0 , W2,1 ≈ sA0,L0T∗∗  0  and using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 once more with  X := s−1  A0,L0T∗∗  0 ∪sA0,L0T∗∗  0 and Y := U2,1∪W2,1 we find T2 in P2 such that T2 ≈ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Application  of Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5 gives s−1  A2,L2T2 � U2,2 and W2,2 � sA2,L2T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' In particular, |T2| ≤ |A2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Repeating in a similar manner, we find Um,m, Um,m−1, Wm,m−1 and Wm,m in Pm such that  Um,m ≈ s−1  Am−1,Lm−1T∗  m−1, Um,m−1 ≈ U∗  m−1,m−1, Wm,m−1 ≈ W∗  m−1,m−1, Wm,m ≈ sAm−1,Lm−1T∗  m−1  and Um,m � Um,m−1 � Wm,m−1 � Wm,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then we successively find Um,m−2, Wm,m−2,  Um,m−3, Wm,m−3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Um,1, Wm,1 such that Um,i ≈ U∗  i,i ≈ s−1  Ai−1,Li−1T∗∗  i−1 and Wm,i ≈ V∗  i,i ≈  sAi−1,Li−1T∗∗  i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Finally, we find Tm in Pm such that Tm ≈ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Application of Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5  gives s−1  Am,LmTm � Um,m and Wm,m � sAm,LmTm which implies |Tm| ≤ |Am|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  We take  T := label(Tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' This completes the construction, The whole procedure is schematically  shown in Figure 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Note that in Pm we have  s−1  Am,LmTm � Um,m � Um,m−1 � · · · � Um,1 � Tm � Wm,1 � · · · � Wm,m � sAm,LmTm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let S be a coarsely Am-periodic segment in Γα and let P be a periodic base  for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote t = ℓAm(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7 we can assume that |P| ≥ t|A|, so P contains  t − 1 translates T, sA,PT, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , st−2  A,PT of a coarsely B-periodic segment T which is a translate  of Tm constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14, S contains coarsely B-periodic segments Z0,  Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , Zt−2 such that Zi ≈ si  A,PT∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We claim, moreover, that for 1 ≤ i ≤ t−3 there exist Vi  94  s−1  A0,L0T0  ˆT0  sA0,L0T0  s−1  A1,L1T1  U11  T1  V11 sA1,L1T1  s−1  A2,L2T2  U22  U21  T2  V21  V22 sA2,L2T2  s−1  A3,L3T3  U33  U32  U31  T3  V31  V32  V33 sA3,L3T3  L0  L1  L2  L3  Figure 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  in S such that Vi ≈ si  A,PT and Vi are all disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since ℓB(Vi) = ℓB(Tm) ≥ p0 this will finish  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Fix an index k in the interval 1 ≤ i ≤ t − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to positioning P and S in Γα we can  assume that P and Pm have the common Am-periodic extension Lm and sk  A,PT = Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, s−1  Am,LmT � Um,m and Wm,m � sAm,LmT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 as in  the procedure above, we successively find pairs (Ui, Wi) for i = m, m − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , 1 such that  Zk−1 � Um � Um−1 � · · · � U1 � Zk � W1 � · · · � Wm � Zk+1 and Ui ≈ U∗  i,i, Wi ≈ W∗  i,i  for i = m, m − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14 again with X := s−1  A0,L0T∗∗  0 ∪ sA0,L0T∗∗  0 ,  Y := U1 ∪ W1 and S = ˆT0 gives Vk with U1 � Vk � W1 and Vk ≈ T0 ≈ sk  A,PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' The proof is  finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let A ∈ Eα+1 and t ≥ 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P be an A-periodic word with  |P| = t|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then  t  n + t < µ(P) <  t  n − t + ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Moreover, for t ≥ 200 we have also  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='89 t  n < µ(P) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 t  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Denote N = |(An)◦|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Recall that µ(P) = |P|α/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Up to cyclic shift of A, we assume  that P ≖ At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' For the lower bound on µ(P) in the first inequality, we observe that the cyclic  word (An)◦ can be covered with ⌈n  t ⌉ copies of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14, this implies  N <  �n  t + 1  �  |P|α  which is equivalent to  t  n+t < µ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, for the upper bound we observe that ⌊n  t ⌋  disjoint copies of P can be placed inside (An)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then again by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='14,  N ≥  �n  t  �  (|P|α − 1) >  �n  t − 1  �  (|P|α − 1)  which implies by (S1) with α := α + 1  µ(P) <  t  n − t + 1  N ≤  t  n − t + ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  95  If t ≥ 200 then we partition At into k subwords Ati with 80 ≤ ti ≤ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We have  �  i  |Ati|α − (k − 1) ≤ |P|α ≤  �  i  |Ati|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  and by the already proved bounds on µ(Ati), for each i we have  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='94ti  n < µ(Ati) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='07ti  n + 1  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Then  µ(P) ≥  �  i  µ(Ati) − k − 1  N  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='94 t  n − k  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4, N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Hence  k  N ≤ t  80  � n  N  � 1  n ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='05 t  n  and we obtain the required bound µ(P) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='89 t  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Similarly, for the upper bound on µ(P)  we get  µ(P) ≤  �  i  µ(Ati) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='07 t  n + k  N ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='12 t  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' (P2) implies (S2)α+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='6, if P is a subword of An with A ∈ Eα+1 and µ(P) ≥ λ then  |P| ≥ t|A| where t satisfies  t  n − t ≥ λ − ω ≥ 1  24 −  1  480  and hence t > 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since 76 > p1, the required implication is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Presentation (15-1) satisfies (P2) and therefore satisfies the iterated  small cancellation condition (S0)–(S3) for all α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Indeed, assume that L1 and L2 are periodic lines in Γα with periods A, B ∈ Eα+1  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let P and Q be close subpath of L1 and L2, respectively, such that |P| ≥ p1|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  If A is conjugate to B in Gα then A = B according to Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='3 and the statement  follows from Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If A is not conjugate to B in Gα then B is suspended of  level 0 as a simple period over Gα and hence cannot belong to Eα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  From this point, we may assume that all statements in Sections 5–16 are true for all values  of rank α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Every element of G is conjugate to a power of some C ∈ �  α≥1 Eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Let g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' If g has finite order then by Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, g is conjugate to a power  of some C ∈ �  α≥1 Eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We assume that g has infinite order and come to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  By Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='8 we represent g by a word X reduced in G such that for some α ≥ 1,  X contains no fragments F of rank β ≥ α with µf(F) ≥ 3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By our assumption, X has  infinite order in all Gβ for β ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, X is conjugate in Gα to  a word of the form At where A is a simple period over Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Using Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13(iii) we  conclude that X is conjugate to At already in Gα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Then applying Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='9 with  96  β := α, α + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' we see that no cyclic shift of A contains a fragment K of rank β ≥ α with  µf(K) ≥ 9ω and that A is cyclically reduced in Gβ for all β > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Moreover, by Propositions  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16(iii) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='11, A is strongly cyclically reduced in Gβ for all β > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Assume that for some β ≥ α, A is conjugate in Gβ to a power Br of a simple period  over Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='16, A and Br are conjugate already in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Since A is a non-  power in Gα, we have r = 1 and then by Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='13 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='5, A is a non-power  in Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' We showed that A is a simple period over Gβ for any β ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' But this is impossible  because by Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='4 we should have |A|β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25 and hence |A| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='25ζ−β for any  β ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  □  As an immediate consequence we get:  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' G satisfies the identity xn = 1 and therefore is isomorphic to the free  Burnside group B(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
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+page_content=' transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
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+page_content=' Г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Лысёнок.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Бесконечные бернсайдовы группы четного периода.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Изв.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' РАН.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Сер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=',  60(3):3–224, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Engl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' : I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Lysenok, Infinite Burnside groups of even exponent, Izvestiya:  Mathematics, 60:3 (1996), 453–654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  [10] И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Лысёнок.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Подход к изучению конечно определенных групп, основанный на понятии  дискретной кривизны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' заметки, 103(4):568–575, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Engl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' : Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Notes, 103:4 (2018),  610–615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  [11] П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Новиков and С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Адян.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' О бесконечных периодических группах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' I-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Известия АН СССР.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  сер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=', 32:212–244;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 251–524;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 709–731, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Engl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' : P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Novikov and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Adian, Infinite  periodic groups, I-III, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' USSR-Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 2 (1968), 209–236;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 241–479;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 665–685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  [12] А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ольшанский.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' О теореме Новикова–Адяна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' сб.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=', 118(160)(2(6)):203–235, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Engl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ol’shanski˘ı, On the Novikov–Adyan theorem, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' USSR-Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 46:2 (1983), 203–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  [13] А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ольшанский.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Геометрия определяющих соотношений в группах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Наука, Москва, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Engl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ol’shanski˘ı, Geometry of defining relations in groups, Kluwer, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  [14] И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Н.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Санов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Решение проблемы Бернсайда для показателя 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Учен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' зап.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' ЛГУ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Сер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' наук.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=',  10:166–170, 1940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' [I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Sanov, Solution of Burnside’s problem for exponent 4, Uchenye Zapiski  Leningrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Gos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=', 1940, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 10, 166–170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  [15] В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Тартаковский.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Решение проблемы тождества для групп с k-сократимым базисом при k > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  Изв.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' АН СССР.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Сер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=', 13(6):483–494, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' [V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Tartakovskii, Solution of the word problem  for groups with a k-reduced basis for k > 6, Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Nauk SSSR Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content=' 13:6 (1949), 483–494].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
+page_content='  97' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE4T4oBgHgl3EQfSgzP/content/2301.05000v1.pdf'}
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+Springer Nature 2021 LATEX template
+Spatiotemporal linear stability of viscoelastic
+subdiffusive channel flows: a fractional
+calculus framework
+Tanisha Chauhan1†, Diksha Bansal1† and Sarthok Sircar1*
+1*Dept. Mathematics, IIIT Delhi, Okhla Phase III, New Delhi,
+110020, Delhi, India.
+*Corresponding author(s). E-mail(s): sarthok@iiitd.ac.in;
+Contributing authors: tanishai@iiitd.ac.in; dikshab@iiitd.ac.in;
+†These authors contributed equally to this work.
+Abstract
+The temporal and spatiotemporal linear stability analyses of viscoelastic,
+subdiffusive, plane Poiseuille and Couette flows obeying the Fractional
+Upper Convected Maxwell (FUCM) equation in the limit of low to
+moderate Reynolds number (Re) and Weissenberg number (W e), is
+reported to identify the regions of topological transition of the advanc-
+ing flow interface. In particular, we demonstrate how the exponent in
+the subdiffusive power-law scaling (tα, with 0<α ≤ 1) of the mean
+square displacement of the tracer particle, in the microscale [Mason and
+Weitz, Phys. Rev. Lett. 74, 1250-1253 (1995)] is related to the fractional
+order of the derivative, α, of the corresponding non-linear stress con-
+stitutive equation in the continuum. The stability studies are limited
+to two exponents: monomer diffusion in Rouse chain melts, α = 1/2,
+and in Zimm chain solutions, α =
+2/3. The temporal stability anal-
+ysis indicates that with decreasing order of the fractional derivative:
+(a) the most unstable mode decreases, (b) the peak of the most unsta-
+ble mode shifts to lower values of Re, and (c) the peak of the most
+unstable mode, for the Rouse model precipitates towards the limit
+Re → 0. The Briggs idea of analytic continuation is deployed to clas-
+sify regions of temporal stability, absolute and convective instabilities
+and evanescent modes. The spatiotemporal phase diagram indicates an
+abnormal region of temporal stability at high fluid inertia, revealing the
+presence of a non-homogeneous environment with hindered flow, thus
+1
+arXiv:2301.02078v1  [physics.flu-dyn]  5 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Article Title
+highlighting the potential of the model to effectively capture certain
+experimentally observed, flow-instability transition in subdiffusive flows.
+Keywords: Spatiotemporal stability, anomalous diffusion, non-Markovian
+processes, Caputo integral, Upper Convected Maxwell model
+1 Introduction
+The subject of anomalous diffusion has received tremendous attention over
+the last half-century, ranging from physics [1–3], biology [4] to quantitative
+finance [5]. Some of the most significant and profoundly published exper-
+imental results are better rationalized within the viscoelastic subdiffusive
+approach in random environments such as the cytosol and the plasma mem-
+brane of biological cells [6], crowded complex fluids and polymer solutions [7],
+dense colloidal suspensions [8] and single-file diffusion in colloidal systems [9].
+The observed (anomalous) subdiffusion often combines features of ergodic
+fractional Brownian motion (reflecting viscoelasticity) and the nonergodic
+jumplike non-Markovian diffusional processes (reflecting disorder) [10, 11]. The
+subdiffusive object is considered primarily as being elastic and structurally
+robust, although it requires ‘fluidity’ and flexibility besides its elasticity for
+a proper functioning, e. g., consider a viscoelastic nanoscaled polymer drop
+armed with a rigid backbone that can take on different macroscopic con-
+formations [12]. In this article, we demonstrate how the dynamics of the
+subdiffusive fluids at microscale (often represented via a generalized Langevin
+equation (GLE) at the molecular level, using a dissipative memory kernel)
+is ‘upscaled’ to a fractional viscoelastic stress constitutive equation at the
+continuum level [13].
+Fractional calculus serves as a powerful tool for modeling the constitutive
+relations in the linear [14] as well as nonlinear viscoelasticity theory [8] and
+to explain certain paradoxical experimental findings [2, 3] (one such experi-
+mentally abnormal feature captured by our model, namely the occurrence of
+temporal stability at high fluid inertia, is mentioned in Section 5). Gemant
+highlighted the relaxation curves for some viscoelastic fluids by employing a
+fractional viscoelastic model for the first time [15]. Scott-Blair developed a
+new constitutive law (known as the ‘fractional Newton model’) to describe the
+experimental outcome of Gemant on stress relaxation [16, 17]. Caputo intro-
+duced the fractional Voigt model to simulate the dissipation in seismology [18].
+Bagley and Torvik [19] showed that there exists a quantitative connection
+between the fractional viscoelastic model (at the macroscopic level) and the
+molecular theory of Rouse’s polymer chain melts [20]. With the development
+of the fractional viscoelastic model, the flow of the fractional viscoelastic fluid
+has been extensively investigated [21–25]. Tan and Xu [21] used Laplace trans-
+forms to obtain the analytical solution for velocity and stress of the plane
+surface flow of a fractional Maxwell fluid. Qi and Xu [22] studied the plane
+
+Springer Nature 2021 LATEX template
+Article Title
+3
+Poiseuille flow and plane Couette flow of a generalized Oldroyd-B fluid with
+fractional derivative. Zheng et al. [24] found the analytical solution for velocity
+and stress of magnetohydrodynamic flow of a generalized Oldroyd-B fluid gen-
+erated with an accelerating plate, with fractional derivatives. Recently, Zhao
+et al. [25] considered the natural convection heat transfer of viscoelastic fluid
+with fractional the Maxwell model over a vertical plate. More recent appli-
+cations of the fractional viscoelastic flows include the study of the stability
+of coastal morphodynamics and seafloor topology [26], regulation of the tis-
+sue morphodynamics [27] and capturing spatiotemporal disorder in anomalous
+transport of viscous flows [28].
+The detailed exploration of the existing literature serves as a clear motiva-
+tion for the work reported here, which is to provide a comprehensive picture of
+the stability of the two-dimensional, viscoelastic, subdiffusive, fully developed,
+Poiseuille and Couette flows. The present work significantly differs from the
+existing studies in the sense that we analyse the linear stability of viscoelastic,
+subdiffusive, channel flows through a combined temporal and spatiotemporal
+stability analysis (rather than only a temporal stability analysis of the classi-
+cal (or integer order) viscoelastic channel flows [29]) and the aim is to address
+the following intriguing questions: What is the critical flow/polymer relaxation
+condition for the onset of instability? And more crucially, what is the linear
+spatiotemporal, time asymptotic response of the flow at the critical value of
+the material parameters, leading to the topological transition of the advancing
+flow interface of the subdiffusive channel flows?
+While the molecular theory of polymer dynamics has already established
+the correspondence between subdiffusive dynamics and linear viscoelastic
+relaxation of polymer melts and solutions. [30, 31], we ‘upscale’ these ideas at
+the continuum mechanical scale. In particular, we highlight how the exponent
+in the subdiffusive power-law timescale, tα [32], is related to the fractional
+order, α, of the corresponding non-linear stress constitutive equations in the
+continuum (refer Section 2.1). The temporal and spatiotemporal stability of
+two specific cases of monomer diffusion in Rouse chain melts (α = 1/2) [20],
+and in Zimm chain solution (α = 2/3) [33] are reported in detail (Section 4,
+Section 5). The Rouse model predicts that the viscoelastic properties of the
+polymer chain can be described by a generalized Maxwell model, where the
+elasticity is governed by a single relaxation time, which is independent of the
+number of Maxwell elements (or the so-called ‘submolecules’). In contrast,
+the Zimm’s model predicts the (‘shear rate and polymer concentration inde-
+pendent’) viscosity of the polymer solution by calculating the hydrodynamic
+interaction of flexible polymers (an idea which was originally proposed by
+Kirkwood [34]) by approximating the chains using a bead-spring setup.
+
+Springer Nature 2021 LATEX template
+4
+Article Title
+2 Problem formulation: Mathematical model,
+linear stability analysis and numerical method
+In this study, the linearized stability analyses of the fully developed, planar
+Poiseuille and Couette flows inside an infinitely long channel of width H (i. e.,
+0 ≤ y ≤ H such that x/y ≫ 1, where x and y are the flow and the shear
+gradient direction, respectively) is reported.
+2.1 Mathematical model
+We consider a viscoelastic fluid subject to a shear deformation. Then, an
+infinitesimal elastic stress, τxy at time t arising from a small strain increment
+dγ at an earlier time t′ is given by,
+dτxy = G(t − t′)˙γ(t′)dt′,
+(1)
+where the relaxation modulus, G(t), represents the influence of the dissipa-
+tive processes of the surrounding concentrated fluid medium [35]. Assuming
+linearity, the Boltzmann superposition principle may be utilized to construct
+the elastic stress at time t by summing up all of the infinitesimal contributions
+over the entire flow history, which is extended into the infinite past [36],
+τxy =
+� t
+−∞
+G(t − t′)˙γ(t′)dt′.
+(2)
+In their seminal work on passive micro-rheology, Mason and co-workers [31]
+have identified an approximate relation between the time-dependent memory
+kernel describing the viscous damping of the tracer particle at micro-scale (and
+which obeys the GLE, e. g., see equation (15) in [32]), ζ(t), and the stress
+relaxation modulus, G(t), i. e.,
+ζ(t) = 6πaG(t),
+(3)
+where a is the radius of the tracer particle (assumed spherical). In the regime of
+linear viscoelasticity, one of the most commonly used three-parameter family
+of memory kernel is the generalized Rouse kernel for an equally weighted sum
+of negatively decaying exponential functions [37],
+ζ(t) = 1
+N
+N−1
+�
+k=0
+e−( k
+N )
+1
+α ( t
+λ0 ),
+(4)
+for a number of kernels determining the length of the subdiffusive phase, N,
+relaxation time, λ0, and a subdiffusive exponent, α ∈ (0 1]. Since the polymeric
+liquids of our interest [38–46] show subdiffusive behavior on all length scales,
+
+Springer Nature 2021 LATEX template
+Article Title
+5
+we consider the case when N → ∞ in the prony series (4). For t > λ0, this
+limiting behavior leads to the relation
+ζ(t) =
+˜G
+Γ(1 − α)
+� t
+λ0
+�−α
+,
+(5)
+where Γ(x) is the complete gamma function and ˜G = Γ(1 + α)Γ(1 − α), is a
+constant. In equation (5), we have used the fact that the Riemann sum on an
+infinite interval,
+lim
+N→∞
+1
+N
+N−1
+�
+k=0
+f
+� k
+N
+�
+=
+� ∞
+0
+f(x)dx.
+(6)
+Using equations (2, 3, 5), one arrives at,
+τxy =
+Gλα
+0
+Γ(1 − α)
+� t
+−∞
+dt′(t − t′)−α dγ(t′)
+dt′
+,
+(7)
+where the constant, G =
+˜
+G
+6πa. We remark that G is no longer a constant
+(typically G = G(t)) when the concentration effects, such as the bond and
+entanglement effects, are considered [8]. The right-hand side of equation (7)
+represents a fractional integral corresponding to the Caputo formalism [47, 48],
+−∞D−β
+t
+f(t) =
+1
+Γ(β)
+� t
+−∞
+dt′
+(t − t′)1−β
+df(t′)
+dt′ .
+(8)
+Utilizing equations (7, 8), we arrive at the basic equation governing stress-
+strain relation in linear viscoelastic subdiffusive media,
+τxy = Gλα
+0
+dα−1
+dtα−1
+dγ(t)
+dt
+= Gλα
+0
+dαγ
+dtα ,
+(9)
+including the limiting cases of a purely elastic solid (α→0 or a Hookean spring)
+and a purely viscous fluid (α = 1 or a dashpot) [16]. Through combinations of
+springs and dashpots, one arrives at standard linear viscoelastic models, includ-
+ing the Maxwell, Kelvin-Voigt, Zener, Poynting-Thomson and Burgers’ model
+and others [17]. The problem is that the corresponding differential equations
+have a relatively restricted class of solutions, which are too limited to provide
+an adequate description for the class of complex fluids discussed in Section 1.
+To overcome this shortcoming, one can relate the stress and strain through
+the fractional equation (9), which allows a smooth interpolation between a
+purely elastic behavior and a purely viscous pattern. In the present analysis, we
+have selected the Fractional Upper Convected Maxwell equation (FUCM) to
+describe the nonlinear viscoelastic response of the subdiffusive media, derived
+next.
+Figure 1a depicts the standard Maxwell model in which a spring and a
+dashpot are connected in series [35]. We generalize this model by replacing
+
+Springer Nature 2021 LATEX template
+6
+Article Title
+Fig. 1 (a) The Maxwell element and (b) its fractional generalization.
+these elements with their corresponding fractional elements: (αi, Gi, λi), i =
+1, 2 (figure 1b). Because of the sequential construction, the stress, τ, is the
+same for both elements and their respective stress-strain relations are
+γi = G−1
+i
+λ−αi
+i
+d−αiτxy
+dt−αi ,
+i = 1, 2,
+(10)
+where both expressions follow from equation (9). Due to the construction of
+the generalized Maxwell model, we have γ(t) = γ1(t) + γ2(t), from which it
+follows,
+τxy + G1λα1
+1
+G2λα2
+2
+dα1−α2τxy
+dtα1−α2
+= G1λα1
+1
+dα1γ
+dtα1 .
+(11)
+Equation (11) can be simplified by setting λ = (G1λα1
+1 /G2λα2
+2 )1/(α1−α2) and
+E = G1(λ1/λ)α1. Without loss of generality, we assume α2 = 0 and α = α1(>
+0), and arrive at
+τxy + λα dατxy
+dtα
+= ηp
+dαγ
+dtα ,
+(12)
+where the constant, ηp = Eλα. We can extend equation (12) to three dimen-
+sions by replacing the elastic stress, τxy, with the stress tensor, τ, and the
+derivative, dαγ
+dtα , with the rate of strain tensor, D = (∇v + (∇v)T ) (where the
+operator ∇(·) =
+∂
+∂x(·)), to arrive at
+τ + λα dατ
+dtα = ηpD,
+(13)
+using the definition of fractional velocity, v = dαx
+dtα [49], which has a dimension
+of
+H
+T α (refer Section 2.2 for the discussion on non-dimensionalization). Frac-
+tional velocities are defined as limits of the difference quotients of a fractional
+
+(α2, G2,2)
+(α1, G1, A1)
+no
+(n)
+(b)Springer Nature 2021 LATEX template
+Article Title
+7
+power and they generalize the notion of a local derivative [50]. These deriva-
+tives are frequently used, for example, to model instantaneous interactions in
+Langevin dynamics [51].
+Equation (13) is the rheological constitutive equation of the fractional
+Maxwell model describing the linear viscoelastic media. The simplest way
+to combine rheological nonlinearity is to replace the (fractional) mate-
+rial time derivative in equation (13) with the (fractional) frame invariant,
+upper-convected time derivative [52, 53], which leads us to FUCM, as follows,
+τ + λα▽τ = ηpD,
+(14)
+where the fractional upper-convected time derivative of the tensor τ is defined
+as,
+▽τ = ∂ατ
+∂tα + v · ∇τ − (∇v)T τ − τ∇v.
+(15)
+The fractional time derivative, ∂α
+∂tα , in equation (14, 15) is based on the Caputo
+definition (8).
+The continuity and the momentum equations for an incompressible,
+subdiffusive flow (consistent with the stress constitutive relation (14)) are,
+∇ · v = 0,
+ρ
+�∂αv
+∂tα + v · ∇v
+�
+= −∇p+ηs∇ · D + ∇ · τ,
+(16)
+where ρ is the density and p is the isotropic pressure. Equations (14,16) repre-
+sent the equations of motion describing the flow-instability of the subdiffusive
+viscoelastic fluids.
+As a result of the dissipative processes, viscoelastic materials have mem-
+ory, that is, their actual mechanical response is modulated by the past [54].
+The fractional derivative operators account for the complete history to obtain
+the derivative at an instant. Unlike the classical Maxwell model [45, 55] which
+accounts for only the elastic (or stored) part of the deformation work, the
+fractional Maxwell model accounts for both forms (stored and dissipated) of
+energy at any time point. Although Mckinley pointed out that the fractional
+Maxwell model generally cannot capture polymer shear-thinning [56], the frac-
+tional version provides a better fit of the relaxation and creep behavior for a
+significantly large class of viscoelastic materials using fewer parameters than
+the classical version [54].
+2.2 Linear stability analysis
+Using the following scales for non-dimensionalizing the governing equations:
+the height of the channel H for length, the timescale T corresponding to max-
+imum base flow velocity (i. e., T = (H/U0)1/α) for time and ρU2
+0 for pressure
+and stresses, we characterize equations (14, 16), rephrased as follows,
+∇ · v = 0,
+(17a)
+
+Springer Nature 2021 LATEX template
+8
+Article Title
+Re
+�∂αv
+∂tα + v · ∇v
+�
+= −∇p + ν∇ · D + (1 − ν)∇ · A,
+(17b)
+∂αA
+∂tα + v · ∇A − (∇v)T A − A∇v = D − A
+We
+,
+(17c)
+using the dimensionless groups, Re = ρU0H
+η0
+(Reynolds number), We = λαU0
+H
+(Weissenberg number) and where ηs, ηp, η0(= ηs + ηp) and ν(= ηs/η0) are
+the solvent viscosity, the polymeric contribution to the shear viscosity, the
+total viscosity and the viscous contribution to the total viscosity of the fluid,
+respectively. In equation (17), the elastic stress is represented as τ = (1−ν)A.
+The current analysis deploys fractional derivative of exponentials [47, 48] given
+as,
+dα(eiat)
+dtα
+= (ia)αeiat,
+(18)
+Let us denote the mean flow variables with capital letters and with a subscript
+‘0’. We assume that the mean flow is two-dimensional, quasiparallel with its
+variation entirely in the shear gradient direction. Then, the (non-dimensional)
+velocity can be written as follows,
+U0 =
+�
+(y − y2) + δy
+�
+ex,
+(19)
+where ex is the unit vector along the x-direction. Flow-instability studies of two
+specific forms of channel flows are considered in this article: plane Poiseuille
+flow (δ = 0) and the plane Couette flow (δ = 1.0). The other mean flow
+variables satisfying equation (17), including the mean pressure, P0, and the
+base state elastic stress tensor, A0 = [A0ij], is given by,
+P0 = −2x − 8We(1 − ν)
+�
+y − y2 + δy
+�
+,
+(20a)
+A011 = 0,
+(20b)
+A012 = A021 = (1 + δ − 2y),
+(20c)
+A022 = 2We (1 − 2y + δ)2 ,
+(20d)
+and whose linearized stability analysis is presented next.
+The viscoelastic version of the Squire’s theorem for plane parallel, classi-
+cal Oldroyd-B fluids [57] indicates that it is possible to restrict our stability
+analysis to the case when the disturbances are two-dimensional. Assuming an
+independent fate of each wavenumber, k (whose real part is chosen to be pos-
+itive) and frequency, ω, it is natural to consider disturbances in the form of a
+normal mode expansion, such that the total velocity, pressure and stress are
+expressed in terms of their mean values and perturbations amplitudes (denoted
+by ˚
+(·)), as follows,
+r = R0 + ϵ˚rei(kx−ωt)
+(21)
+where ϵ ≪ 1 and r = [v p A11 A12 A22]T , R0 = [U0 P0 A011 A012 A022]T and
+˚r = [X0y(1 − y) X1y(1 − y) X2 X3 X4 X5]T represent the total, the mean flow
+
+Springer Nature 2021 LATEX template
+Article Title
+9
+variables and the disturbance amplitudes, respectively. The disturbance ampli-
+tude, ˚r, is chosen such that it satisfies the no-slip condition on the channel
+walls. Substituting the solution form (21) in equations (17a-17c) and retaining
+the O(ϵ) terms to arrive at the linearized equation governing conservation of
+mass,
+X0
+�
+ik(y − y2)
+�
++ X1 [1 − 2y] = 0,
+(22)
+the linearized equation describing the conservation of momentum in the
+x−direction,
+X0
+�
+(y − y2)
+�
+Re(−iω)α + ikRe(y − y2 + δy) − 2ν(ik)2�
++
+2 ν] + X1
+�
+Re(1 − 2y + δ)(y − y2) − ν(1 − 2y)(ik)
+�
++ ik
+X2 − ik(1 − ν)X3 = 0,
+(23)
+and the one governing the conservation of momentum in the y−direction,
+X0 [−ikν (1 − 2y)] + X1
+�
+(y − y2) (Re(−iω)α + ikRe
+(y − y2 + δy) − ν(ik)2�
++ 4ν
+�
+− (1 − ν)ikX4 = 0.
+(24)
+The linearized equation for the elastic stress component A11,
+X0
+� −2
+We(y − y2)(ik)
+�
++ X1
+�
+−2ik (1 − 2y + δ) (y − y2)
+�
++
+X3
+�
+(−iω)α + ik(y − y2 + δy) +
+1
+We
+�
+= 0,
+(25)
+for the component A12 (or A21),
+X0
+�
+−ik (1 − 2y + δ) (y − y2) −
+1
+We (1 − 2y)
+�
++ X1
+�
+−2(y − y2) − 2ikWe(1 − 2y + δ)2(y − y2) − (1 − 2y + δ)
+(1 − 2y) −
+1
+We(y − y2)(ik)
+�
++ X3 [−(1 − 2y + δ)] + X4
+�
+(−iω)α + ik(y − y2 + δy) +
+1
+We
+�
+= 0,
+(26)
+and for the component A22,
+X0 [2 (1 − 2y + δ) (2y − 1)]+X1
+�
+−8We (1 − 2y + δ) (y−y2)
+−4We(1 − 2y + δ)2(1 − 2y) − 2(1 − 2y)
+We
+�
++ X4 [−2(1 − 2y
++δ)] + X5
+�
+(−iω)α + ik(y − y2 + δy) +
+1
+We
+�
+= 0.
+(27)
+
+Springer Nature 2021 LATEX template
+10
+Article Title
+Equations (22-27) may be written in a matrix-vector format as follows,
+�
+�����������
+ik(y−y2)
+1−2y
+0
+0
+0
+0
+M1
+M2
+ik −ik(1−ν)
+0
+0
+νik(2y−1)
+M3
+0
+0
+−ik(1−ν)
+0
+2ik
+W e(y2−y)
+2ik(1−2y+δ)(y2−y) 0
+M4
+0
+0
+M5
+M6
+0 −(1−2y+δ)
+M4
+0
+2(1−2y+δ)(2y−1)
+M7
+0
+0
+−2(1−2y+δ) M4
+�
+�����������
+�
+�����������
+X0
+X1
+X2
+X3
+X4
+X5
+�
+�����������
+=
+�
+�����������
+0
+0
+0
+0
+0
+0
+�
+�����������
+,
+(28)
+where the expressions Mi (i = 1, . . . 7) are listed in Section A. A nontrivial
+solution for the system (28), imposes a zero determinant condition on the
+coefficient matrix which leads to the dispersion relation, D(k, ω) = 0, given by,
+1
+Wek2M4 (2ik (−1 + ν) (1 + δ − 2y) (−1 + y) y (1 + (−2
++ikWe (1 + δ − 2y) (−1 + y)) y) + M 2
+4 We
+�
+ν (1 − 2y)2
+−M3 (−1 + y) y) + M4 (−1 + ν) We (M5 − 2M5y + ik
+M6 (−1 + y) y)) = 0.
+(29)
+2.3 Numerical method
+In the ensuing description, we denote real/imaginary components with sub-
+script r/i, respectively. The zeros of the dispersion relation (equation (29)) were
+explored within the complex k − ω plane inside the region ωr ∈ [−1700, 0.2],
+ωi ∈ [−5000, 100], kr ∈ [0, 5] and ki ∈ [−0.1, 0.2]. For a real wavenumber k,
+the procedure for finding the most unstable mode (which is the largest positive
+imaginary component of any root of the dispersion relation or the temporal
+growth rate, ωTemp, refer Section 4), consists of detecting the admissible saddle
+points (ω ∈ C, k ∈ R) satisfying the equations [58],
+D(k, ω) = 0,
+(30a)
+∂ωi
+∂k =
+∂D/∂k
+∂D/∂ωi = 0,
+(30b)
+and then (among all the possible roots of equation (30)) identifying those roots
+with the largest positive imaginary component of the frequency. Equation (30)
+is solved using a multivariate Newton-Raphson algorithm (refer Author’s
+previously published results [45, 55] for a detailed outline of this method).
+Next, in the spatiotemporal analysis, eigenpairs with complex wavenumbers
+and frequencies are permitted in the solution of equation (30). The necessary
+(but not sufficient) condition for the presence of absolute instability is the van-
+ishing characteristic of the group velocity of the flow, v g, at the saddle point
+in the k-plane or the branch point in the ω-plane (vg = ∂ω
+∂k = ( ∂D
+∂k )/( ∂D
+∂ω ) = 0,
+
+Springer Nature 2021 LATEX template
+Article Title
+11
+such that ω = D(k) satisfies the dispersion relation). But the group velocity
+is zero at every saddle point, in particular where the two k-branches meet,
+independent of whether the branches originate from the same half of the k-
+plane (i. e., when evanescent modes are detected) or not. To overcome this
+inadequacy, Briggs [59] devised the idea of analytic continuation in which the
+Laplace contour is deformed towards the ωr axis of the complex ω-plane, with
+the simultaneous adjustment of the Fourier contour in the k-plane to main-
+tain the separation of the k-branches; those which originate from the top half
+(the upstream modes with ki > 0) from those which originate from the bot-
+tom half of the k-plane (or the downstream modes). The deformation of the
+Fourier contour (while preserving causality) is inhibited, however, when the
+paths of the two k-branches originating from the opposite halves of the k-
+plane intersect each other, leading to the appearance of saddle points which
+are the pinch point, kpinch. The concurrent branch point appearance in the
+ω-plane is the cusp point, ωcusp (i. e., D(kpinch, ωcusp) = ∂D(kpinch,ωcusp)
+∂k
+= 0
+but
+∂2D(kpinch,ωcusp)
+∂k2
+̸= 0). Kupfer [60] employed a local mapping procedure
+to conceptualize the stability characteristics of this branch point. Near a
+‘reasonably close’ neighborhood of the pinch point, a local Taylor series expan-
+sion yields a dispersion relation that has a second-order algebraic form in
+the ω-plane (and which is a first-order saddle point in the k-plane), i. e.,
+(ω − ωcusp) ∼ (k − kpinch)2. This period-doubling characteristic of the map
+causes the ki-contours to ‘rotate’ around ωcusp, forming a cusp. In the ω-plane,
+we draw a ray parallel to the ωi-axis from the cusp point such that it intersects
+the image of the Fourier contour (or ki = 0 curve) and count the number of
+intersections (consequently, count the number of times both k-branches cross
+the kr-axis before forming a pinch point in the k-plane. If the ray drawn from
+the cusp point intersects the image of the Fourier contour in the ω-plane (or
+if either one or both the k-branches cross the kr-axis) even number of times,
+then the flow dynamics correspond to an evanescent mode. Otherwise, in the
+case of odd intersections, the observed cusp point is genuine, leading to either
+an absolutely unstable system (in the upper half of the ω-plane) or a convec-
+tively unstable system (in the lower half of the ω-plane); provided the system
+is temporally unstable.
+Under the assumption that dispersion relation is a complex analytic
+function satisfying Cauchy-Riemann relations, the expressions are chosen pref-
+erentially to numerically evaluate the derivatives in equation (30b). The
+numerical continuation of the temporal growth rate (Section 4) and the abso-
+lute growth rate curves (Section 5) were realized within the range Re ∈
+[10−6, 102], We ∈ [0, 103] and at two specific values of ν = 0.05 (the elastic
+stress-dominated case) and ν = 0.3 (the viscous stress-dominated case), using
+a discrete step-size of △Re = 10−6 and △We = 10−4, respectively. While the
+(non-dimensional) physical domain spans within the range, x ∈ (−∞, ∞); y ∈
+[0, 1], the temporal growth rate (ωTemp
+i
+) and the absolute growth rate (ωcusp
+i
+)
+of the perturbations are probed at four discrete, transverse spatial locations
+of the advancing interface: y = 0.2, 0.9, 0.7, 0.5. While the former two values
+
+Springer Nature 2021 LATEX template
+12
+Article Title
+of y are chosen qualitatively to probe the near-wall effects, the last value is
+selected to understand the development of the centerline instability.
+3 Model validation
+The model and the numerical method outlined in Section 2 is validated by
+reproducing the neutral stability curves for a plane Poiseuille of a classical
+Oldroyd-B fluid, as investigated by Atalik [61] (α = 1.0 or the red curves in
+figure 2, see figure 6 in [61]). The neutral stability curves for two fractional
+orders (α = 0.99 (blue curves) and α = 0.95 (green curves)) are also shown
+for comparison. The locus of neutrally stable points are found after selecting
+ωi = 0 in the dispersion relation (29) and solving for the unknowns (ωr, k), at
+fixed values of Reynolds and elasticity number (E = W e
+Re ).
+Two conclusions can be deduced from figure 2. First, notice that the min-
+imum value of the critical Reynolds number predicting a temporal instability
+increases, with increasing viscosity ratio, both for the classical case (a result
+identical to the one predicted by Atalik [61]) as well as the subdiffusive case.
+Second, observe that this minimum value of Re is significantly lower and
+appears at significantly larger values of E, for the subdiffusive fluid. These two
+observations indicate that the transition to instability are primarily driven by
+elasticity (rather than fluid inertia) for subdiffusive fluids. A more detailed
+outlook of the influence of elasticity is acquired by examining the temporal
+growth rates, described next.
+Fig. 2 Neutral stability curves at the centerline (y = 0) for plane Poiseulle viscoelastic flow
+of a classical fluid (red curves, source: figure 6 in [61]) versus subdiffusive fluid at α = 0.99
+(blue curves), α = 0.95 (green curves) and at viscosity ratio, ν = 0.01 (solid curves), ν = 0.9
+(dashed curves), projected onto the Re − E plane.
+
+2800
+α=1
+2750
+2100
+e
+2000
+130
+R
+1400
+0
+2
+4
+6
+×10-4
+700
+43
+0.01
+0.06
+0
+0
+0.02
+E
+0.04
+0.06Springer Nature 2021 LATEX template
+Article Title
+13
+4 Temporal stability analysis
+First, we explore the linear stability of the system (28) by exclusively assigning
+ω to be a complex number. In earlier studies on wall-bounded viscoelastic flows,
+elasticity (characterized by the parameter, We) was found to have a destabi-
+lizing effect (for example, see [29] and the references within). In this study,
+we partially extend some of these ideas for the subdiffusive, two-dimensional
+Poiseuille and Couette flows within a selected range of parameters, Re, We, ν
+and specially for the case of the Rouse chain melts and the Zimm chain
+solution, which corresponds to the fractional order derivatives, α = 1/2, 2/3,
+respectively. Figures 3 and 4 present the variation of the most unstable mode
+versus Re, and at fixed We and ν for viscoelastic Poiseuille and Couette flows,
+respectively.
+Observe that the elastic stress-dominated case (or ν = 0.05 case) is tem-
+porally more unstable (i. e., compare the maximum ‘y’ value on the ordinate
+axis of the figures on the left column versus those on the right column in
+figures 3 and 4). Also, observe especially for the Zimm’s case in Poiseuille
+flow, that not only the peak of the most unstable mode increases, but also
+the range of Reynolds number exhibiting temporal instability increases with
+increasing values of We (i. e., notice the dashed green, blue and the red curves
+in figure 3). Also, analogous with the traditional (or integer order) viscoelas-
+tic channel flows, we find that for intermediate values of Reynolds number
+(or 1 ≤ Re ≤ 55), elasticity is destabilizing (notice, from the dashed curves
+in figure insets in figures 3 and 4, that the most unstable mode is larger
+for larger values of We). These observations lead us to conclude that elas-
+ticity has a destabilizing impact, within the intermediate range of Re. This
+destabilization mechanism is the result of a complex interaction between the
+inertial forces (typically operative at larger Reynolds number) and the nor-
+mal stress anisotropy through elasticity (proportional to We) and can be
+explained via an energy formalism: the stretching of the polymers with increas-
+ing elasticity brings about a normal stress anisotropy, leading to an elastically
+loaded fluid, that is, when the polymers stretch, elastic energy is stored in the
+sheared fluid. This energy is transferred and released after the fluid element
+has been adverted to other regions where the shear-induced stretching forces
+are smaller [53]. However, for sufficiently larger values of Re (or Re > 55),
+we find the emergence of the temporally stable state. The appearance of the
+temporally stable state at high fluid inertia, is a hallmark of subdiffusive flows
+and the details of the same are elaborated in Section 5.
+Regarding the near-wall effects, notice that the Zimm’s case in Poiseuille
+flow is more unstable near wall (i. e., comparing the maximum ‘y’ value on
+the ordinate axis for y = 0.2, 0.9, figures 3a,b and 3g,h respectively) in com-
+parison with the corresponding instability on channel centerline (y = 0.5
+case, figures 3c,d). Further, within the intermediate range of Reynolds num-
+ber (i. e., 1 ≤ Re ≤ 55), elasticity is destabilizing near the walls (e. g., see the
+dashed curves in figure insets in figure 3a,b,g,h). All these near-wall effects can
+be understood via a mechanism similar to the one proposed by Rabaud [62]
+
+Springer Nature 2021 LATEX template
+14
+Article Title
+Fig. 3 Most unstable mode, ωTemp
+i
+for the Poiseuille flow case, vs. Reynolds number for
+the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at
+We = 15.0, 25.0, 35.0 (red, blue and green curves, respectively), viscosity ratios, ν = 0.05
+(left column) and ν = 0.3 (right column) and at transverse spatial locations: (a, b) y = 0.2,
+(c, d) y = 0.5, (e, f) y = 0.7 and (g, h) y = 0.9.
+for wall-bounded Newtonian flows: the boundary effects induce a local per-
+turbation on the advancing interface which (when coupled with elasticity)
+destabilizes the flow.
+Finally, we find that the order of the fractional derivative, α has a strong
+correlation with the temporal stability of the channel flows. For both types
+of flows, we deduce that the Zimm’s model is temporally more unstable than
+
+3.5
+(h)
+2.3
+Q=2/3
+Temp
+0.03
+3
+1.2
+-0.8
+0.6
+0
+-3.5 l0g1(Re) -1.25
+-6V = 0.05
+54
+(a)
+α=1/2
+α=1/2
+0.12
+36
+10.2
+0.015
+Temp
+-5.2
+-4.35
+1-4.1
+-3.4
+Q=2/3
+3
+×103
+13
+18
+3
+-0.6
+-0.15
+0
+-3.5 log1o(Re) -1.25
+-5.95
+1V = 0.3
+2.6
+(q)
+2
+Q=2/3
+0.03
+Temp
+3
+0
+-0.95
+0.21
+0.65
+0
+-3.5 l0g1o(Re)-1.25
+-5.953
+(c)
+α=2/3
+0.3
+α=1/2
+0.6
+2
+0.65
+Temp
+0.2
+-4.3
+-4.2
+I
+3
+0
+l0g10(Re) -0.5
+-4.35
+-2.5
+1.50.6
+(d)
+α=2/3
+0.035
+Q=1/2
+0.035
+0.4
+1.2
+Temp
+0.01
+T
+-4.32
+-4.3
+3
+0.2
+0
+log10(Re) -0.5
+-4.35
+-2.5
+1.54.5
+(e)
+3.3
+Temp
+3
+0
+log10 (Re)
+-4.35
+-2.3
+-0.25
+1.750.6
+α=2/3
+0.02
+f)
+Qα=2/3
+0.0065
+0.4
+0.1
+0.73
+0.0055
+Temp
+1.38
+1.47
+3
+0.2
+0
+-2.3
+log1. (Re)
+-4.35
+-0.25
+1.7575
+(g)
+Q=1/2
+1.5
+60
+Temp
+-4.7
+&
+Q=1/2
+0.15
+3
+20
+0
+-3.5
+-2.2
+-3.5 l0g1.(Re) -1.25
+-6Springer Nature 2021 LATEX template
+Article Title
+15
+the Rouse case. In a series of in silico studies, an investigation of the most
+unstable mode within the range, α ∈ [0.5 1.0], reveals: (a) the most unstable
+mode decreases with decreasing order of the fractional derivative, α, (b) the
+peak of the most unstable mode shifts to lower values of Re with decreasing
+values of α, and (c) in particular, the peak of the most unstable mode, for
+the Rouse model (i. e., the solid curves in figures 3, 4), precipitates towards
+the limit Re → 0. In other words, the transition pathway to flow turbulence
+in the Rouse polymer flows is characterized via elastic turbulence (appearing
+at vanishingly low values of Re and at moderate to high values of We) [63].
+To summarize, as α decreases, the nature of the transition pathway to flow
+turbulence changes from that of the elastoinertial turbulence (characterized
+by moderate values of Re and We) to elastic turbulence.
+We recapitulate the interplay of the inertial forces (characterized by the
+parameter Re), the elastic forces (represented by the parameter We) as well as
+the boundary effects and the order of the fractional derivative on the progres-
+sion of the temporal instability (exemplified by the most unstable mode) of
+the viscoelastic subdiffusive channel flows as follows: elasticity combined with
+reasonably large fluid inertia has a destabilizing impact on the evolving flow
+front. The finite boundary is shown to have a destabilizing influence. Finally,
+the order of the subdiffusive timescale (alternatively, the order of the fractional
+derivative) impacts the nature of the transition pathway to turbulence (if any).
+In the next section, we outline a deeper characterization of these instabilities
+via the spatiotemporal analysis.
+5 Spatiotemporal stability analysis
+Spatiotemporal analysis is typically relevant when one introduces an impulse
+excitation locally in a flow and observes how that disturbance evolves in
+time [58]. More significantly, we evaluate the absolute growth rate (or ωCusp
+i
+,
+details on computing these points are elaborated in Section 2.3) to identify the
+region of absolute instability, or the region indicating the topological recon-
+figuration and subsequent pinch-off of the advancing interface [64]. However,
+evanescent modes are also encountered in our analysis [55]. These modes do
+not merely depend on the sign of the absolute growth rate and have to be
+found via the sufficient conditions proposed by Briggs [60] (refer Section 2.3).
+Evanescent modes are briefly refered in the description of the phase diagrams
+(figures 7 and 8).
+Figure 5 represents the absolute growth rate curves versus Re for Poiseuille
+flows, at three fixed values of Weissenberg number, We = 15.0, 25.0, 35.0,
+and at ν = 0.05 (the elastic stress-dominated case) and ν = 0.3 (the vis-
+cous stress-dominated case). For the selected values of We and for the elastic
+stress-dominated case, we find that the Rouse model exhibits a transition from
+absolute instability towards temporal stability at a critical value of Reynolds
+number, Rec < 10−3, at y = 0.2 (figure 5a). This critical Reynolds number
+
+Springer Nature 2021 LATEX template
+16
+Article Title
+Fig. 4 Most unstable mode, ωTemp
+i
+for the Couette flow case, vs. Reynolds number for
+the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at
+We = 15.0, 25.0, 35.0 (red, blue and green curves, respectively), viscosity ratios, ν = 0.05
+(left column) and ν = 0.3 (right column) and at transverse spatial locations: (a, b) y = 0.2,
+(c, d) y = 0.5, (e, f) y = 0.7 and (g, h) y = 0.9.
+increases as one moves closer to the upper plate and along the transverse spa-
+tial location, y (i. e., compare the Rec values from the inset in figures 5a,c,e,g).
+For the viscous stress-dominated case, the Rouse model indicates the follow-
+ing transition with increasing values of Re: convective instability→ absolute
+instability→ temporal stability. Again, the critical value of Reynolds number
+at these transition points, increases as one progressively moves towards the
+
+v = 0.05
+80
+(a)
+Q=2/3
+0.02
+60
+Temp
+α=1/2
+0
+0.09
+-1.6
+-0.6
+T
+3
+20
+0
+-4.5
+-3.4
+0
+log1o (Re)
+-6
+-4
+-2
+0V = 0.3
+22
+(b)
+Q=2/3
+0.02
+16.5
+Q=1/2
+0.06
+0
+Temp
+-1.7
+-0.7
+3
+0
+-4.6
+-3.5
+5.5
+0
+log1. (Re)
+-6
+-4
+-2
+0108
+(c)
+Qα=2/3
+0.025
+81
+0
+Temp
+Q=1/2
+-1.1
+0.05
+3
+27
+0
+-3.7
+-2.9
+0
+log1o(Re)
+-6
+-4
+-2
+0(d)
+Q=2/3
+0.03
+0.02
+8.25
+0.01
+0
+-1.3
+-0.1
+Temp
+Q=1/2
+0.18
+3
+2.75
+0
+4.5
+3.5
+-6
+-4
+-2
+0100
+(e)
+Q=2/3
+0.01
+75
+0
+Temp
+α=1/2
+-0.4
+0.22
+3
+25
+0
+-3.8
+-2.4
+-3.5 log1o(Re) -1.25
+-64.5
+(f)
+Q=2/3
+0.01
+3.38
+0
+-0.4
+0.65
+Temp
+Q=1/2
+0.2
+3
+1.12
+0
+4.3
+-2.5
+0
+-3.5 log1o(Re) -1.25
+-616.2
+(g)
+Qα=2/3
+0.04
+12
+0
+0.43
+1.73
+Temp
+α=1/2
+0.1
+3
+0
+2.95
+-1.47
+0
+8 l0g1o(Re) -0.82
+-6
+-3.38
+1.730.9
+h
+α=2/3
+0.027
+0.68
+Temp
+0.43
+1.39
+3
+0.25
+log1.(Re) -0.75
+-6
+-3.3
+1.75Springer Nature 2021 LATEX template
+Article Title
+17
+upper plate (refer figures 5b,d,f,h). In contrast, the Zimm’s model highlights
+a direct transition from absolute instability towards temporal stability, in the
+range of low to moderate values of Re and for both the elastic as well as the
+viscous stress-dominated case. However, both of these model reveals temporal
+stability in the limit of vanishingly small Reynolds number (or in the strongly
+elastic limit).
+For Couette flows (figure 6), we find that the absolute growth rate curves
+display a transition from convective instability towards temporal stability ver-
+sus Re, such that the critical Reynolds number at the transition point increases
+with increasing transverse spatial coordinate, y, with an exception at y = 0.9),
+irrespective of the selected values of We, ν or α. At y = 0.9, a transition
+sequence in the order: temporal stability→ convective instability→ temporal
+stability (temporal stability→ convective instability→ absolute instability→
+temporal stability) appears for the Rouse (Zimm’s) model, with increasing val-
+ues of Re. We remark that while some observations listed above follow from
+the well-established mechanisms seen in classical (or integer order) viscoelas-
+tic flows, namely the lack of symmetry in the flow-instability transition across
+the centerline (due to the anisotropy of the elastic stresses) as well as the
+appearance of absolute/convective instabilities at intermediate values of Re
+(generated due to the instability via the polymer elasticity), other observa-
+tions are relatively novel, specifically the flow induced (temporal) stabilization
+at higher values of Re.
+Next, we classify the nature of these instabilities by computing the
+boundaries of the temporally stable regions (S), evanescent modes (E), the
+convectively unstable (C) and the absolutely unstable regions (A) within
+a selected range of the flow-elasticity-viscosity parameter space, i. e., Re ∈
+[10−6, 100], We ∈ [0, 103] and ν ∈ {0.05, 0.30}. While convective instability
+grows in amplitude as it is swept along by the flow, absolute instability occurs
+at fixed spatial locations, leading to surface transitions (or pinch-off) of the
+advancing interface [58]. The flow stability phase diagram for Poiseuille flows,
+projected onto the Re−We parameter space (figure 7) divulge the presence of
+absolutely unstable and convectively unstable region at low to moderate values
+of Re and We (Re ∈ [10−5, 10] and We ∈ [3, 350] for absolute instability, and
+Re ∈ [10−5, 10−2] and We ∈ [50, 150] for convective instability, respectively),
+as result of a complex tug-of-war between the inertial forces (proportional to
+Re) and the normal stress anisotropy through elasticity (proportional to We).
+Similarly, the flow stability phase diagram for Couette flows (figure 8) disclose
+convectively unstable region at low to moderate values of Re and moderately
+high values of We (Re ∈ [10−5, 55] and We ∈ [0, 900]) and absolutely unsta-
+ble region for moderate values of Re and We, only near the upper plate (i. e.,
+Re ∈ [0.1, 10] and We ∈ [0, 100] at y = 0.9, refer figure 8h). To summarize,
+the parameter regions susceptible to topological transitions (or the parame-
+ter space which indicate absolute instability) in subdiffusive channel flows, are
+those driven by moderate inertia coupled with moderate to high elasticity.
+
+Springer Nature 2021 LATEX template
+18
+Article Title
+Fig. 5 Absolute growth rate, ωCusp
+i
+for the Poiseuille flow case, vs. Reynolds number for
+the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at
+We = 15.0, 25.0, 35.0 (red, blue and green curves, respectively), viscosity ratios, ν = 0.05
+(left column) and ν = 0.3 (right column) and at transverse spatial locations: (a, b) y = 0.2,
+(c, d) y = 0.5, (e, f) y = 0.7 and (g, h) y = 0.9.
+A notably ‘abnormal’ feature in the phase diagrams (7, 8) is the presence
+of temporal stability at high inertia (i. e., Re ≥ 55). While the in silico studies
+of the classical Oldroyd-B channel flows indicate the appearance of temporal
+instability for Reynolds number as low as Re ∼ 50 [29], temporal stability at
+high fluid inertia for viscoelastic flows is only recognized in experimental real-
+izations (until now). For example, Riley [65] reported an elasticity induced flow
+
+v = 0.05
+0.1
+(a)
+α=1/2
+0.001
+0.075
+Cusp
+3
+X10-6
+α=2/3
+4
+3
+2.5
+0.025
+2
+-1.25
+0.2
+0
+-3.5 log1o(Re) -1.25
+-5.95
+1v = 0.3
+0.01
+(b)
+-0.003
+α=2/3
+Cusp
+2.5
+3
+X10-5,
+Qα=1/2
+2
+-1
+-0.5
+0
+-0.025
+0
+-3.85
+-2.78
+-0.04
+-3.5 log10(Re)
+-1.25
+-5.95X 10-4
+9
+(c)
+6.8
+α=1/2
+2
+Cusp
+α=2/3
+8
+3
+0.15
+-3
+-1.2
+1
+2.2
+4
+-1.75
+0
+log1.(Re) -0.5
+-4.35
+-2.5
+1.50.1
+(d)
+X10-6
+Q=2/3
+-2
+X10-7
+α=1/2 V
+2
+Cusp
+0
+3.5
+-2.5
+1.5
+-10
+-5
+-2.6
+-1.8
+7
+log1. (Re)
+-4.35
+-2.5
+-0.5
+1.5X10
+(e)
+α=1/2
+X10~6
+3
+4
+Cusp
+α=2/3
+3.14
+0
+3
+2
+3.06
+0
+1.75
+0
+log1o(Re) -0.25
+-4.35
+-2.3
+1.750.1
+(f)
+×105
+X0=2/3
+-2
+X10-7
+Q=1/2 V
+2
+5
+Cusp
+0
+3
+0.2
+5
+-2.3
+1.75
+-2.5
+-1.7
+-5
+-7
+log10(Re)
+-0.25
+-4.35
+-2.3
+1.750.18
+(g)
+X10-7
+Q=2/3
+10
+0.135
+X10-4
+α=1/2
+8.6
+Cusp
+-0.5
+0.7
+3
+0
+4
+-2.1
+0.045
+0
+-3.5 log1o(Re) -1.25
+-60.007
+(h)
+X10-6
+Q=2/3 
+5
+Cusp
+3
+ α=1/2
+D
+0.5
+-2
+0.6
+-0.014
+-010001
+-3.5
+-2
+-0.021
+-3.5 log1o(Re) -1.25
+-6Springer Nature 2021 LATEX template
+Article Title
+19
+Fig. 6 Absolute growth rate, ωCusp
+i
+for the Couette flow case, vs. Reynolds number for
+the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at
+We = 15.0, 25.0, 35.0 (red, blue and green curves, respectively), viscosity ratios, ν = 0.05
+(left column) and ν = 0.3 (right column) and at transverse spatial locations: (a, b) y = 0.2,
+(c, d) y = 0.5, (e, f) y = 0.7 and (g, h) y = 0.9.
+stabilization of viscoelastic fluids coated over complaint surfaces at a fairly
+high Reynolds number (Re ∼ 4000). In a separate study involving ethanol gel
+fuels, elastic stabilization at a high shear rate was attributed due to an abnor-
+mally high second normal stress difference [66]. Viscoelastic flow stabilization
+at higher values of Re, in tapered microchannels, was explained due to the
+presence of wall effects [67]. In another in vitro study, a biofilm deacidification
+
+v = 0.05
+0
+(a)
+Q=2/3
+-325
+Cusp
+α=1/2
+-0.5
+3
+-975
+-70
+-4.7
+-3.3
+-1300
+log1o (Re)
+-6
+-4
+-2
+0V = 0.3
+0
+(b)
+Qα=2/3
+0
+-65
+Cusp
+-0.5
+α=1/2
+-2.9
+-0.65
+3
+-195
+-20
+-4.5
+-3.5
+-260
+log10 (Re)
+-2
+-6
+-4
+0(c)
+Qα=2/3
+0
+-875
+Cusp
+-0.7
+α=1/2
+2
+0
+0
+-2625
+-100
+-4.5
+-2.8
+-3500
+log10 (Re)
+-6
+.4
+-2
+0(p)
+Q=2/3
+-85
+Cusp
+-0.5
+α=1/2
+-2.5
+0
+3
+-255
+-15
+-3.9
+3
+-340
+log10 (Re)
+-6
+4
+-2
+0(e)
+Q=2/3 V
+0
+-1250
+Cusp
+-0.1
+α=1/2
+-0.9
+0.9
+0
+-3750
+-50
+-3.5
+-2.3
+-5000
+3.5 log1(Re) -1.25
+-60
+(f)
+Q=2/3 V
+0
+-90
+Cusp
+-0.1
+0.75
+α=1/2
+3
+-270
+-12
+-3.5
+-2.6
+-360
+-3.5 l0g1o(Re) -1.25
+-60.005
+(g)
+Q=2/3 V
+0.005
+-975
+0
+Cusp
+-0.015
+α=1/2
+-0.7
+1.7
+3
+-2925
+-10
+-2.5
+-1.5
+-3900
+-3.3 log10(Re)
+-6
+-0.75
+1.750.004
+(y)
+/ Qα=2/3
+0.004
+0
+-100
+Cusp
+-0.015
+-0.8
+Vα=1/2
+3
+-300
+-15
+-3.2
+-2.1
+-400
+-3.3 log1o(Re)
+-6
+)-0.75
+1.75Springer Nature 2021 LATEX template
+20
+Article Title
+Fig. 7 Viscoelastic subdiffusive Poiseuille flow stability phase diagram in the Re − We
+parametric space for the Rouse model (solid curves) and for the Zimm’s model (dashed
+curves), evaluated at transverse spatial locations: (a, b) y = 0.2, (c, d) y = 0.5, (e, f) y = 0.7
+and (g, h) y = 0.9 and at fixed values of viscosity ratios, ν = 0.05 (left column) and ν = 0.3
+(right column).
+created a non-homogeneous environment for molecular diffusion, leading to a
+‘subdiffusive effect’ with hindered flow rates [68]. These in vitro studies not
+only corroborate our numerical outcome, especially establishing the emergence
+of temporally stable region at high inertia, but also highlight the potential
+of fractional calculus in effectively capturing the flow-instability transition in
+subdiffusive flows.
+
+V= 0.05
+3
+(a)
+s
+2
+A
+A
+-
+A, E
+1
+0
+-6
+-4
+log,.(Re)
+0
+2V = 0.3
+3
+(q)
+S
+2
+A, E
+A
+-
+C, E
+-
+A, E
+1
+0
+-6
+-4
+log,.(Re)
+0
+23
+(c)
+S
+A
+A
+2
+A, E
+-
+A, E
+!
+-
+-
+1
+0
+-6
+-4
+log1.(Re)
+0
+23
+(d)
+S
+A, E
+2
+-
+C, E
+A
+A, E
+1
+-
+0
+-6
+-4
+log,.(Re)
+0
+23
+(e)
+S
+2
+A, E
+A, E
+1
+0
+-6
+-4
+log,.(Re)
+0
+23
+(f)
+S
+2
+A, E
+A
+C, E
+A, E
+1
+0
+-6
+-4
+log,.(Re)
+0
+23
+(g)
+S
+2
+A,E
+A, E
+1
+-
+0
+-6
+-4
+log,.(Re)
+0
+23
+(q)
+A
+A, E
+-
+A, E
+1
+S
+0
+-6
+-4
+log,.(Re)
+0
+2Springer Nature 2021 LATEX template
+Article Title
+21
+Fig. 8 Viscoelastic subdiffusive Couette flow stability phase diagram in the Re − We para-
+metric space for the Rouse model (solid curves) and for the Zimm’s model (dashed curves),
+evaluated at transverse spatial locations: (a, b) y = 0.2, (c, d) y = 0.5, (e, f) y = 0.7 and (g,
+h) y = 0.9 and at fixed values of viscosity ratios, ν = 0.05 (left column) and ν = 0.3 (right
+column).
+6 Concluding remarks
+This investigation addresses the temporal and the spatiotemporal linear sta-
+bility analyses of viscoelastic, subdiffusive, plane Poiseuille and Couette flows
+in the limit of low to moderate Reynolds number and moderate to high Weis-
+senberg number. Section 2 presented the viscoelastic, subdiffusive channel flow
+
+V = 0.05
+3
+(a)
+S
+2
+C
+1
+1
+-
+0
+-4
+log, (Re)
+-6
+0
+2V = 0.3
+3
+(b)
+2
+S
+1
+C
+-
+C
+-
+1
+-
+1
+-
+-
+1
+-
+n
+-6
+-4
+log,.(Re)
+0
+23
+(c)
+S
+2
+C
+1
+-
+0
+-6
+-4
+log,.(Re)
+0
+23
+(d)
+S
+2
+-
+-
+-
+1
+-
+-
+-
+-
+1
+-
+I
+-
+-
+0
+-6
+-4
+log,.(Re)
+0
+23
+(e)
+S
+2
+.(We),
+-
+1
+-
+C
+-
+-
+S
+-
+-
+-
+-6
+-4
+log,.(Re)
+0
+23
+(f)
+S
+2
+C.
+E
+1
+C
+-
+C
+-
+-
+-
+-
+-6
+-4
+log,.(Re)
+0
+2(g)
+S
+-
+-
+1
+F
+-
+C
+1
+-
+-
+A, E
+1
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+0
+-6
+-4
+log,.(Re)
+0
+23
+(h)
+-
+S
+-
+-
+S
+C, E
+c
+2
+-
+1
+-.
+c
+-
+-
+1
+-
+-
+1
+-
+-
+-
+-
+-
+-
+-
+1
+-
+-
+1
+-
+0
+-6
+-4
+log,.(Re)
+0
+2Springer Nature 2021 LATEX template
+22
+Article Title
+model, the elements of linear stability analysis as well as the numerical method
+needed to solve the resulting dispersion relation. Section 3 validated the model
+for the classical planar Poiseuille flow obeying the Oldroyd-B stress consti-
+tutive equation [61]. The temporal stability analysis in Section 4 indicates
+that with decreasing order of the fractional derivative: (a) the most unstable
+mode decreases, (b) the peak of the most unstable mode shifts to lower val-
+ues of Re, and (c) in particular, the peak of the most unstable mode, for the
+Rouse model converges towards the limit Re → 0. The spatiotemporal phase
+diagram in Section 5 indicates an abnormal region of temporal stability at
+high fluid inertia coupled with high elasticity, due to the presence of a non-
+homogeneous environment with hindered flow. Although we have shown how
+the exponent in the subdiffusive power-law scaling of the mean square dis-
+placement of the tracer particle in the microscale is related to the fractional
+order of the corresponding non-linear stress constitutive equations in the con-
+tinuum, the arguments presented herein are ‘phenomenological’ in nature. A
+more rigorous effort involving the micro-to-macro upscaling via kinetic theory
+arguments [53], is currently underway.
+Acknowledgements
+T. C., D. B. and S.S. acknowledges the financial support of the Grant CSIR
+09 /1117 (0012) /2020-EMR-I and DST ECR/2017/000632, respectively.
+Appendix A
+Viscoelastic dispersion relation
+The expressions, Mi, utilized in the viscoelastic dispersion relation outlined in
+Section 2.2, is given as,
+M1 = (y−y2)
+�
+Re(−iω)α+ikRe(y−y2+δy)−2ν(ik)2�
++2ν,
+M2 = Re(1 − 2y + δ)(y − y2) − ν(1 − 2y)(ik),
+M3 = (y−y2)
+�
+Re(−iω)α+ikRe(y−y2+δy)−ν(ik)2�
++4ν,
+M4 = (−iω)α + ik(y − y2 + δy) +
+1
+We,
+M5 = −ik (1 − 2y + δ) (y − y2) −
+1
+We (1 − 2y) ,
+M6 = −2(y − y2) − 2ikWe(1 − 2y + δ)2(y − y2) − (1−
+2y + δ)(1 − 2y) −
+1
+We(y − y2)(ik),
+M7 = −8We (1 − 2y + δ) (y − y2) − 4We(1 − 2y + δ)2
+(1 − 2y) − 2(1 − 2y)
+We
+.
+(A1)
+
+Springer Nature 2021 LATEX template
+Article Title
+23
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+
diff --git a/jNA0T4oBgHgl3EQfIv8h/content/tmp_files/load_file.txt b/jNA0T4oBgHgl3EQfIv8h/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='Springer Nature 2021 LATEX template  Spatiotemporal linear stability of viscoelastic  subdiffusive channel flows: a fractional  calculus framework  Tanisha Chauhan1†, Diksha Bansal1† and Sarthok Sircar1*  1*Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='  Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' E-mail(s): sarthok@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Contributing authors: tanishai@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' dikshab@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Abstract  The temporal and spatiotemporal linear stability analyses of viscoelastic,  subdiffusive, plane Poiseuille and Couette flows obeying the Fractional  Upper Convected Maxwell (FUCM) equation in the limit of low to  moderate Reynolds number (Re) and Weissenberg number (W e), is  reported to identify the regions of topological transition of the advanc-  ing flow interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In particular, we demonstrate how the exponent in  the subdiffusive power-law scaling (tα, with 0<α ≤ 1) of the mean  square displacement of the tracer particle, in the microscale [Mason and  Weitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 74, 1250-1253 (1995)] is related to the fractional  order of the derivative, α, of the corresponding non-linear stress con-  stitutive equation in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The stability studies are limited  to two exponents: monomer diffusion in Rouse chain melts, α = 1/2,  and in Zimm chain solutions, α =  2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The temporal stability anal-  ysis indicates that with decreasing order of the fractional derivative:  (a) the most unstable mode decreases, (b) the peak of the most unsta-  ble mode shifts to lower values of Re, and (c) the peak of the most  unstable mode, for the Rouse model precipitates towards the limit  Re → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The Briggs idea of analytic continuation is deployed to clas-  sify regions of temporal stability, absolute and convective instabilities  and evanescent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The spatiotemporal phase diagram indicates an  abnormal region of temporal stability at high fluid inertia, revealing the  presence of a non-homogeneous environment with hindered flow, thus  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='02078v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='flu-dyn]  5 Jan 2023  Springer Nature 2021 LATEX template  2  Article Title  highlighting the potential of the model to effectively capture certain  experimentally observed, flow-instability transition in subdiffusive flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Keywords: Spatiotemporal stability, anomalous diffusion, non-Markovian  processes, Caputo integral, Upper Convected Maxwell model  1 Introduction  The subject of anomalous diffusion has received tremendous attention over  the last half-century, ranging from physics [1–3], biology [4] to quantitative  finance [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Some of the most significant and profoundly published exper-  imental results are better rationalized within the viscoelastic subdiffusive  approach in random environments such as the cytosol and the plasma mem-  brane of biological cells [6], crowded complex fluids and polymer solutions [7],  dense colloidal suspensions [8] and single-file diffusion in colloidal systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  The observed (anomalous) subdiffusion often combines features of ergodic  fractional Brownian motion (reflecting viscoelasticity) and the nonergodic  jumplike non-Markovian diffusional processes (reflecting disorder) [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The  subdiffusive object is considered primarily as being elastic and structurally  robust, although it requires ‘fluidity’ and flexibility besides its elasticity for  a proper functioning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', consider a viscoelastic nanoscaled polymer drop  armed with a rigid backbone that can take on different macroscopic con-  formations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In this article, we demonstrate how the dynamics of the  subdiffusive fluids at microscale (often represented via a generalized Langevin  equation (GLE) at the molecular level, using a dissipative memory kernel)  is ‘upscaled’ to a fractional viscoelastic stress constitutive equation at the  continuum level [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Fractional calculus serves as a powerful tool for modeling the constitutive  relations in the linear [14] as well as nonlinear viscoelasticity theory [8] and  to explain certain paradoxical experimental findings [2, 3] (one such experi-  mentally abnormal feature captured by our model, namely the occurrence of  temporal stability at high fluid inertia, is mentioned in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Gemant  highlighted the relaxation curves for some viscoelastic fluids by employing a  fractional viscoelastic model for the first time [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Scott-Blair developed a  new constitutive law (known as the ‘fractional Newton model’) to describe the  experimental outcome of Gemant on stress relaxation [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Caputo intro-  duced the fractional Voigt model to simulate the dissipation in seismology [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Bagley and Torvik [19] showed that there exists a quantitative connection  between the fractional viscoelastic model (at the macroscopic level) and the  molecular theory of Rouse’s polymer chain melts [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' With the development  of the fractional viscoelastic model, the flow of the fractional viscoelastic fluid  has been extensively investigated [21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Tan and Xu [21] used Laplace trans-  forms to obtain the analytical solution for velocity and stress of the plane  surface flow of a fractional Maxwell fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Qi and Xu [22] studied the plane  Springer Nature 2021 LATEX template  Article Title  3  Poiseuille flow and plane Couette flow of a generalized Oldroyd-B fluid with  fractional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' [24] found the analytical solution for velocity  and stress of magnetohydrodynamic flow of a generalized Oldroyd-B fluid gen-  erated with an accelerating plate, with fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Recently, Zhao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' [25] considered the natural convection heat transfer of viscoelastic fluid  with fractional the Maxwell model over a vertical plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' More recent appli-  cations of the fractional viscoelastic flows include the study of the stability  of coastal morphodynamics and seafloor topology [26], regulation of the tis-  sue morphodynamics [27] and capturing spatiotemporal disorder in anomalous  transport of viscous flows [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  The detailed exploration of the existing literature serves as a clear motiva-  tion for the work reported here, which is to provide a comprehensive picture of  the stability of the two-dimensional, viscoelastic, subdiffusive, fully developed,  Poiseuille and Couette flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The present work significantly differs from the  existing studies in the sense that we analyse the linear stability of viscoelastic,  subdiffusive, channel flows through a combined temporal and spatiotemporal  stability analysis (rather than only a temporal stability analysis of the classi-  cal (or integer order) viscoelastic channel flows [29]) and the aim is to address  the following intriguing questions: What is the critical flow/polymer relaxation  condition for the onset of instability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' And more crucially, what is the linear  spatiotemporal, time asymptotic response of the flow at the critical value of  the material parameters, leading to the topological transition of the advancing  flow interface of the subdiffusive channel flows?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  While the molecular theory of polymer dynamics has already established  the correspondence between subdiffusive dynamics and linear viscoelastic  relaxation of polymer melts and solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' [30, 31], we ‘upscale’ these ideas at  the continuum mechanical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In particular, we highlight how the exponent  in the subdiffusive power-law timescale, tα [32], is related to the fractional  order, α, of the corresponding non-linear stress constitutive equations in the  continuum (refer Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The temporal and spatiotemporal stability of  two specific cases of monomer diffusion in Rouse chain melts (α = 1/2) [20],  and in Zimm chain solution (α = 2/3) [33] are reported in detail (Section 4,  Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The Rouse model predicts that the viscoelastic properties of the  polymer chain can be described by a generalized Maxwell model, where the  elasticity is governed by a single relaxation time, which is independent of the  number of Maxwell elements (or the so-called ‘submolecules’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In contrast,  the Zimm’s model predicts the (‘shear rate and polymer concentration inde-  pendent’) viscosity of the polymer solution by calculating the hydrodynamic  interaction of flexible polymers (an idea which was originally proposed by  Kirkwood [34]) by approximating the chains using a bead-spring setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  4  Article Title  2 Problem formulation: Mathematical model,  linear stability analysis and numerical method  In this study, the linearized stability analyses of the fully developed, planar  Poiseuille and Couette flows inside an infinitely long channel of width H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=',  0 ≤ y ≤ H such that x/y ≫ 1, where x and y are the flow and the shear  gradient direction, respectively) is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1 Mathematical model  We consider a viscoelastic fluid subject to a shear deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Then, an  infinitesimal elastic stress, τxy at time t arising from a small strain increment  dγ at an earlier time t′ is given by,  dτxy = G(t − t′)˙γ(t′)dt′,  (1)  where the relaxation modulus, G(t), represents the influence of the dissipa-  tive processes of the surrounding concentrated fluid medium [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Assuming  linearity, the Boltzmann superposition principle may be utilized to construct  the elastic stress at time t by summing up all of the infinitesimal contributions  over the entire flow history, which is extended into the infinite past [36],  τxy =  � t  −∞  G(t − t′)˙γ(t′)dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (2)  In their seminal work on passive micro-rheology, Mason and co-workers [31]  have identified an approximate relation between the time-dependent memory  kernel describing the viscous damping of the tracer particle at micro-scale (and  which obeys the GLE, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', see equation (15) in [32]), ζ(t), and the stress  relaxation modulus, G(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=',  ζ(t) = 6πaG(t),  (3)  where a is the radius of the tracer particle (assumed spherical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In the regime of  linear viscoelasticity, one of the most commonly used three-parameter family  of memory kernel is the generalized Rouse kernel for an equally weighted sum  of negatively decaying exponential functions [37],  ζ(t) = 1  N  N−1  �  k=0  e−( k  N )  1  α ( t  λ0 ),  (4)  for a number of kernels determining the length of the subdiffusive phase, N,  relaxation time, λ0, and a subdiffusive exponent, α ∈ (0 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Since the polymeric  liquids of our interest [38–46] show subdiffusive behavior on all length scales,  Springer Nature 2021 LATEX template  Article Title  5  we consider the case when N → ∞ in the prony series (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' For t > λ0, this  limiting behavior leads to the relation  ζ(t) =  ˜G  Γ(1 − α)  � t  λ0  �−α  ,  (5)  where Γ(x) is the complete gamma function and ˜G = Γ(1 + α)Γ(1 − α), is a  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In equation (5), we have used the fact that the Riemann sum on an  infinite interval,  lim  N→∞  1  N  N−1  �  k=0  f  � k  N  �  =  � ∞  0  f(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (6)  Using equations (2, 3, 5), one arrives at,  τxy =  Gλα  0  Γ(1 − α)  � t  −∞  dt′(t − t′)−α dγ(t′)  dt′  ,  (7)  where the constant, G =  ˜  G  6πa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' We remark that G is no longer a constant  (typically G = G(t)) when the concentration effects, such as the bond and  entanglement effects, are considered [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The right-hand side of equation (7)  represents a fractional integral corresponding to the Caputo formalism [47, 48],  −∞D−β  t  f(t) =  1  Γ(β)  � t  −∞  dt′  (t − t′)1−β  df(t′)  dt′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (8)  Utilizing equations (7, 8), we arrive at the basic equation governing stress-  strain relation in linear viscoelastic subdiffusive media,  τxy = Gλα  0  dα−1  dtα−1  dγ(t)  dt  = Gλα  0  dαγ  dtα ,  (9)  including the limiting cases of a purely elastic solid (α→0 or a Hookean spring)  and a purely viscous fluid (α = 1 or a dashpot) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Through combinations of  springs and dashpots, one arrives at standard linear viscoelastic models, includ-  ing the Maxwell, Kelvin-Voigt, Zener, Poynting-Thomson and Burgers’ model  and others [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The problem is that the corresponding differential equations  have a relatively restricted class of solutions, which are too limited to provide  an adequate description for the class of complex fluids discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  To overcome this shortcoming, one can relate the stress and strain through  the fractional equation (9), which allows a smooth interpolation between a  purely elastic behavior and a purely viscous pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In the present analysis, we  have selected the Fractional Upper Convected Maxwell equation (FUCM) to  describe the nonlinear viscoelastic response of the subdiffusive media, derived  next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Figure 1a depicts the standard Maxwell model in which a spring and a  dashpot are connected in series [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' We generalize this model by replacing  Springer Nature 2021 LATEX template  6  Article Title  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 1 (a) The Maxwell element and (b) its fractional generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  these elements with their corresponding fractional elements: (αi, Gi, λi), i =  1, 2 (figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Because of the sequential construction, the stress, τ, is the  same for both elements and their respective stress-strain relations are  γi = G−1  i  λ−αi  i  d−αiτxy  dt−αi ,  i = 1, 2,  (10)  where both expressions follow from equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Due to the construction of  the generalized Maxwell model, we have γ(t) = γ1(t) + γ2(t), from which it  follows,  τxy + G1λα1  1  G2λα2  2  dα1−α2τxy  dtα1−α2  = G1λα1  1  dα1γ  dtα1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (11)  Equation (11) can be simplified by setting λ = (G1λα1  1 /G2λα2  2 )1/(α1−α2) and  E = G1(λ1/λ)α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Without loss of generality, we assume α2 = 0 and α = α1(>  0), and arrive at  τxy + λα dατxy  dtα  = ηp  dαγ  dtα ,  (12)  where the constant, ηp = Eλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' We can extend equation (12) to three dimen-  sions by replacing the elastic stress, τxy, with the stress tensor, τ, and the  derivative, dαγ  dtα , with the rate of strain tensor, D = (∇v + (∇v)T ) (where the  operator ∇(·) =  ∂  ∂x(·)), to arrive at  τ + λα dατ  dtα = ηpD,  (13)  using the definition of fractional velocity, v = dαx  dtα [49], which has a dimension  of  H  T α (refer Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2 for the discussion on non-dimensionalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Frac-  tional velocities are defined as limits of the difference quotients of a fractional  (α2, G2,2)  (α1, G1, A1)  no  (n)  (b)Springer Nature 2021 LATEX template  Article Title  7  power and they generalize the notion of a local derivative [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' These deriva-  tives are frequently used, for example, to model instantaneous interactions in  Langevin dynamics [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Equation (13) is the rheological constitutive equation of the fractional  Maxwell model describing the linear viscoelastic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The simplest way  to combine rheological nonlinearity is to replace the (fractional) mate-  rial time derivative in equation (13) with the (fractional) frame invariant,  upper-convected time derivative [52, 53], which leads us to FUCM, as follows,  τ + λα▽τ = ηpD,  (14)  where the fractional upper-convected time derivative of the tensor τ is defined  as,  ▽τ = ∂ατ  ∂tα + v · ∇τ − (∇v)T τ − τ∇v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (15)  The fractional time derivative, ∂α  ∂tα , in equation (14, 15) is based on the Caputo  definition (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  The continuity and the momentum equations for an incompressible,  subdiffusive flow (consistent with the stress constitutive relation (14)) are,  ∇ · v = 0,  ρ  �∂αv  ∂tα + v · ∇v  �  = −∇p+ηs∇ · D + ∇ · τ,  (16)  where ρ is the density and p is the isotropic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Equations (14,16) repre-  sent the equations of motion describing the flow-instability of the subdiffusive  viscoelastic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  As a result of the dissipative processes, viscoelastic materials have mem-  ory, that is, their actual mechanical response is modulated by the past [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  The fractional derivative operators account for the complete history to obtain  the derivative at an instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Unlike the classical Maxwell model [45, 55] which  accounts for only the elastic (or stored) part of the deformation work, the  fractional Maxwell model accounts for both forms (stored and dissipated) of  energy at any time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Although Mckinley pointed out that the fractional  Maxwell model generally cannot capture polymer shear-thinning [56], the frac-  tional version provides a better fit of the relaxation and creep behavior for a  significantly large class of viscoelastic materials using fewer parameters than  the classical version [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2 Linear stability analysis  Using the following scales for non-dimensionalizing the governing equations:  the height of the channel H for length, the timescale T corresponding to max-  imum base flow velocity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' T = (H/U0)1/α) for time and ρU2  0 for pressure  and stresses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' we characterize equations (14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' rephrased as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  ∇ · v = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (17a)  Springer Nature 2021 LATEX template  8  Article Title  Re  �∂αv  ∂tα + v · ∇v  �  = −∇p + ν∇ · D + (1 − ν)∇ · A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (17b)  ∂αA  ∂tα + v · ∇A − (∇v)T A − A∇v = D − A  We  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (17c)  using the dimensionless groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Re = ρU0H  η0  (Reynolds number),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' We = λαU0  H  (Weissenberg number) and where ηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' ηp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' η0(= ηs + ηp) and ν(= ηs/η0) are  the solvent viscosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' the polymeric contribution to the shear viscosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' the  total viscosity and the viscous contribution to the total viscosity of the fluid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In equation (17), the elastic stress is represented as τ = (1−ν)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  The current analysis deploys fractional derivative of exponentials [47, 48] given  as,  dα(eiat)  dtα  = (ia)αeiat,  (18)  Let us denote the mean flow variables with capital letters and with a subscript  ‘0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' We assume that the mean flow is two-dimensional, quasiparallel with its  variation entirely in the shear gradient direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Then, the (non-dimensional)  velocity can be written as follows,  U0 =  �  (y − y2) + δy  �  ex,  (19)  where ex is the unit vector along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Flow-instability studies of two  specific forms of channel flows are considered in this article: plane Poiseuille  flow (δ = 0) and the plane Couette flow (δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The other mean flow  variables satisfying equation (17), including the mean pressure, P0, and the  base state elastic stress tensor, A0 = [A0ij], is given by,  P0 = −2x − 8We(1 − ν)  �  y − y2 + δy  �  ,  (20a)  A011 = 0,  (20b)  A012 = A021 = (1 + δ − 2y),  (20c)  A022 = 2We (1 − 2y + δ)2 ,  (20d)  and whose linearized stability analysis is presented next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  The viscoelastic version of the Squire’s theorem for plane parallel, classi-  cal Oldroyd-B fluids [57] indicates that it is possible to restrict our stability  analysis to the case when the disturbances are two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Assuming an  independent fate of each wavenumber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' k (whose real part is chosen to be pos-  itive) and frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' it is natural to consider disturbances in the form of a  normal mode expansion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' such that the total velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' pressure and stress are  expressed in terms of their mean values and perturbations amplitudes (denoted  by ˚  (·)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  r = R0 + ϵ˚rei(kx−ωt)  (21)  where ϵ ≪ 1 and r = [v p A11 A12 A22]T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' R0 = [U0 P0 A011 A012 A022]T and  ˚r = [X0y(1 − y) X1y(1 − y) X2 X3 X4 X5]T represent the total,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' the mean flow  Springer Nature 2021 LATEX template  Article Title  9  variables and the disturbance amplitudes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The disturbance ampli-  tude, ˚r, is chosen such that it satisfies the no-slip condition on the channel  walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Substituting the solution form (21) in equations (17a-17c) and retaining  the O(ϵ) terms to arrive at the linearized equation governing conservation of  mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  X0  �  ik(y − y2)  �  + X1 [1 − 2y] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (22)  the linearized equation describing the conservation of momentum in the  x−direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  X0  �  (y − y2)  �  Re(−iω)α + ikRe(y − y2 + δy) − 2ν(ik)2�  +  2 ν] + X1  �  Re(1 − 2y + δ)(y − y2) − ν(1 − 2y)(ik)  �  + ik  X2 − ik(1 − ν)X3 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (23)  and the one governing the conservation of momentum in the y−direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  X0 [−ikν (1 − 2y)] + X1  �  (y − y2) (Re(−iω)α + ikRe  (y − y2 + δy) − ν(ik)2�  + 4ν  �  − (1 − ν)ikX4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (24)  The linearized equation for the elastic stress component A11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  X0  � −2  We(y − y2)(ik)  �  + X1  �  −2ik (1 − 2y + δ) (y − y2)  �  +  X3  �  (−iω)α + ik(y − y2 + δy) +  1  We  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (25)  for the component A12 (or A21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  X0  �  −ik (1 − 2y + δ) (y − y2) −  1  We (1 − 2y)  �  + X1  �  −2(y − y2) − 2ikWe(1 − 2y + δ)2(y − y2) − (1 − 2y + δ)  (1 − 2y) −  1  We(y − y2)(ik)  �  + X3 [−(1 − 2y + δ)] + X4  �  (−iω)α + ik(y − y2 + δy) +  1  We  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (26)  and for the component A22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  X0 [2 (1 − 2y + δ) (2y − 1)]+X1  �  −8We (1 − 2y + δ) (y−y2)  −4We(1 − 2y + δ)2(1 − 2y) − 2(1 − 2y)  We  �  + X4 [−2(1 − 2y  +δ)] + X5  �  (−iω)α + ik(y − y2 + δy) +  1  We  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (27)  Springer Nature 2021 LATEX template  10  Article Title  Equations (22-27) may be written in a matrix-vector format as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  �  �����������  ik(y−y2)  1−2y  0  0  0  0  M1  M2  ik −ik(1−ν)  0  0  νik(2y−1)  M3  0  0  −ik(1−ν)  0  2ik  W e(y2−y)  2ik(1−2y+δ)(y2−y) 0  M4  0  0  M5  M6  0 −(1−2y+δ)  M4  0  2(1−2y+δ)(2y−1)  M7  0  0  −2(1−2y+δ) M4  �  �����������  �  �����������  X0  X1  X2  X3  X4  X5  �  �����������  =  �  �����������  0  0  0  0  0  0  �  �����������  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (28)  where the expressions Mi (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 7) are listed in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' A nontrivial  solution for the system (28), imposes a zero determinant condition on the  coefficient matrix which leads to the dispersion relation, D(k, ω) = 0, given by,  1  Wek2M4 (2ik (−1 + ν) (1 + δ − 2y) (−1 + y) y (1 + (−2  +ikWe (1 + δ − 2y) (−1 + y)) y) + M 2  4 We  �  ν (1 − 2y)2  −M3 (−1 + y) y) + M4 (−1 + ν) We (M5 − 2M5y + ik  M6 (−1 + y) y)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (29)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3 Numerical method  In the ensuing description, we denote real/imaginary components with sub-  script r/i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The zeros of the dispersion relation (equation (29)) were  explored within the complex k − ω plane inside the region ωr ∈ [−1700, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2],  ωi ∈ [−5000, 100], kr ∈ [0, 5] and ki ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' For a real wavenumber k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  the procedure for finding the most unstable mode (which is the largest positive  imaginary component of any root of the dispersion relation or the temporal  growth rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' ωTemp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' refer Section 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' consists of detecting the admissible saddle  points (ω ∈ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' k ∈ R) satisfying the equations [58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  D(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' ω) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (30a)  ∂ωi  ∂k =  ∂D/∂k  ∂D/∂ωi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (30b)  and then (among all the possible roots of equation (30)) identifying those roots  with the largest positive imaginary component of the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Equation (30)  is solved using a multivariate Newton-Raphson algorithm (refer Author’s  previously published results [45, 55] for a detailed outline of this method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Next, in the spatiotemporal analysis, eigenpairs with complex wavenumbers  and frequencies are permitted in the solution of equation (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The necessary  (but not sufficient) condition for the presence of absolute instability is the van-  ishing characteristic of the group velocity of the flow, v g, at the saddle point  in the k-plane or the branch point in the ω-plane (vg = ∂ω  ∂k = ( ∂D  ∂k )/( ∂D  ∂ω ) = 0,  Springer Nature 2021 LATEX template  Article Title  11  such that ω = D(k) satisfies the dispersion relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' But the group velocity  is zero at every saddle point, in particular where the two k-branches meet,  independent of whether the branches originate from the same half of the k-  plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', when evanescent modes are detected) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' To overcome this  inadequacy, Briggs [59] devised the idea of analytic continuation in which the  Laplace contour is deformed towards the ωr axis of the complex ω-plane, with  the simultaneous adjustment of the Fourier contour in the k-plane to main-  tain the separation of the k-branches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' those which originate from the top half  (the upstream modes with ki > 0) from those which originate from the bot-  tom half of the k-plane (or the downstream modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The deformation of the  Fourier contour (while preserving causality) is inhibited, however, when the  paths of the two k-branches originating from the opposite halves of the k-  plane intersect each other, leading to the appearance of saddle points which  are the pinch point, kpinch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The concurrent branch point appearance in the  ω-plane is the cusp point, ωcusp (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', D(kpinch, ωcusp) = ∂D(kpinch,ωcusp)  ∂k  = 0  but  ∂2D(kpinch,ωcusp)  ∂k2  ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Kupfer [60] employed a local mapping procedure  to conceptualize the stability characteristics of this branch point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Near a  ‘reasonably close’ neighborhood of the pinch point, a local Taylor series expan-  sion yields a dispersion relation that has a second-order algebraic form in  the ω-plane (and which is a first-order saddle point in the k-plane), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=',  (ω − ωcusp) ∼ (k − kpinch)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' This period-doubling characteristic of the map  causes the ki-contours to ‘rotate’ around ωcusp, forming a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In the ω-plane,  we draw a ray parallel to the ωi-axis from the cusp point such that it intersects  the image of the Fourier contour (or ki = 0 curve) and count the number of  intersections (consequently, count the number of times both k-branches cross  the kr-axis before forming a pinch point in the k-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' If the ray drawn from  the cusp point intersects the image of the Fourier contour in the ω-plane (or  if either one or both the k-branches cross the kr-axis) even number of times,  then the flow dynamics correspond to an evanescent mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Otherwise, in the  case of odd intersections, the observed cusp point is genuine, leading to either  an absolutely unstable system (in the upper half of the ω-plane) or a convec-  tively unstable system (in the lower half of the ω-plane);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' provided the system  is temporally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Under the assumption that dispersion relation is a complex analytic  function satisfying Cauchy-Riemann relations, the expressions are chosen pref-  erentially to numerically evaluate the derivatives in equation (30b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The  numerical continuation of the temporal growth rate (Section 4) and the abso-  lute growth rate curves (Section 5) were realized within the range Re ∈  [10−6, 102], We ∈ [0, 103] and at two specific values of ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05 (the elastic  stress-dominated case) and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3 (the viscous stress-dominated case), using  a discrete step-size of △Re = 10−6 and △We = 10−4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' While the  (non-dimensional) physical domain spans within the range, x ∈ (−∞, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' y ∈  [0, 1], the temporal growth rate (ωTemp  i  ) and the absolute growth rate (ωcusp  i  )  of the perturbations are probed at four discrete, transverse spatial locations  of the advancing interface: y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' While the former two values  Springer Nature 2021 LATEX template  12  Article Title  of y are chosen qualitatively to probe the near-wall effects, the last value is  selected to understand the development of the centerline instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  3 Model validation  The model and the numerical method outlined in Section 2 is validated by  reproducing the neutral stability curves for a plane Poiseuille of a classical  Oldroyd-B fluid, as investigated by Atalik [61] (α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0 or the red curves in  figure 2, see figure 6 in [61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The neutral stability curves for two fractional  orders (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='99 (blue curves) and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='95 (green curves)) are also shown  for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The locus of neutrally stable points are found after selecting  ωi = 0 in the dispersion relation (29) and solving for the unknowns (ωr, k), at  fixed values of Reynolds and elasticity number (E = W e  Re ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Two conclusions can be deduced from figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' First, notice that the min-  imum value of the critical Reynolds number predicting a temporal instability  increases, with increasing viscosity ratio, both for the classical case (a result  identical to the one predicted by Atalik [61]) as well as the subdiffusive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Second, observe that this minimum value of Re is significantly lower and  appears at significantly larger values of E, for the subdiffusive fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' These two  observations indicate that the transition to instability are primarily driven by  elasticity (rather than fluid inertia) for subdiffusive fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' A more detailed  outlook of the influence of elasticity is acquired by examining the temporal  growth rates, described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 2 Neutral stability curves at the centerline (y = 0) for plane Poiseulle viscoelastic flow  of a classical fluid (red curves, source: figure 6 in [61]) versus subdiffusive fluid at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='99  (blue curves), α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='95 (green curves) and at viscosity ratio, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='01 (solid curves), ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9  (dashed curves), projected onto the Re − E plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  2800  α=1  2750  2100  e  2000  130  R  1400  0  2  4  6  ×10-4  700  43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='06  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='02  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='06Springer Nature 2021 LATEX template  Article Title  13  4 Temporal stability analysis  First, we explore the linear stability of the system (28) by exclusively assigning  ω to be a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In earlier studies on wall-bounded viscoelastic flows,  elasticity (characterized by the parameter, We) was found to have a destabi-  lizing effect (for example, see [29] and the references within).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In this study,  we partially extend some of these ideas for the subdiffusive, two-dimensional  Poiseuille and Couette flows within a selected range of parameters, Re, We, ν  and specially for the case of the Rouse chain melts and the Zimm chain  solution, which corresponds to the fractional order derivatives, α = 1/2, 2/3,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Figures 3 and 4 present the variation of the most unstable mode  versus Re, and at fixed We and ν for viscoelastic Poiseuille and Couette flows,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Observe that the elastic stress-dominated case (or ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05 case) is tem-  porally more unstable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', compare the maximum ‘y’ value on the ordinate  axis of the figures on the left column versus those on the right column in  figures 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Also, observe especially for the Zimm’s case in Poiseuille  flow, that not only the peak of the most unstable mode increases, but also  the range of Reynolds number exhibiting temporal instability increases with  increasing values of We (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', notice the dashed green, blue and the red curves  in figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Also, analogous with the traditional (or integer order) viscoelas-  tic channel flows, we find that for intermediate values of Reynolds number  (or 1 ≤ Re ≤ 55), elasticity is destabilizing (notice, from the dashed curves  in figure insets in figures 3 and 4, that the most unstable mode is larger  for larger values of We).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' These observations lead us to conclude that elas-  ticity has a destabilizing impact, within the intermediate range of Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' This  destabilization mechanism is the result of a complex interaction between the  inertial forces (typically operative at larger Reynolds number) and the nor-  mal stress anisotropy through elasticity (proportional to We) and can be  explained via an energy formalism: the stretching of the polymers with increas-  ing elasticity brings about a normal stress anisotropy, leading to an elastically  loaded fluid, that is, when the polymers stretch, elastic energy is stored in the  sheared fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' This energy is transferred and released after the fluid element  has been adverted to other regions where the shear-induced stretching forces  are smaller [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' However, for sufficiently larger values of Re (or Re > 55),  we find the emergence of the temporally stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The appearance of the  temporally stable state at high fluid inertia, is a hallmark of subdiffusive flows  and the details of the same are elaborated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Regarding the near-wall effects, notice that the Zimm’s case in Poiseuille  flow is more unstable near wall (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', comparing the maximum ‘y’ value on  the ordinate axis for y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9, figures 3a,b and 3g,h respectively) in com-  parison with the corresponding instability on channel centerline (y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  case, figures 3c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Further, within the intermediate range of Reynolds num-  ber (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', 1 ≤ Re ≤ 55), elasticity is destabilizing near the walls (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', see the  dashed curves in figure insets in figure 3a,b,g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' All these near-wall effects can  be understood via a mechanism similar to the one proposed by Rabaud [62]  Springer Nature 2021 LATEX template  14  Article Title  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 3 Most unstable mode, ωTemp  i  for the Poiseuille flow case, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Reynolds number for  the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at  We = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0 (red, blue and green curves, respectively), viscosity ratios, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  (left column) and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3 (right column) and at transverse spatial locations: (a, b) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2,  (c, d) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5, (e, f) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7 and (g, h) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  for wall-bounded Newtonian flows: the boundary effects induce a local per-  turbation on the advancing interface which (when coupled with elasticity)  destabilizes the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Finally, we find that the order of the fractional derivative, α has a strong  correlation with the temporal stability of the channel flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' For both types  of flows, we deduce that the Zimm’s model is temporally more unstable than  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  (h)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  Q=2/3  Temp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='03  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 l0g1(Re) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  6V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  54  (a)  α=1/2  α=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='12  36  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='015  Temp  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='4  Q=2/3  3  ×103  13  18  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='15  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 log1o(Re) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='95  1V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  (q)  2  Q=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='03  Temp  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='65  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 l0g1o(Re)-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='953  (c)  α=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  α=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='65  Temp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  I  3  0  l0g10(Re) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  (d)  α=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='035  Q=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  Temp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='01  T  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='32  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  0  log10(Re) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  (e)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  Temp  3  0  log10 (Re)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  α=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='02  f)  Qα=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0055  Temp  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='47  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='7575  (g)  Q=1/2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  60  Temp  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7  &  Q=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='15  3  20  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 l0g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  6Springer Nature 2021 LATEX template  Article Title  15  the Rouse case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In a series of in silico studies, an investigation of the most  unstable mode within the range, α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0], reveals: (a) the most unstable  mode decreases with decreasing order of the fractional derivative, α, (b) the  peak of the most unstable mode shifts to lower values of Re with decreasing  values of α, and (c) in particular, the peak of the most unstable mode, for  the Rouse model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', the solid curves in figures 3, 4), precipitates towards  the limit Re → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In other words, the transition pathway to flow turbulence  in the Rouse polymer flows is characterized via elastic turbulence (appearing  at vanishingly low values of Re and at moderate to high values of We) [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  To summarize, as α decreases, the nature of the transition pathway to flow  turbulence changes from that of the elastoinertial turbulence (characterized  by moderate values of Re and We) to elastic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  We recapitulate the interplay of the inertial forces (characterized by the  parameter Re), the elastic forces (represented by the parameter We) as well as  the boundary effects and the order of the fractional derivative on the progres-  sion of the temporal instability (exemplified by the most unstable mode) of  the viscoelastic subdiffusive channel flows as follows: elasticity combined with  reasonably large fluid inertia has a destabilizing impact on the evolving flow  front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The finite boundary is shown to have a destabilizing influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Finally,  the order of the subdiffusive timescale (alternatively, the order of the fractional  derivative) impacts the nature of the transition pathway to turbulence (if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  In the next section, we outline a deeper characterization of these instabilities  via the spatiotemporal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  5 Spatiotemporal stability analysis  Spatiotemporal analysis is typically relevant when one introduces an impulse  excitation locally in a flow and observes how that disturbance evolves in  time [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' More significantly, we evaluate the absolute growth rate (or ωCusp  i  ,  details on computing these points are elaborated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3) to identify the  region of absolute instability, or the region indicating the topological recon-  figuration and subsequent pinch-off of the advancing interface [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' However,  evanescent modes are also encountered in our analysis [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' These modes do  not merely depend on the sign of the absolute growth rate and have to be  found via the sufficient conditions proposed by Briggs [60] (refer Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Evanescent modes are briefly refered in the description of the phase diagrams  (figures 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Figure 5 represents the absolute growth rate curves versus Re for Poiseuille  flows, at three fixed values of Weissenberg number, We = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0,  and at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05 (the elastic stress-dominated case) and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3 (the vis-  cous stress-dominated case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' For the selected values of We and for the elastic  stress-dominated case, we find that the Rouse model exhibits a transition from  absolute instability towards temporal stability at a critical value of Reynolds  number, Rec < 10−3, at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2 (figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' This critical Reynolds number  Springer Nature 2021 LATEX template  16  Article Title  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 4 Most unstable mode, ωTemp  i  for the Couette flow case, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Reynolds number for  the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at  We = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0 (red, blue and green curves, respectively), viscosity ratios, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  (left column) and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3 (right column) and at transverse spatial locations: (a, b) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2,  (c, d) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5, (e, f) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7 and (g, h) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  increases as one moves closer to the upper plate and along the transverse spa-  tial location, y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', compare the Rec values from the inset in figures 5a,c,e,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  For the viscous stress-dominated case, the Rouse model indicates the follow-  ing transition with increasing values of Re: convective instability→ absolute  instability→ temporal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Again, the critical value of Reynolds number  at these transition points, increases as one progressively moves towards the  v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  80  (a)  Q=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='02  60  Temp  α=1/2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  T  3  20  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='4  0  log1o (Re)  6  4  2  0V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  22  (b)  Q=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='02  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  Q=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='06  0  Temp  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7  3  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  0  log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  6  4  2  0108  (c)  Qα=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='025  81  0  Temp  Q=1/2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  3  27  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9  0  log1o(Re)  6  4  2  0(d)  Q=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='02  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='1  Temp  Q=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='18  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='75  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  6  4  2  0100  (e)  Q=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='01  75  0  Temp  α=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='22  3  25  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 log1o(Re) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  (f)  Q=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='01  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='38  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='65  Temp  Q=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='12  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='5 log1o(Re) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  (g)  Qα=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='04  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='73  Temp  α=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1  3  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='47  0  8 l0g1o(Re) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='82  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9  h  α=2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='027  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='68  Temp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='25  log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='75  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='75Springer Nature 2021 LATEX template  Article Title  17  upper plate (refer figures 5b,d,f,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In contrast, the Zimm’s model highlights  a direct transition from absolute instability towards temporal stability, in the  range of low to moderate values of Re and for both the elastic as well as the  viscous stress-dominated case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' However, both of these model reveals temporal  stability in the limit of vanishingly small Reynolds number (or in the strongly  elastic limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  For Couette flows (figure 6), we find that the absolute growth rate curves  display a transition from convective instability towards temporal stability ver-  sus Re, such that the critical Reynolds number at the transition point increases  with increasing transverse spatial coordinate, y, with an exception at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9),  irrespective of the selected values of We, ν or α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' At y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9, a transition  sequence in the order: temporal stability→ convective instability→ temporal  stability (temporal stability→ convective instability→ absolute instability→  temporal stability) appears for the Rouse (Zimm’s) model, with increasing val-  ues of Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' We remark that while some observations listed above follow from  the well-established mechanisms seen in classical (or integer order) viscoelas-  tic flows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' namely the lack of symmetry in the flow-instability transition across  the centerline (due to the anisotropy of the elastic stresses) as well as the  appearance of absolute/convective instabilities at intermediate values of Re  (generated due to the instability via the polymer elasticity),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' other observa-  tions are relatively novel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' specifically the flow induced (temporal) stabilization  at higher values of Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Next, we classify the nature of these instabilities by computing the  boundaries of the temporally stable regions (S), evanescent modes (E), the  convectively unstable (C) and the absolutely unstable regions (A) within  a selected range of the flow-elasticity-viscosity parameter space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', Re ∈  [10−6, 100], We ∈ [0, 103] and ν ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' While convective instability  grows in amplitude as it is swept along by the flow, absolute instability occurs  at fixed spatial locations, leading to surface transitions (or pinch-off) of the  advancing interface [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The flow stability phase diagram for Poiseuille flows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  projected onto the Re−We parameter space (figure 7) divulge the presence of  absolutely unstable and convectively unstable region at low to moderate values  of Re and We (Re ∈ [10−5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 10] and We ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 350] for absolute instability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' and  Re ∈ [10−5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 10−2] and We ∈ [50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 150] for convective instability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' respectively),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  as result of a complex tug-of-war between the inertial forces (proportional to  Re) and the normal stress anisotropy through elasticity (proportional to We).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Similarly, the flow stability phase diagram for Couette flows (figure 8) disclose  convectively unstable region at low to moderate values of Re and moderately  high values of We (Re ∈ [10−5, 55] and We ∈ [0, 900]) and absolutely unsta-  ble region for moderate values of Re and We, only near the upper plate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=',  Re ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1, 10] and We ∈ [0, 100] at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9, refer figure 8h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' To summarize,  the parameter regions susceptible to topological transitions (or the parame-  ter space which indicate absolute instability) in subdiffusive channel flows, are  those driven by moderate inertia coupled with moderate to high elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  18  Article Title  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 5 Absolute growth rate, ωCusp  i  for the Poiseuille flow case, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Reynolds number for  the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at  We = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='0 (red, blue and green curves, respectively), viscosity ratios, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  (left column) and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3 (right column) and at transverse spatial locations: (a, b) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2,  (c, d) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5, (e, f) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='7 and (g, h) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  A notably ‘abnormal’ feature in the phase diagrams (7, 8) is the presence  of temporal stability at high inertia (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', Re ≥ 55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' While the in silico studies  of the classical Oldroyd-B channel flows indicate the appearance of temporal  instability for Reynolds number as low as Re ∼ 50 [29], temporal stability at  high fluid inertia for viscoelastic flows is only recognized in experimental real-  izations (until now).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' For example, Riley [65] reported an elasticity induced flow  v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1  (a)  α=1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='075  Cusp  3  X10-6  α=2/3  4  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='025  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 log1o(Re) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='95  1v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='01  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='003  α=2/3  Cusp  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  3  X10-5,  Qα=1/2  2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='025  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='85  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='04  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5 log10(Re)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='95X 10-4  9  (c)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='8  α=1/2  2  Cusp  α=2/3  8  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='15  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='75  0  log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1  (d)  X10-6  Q=2/3  2  X10-7  α=1/2 V  2  Cusp  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='5  10  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='8  7  log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='5X10  (e)  α=1/2  X10~6  3  4  Cusp  α=2/3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='14  0  3  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='75  0  log1o(Re) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='1  (f)  ×105  X0=2/3  2  X10-7  Q=1/2 V  2  5  Cusp  0  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='7  5  7  log10(Re)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='18  (g)  X10-7  Q=2/3  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content=' 6 Absolute growth rate, ωCusp  i  for the Couette flow case, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Reynolds number for  the Rouse model (solid curves) and for the Zimm’s model (dashed curves), evaluated at  We = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  stabilization of viscoelastic fluids coated over complaint surfaces at a fairly  high Reynolds number (Re ∼ 4000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' In a separate study involving ethanol gel  fuels, elastic stabilization at a high shear rate was attributed due to an abnor-  mally high second normal stress difference [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='004  0  100  Cusp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content=' 7 Viscoelastic subdiffusive Poiseuille flow stability phase diagram in the Re − We  parametric space for the Rouse model (solid curves) and for the Zimm’s model (dashed  curves), evaluated at transverse spatial locations: (a, b) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='3  (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  created a non-homogeneous environment for molecular diffusion, leading to a  ‘subdiffusive effect’ with hindered flow rates [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' These in vitro studies not  only corroborate our numerical outcome, especially establishing the emergence  of temporally stable region at high inertia, but also highlight the potential  of fractional calculus in effectively capturing the flow-instability transition in  subdiffusive flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='05  3  (a)  s  2  A  A A, E  1  0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  2V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  3  (q)  S  2  A, E  A C, E A, E  1  0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content=' (Re)  0  23  (d)  S  A, E  2 C, E  A  A, E  1 0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content=' (Re)  0  23  (g)  S  2  A,E  A, E  1 0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content=' 8 Viscoelastic subdiffusive Couette flow stability phase diagram in the Re − We para-  metric space for the Rouse model (solid curves) and for the Zimm’s model (dashed curves),  evaluated at transverse spatial locations: (a, b) y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content='9 and at fixed values of viscosity ratios, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05 (left column) and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3 (right  column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  6 Concluding remarks  This investigation addresses the temporal and the spatiotemporal linear sta-  bility analyses of viscoelastic, subdiffusive, plane Poiseuille and Couette flows  in the limit of low to moderate Reynolds number and moderate to high Weis-  senberg number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Section 2 presented the viscoelastic, subdiffusive channel flow  V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='05  3  (a)  S  2  C  1  1 0  4  log, (Re)  6  0  2V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='3  3  (b)  2  S  1  C C 1 1 1 n  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  23  (c)  S  2  C  1 0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  23  (d)  S  2 1 1 I 0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  23  (e)  S  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (We), 1 C S 6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  23  (f)  S  2  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  E  1  C C 6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  2(g)  S 1  F C  1 A, E  1 0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  23  (h) S S  C, E  c  2 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  c 1 1 1 1 0  6  4  log,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' (Re)  0  2Springer Nature 2021 LATEX template  22  Article Title  model, the elements of linear stability analysis as well as the numerical method  needed to solve the resulting dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Section 3 validated the model  for the classical planar Poiseuille flow obeying the Oldroyd-B stress consti-  tutive equation [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The temporal stability analysis in Section 4 indicates  that with decreasing order of the fractional derivative: (a) the most unstable  mode decreases, (b) the peak of the most unstable mode shifts to lower val-  ues of Re, and (c) in particular, the peak of the most unstable mode, for the  Rouse model converges towards the limit Re → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' The spatiotemporal phase  diagram in Section 5 indicates an abnormal region of temporal stability at  high fluid inertia coupled with high elasticity, due to the presence of a non-  homogeneous environment with hindered flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Although we have shown how  the exponent in the subdiffusive power-law scaling of the mean square dis-  placement of the tracer particle in the microscale is related to the fractional  order of the corresponding non-linear stress constitutive equations in the con-  tinuum, the arguments presented herein are ‘phenomenological’ in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' A  more rigorous effort involving the micro-to-macro upscaling via kinetic theory  arguments [53], is currently underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Acknowledgements  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' acknowledges the financial support of the Grant CSIR  09 /1117 (0012) /2020-EMR-I and DST ECR/2017/000632, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  Appendix A  Viscoelastic dispersion relation  The expressions, Mi, utilized in the viscoelastic dispersion relation outlined in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='2, is given as,  M1 = (y−y2)  �  Re(−iω)α+ikRe(y−y2+δy)−2ν(ik)2�  +2ν,  M2 = Re(1 − 2y + δ)(y − y2) − ν(1 − 2y)(ik),  M3 = (y−y2)  �  Re(−iω)α+ikRe(y−y2+δy)−ν(ik)2�  +4ν,  M4 = (−iω)α + ik(y − y2 + δy) +  1  We,  M5 = −ik (1 − 2y + δ) (y − y2) −  1  We (1 − 2y) ,  M6 = −2(y − y2) − 2ikWe(1 − 2y + δ)2(y − y2) − (1−  2y + δ)(1 − 2y) −  1  We(y − y2)(ik),  M7 = −8We (1 − 2y + δ) (y − y2) − 4We(1 − 2y + δ)2  (1 − 2y) − 2(1 − 2y)  We  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content='  (A1)  Springer Nature 2021 LATEX template  Article Title  23  References  [1] Goychuk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' Electrochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
+page_content=' 5(23), 3645–  3653 (2018)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf'}
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+MNRAS 000, 1–13 (2022)
+Preprint 11 January 2023
+Compiled using MNRAS LATEX style file v3.0
+AGN in post-mergers from the Ultraviolet Near Infrared Optical Northern
+Survey
+Robert W. Bickley,1★ Sara L. Ellison,1 David R. Patton,2 Scott Wilkinson1
+1Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8P 1A1, Canada
+2Department of Physics and Astronomy, Trent University, 1600 West Bank Drive, Peterborough, ON K9L 0G2, Canada
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The kinematic disturbances associated with major galaxy mergers are known to produce gas inflows, which in turn may trigger
+accretion onto the supermassive black holes (SMBH) of the participant galaxies. While this effect has been studied in galaxy
+pairs, the frequency of active galactic nuclei (AGN) in fully coalesced post-merger systems is poorly constrained due to the
+limited size or impurity of extant post-merger samples. Previously, we combined convolutional neural network (CNN) predictions
+with visual classifications to identify a highly pure sample of 699 post-mergers in deep 𝑟-band imaging. In the work presented
+here, we quantify the frequency of AGN in this sample using three metrics: optical emission lines, mid-infrared (mid-IR) colour,
+and radio detection of low-excitation radio galaxies (LERGs). We also compare the frequency of AGN in post-mergers to that in
+a sample of spectroscopically identified galaxy pairs. We find that AGN identified by narrow-line optical emission and mid-IR
+colour have an increased incidence rate in post-mergers, with excesses of ~4 over mass- and redshift-matched controls. The
+optical and mid-IR AGN excesses in post-mergers exceed the values found for galaxy pairs, indicating that AGN activity in
+mergers peaks after coalescence. Conversely, we recover no significant excess of LERGs in post-mergers or pairs. Finally, we
+find that the [OIII] luminosity (a proxy for SMBH accretion rate) in post-mergers that host an optical AGN is ~0.3 dex higher
+on average than in non-interacting galaxies with an optical AGN, suggesting that mergers generate higher accretion rates than
+secular triggering mechanisms.
+Key words: galaxies: evolution – galaxies: interactions – galaxies: peculiar – methods: statistical – techniques: image processing
+1 INTRODUCTION
+Galaxy mergers are unique within the framework of hierarchical
+assembly in that they simultaneously transform the kinematics, mor-
+phologies, and intrinsic properties of the participant galaxies (White
+& Rees 1978; Lacey & Cole 1993; Boylan-Kolchin et al. 2008; Jiang
+et al. 2008). Simulations (e.g. Toomre & Toomre 1972; Conselice
+2006; Lotz et al. 2008) reproduce the observed signatures — stellar
+shells, streams, and tails — of interacting and post-merger galax-
+ies (Darg et al. 2010; Kartaltepe et al. 2015; Simmons et al. 2017).
+Numerous simulations are also in qualitative agreement about the
+chemical evolution and new star formation experienced by galaxies
+after they experience a disruptive, gas-rich merger (e.g. Toomre &
+Toomre 1972; Springel et al. 2005; Di Matteo et al. 2008; Moreno
+et al. 2019; Patton et al. 2020; Hani et al. 2020). Cold gas that might
+have been in a stable orbit in one or both progenitors can be dis-
+rupted by mergers, funneled towards the centres of the participant
+galaxies, and may be responsible for central starbursts suggested in
+observations (Ellison et al. 2008, 2013; Nikolic et al. 2004; Scud-
+der et al. 2012; Scott & Kaviraj 2014; Knapen 2015; Thorp et al.
+2019). The cold gas that is responsible for central starbursts may
+also accrete onto a galaxy’s super-massive black hole (SMBH) and
+★ E-mail: rbickley@uvic.ca
+produce the observational signatures of an active galactic nucleus
+(AGN; e.g. Satyapal et al. 2014; Lackner et al. 2014; Weston et al.
+2016; Goulding et al. 2018; Ellison et al. 2011, 2019). Together,
+increased star formation and energetic feedback from the AGN may
+enrich the circum-galactic medium through ejective means (Johnson
+et al. 2015; Hani et al. 2018). The stellar and gas-phase kinematics
+of merger progenitors are also disrupted in the process (Lynden-Bell
+1967; Toomre 1977; Negroponte & White 1983; Hernquist 1992;
+Naab & Burkert 2003; Robertson et al. 2006; Jesseit et al. 2009;
+Berg et al. 2014; Clauwens et al. 2018; Hani et al. 2018), and star
+formation may truncate rapidly due to either gas ejection or heating
+after the epoch of intense star formation and AGN feedback is com-
+plete (Sanders et al. 1988; Hopkins et al. 2006; Yesuf et al. 2014;
+Quai et al. 2021). Indeed, Ellison et al. (2022) recently identified an
+observational link between the post-merger phase and the signatures
+of rapid quenching using the same post-merger sample studied in
+Bickley et al. (2021) and this paper.
+While a merger-AGN connection is therefore well established in
+pre-coalescence galaxy pairs, the precise role of post-coalescence
+mergers in switching on AGN and feeding them is not well con-
+strained. Until recently, merger samples in the literature have been
+either too small to perform precise statistics, or dubious in purity.
+Without a large and highly pure merger sample, the quantitative role
+of mergers in SMBH evolution cannot be studied effectively.
+© 2022 The Authors
+arXiv:2301.03681v1  [astro-ph.GA]  9 Jan 2023
+
+2
+R. W. Bickley et al.
+Any effort to study the observed signatures of merger-induced
+phenomena across the entire merger sequence requires a post-merger
+sample that is both pure (containing as high a fraction as possible
+of genuine mergers) and representative. Merger identification is rel-
+atively straightforward in spectroscopic galaxy pairs, which can be
+identified by their visual appearances (Kampczyk et al. 2007; Bundy
+et al. 2005; Brinchmann et al. 1998; Darg et al. 2010) or statistically,
+by grouping galaxies together in angular position and line-of-sight
+radial velocity in order to mitigate potential contamination by false
+pairs1. Thanks to the large number of spectroscopically-identified
+galaxy pairs in redshift surveys, the statistical influence of the pair
+phase has already been explored in great detail (e.g., Patton et al.
+2000; Barton et al. 2000; Lin et al. 2004; De Propris et al. 2005; Lin
+et al. 2008).
+Small samples (e.g. Ellison et al. 2013), and detailed spatially re-
+solved case studies of individual post-mergers (e.g. Thorp et al. 2019;
+Barrera-Ballesteros et al. 2015; Pan et al. 2019) offer a provisional
+understanding of post-coalescence galaxies. While these results have
+hinted at changes in star formation (e.g. Ellison et al. 2013, Elli-
+son et al. 2020), chemical evolution (e.g. Bustamante et al. 2020),
+and intense SMBH activity (e.g. Carpineti et al. 2012) in the post-
+merger epoch, fully coalesced galaxies are much more difficult to
+identify since they are no longer spectroscopically distinct from their
+companion(s). Consequently, the specific quantitative contribution
+of coalescence in producing these phenomena (especially SMBH
+triggering and accretion) is still being evaluated in simulations (e.g.
+Sivasankaran et al. 2022) and observations (e.g. Ellison et al. 2019;
+Gao et al. 2020).
+Because the characteristic features of the post-merger phase are
+relatively faint, morphological merger identification methods require
+imaging of adequate depth and resolution (as demonstrated by Bot-
+trell et al. 2019a, Huertas-Company et al. 2019, Ćiprijanović et al.
+2021). The Canada France Imaging Survey (CFIS), part of the Ul-
+traviolet Near Infrared Optical Northern Survey (UNIONS) collab-
+oration, offers a useful combination of imaging quality and volume,
+with ~0.7 arcsecond seeing, and 𝑟-band imaging that will eventually
+cover 5,000 square degrees of the sky. The survey’s 5-𝜎 point-source
+depth (24.85 mag in the 𝑟-band for the MegaCam wide-field optical
+imager) is sufficient to capture the low-surface brightness features
+necessary for merger identification in bright, low-redshift galaxies
+(e.g. Sola et al. 2022). Estimates of the low-𝑧 merger rate suggest
+that the UNIONS footprint will include thousands of post-mergers
+(Lacey & Cole 1993; Lotz et al. 2011; Bluck et al. 2012; Casteels
+et al. 2014; Rodriguez-Gomez et al. 2015; Martin et al. 2018).
+Convolutional neural networks (CNNs) have already been suc-
+cessfully applied to a number of tasks in astronomy (e.g. Huertas-
+Company et al. 2015; Domínguez Sánchez et al. 2018; Jacobs
+et al. 2019; Domínguez Sánchez et al. 2019; Ntampaka et al. 2019;
+Huertas-Company et al. 2019; Hausen & Robertson 2020), and are
+a natural candidate for merger identification in imaging (e.g. Acker-
+mann et al. 2018; Walmsley et al. 2019, Pearson et al. 2019a, Ferreira
+et al. 2020, Wang et al. 2020) and in stellar velocity fields (e.g. Hung
+et al. 2016, McElroy et al. 2022, Bottrell et al. 2022). The specific task
+of identifying a pure and complete post-merger sample in UNIONS
+imaging with a simulation-trained CNN is discussed in principle in
+Bickley et al. (2021), and carried out in Bickley et al. (2022). After
+training and evaluation on realism-added mock CFIS observations of
+1 Galaxies with small angular separations on the sky, but which are not
+destined or likely to merge on account of large separations in radial distance
+and/or velocity.
+galaxies from the 100-1 run of the IllustrisTNG simulations (Mari-
+nacci et al. 2018; Naiman et al. 2018; Nelson et al. 2018; Pillepich
+et al. 2018; Springel et al. 2018; Nelson et al. 2019) the CNN was
+used to classify all CFIS galaxies with available SDSS Data Release
+7 (DR7) spectra.
+Bickley et al. (2021) noted that a CNN (or any automated merger
+classification method) with an accuracy of < 100% (the CNN de-
+ployed in this work has an accuracy of ~88% on test set galaxies)
+would invariably fail to produce a pure post-merger sample on ac-
+count of Bayesian statistics (Bayes & Price 1763) and the minuscule
+prior probability that any given galaxy in the low-redshift universe
+will be a post-merger. To address this issue, Bickley et al. (2021)
+suggest that a hybrid method, in which a subset of galaxies predicted
+by the CNN to be post-mergers are subsequently inspected visually,
+would offer a reasonable combination of efficiency and reproducibil-
+ity without sacrificing the purity of the final post-merger sample.
+The strengths of CNNs and human classifiers are complementary: a
+simulation-trained CNN can classify galaxies quickly, rule out a large
+number of galaxies unlikely to be mergers, and capture a breadth of
+observed merger characteristics spanning the range of remnant stellar
+masses, merger mass ratios, redshifts, orbital parameters, and obser-
+vational conditions included in the training set. Conversely, human
+classifiers are capable of meticulous classification with the goal of
+sample purity in mind, and are capable of explaining and defending
+their decisions.
+As long as the CNN’s training data is observationally realistic (e.g.
+Bottrell et al. 2019b; Huertas-Company et al. 2019; Ćiprijanović
+et al. 2021) and the simulated galaxies exhibit the same morpho-
+logical characteristics as galaxies in the low-redshift universe (e.g.
+Rodriguez-Gomez et al. 2019; Zanisi et al. 2021), visual classifiers
+ought to inherit a sample with a high post-merger fraction from
+the CNN. Critically, this particular combination of classifiers also
+improves over previous post-merger identification efforts in the di-
+versity of post-mergers included in the final sample. If less disturbed
+post-merger morphologies are included in the CNN’s training set
+and given appropriate consideration in the visual classification phase
+of the hybrid method, they can be preserved and studied alongside
+more visually obvious major mergers as long as their disturbed mor-
+phologies are sufficiently bright to be captured at the depth of CFIS.
+Methods using shallower (e.g., SDSS) imaging, visual classifications
+(e.g., Ellison et al. 2013) or CNNs trained on visual classifications
+(e.g., Pearson et al. 2019b; Gao et al. 2020) do not reap the same
+benefit — even when galaxies are inspected with great care, the most
+dramatic merger morphologies always inspire more confidence in
+visual classifications.
+In this work, we briefly review the simulation-trained CNN and
+our method for its deployment in a hybrid classification scheme, as
+well as three observational methods (optical spectroscopy, mid-IR
+photometry, and radio classification) of AGN identification (Sec-
+tion 2). We next use the visually confirmed post-merger galaxies
+from Bickley et al. (2022) to investigate the link between the merger
+sequence and the triggering of each AGN type (Section 3.1), and in-
+vestigate the reasons for differences between the Ellison et al. (2013)
+post-merger results and our own. Finally, we select post-merger and
+SDSS pair samples of optical AGN, and compare their [OIII] lu-
+minosities with those of control AGN in order to approximate the
+accretion rate enhancements produced by ongoing and completed
+mergers. We assume cosmological parameters (Ωm0 = 0.3, ΩΛ0 =
+0.7, ℎ= 0.7) when calculating luminosity distances, and for any other
+cosmology-dependent quantities appearing in this work.
+MNRAS 000, 1–13 (2022)
+
+AGN in UNIONS post-mergers
+3
+2 METHODS
+2.1 UNIONS data
+Our census of AGN in post-mergers would not be possible without
+high quality imaging resolution and depth over a large sky area. The
+UNIONS collaboration is a new consortium of wide field imaging
+surveys of the northern hemisphere and represents an excellent oppor-
+tunity for merger searches. UNIONS consists of 1) CFIS conducted at
+the 3.6-meter CFHT on Maunakea, 2) members of the Pan-STARRS
+team, and 3) the Wide Imaging with Subaru HyperSuprimeCam of
+the Euclid Sky (WISHES) team. Bickley et al. (2021) used only the 𝑟-
+band imaging in creating synthetic images, and the CNN predictions
+used in this work are made exclusively on CFIS 𝑟-band imaging.
+The observing pattern employed by CFIS uses three single-
+exposure visits with field-of-view (FOV) offsets in between for op-
+timal astrometric and photometric calibration with respect to ob-
+serving conditions. This also ensures that the entire survey footprint,
+including areas in the "chip gaps" between MegaCam’s multiple
+CCDs for a given exposure, will be visited for at least two exposures.
+After raw images are collected by CFHT, they are detrended (i.e.,
+the bias is removed and the images are flat-fielded using night sky
+flats) with the software package MegaPipe (Gwyn 2008, 2019). The
+images are next astrometrically calibrated using Gaia data release
+2 (Gaia Collaboration et al. 2016, 2018) as a reference frame. Pan-
+STARRS 3𝜋 𝑟-band photometry (Chambers & Pan-STARRS Team
+2016) is used to generate a run-by-run differential calibration across
+the MegaCam mosaic, and an image-by-image absolute calibration.
+Finally, the individual images are stacked onto an evenly spaced grid
+of 0.5-degree-square tiles using Pan-STARRS PS1 stars as in-field
+standards for photometric calibration. The resulting 𝑟-band images
+have a typical 5-𝜎 point-source depth of 24.85 mag, ~0.6-arcsecond
+seeing, and a pixel size of 0.187 arcseconds.
+In order to prepare CFIS galaxy images for classification, we
+cropped them to a physical scale of 100 kpc on a side based on
+their redshifts, resized to 138 × 138 pixels, and normalized on a
+linear scale with brightnesses of 0 and 1 assigned to the faintest and
+brightest pixels, respectively. Since this method requires an accurate
+spectroscopic redshift in order to crop and scale each galaxy image
+appropriately, we restrict our post-merger search to galaxies with 𝑧 <
+0.5 observed by both the CFIS data release 2 (DR2) and SDSS DR7,
+and match the two catalogs with a 2-arcsecond tolerance, yielding
+168,597 galaxies. Appearances of the same galaxies in the archival
+Wide-field Infrared Survey Explorer (WISE) space telescope2 pho-
+tometric catalog as well as the NRAO VLA Sky Survey (NVSS) and
+Faint Images of the Radio Sky at Twenty-centimeters (FIRST) radio
+catalogs will facilitate subsequent characterization of their SMBHs;
+see Sections 2.6 and 2.7. After processing, the galaxy images have
+the same physical and pixel dimensions as the synthetic images used
+for training in Bickley et al. (2021).
+The 𝑟-band imaging data is supplemented by derived measure-
+ments (e.g., redshifts) from the MPA-JHU catalog of derived proper-
+ties3 from SDSS DR7, stellar mass estimates from Kauffmann et al.
+(2003a), and corrected optical emission line fluxes from Scudder
+et al. (2012). In Section 3.1.2 we use non-parametric morpholog-
+ical statistics from Pawlik et al. (2016) to characterize degrees of
+merger-induced disturbance.
+2 wise2.ipac.caltech.edu/docs/release/allsky/
+3 wwwmpa.mpa-garching.mpg.de/SDSS/DR7/
+2.2 The visually confirmed post-merger sample
+In Bickley et al. (2021), the CNN was trained on realism-added
+images of IllustrisTNG post-merger remnants with mass ratios of
+at least 1:10, stellar masses between 1010−12 M⊙, and which had
+experienced coalescence between the previous and present simula-
+tion snapshots (corresponding to a timescale of ≤160 Myr) every
+snapshot since simulation 𝑧=1, and control galaxies that had not ex-
+perienced a merger in at least 2 Gyr of simulation time. Bickley et al.
+(2021) demonstrated that the network classified reserved simulated
+post-mergers and control images with ~88% accuracy, but cautioned
+that high accuracy would not directly lead to a pure predicted post-
+merger sample on account of the rarity of post-mergers in the real
+Universe (and simulated universes as well). As a result, Bickley et al.
+(2021) explored the utility of progressive cuts in CNN p(x), or the
+floating point prediction between 0−1 returned by the network as a
+prediction of merger status. In a sample with a realistic proportion
+of post-mergers, using p(x) > 0.5 to select mergers would inevitably
+lead to a sample of inadequate purity. Bickley et al. (2021) argued
+that a hybrid approach, in which a more stringent p(x) cut is com-
+bined with subsequent visual inspection, could be an efficient way
+forward.
+Our CNN was first used to classify real CFIS galaxies in Bickley
+et al. (2022), and the same classifications are prerequisite to this
+work. The CNN assigned p(x) predictions > 0.5 to 6,778 galaxies.
+Because of the natural rarity of post-merger galaxies, visual inspec-
+tion showed the galaxy sample with p(x) > 0.5 to be significantly
+contaminated with non-interacting galaxies, a result consistent with
+our expectations based on Bayesian statistics (see Section 3.6 in
+Bickley et al. 2021). We therefore chose to visually inspect the 2,000
+galaxies with classifications greater than p(x) > 0.75. There is no spe-
+cial significance to p(x) > 0.75 other than the fact that it produces a
+reasonable number of predicted post-mergers for subsequent inspec-
+tion by a classification team. In order to identify a pure post-merger
+sample, we used a conservative classification method that required a
+consensus post-merger classification with mutual agreement by the
+authors RWB, SLE, and DRP after careful inspection of the CFIS
+galaxy image, the wider field, and spectroscopic companions via
+SDSS. Galaxies for which a consensus could not be reached were
+discarded. Of the 2,000 galaxies inspected, 1,201 were rejected by
+the classification team on account of doubtful merger status, and a
+consensus was not reached for 100 additional galaxies. After visual
+inspection, a sample of 699 galaxies with unambiguous post-merger
+morphologies (described fully in Bickley et al. 2022) were confirmed
+unanimously by all three authors.
+The merger sample in Darg et al. (2010) was selected based on
+the merger vote fraction of citizen scientist volunteers on SDSS
+imaging, and consisted of 3003 visually selected mergers (although
+not all are post-coalescence). Ellison et al. (2013) inspected 370
+galaxies from the Darg et al. (2010) merger catalog which were
+strongly perturbed but lacked a spectroscopic companion (suggesting
+either a flyby or a completed merger) by eye and selected a smaller
+sample of 97 post-mergers. Our post-merger catalog, first presented
+in Bickley et al. (2022), offers an improvement in post-merger purity
+over Darg et al. (2010) and a substantial increase in size compared to
+Ellison et al. (2013). The post-mergers studied in this work were also
+selected using deeper imaging, and as a result include fainter and
+more subdued merger morphologies than could be detected using
+SDSS imaging.
+36 post-mergers from this visually confirmed sample are shown
+with logarithmic brightness scaling in Figure 1. As with all con-
+firmed post-mergers in the sample, the galaxies exhibit unambiguous
+MNRAS 000, 1–13 (2022)
+
+4
+R. W. Bickley et al.
+merger-induced morphologies: tidal tails, streams, or shells. While
+absolute knowledge of the status of each galaxy’s nucleus is limited
+by the resolution and seeing of the imaging, each galaxy in the sample
+appears to have only a single bright nucleus. In addition, we ruled
+out any galaxies whose merger-like features could plausibly have
+been induced by any nearby object imaged by CFIS, or identified by
+projected separation in SDSS. By eliminating companions both near
+and far, we can confidently deduce that the merger features in each
+of our 699 galaxies were produced by a partner that has been wholly
+accreted at the time of imaging.
+The study of star formation in the visually confirmed post-merger
+sample in Bickley et al. (2022) lent credence to the suggestion in
+Bickley et al. (2021) that a hybrid approach to post-merger identifi-
+cation, starting with a preliminary filtering by the CNN and ending
+with rigorous human visual classifications, would be an efficient way
+to identify a pure sample of galaxies belonging to the exceedingly
+rare post-merger class in a large observational sample. Even though
+the visually confirmed sample is not free of biases, we maintain that
+the Bickley et al. (2022) visually confirmed post-merger sample is
+likely representative of the remnants of major mergers.
+We also compare the characteristics of the visually confirmed,
+larger post-merger sample to those of the Ellison et al. (2013) post-
+mergers, which were visually selected from the larger Darg et al.
+(2010) post-merger catalog by the author SLE. Though lesser in
+number (97 in the sample before any experiment-specific cuts are
+applied), they provide useful context: they represent a highly pure
+sample as well, albeit one that is likely biased towards mergers of
+greater visual strength due to their selection from shallower SDSS
+imaging.
+Figure 2 shows the stellar masses and redshifts of both post-merger
+samples compared to the parent sample of galaxies from CFIS DR2.
+Both merger samples span the dynamic range of CFIS DR2 galaxy
+stellar masses, but the new visually confirmed post-merger sample
+identifies more distant post-merger galaxies out to 𝑧~0.3, whereas
+the Ellison et al. (2013) post-mergers lie below 𝑧 < 0.1. The success
+of the CNN at identifying galaxies over the full CFIS redshift range
+likely stems from its training set, which was composed of simulated
+galaxies inserted at redshifts drawn at random from the real CFIS
+distribution. The visually confirmed post-mergers also preferentially
+lie at high stellar masses compared to the parent sample. This bias
+is discussed at length in Bickley et al. (2022), and we posit two
+candidates for the cause. First, the CNN was trained on galaxies with
+masses M★ > 1010 M⊙ in order to ensure that their morphologies
+were adequately resolved by the simulation, and it is natural for the
+CNN to preferentially select galaxies similar to those on which it was
+trained. Second, the observability of merger-induced morphologies
+depends on the brightness of a galaxy relative to that of the sky, and
+more massive galaxies are likely to be brighter in appearance. This
+bias was affected during the CNN phase of the Bickley et al. (2022)
+hybrid merger identification effort, and therefore propagates through
+to the visually confirmed post-merger sample as well.
+2.3 Galaxy pairs
+Though this work only addresses the AGN characteristics of ob-
+served post-mergers, the characteristics of the post-merger epoch of
+galaxy evolution is best contextualized within the merger sequence.
+A sample of pre-coalescence galaxy pairs is therefore required. We
+use the galaxy pair sample compiled by Patton et al. (2016), and
+select objects for comparison emulating the method employed in El-
+lison et al. (2013). Whenever they are shown for context in this work,
+SDSS pairs are required to have projected separations 𝑟 𝑝 ≤ 80ℎ−1
+70
+kpc, line of sight velocity differences ΔV ≤ 300 km s−1 (in order
+to minimize accidental projections of a false pair), and stellar mass
+ratios of 0.1 ≤ M1/M2 ≤ 10.
+Note that we are applying symmetrical mass ratio criteria to our
+pair samples and the post-merger samples used to train our CNN (1:10
+or more, see Section 2.2). Although merger mass ratio may scale
+with the degree of morphological disturbance of merger remnants,
+we do not find that the inclusion / exclusion of minor galaxy pairs
+(e.g., changing the mass ratio criterion to 0.25 ≤ M1/M2 ≤ 4)
+has a significant influence on the excesses we calculate. We also
+expect the median progenitor mass ratio of our post-merger sample
+to increase following the visual inspection effort detailed in Bickley
+et al. (2022), since merger candidates of unconvincing strength were
+removed. We therefore present the physical characteristics of merger
+remnants alongside galaxy pairs with the caveat that the pairs studied
+may not be precise pre-merger analogues to the post-mergers.
+A larger ΔV requirement, e.g., ≤ 1000 km s−1 could capture
+a larger sample of galaxies that could conceivably be experiencing
+interactions, but the risk of contamination by non-interacting galaxies
+also increases. In order to restrict our analysis to galaxies that are
+very likely to be pre-mergers, we therefore follow Patton et al. (2016)
+and use ΔV ≤ 300 km s−1. Regardless of the choice of ΔV, we expect
+some contamination from interlopers to persist.
+2.4 Control pools
+Wherever post-mergers are studied in this paper, we will compare
+their physical characteristics to those of control galaxies that are sim-
+ilar in mass and redshift, but which are not themselves post-mergers.
+To this end, we again use the Bickley et al. (2022) CNN classifica-
+tions to our advantage. The vast majority (131,168 of 168,597) of
+galaxies in the CFIS sample are assigned CNN p(x) < 0.1 by the
+network, nominally indicating non-post-merger status. Even with an
+accurate classifier, Bayesian statistics suggest that a small number
+of genuine post-mergers will fall below this threshold. Thanks to the
+initial rarity of post-mergers, however, we can reasonably assume that
+the extreme dilution of post-mergers below p(x) < 0.1 will effectively
+eliminate their influence on our results.
+Wherever the Ellison et al. (2013) sample is shown, their physical
+characteristics are compared to those of their own mass- and 𝑧-
+matched corresponding control sample with a Galaxy Zoo merger
+vote fraction of zero. As with the visually confirmed post-merger
+sample, we expect that the signal from misclassified post-mergers
+contaminating this control sample should be effectively zero.
+The controls for galaxy pairs must have projected separations of
+𝑟 𝑝 > 80ℎ−1
+70 kpc, and Galaxy Zoo (Darg et al. 2010) merger vote
+fractions of zero in order to ensure that they do not belong to an
+interacting pair, and have not merged recently. The particular control
+matching methodologies for each of our experiments are described
+in detail in Section 3.
+2.5 Optical (Seyfert II) AGN
+Since spectroscopic redshifts are required for appropriate treatment
+of our CFIS galaxy images, all post-mergers identified in CFIS 𝑟-
+band imaging by the CNN have available SDSS DR7 optical spectra
+as well. Conveniently, we can use emission line measurements from
+these same optical spectra to search for the luminous echoes of AGN
+in our galaxies’ gaseous narrow-line regions (NLR). Galaxies ex-
+hibiting the NLR optical emission line characteristics of the Seyfert
+II class will hereafter be referred to as "optical AGN".
+MNRAS 000, 1–13 (2022)
+
+AGN in UNIONS post-mergers
+5
+Figure 1. A mosaic of 36 post-mergers from our hybrid (CNN-identified and visually confirmed) post-merger sample, cropped to a physical size of 100 kpc
+on a side. All galaxies in the sample have distinctive merger-induced morphologies that could not be plausibly produced by any discernible object in the CFIS
+imaging, or by any spectroscopic companions as identified using SDSS. Furthermore, each of the post-mergers has only a single post-coalescence bright nucleus.
+Galaxies are shown in log-scale with the contrast adjusted for consistency.
+In order to quantify the incidence of optical AGN in post-mergers,
+as well as estimate the accretion rates of their black holes, we use the
+"maximum starburst line" on the Baldwin, Phillips, Terlevich (BPT;
+Baldwin et al. 1981) diagram computed by Kewley et al. (2001).
+The line represents the theoretical boundary separating AGN from
+starbursts, determined using grids of models with varied metallicities
+and ionization parameters, and is more stringent than the criterion of
+Kauffmann et al. (2003b) which is often used to separate purely star-
+forming galaxies from those with potential contributions from AGN.
+We use Milky Way and host extinction-corrected optical emission
+line measurements (from the SDSS MPA-JHU catalog and Scudder
+et al. 2012) to place our target and control galaxies on the BPT
+diagram. In order to avoid contamination from shocks, which are
+expected in gas-rich galaxy mergers, we do not count objects on the
+AGN side of the Kewley et al. (2001) diagram as AGN if they fall
+MNRAS 000, 1–13 (2022)
+
+z: 0.07
+z: 0.18
+z: 0.18
+z: 0.11
+z: 0.19
+z: 0.16
+z: 0.25
+z: 0.21
+z: 0.13
+z: 0.13
+z: 0.15
+z: 0.19
+z: 0.08
+z: 0.17
+z: 0.07
+z: 0.16
+z: 0.11
+z: 0.18
+z: 0.12
+z: 0.11
+z: 0.24
+z: 0.16
+z: 0.15
+z: 0.09
+z: 0.1
+z: 0.13
+z: 0.1
+z: 0.11
+z: 0.21
+z: 0.1
+z: 0.17
+z: 0.23
+z: 0.15
+z: 0.25
+z: 0.116
+R. W. Bickley et al.
+Figure 2. The stellar masses and redshifts of the 699 visually confirmed post-
+merger galaxies (magenta crosses) studied in the present paper and 97 Ellison
+et al. (2013) post-mergers (black stars) superimposed over CFIS DR2 parent
+sample (gradient histogram). CFIS 𝑟-band imaging was processed and used
+as the input for CNN classification as well as for visual inspection in (Bickley
+et al. 2022).
+below either the [SII] or [OI] BPT diagram low-ionization nuclear
+emission region (LINER) criteria described in Kewley et al. (2006).
+When quantifying AGN excesses (the ratio of the AGN fractions in
+mergers and controls) in Section 3.1, we require S/N ≥ 5 on all four
+emission lines used in the BPT diagram. Consequently, the nebular
+emission lines of galaxies that we count in our excess calculations are
+dominated by the high-energy ionization associated with the AGN,
+and our excess calculations capture the frequency with which mergers
+induce strong, unambiguous optical evidence of AGN.
+Later, when we approximate the SMBH accretion rate with the
+luminosity of the [OIII] emission line in Section 3.2, we require an
+ensemble of 5 or more controls (which must also be optical AGN) for
+each merger in order to calculate a robust accretion rate enhancement.
+The [OIII] emission line has been used many times to approximate
+the accretion rates of optically identified AGN, (e.g. Kauffmann et al.
+2003b; Brinchmann et al. 2004; Chen et al. 2009; Liu et al. 2012),
+but [OIII] flux can also be contaminated by star formation. The link
+between [OIII] luminosity and star formation rate is complex, and its
+dependence on several parameters is explored in Kewley et al. (2001).
+Nonetheless, the Kewley et al. (2001) criterion we use is intended
+to be a maximal starburst division, such that galaxies above it can
+only be produced (in their models) by AGN contributions. We choose
+galaxies for this experiment whose emission is unambiguously AGN-
+dominated according to the same Kewley et al. (2001) criterion used
+in our excess calculations, and compare the AGN found in mergers to
+those found in non-mergers. The novel size of the visually confirmed
+post-merger sample allows us to use the same S/N ≥ 5 cut as we
+do when measuring optical AGN excesses, and additionally remove
+LINERs using the [SII] and [OI] Kewley et al. (2006) criteria in order
+to ensure a robust connection between observed [OIII] luminosity and
+SMBH accretion.
+2.6 Infrared AGN
+Though they are useful for identifying a particular subset of AGN,
+optical emission lines do not represent a complete census of SMBH
+activity in low-redshift galaxies (Hickox & Alexander 2018). A num-
+ber of studies have demonstrated the utility of mid-infrared (mid-IR)
+observations to identify dust-obscured AGN (e.g. Lacy et al. 2004;
+Stern et al. 2005; Donley et al. 2007; Hickox et al. 2007; Donley
+et al. 2008; Eckart et al. 2009; Stern et al. 2012; Mateos et al. 2013).
+Many galaxies with ongoing SMBH accretion are obscured by dust
+such that their optical emission line strengths cannot be reliably mea-
+sured, while at the same time reddening their mid-IR colours (e.g.
+Assef et al. 2013). In fact, mergers may preferentially produce dust-
+obscured AGN (Satyapal et al. 2014, Weston et al. 2017, Blecha et al.
+2018). Models predict that when one or both companions involved in
+the interaction is dusty, the kinematic disturbances induced by the fi-
+nal stages of a merger may distribute the dust in the centre, obscuring
+optical emission that might be observable in a dynamically settled
+galaxy (Yutani et al. 2022).
+In order to identify dust-obscured AGN, we turn to legacy all-
+sky mid-IR observations from the WISE space telescope, and follow
+Satyapal et al. (2014) and Ellison et al. (2013) in deploying a WISE
+W1−W2 > 0.5 colour cut. Because we are interested in studying
+the proportion of galaxies with WISE photometry whose W1−W2
+colours exceed 0.5, we also require that all galaxies (post-mergers as
+identified by any method, galaxy pairs, and controls) included in our
+WISE-related results have been detected by WISE. WISE sources
+considered in our analysis have S/N of at least 2 for both W1 and
+W2, but a more stringent S/N criterion does not change our results
+since > 99% of our sources have S/N > 10 for both W1 and W2. Mid-
+IR observations can be affected by star formation, but even template
+spectra for galaxies with extreme star formation have been shown
+to fall below W1−W2 = 0.5 at low 𝑧 (Assef et al. 2013; Satyapal
+et al. 2014). Still, our visually-confirmed post-mergers are known to
+be enhanced in star formation (Bickley et al. 2022), so it remains
+conceivable that a small contribution from star formation may affect
+the signal in Section 3.1.2. Caution should therefore be exercised in
+direct interpretation of the quantities reported therein.
+2.7 LERGs
+Optical and mid-IR AGN detections together provide a reasonably
+complete census of the observational phenomena associated with ra-
+diatively efficient, rapidly accreting SMBH with a luminous accretion
+disk. Indeed, these AGN are most closely linked with the archetypal
+understanding of the role of the merger sequence in galaxy evolution,
+in which gas inflows simultaneously produce upticks in star forma-
+tion and SMBH accretion (e.g. Hopkins et al. 2006). While useful,
+this narrative largely excludes mergers between gas-poor systems.
+Although low-excitation AGN states are most often associated with
+relatively isotropic accretion of hot gas from galactic halos (e.g. Best
+et al. 2005, 2006; Allen et al. 2006; Hardcastle et al. 2007; Gaspari
+et al. 2013), there has been some evidence of a role for mergers in
+triggering radiatively inefficient accretion (e.g. Sabater et al. 2013;
+Garofalo 2019).
+In order to investigate the role of mergers in this context, we also
+quantify the proportion of low-excitation radio galaxies (LERGs) in
+our merger and pair samples relative to controls selected from the
+same SDSS parent sample as in Section 2.5. These low-excitation
+radio-selected AGN are taken from the compilation of Best & Heck-
+man (2012), who match SDSS to a pair of radio catalogs: NVSS
+and FIRST. After classifying their detections as either star form-
+MNRAS 000, 1–13 (2022)
+
+12.0
+11.5
++
++
+Stellar mass, log(Mo)
++
++
+11.0
+十 
++
++
+++++
++
+10.5
+10.0
+9.5
+visually confirmed PMs
++
+9.0
+Ellison et al. 2013 PMs
+*
++★
+8.5
+0.0
+0.1
+0.2
+0.3
+zAGN in UNIONS post-mergers
+7
+ing or AGN via a combination of 4000Å break strengths, radio-to-
+emission-line luminosities, and BPT diagnostics, they use optical
+emission lines and a second decision tree to distinguish the latter
+into high-excitation radio galaxies (HERGs) and LERGs. Because
+the high-excitation AGN state is well described by our optical and
+mid-IR observations, we consider only the LERGs in this work.
+2.8 Overlap of AGN types
+The visually confirmed post-merger sample, Ellison et al. (2013)
+post-merger sample, and SDSS galaxy pair sample contain 699,
+96, and 17,566 galaxies, respectively. The visually confirmed post-
+merger sample contains the following:
+• 32 optical AGN meeting the Kewley et al. (2001) and Kewley
+et al. (2006) criteria (excluding star-forming, composite, and LIN-
+ERs) with S/N > 5 for the emission lines used in BPT placement
+• 66 mid-IR AGN with W1−W2 > 0.5
+• 14 LERGs
+• 17 galaxies that host AGN identifiable using both optical and
+mid-IR criteria
+• No overlap between either the optical or mid-IR AGN and
+LERGs
+On average, the optical AGN have a WISE colour of ~0.35, and
+the LERGs have a typical WISE colour of ~0.12. Our choices of
+criteria for optical and mid-IR AGN do not by definition preclude the
+possibility of a galaxy being a LERG, and such dual-status objects
+do appear in the SDSS DR7 and WISE catalogues. They are rare,
+however, and as a result none appear in the visually confirmed post-
+merger sample.
+3 RESULTS
+3.1 AGN in post-mergers
+After identifying our post-merger samples and establishing AGN cri-
+teria, we can quantify the strength of the link between merger status
+and SMBH incidence by calculating AGN excesses — the ratio of
+the AGN fractions in matched samples of post-mergers (or pairs)
+and controls. When controls are properly matched and statistics are
+of adequate quality, any excess > 1 indicates that the differentiating
+characteristic between the two samples (i.e., post-merger or pair sta-
+tus) is responsible for (or correlated with a factor that is responsible
+for) the increased AGN frequency.
+In calculating each excess, we match controls to the post-mergers
+or pairs (hereafter referred to collectively as "target galaxies") on stel-
+lar mass and redshift. Controlling for environment using additional
+parameters (halo mass, environment density) does not qualitatively
+change our results, suggesting that the influence of the most rele-
+vant interacting companion (in the case of galaxy pairs) or coalesced
+companion (for the post-mergers) is largely responsible for the signal
+uncovered in this Section. In order to ensure we have the same num-
+ber of controls per target galaxy, we identify the closest non-merger
+match (without replacement) in 2-D parameter space for each tar-
+get galaxy, assigning equal weight to each parameter. After finding
+one control for each galaxy, a Kolmogorov–Smirnov (K-S; Smirnov
+1948) statistic is calculated for the M★ and 𝑧 distributions of the
+target galaxies and controls. If the K-S 𝑝 value is below 0.9, the con-
+trol pool is finalized and we calculate the AGN fractions and excess
+for the relevant observational type (optical, mid-IR, and LERG). If
+Figure 3. Optical (top), mid-IR (centre), and LERG (bottom) AGN excesses
+in SDSS spectroscopic pairs (blue), the Ellison et al. (2013) post-merger
+sample (teal), and the visually confirmed post-merger sample (magenta).
+Vertical errors are calculated by adding the inverses of the binomial errors
+on each fraction
+√︁
+𝑓 (1 − 𝑓 )/𝑁 , where f is the AGN fraction in the target
+or control sample, and N is the size of that sample. Horizontal errors are the
+bin widths. Our visually confirmed, more inclusive post-merger sample finds
+a stronger optical AGN excess, and a weaker mid-IR AGN excess. This is
+consistent with the hypothesis that more minor mergers, which are present
+in the visually confirmed sample and largely absent from the Ellison et al.
+(2013) sample, are less likely to be dust-obscured after the merger. We find
+LERG excesses for post-mergers and galaxy pairs consistent with unity. Even
+with a relatively simple control-matching scheme, the merger sequence does
+not appear to be strongly connected to LERG status.
+the 𝑝 value remains above 0.9, we attempt to match additional equal-
+sized batches of controls, checking the K-S 𝑝 value every time before
+adding an additional batch. We cap the number of controls per post-
+merger at 10, as the counting statistics cease to improve significantly
+by that time. We use the same statistical control matching methodol-
+ogy for each target sample (Bickley et al. 2022 post-mergers, Ellison
+et al. 2013 post-mergers, and galaxy pairs), allowing for the control
+pool (see Section 2.4) and number of controls per target galaxy to
+vary (between 1−10 depending on the continued success of K-S tests)
+between each subset of galaxy pairs (binned by projected separation)
+and post-merger sample we study.
+In the target samples and the matched control samples, the optical
+AGN fraction is the number of BPT AGN above the Kewley et al.
+(2001) and Kewley et al. (2006) lines with S/N ≥ 5 divided by the total
+number of target or control galaxies. The WISE AGN fraction is the
+number of galaxies with W1−W2 > 0.5 divided by the total number of
+target or control galaxies. For the WISE excess, the target samples and
+control pools are restricted to galaxies with WISE detections, even
+though the matching parameters are derived from SDSS spectra. The
+MNRAS 000, 1–13 (2022)
+
+4
+Optical Excess
+visually confirmed PMs
+Ellison et al. 2013 PMs
+3
+spectroscopic pairs
+2
+S
+S
+WISE Exces
+10
+5
+0
+1.5
+ERG Excess
+1.0
+0.5
+0.0
+20
+40
+60
+80
+0
+rp, kpc8
+R. W. Bickley et al.
+LERG fraction is the number of LERGs in each sample divided by
+the total number of target or control galaxies. For the LERG excess
+we require that the target and control pool galaxies are classified in
+the Best & Heckman (2012) catalog as either star forming galaxies,
+HERGs, or LERGs. All excesses are the ratio of the AGN fraction
+in the target sample and the AGN fraction in the relevant matched
+control sample.
+3.1.1 Optical AGN excess
+The top panel of Figure 3 shows the optical excesses for spectroscopic
+pairs (blue) in 8 bins of projected separation between 0−80ℎ−1
+70 kpc,
+as well as for Ellison et al. (2013) post-mergers (teal) and the visually
+confirmed post-merger sample (magenta). In matching controls for
+Figure 3, we reached the cap of 10 controls per target (see Section 3.1
+above) in both post-merger samples, and in each bin of projected sep-
+aration. We uncover a decreasing trend of optical AGN excess with
+increasing 𝑟 𝑝, peaking with an excess of ~1.7 for galaxy pairs sepa-
+rated by ≤ 10ℎ−1
+70 kpc. Using a more generous optical AGN criterion
+(Stasińska et al. 2006, which allows for the inclusion of composite
+galaxies with significant contributions from both star formation and
+AGN), and a different control matching algorithm (allowing for dif-
+ferent numbers of controls to be selected for galaxies belonging to
+the same target sample), Ellison et al. (2011) still reported quantita-
+tively consistent pair-phase excesses, from ~0−2.5 for galaxy pairs
+spanning separations of 80−0 ℎ−1
+70 kpc. We recalculate the post-
+merger optical AGN excess in the Ellison et al. (2013) sample in
+order to compare using our updated control-matching method, and
+find ~2 times as many optical AGN compared to controls. We find
+that optical AGN are even more common in the visually confirmed
+post-merger sample, with an excess of 3.7.
+3.1.2 WISE AGN excess
+There are 6,106 SDSS pairs, 364 visually confirmed post-mergers,
+and 78 Ellison et al. (2013) post-mergers for which we can calculate a
+WISE mid-IR colour. The middle panel of Figure 3 shows the excess
+of mid-IR AGN with W1−W2 > 0.5 in the same three main galaxy
+samples as in Section 3.1.1 compared to matched control samples,
+which are themselves required to have WISE detections so that their
+WISE W1−W2 colours are calculable. Much like the optical AGN
+excess, we find a decreasing excess of mid-IR AGN in bins of 𝑟 𝑝
+with a peak value of ~3.6 for the closest spectroscopic pairs. These
+results are statistically consistent with the pair phase W1−W2 > 0.5
+results reported in Satyapal et al. (2014), even though a different
+control matching methodology was used. Our visually confirmed
+post-merger sample shows a mid-IR AGN excess consistent with the
+closest pairs, but significantly below the Ellison et al. (2013) sample,
+which contains 13.3 times as many WISE AGN per capita compared
+to the control sample. Our pair phase and visually confirmed post-
+merger sample excesses for both optical AGN and mid-IR AGN are in
+reasonable concordance (spanning ~0−4 in the pair phase, and 3−4 in
+the post-mergers). This agreement supports the view that the optical
+and mid-IR AGN diagnostics are identifying different observational
+phenomena associated with the same SMBH engines.
+In order to investigate the discrepancies in optical and mid-IR
+AGN excesses between the visually confirmed post-mergers and the
+Ellison et al. (2013) post-mergers, we investigate differences in the
+selection methods used for their identification. Because the CNN,
+Galaxy Zoo classifications, and expert visual classifications all rely
+Figure 4. Shape asymmetry statistics derived from SDSS imaging for SDSS
+galaxies with 𝑧 < 0.35 and M★ > 108.5 M⊙ (grey), the Ellison et al. (2013)
+post-mergers (teal), and the visually confirmed post-mergers (magenta, top
+panel). The vertical bars spanning both panels represent the median of each
+distribution. While both more disturbed than the typical SDSS galaxy, the
+visually confirmed post-merger samples are less disturbed on average than
+the Ellison et al. (2013) mergers, suggesting that a greater number of minor
+mergers are included. This would be a natural consequence of the CNN’s
+inclusive simulated training set. The bottom panel shows the AGN fractions
+in the SDSS sample as identified by optical (X-markers) and mid-IR (dia-
+monds) criteria, see Section 2 for details. The vertical errors are the binomial
+errors on each fraction
+√︁
+𝑓 (1 − 𝑓 )/𝑁 , and the horizontal errors are the bin
+widths. Since the hybrid method typically selects galaxies with smaller shape
+asymmetries, it also selects a proportionally lower fraction of dust-obscured
+AGN. This accounts for the discrepancy between the two post-merger data
+points in Figure 3.
+on morphology, we posit that a morphological bias may be respon-
+sible. In order to investigate differences in sample morphology, we
+use shape asymmetry. Shape asymmetry is a non-parametric mor-
+phological measurement that takes the asymmetry of a binary mask
+that denotes a boundary between the galaxy and the background, de-
+tailed in Pawlik et al. (2016). The binary mask used to measure shape
+asymmetry is generated following the 8-connected structure detec-
+tion method described in Pawlik et al. (2016), in which a galaxy image
+is smoothed using a 3×3 running average filter, and pixels above a
+limiting surface brightness of approximately 24.7 mag arcsec −2,
+equivalent to one standard deviation above the typical sky noise
+level in SDSS imaging, are ascribed to the galaxy. It was devel-
+oped for the purpose of automated merger identification because
+of its emphasis on the particular asymmetry of low-surface bright-
+ness features whose importance would be overlooked by a traditional
+asymmetry measurement. The merit of this metric for merger identi-
+fication is explored in detail in Wilkinson et al. (2022). By deploying
+shape asymmetry on galaxies whose post-merger status is already
+confirmed, shape asymmetry instead measures the morphological
+strength of the post-merger. Tidally disturbed post-mergers who have
+experienced dramatic, major mergers are more likely to have more
+extended morphologies, and higher shape asymmetries compared to
+those whose mergers have been relatively tame. Shape asymmetry
+has also been shown to fade in the several hundred Myr that fol-
+MNRAS 000, 1–13 (2022)
+
+5
+all SDSS
+Ellison et al. 2013
+p(As)
+visually confirmed
+WISE
+optical
+0.10
+AGN fractior
+0.05
+0.00
+0.25
+0.50
+0.75
+1.00
+AsAGN in UNIONS post-mergers
+9
+low coalescence (Pawlik et al. 2016), but so too has it been shown
+that the W1−W2 colour decreases after coalescence (Blecha et al.
+2018). Shape asymmetry therefore also includes information about
+the recency of the merger, along with the initial intensity of the
+morphological disruption.
+Figure 4 investigates the shape asymmetry demographics of our
+two post-merger samples. The top panel shows the normalized dis-
+tributions of shape asymmetry derived from SDSS imaging for the
+entire SDSS DR7 galaxy population with spectroscopic redshifts <
+0.35, and masses > 108.5 M⊙ (grey, representing the area of M★−𝑧
+parameter space encompassing both the Ellison et al. (2013) and
+visually-confirmed post-merger samples, see Figure 2), the Ellison
+et al. (2013) post-merger sample (teal), and the new visually con-
+firmed sample (magenta). It is important to note that shape asym-
+metry does not trend strongly with either stellar mass or redshift,
+and that the qualitative results of our shape asymmetry study do
+not change when we compare our merger samples to their matched
+control galaxies from Sections 3.1.1 and 3.1.2 instead of the SDSS
+parent sample used here. The median SDSS-derived shape asymme-
+try of each sample is plotted over both panels as a dashed line of the
+same colour. Shape asymmetries derived from CFIS 𝑟-band imaging
+are available for the visually confirmed post-mergers, but we present
+only SDSS shape asymmetries in order to allow for direct compar-
+ison to the Ellison et al. (2013) sample, which does not appear in
+full in CFIS. Note that the shapes of the visually confirmed post-
+merger sample, while of course more asymmetrical than SDSS in
+general, are significantly less disturbed than the Ellison et al. (2013)
+post-mergers, with ¯Δ𝐴𝑆 ~0.13. We posit that the typical difference in
+SDSS-derived shape asymmetry between the two post-merger sam-
+ples is the result of the fact that the Ellison et al. (2013) post-mergers
+were identified by strictly visual means in shallow imaging. Con-
+versely, the visually confirmed post-mergers were first identified by a
+CNN trained on CFIS-depth simulated imaging of post-mergers with
+mass ratios as small as 10:1. It is therefore plausible that a number
+of relatively minor post-mergers were preserved by the CNN and
+confirmed during visual classification.
+The bottom panel of Figure 4 shows the local optical and WISE
+AGN fractions of the SDSS parent sample (with 𝑧 < 0.35 and M★ >
+108.5 M⊙) in 10 bins of 𝐴𝑆 between 0 and 1. We find that the optical
+AGN fraction is generally low and consistent with increasing shape
+asymmetry. While the WISE AGN fraction does not trend monoton-
+ically with 𝐴𝑆, the data show that more morphologically disturbed
+galaxies in SDSS are indeed more likely to host an AGN that is iden-
+tifiable by its mid-IR colour. These results indicate that the degree of
+disturbance is unlikely to have a strong impact on the optical AGN
+fraction, but this is not true for WISE AGN, since more disturbed
+galaxies typically have higher mid-IR AGN fractions. Consequently,
+the Ellison et al. (2013) sample is more likely to contain highly dis-
+turbed post-mergers, which host proportionally more dust-obscured
+AGN and a consistent number of optical AGN, while the visually
+confirmed sample is more inclusive of less-disturbed mergers, which
+are less likely to host mid-IR AGN.
+If the degree of morphological disturbance is responsible for the
+difference in mid-IR AGN excess between the two post-merger sam-
+ples, a subset of the visually confirmed post-mergers with the same
+SDSS shape asymmetry demographics as the Ellison et al. (2013)
+post-mergers ought to exhibit an excess that is in better agreement.
+In order to test this hypothesis, we match exactly one galaxy (with-
+out replacement) from the visually confirmed post-merger sample to
+each Ellison et al. (2013) post-merger on shape asymmetry. Where
+multiple visually confirmed post-mergers have shape asymmetries
+within ±5% of an Ellison et al. (2013) post-merger, we select the sin-
+gle best match. Where there are no matches, we grow the tolerance
+from 5% until a single match can be found. Of the 85 galaxies in the
+Ellison et al. (2013) sample with shape asymmetries available, 82
+have a match within 5% of their shape asymmetry in the visually con-
+firmed sample. The remaining 3 galaxies require 2, 3, and 5 growths,
+respectively; they are included for completeness but their exclusion
+does not affect our results. The shape asymmetry-matched visually
+confirmed post-mergers have an optical AGN excess consistent with
+the visually confirmed sample taken as a whole, but their mid-IR
+AGN excess is increased from 3.6 up to 6.5. While still not in perfect
+agreement with the Ellison et al. (2013) sample, this experiment con-
+firms that the degree of morphological disturbance ( ¯Δ𝐴𝑆 ~0.13) and
+sample selection are linked to an increased likelihood of a mid-IR
+AGN detection. This result is consistent with the physical narrative
+presented in Yutani et al. (2022), wherein rapidly accreting AGN in
+extremely recent (within ~4Myr) post-coalescence systems are more
+likely to be observed as dust-obscured galaxies (DOGs) on account
+of central and/or galaxy-scale dispersion of dust from the progenitor
+galaxies. We posit that the longevity and intensity of the dust ob-
+scuration may scale with the dynamic intensity of the merger. As a
+result, the specific method used to identify mergers based on their
+morphology has a significant impact on the quantitative excesses we
+calculate.
+3.1.3 LERGs in post-mergers
+The bottom panel of Figure 3 investigates the role of mergers in
+triggering LERGs. In our mass- and redshift-matched study, we find
+that LERGs are no more likely to exist in our post-merger or pair
+samples than in controls. This result is qualitatively discrepant with
+the Pace & Salim (2014) finding that galaxies hosting radio AGN have
+a 50% excess in the number of satellites, and the link between tidal
+forces associated with pair phase interactions and LERG incidence
+suggested by Sabater et al. (2013). Ellison et al. (2015) find a modest
+pair phase LERG excess of 3.8±0.4, and a small excess of ~4±2
+(nearly consistent with unity as well) in the post-merger phase when
+they match controls on stellar mass and redshift. The lack of an
+elevated LERG incidence rate in post-mergers or galaxy pairs in this
+work is therefore in mild tension with the literature, even though the
+conditions in a post-merger system are certainly not required for the
+triggering of LERGs.
+3.2 Optical AGN accretion rate enhancements
+In addition to the initial triggering of AGN, we can use our post-
+merger sample to determine typical merger-induced accretion rate
+enhancements in optically-identified AGN, using [OIII] luminos-
+ity as a proxy for accretion rate (see also Kauffmann et al. 2003b;
+Brinchmann et al. 2004; Chen et al. 2009; Liu et al. 2012). As stated
+in Section 2.5, we again use a S/N criterion of at least 5 for the four
+BPT emission lines, and explicitly disallow LINERS using the [SII]
+and [OI] criteria of Kewley et al. (2006). Because we are computing
+a luminosity enhancement, we require an ensemble of at least 5 AGN
+controls for each target galaxy in order to compare the luminosity of
+each AGN post-merger or pair to a group of non-merger or non-pair
+counterparts. Rather than finding the nearest controls in parameter
+space, we set initial tolerances of ±0.1 dex in M★ and ±0.05 in 𝑧.
+In practice, all of our target galaxies (pairs and post-mergers alike)
+find at least 5 controls without any growths in parameter space. 31
+post-mergers from the visually confirmed sample, 3 from the Elli-
+son et al. (2013) sample, and 263 SDSS pairs in bins of separation
+MNRAS 000, 1–13 (2022)
+
+10
+R. W. Bickley et al.
+Figure 5. [OIII] luminosities and luminosity enhancements in post-mergers
+and pairs. The top panel shows log-scale [OIII] luminosity histograms for
+optical AGN in the the galaxy pair sample described in Section 2.3 (blue), the
+visually confirmed post-merger sample (magenta), the Ellison et al. (2013)
+post-merger sample (teal), and the optical AGN control pool for the visually
+confirmed post-merger sample (grey). The bottom panel shows Δlog(L[OIII])
+for the same three target samples (galaxy pairs, visually confirmed post-
+mergers, and Ellison et al. 2013 post-mergers). Vertical error bars are the
+statistical error on the median, 1.253𝜎/
+√
+𝑁 , and horizontal error bars are
+the bin widths. We find enhancements approximately consistent with zero in
+the pair phase, with some small local suppressions past 40ℎ−1
+70 kpc. In both
+post-merger samples, we find significant positive excesses. Post-mergers in
+the visually confirmed sample are ~2 times as luminous in [OIII] as their
+non-post-merger controls.
+between 0−80ℎ−1
+70 kpc with mass ratios 0.1 ≤ M1/M2 ≤ 10 selected
+from the Patton et al. (2016) catalog with non-LINER optical AGN
+and S/N of at least 5 on all emission lines used for placement on
+the BPT diagram are ultimately included. The [OIII] luminosity en-
+hancement, Δlog(L[OIII]), is calculated as the difference between the
+logged [OIII] luminosity (in units of erg s−1) of the target galaxy and
+the median logged luminosity of the control ensemble, and hence
+captures the typical accretion rate difference between AGN triggered
+my mergers and secular AGN.
+Figure 5, which shows L[OIII] (top panel) and Δlog(L[OIII]) (bot-
+tom panel) for optical AGN in spectroscopic pairs, the Ellison et al.
+(2013) post-mergers, and the visually confirmed post-merger sample,
+suggests that accretion rates in AGN hosted by galaxies with a close
+companion are consistent with isolated AGN. Conversely, both post-
+merger samples are significantly enhanced compared to the matched
+non-merger AGN control ensembles. The visually confirmed post-
+merger sample is in fact ~2 times as luminous in [OIII] on average
+compared to non-post-merger controls. The accretion rate enhance-
+ments for optical AGN ushered in by coalescence therefore appear
+to be significant, while those produced by pair-phase interactions
+may just as likely be produced by secular or ambient processes. A
+higher positive enhancement of ~1.4 dex is recovered for the three
+post-mergers that remain after applying our quality control cuts to
+the Ellison et al. (2013) post-mergers. While the sample size does
+not invite extensive interpretation, it is possible that the increased
+morphological disturbance that is typical of the Ellison et al. (2013)
+sample is linked to more rapid gas infall and elevated accretion rates
+in these systems.
+The relationship between star formation rate and L[OIII] is of order
+unity (with significant scatter) in BPT star forming galaxies beneath
+the Kauffmann et al. (2003b) criterion with S/N>5. Therefore, in
+order to achieve similar Δlog(L[OIII]), star forming galaxies would
+need to have approximately doubled star formation rates (SFRs).
+Still, since we select galaxies whose nebular emission could not
+plausibly be produced by star formation alone, the [OIII] luminosity
+enhancements uncovered in this Section are primarily indicative of
+SMBH accretion-driven ionization.
+The result does not appear to be an effect of the SDSS fiber aper-
+ture on the Ellison et al. (2013) sample, which lies at low-𝑧 relative
+to the visually confirmed sample (see Figure 2) as we find no corre-
+lation of Δlog(L[OIII]) with 𝑧 for individual galaxies in the visually
+confirmed sample. In the sample of Kewley et al. (2001) AGN with
+S/N ≥ 5 for the four BPT emission lines and stellar masses be-
+tween 1010−12 M⊙, the median measured [OIII] luminosity actually
+increases from 1039.2 to 1041.6 between 0 ≤ 𝑧 ≤ 0.25. The fiber
+aperture effect therefore gives rise to higher [OIII] luminosities at
+higher 𝑧, and moreover, our control-matching methodology accounts
+for systematic changes in [OIII] luminosity with stellar mass and
+redshift. Aperture effects (or more broadly, any redshift or stellar
+mass effects) are not responsible for the difference in Δlog(L[OIII])
+calculated between the Ellison et al. (2013) and visually confirmed
+post-mergers.
+4 SUMMARY
+In this work, we have used the CNN-identified and visually confirmed
+post-merger sample introduced in Bickley et al. (2022) to study the
+triggering and accretion of supermassive black holes in post-merger
+galaxies. We also offer pair phase results in order to contextualize the
+post-merger results within the merger sequence. We match control
+(non-post-merger or non-pair) galaxies on M★ and 𝑧 to our target
+(either post-merger or pair) galaxies in order to study the AGN excess
+— that is, the ratio of the AGN fractions in the target sample and the
+control sample — using optical narrow-line region (via SDSS, the
+BPT diagram, and the Kewley et al. 2001 AGN selection), mid-IR
+(via WISE and the colour criterion W1−W2 > 0.5 used by Satyapal
+et al. 2014 to select dust-obscured AGN), and LERG classifications
+(capturing the low-excitation, isotropic SMBH accretion mode, as
+determined by Best & Heckman 2012). We report the following:
+• We find that optical and mid-IR AGN excesses in the pair sample
+increase from ~1 (i.e., no excess) to ~2−4 as their projected separa-
+tions decrease from 𝑟 𝑝~80 ℎ−1
+70 kpc down to zero. We find that galaxy
+pairs and mergers do not preferentially exhibit the characteristics of
+LERGs, in mild tension with the literature (Figure 3).
+• We find optical and mid-IR AGN excesses in a new sample of
+visually confirmed post-mergers of ~3 − 4 over controls matched on
+mass and redshift, suggesting that the nuclear conditions ushered in
+by post-mergers increase the likelihood that an energetic AGN will
+be triggered.
+• We find a significant connection between high shape asymme-
+try derived from SDSS imaging (roughly analogous to the degree of
+merger disturbance) and a galaxy’s likelihood to host a dust-obscured
+MNRAS 000, 1–13 (2022)
+
+PM control sample
+ galaxies
+102
+101
+#
+100
+38
+40
+42
+log(L[ol, erg/s)
+Alog(L[ol), dex
+visually confirmed PMs
+Ellison et al. 2013 PMs
+spectroscopic pairs
+20
+40
+60
+80
+rp, kpcAGN in UNIONS post-mergers
+11
+AGN in an inclusive sample of SDSS galaxies. This is most likely
+related to the tendency of mergers to disperse the central gas and dust
+belonging to their participant galaxies. This connection contributes
+to the quantitative differences in the optical and mid-IR AGN ex-
+cesses of our visually confirmed merger sample and that of Ellison
+et al. (2013), which is composed of more visually dramatic merger
+examples with higher shape asymmetries on average (Figure 4).
+• Following a number of efforts in the literature (see Section 3.2),
+we use [OIII] luminosity as a proxy for SMBH accretion rate in a
+sample of optical AGN. We find that optical AGN hosted by inter-
+acting galaxy pairs are not preferentially enhanced in their accretion
+rates (as measured by L[OIII]) compared with secularly driven AGN
+in isolated galaxies. Conversely, we find our visually confirmed post-
+merger sample to be ~2 times as bright in [OIII] than the AGN in
+isolated galaxies. This suggests that the typical accretion rate en-
+hancements produced during the pair phase of the merger sequence
+are just as likely to be produced by secular or ambient processes
+(e.g. halo gas accretion, secular gas accretion from stellar winds or
+supernovae), while the post-merger phase produces significant ac-
+cretion rate enhancements. This result, as well as our excess results,
+support an important role for the post-merger epoch in triggering and
+growing the SMBHs residing at the core of every galaxy (Figure 5).
+In addition to the above detailed census of AGN in post-
+coalescence galaxies, we can also revisit the merit of a hybrid (CNN
+plus human visual classification) post-merger identification frame-
+work, which has allowed us to improve on the statistics of literature
+studies of post-mergers, and propose revisions to other results whose
+quantities were more heavily influenced by the selection functions of
+their merger identification method. Because the visually confirmed
+CFIS merger sample is biased only by the training of the CNN and
+the decisions of the visual classification team, we believe the galaxies
+themselves (catalogued in Bickley et al. 2022) will continue to pro-
+vide value in the form of subsequent cross-survey characterization.
+Moreover, the hybrid classification framework itself shows promise
+for future questions in astronomy surrounding rare and elusive ob-
+servational phenomena.
+ACKNOWLEDGEMENTS
+The work detailed above was conducted at the University of Victoria
+in Victoria, British Columbia, as well as in the Township of Esquimalt
+in Greater Victoria. We acknowledge with respect the Lekwungen
+peoples on whose unceded traditional territory the university stands,
+and the Songhees, Esquimalt and ¯WS ´ANE ´C peoples who have stew-
+arded the land for centuries and continue to do so today.
+CFIS is conducted at the Canada-France-Hawaii Telescope on
+Maunakea in Hawaii. We also recognize and acknowledge with re-
+spect the cultural importance of the summit of Maunakea to a broad
+cross section of the Native Hawaiian community.
+We thank Samir Salim, Christopher Agostino, and Connor Bottrell
+for their indespensible feedback on this work.
+This work is based on data obtained as part of the Canada-
+France Imaging Survey, a CFHT large program of the National
+Research Council of Canada and the French Centre National
+de la Recherche Scientifique, and on observations obtained with
+MegaPrime/MegaCam, a joint project of CFHT and CEA Saclay, at
+the Canada-France-Hawaii Telescope (CFHT) which is operated by
+the National Research Council (NRC) of Canada, the Institut Na-
+tional des Science de l’Univers (INSU) of the Centre National de
+la Recherche Scientifique (CNRS) of France, and the University of
+Hawaii. This research used the facilities of the Canadian Astronomy
+Data Centre operated by the National Research Council of Canada
+with the support of the Canadian Space Agency.
+Data from the IllustrisTNG simulations are integral to this work.
+We thank the Illustris Collaboration for making these data available
+to the public.
+Funding for the SDSS and SDSS-II has been provided by the Alfred
+P. Sloan Foundation, the Participating Institutions, the National Sci-
+ence Foundation, the U.S. Department of Energy, the National Aero-
+nautics and Space Administration, the Japanese Monbukagakusho,
+the Max Planck Society, and the Higher Education Funding Council
+for England. The SDSS Web Site is http://www.sdss.org/. The SDSS
+is managed by the Astrophysical Research Consortium for the Partic-
+ipating Institutions. The Participating Institutions are the American
+Museum of Natural History, Astrophysical Institute Potsdam, Uni-
+versity of Basel, University of Cambridge, Case Western Reserve
+University, University of Chicago, Drexel University, Fermilab, the
+Institute for Advanced Study, the Japan Participation Group, Johns
+Hopkins University, the Joint Institute for Nuclear Astrophysics, the
+Kavli Institute for Particle Astrophysics and Cosmology, the Korean
+Scientist Group, the Chinese Academy of Sciences (LAMOST), Los
+Alamos National Laboratory, the Max-Planck-Institute for Astron-
+omy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New
+Mexico State University, Ohio State University, University of Pitts-
+burgh, University of Portsmouth, Princeton University, the United
+States Naval Observatory, and the University of Washington.
+This research was enabled, in part, by the computing resources
+provided by Compute Canada.
+DATA AVAILABILITY
+Simulation data from TNG100-1 used in the generation of training
+images for this work are openly available on the IllustrisTNG website,
+at tng-project.org/data. Template versions of RealSim and RealSim-
+CFIS, developed by Connor Bottrell with modifications by RWB are
+publicly available via GitHub at github.com/cbottrell/RealSim and
+github.com/cbottrell/RealSim-CFIS. Specific image training data
+used to develop the findings of this study are available by request
+from RWB.
+This publication makes use of data products from the Wide-field
+Infrared Survey Explorer, which is a joint project of the University
+of California, Los Angeles, and the Jet Propulsion Laboratory / Cal-
+ifornia Institute of Technology, funded by the National Aeronautics
+and Space Administration. Data from the Sloan Digital Sky Survey
+are available at sdss.org.
+The public visually confirmed post-merger catalog is available via
+MNRAS as a digital resource along with Bickley et al. (2022).
+The Canada France Imaging Survey is a legacy survey for the
+Canadian and French communities. All data are proprietary within
+the Canadian and French communities until Aug. 1st, 2023 when the
+individual frames will become available to the worldwide commu-
+nity. Advanced curated CFIS data products will be made available to
+the world as part of UNIONS public data releases.
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diff --git a/kdE2T4oBgHgl3EQfIgZc/content/tmp_files/load_file.txt b/kdE2T4oBgHgl3EQfIgZc/content/tmp_files/load_file.txt
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@@ -0,0 +1,1674 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf,len=1673
+page_content='MNRAS 000, 1–13 (2022)  Preprint 11 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  AGN in post-mergers from the Ultraviolet Near Infrared Optical Northern  Survey  Robert W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley,1★ Sara L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ellison,1 David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Patton,2 Scott Wilkinson1  1Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8P 1A1, Canada  2Department of Physics and Astronomy, Trent University, 1600 West Bank Drive, Peterborough, ON K9L 0G2, Canada  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  The kinematic disturbances associated with major galaxy mergers are known to produce gas inflows, which in turn may trigger  accretion onto the supermassive black holes (SMBH) of the participant galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While this effect has been studied in galaxy  pairs, the frequency of active galactic nuclei (AGN) in fully coalesced post-merger systems is poorly constrained due to the  limited size or impurity of extant post-merger samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Previously, we combined convolutional neural network (CNN) predictions  with visual classifications to identify a highly pure sample of 699 post-mergers in deep 𝑟-band imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In the work presented  here, we quantify the frequency of AGN in this sample using three metrics: optical emission lines, mid-infrared (mid-IR) colour,  and radio detection of low-excitation radio galaxies (LERGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We also compare the frequency of AGN in post-mergers to that in  a sample of spectroscopically identified galaxy pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We find that AGN identified by narrow-line optical emission and mid-IR  colour have an increased incidence rate in post-mergers, with excesses of ~4 over mass- and redshift-matched controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The  optical and mid-IR AGN excesses in post-mergers exceed the values found for galaxy pairs, indicating that AGN activity in  mergers peaks after coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Conversely, we recover no significant excess of LERGs in post-mergers or pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Finally, we  find that the [OIII] luminosity (a proxy for SMBH accretion rate) in post-mergers that host an optical AGN is ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3 dex higher  on average than in non-interacting galaxies with an optical AGN, suggesting that mergers generate higher accretion rates than  secular triggering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Key words: galaxies: evolution – galaxies: interactions – galaxies: peculiar – methods: statistical – techniques: image processing  1 INTRODUCTION  Galaxy mergers are unique within the framework of hierarchical  assembly in that they simultaneously transform the kinematics, mor-  phologies, and intrinsic properties of the participant galaxies (White  & Rees 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Lacey & Cole 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Boylan-Kolchin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Jiang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Toomre & Toomre 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Conselice  2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Lotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2008) reproduce the observed signatures — stellar  shells, streams, and tails — of interacting and post-merger galax-  ies (Darg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Kartaltepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Simmons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Numerous simulations are also in qualitative agreement about the  chemical evolution and new star formation experienced by galaxies  after they experience a disruptive, gas-rich merger (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Toomre &  Toomre 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Moreno  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Cold gas that might  have been in a stable orbit in one or both progenitors can be dis-  rupted by mergers, funneled towards the centres of the participant  galaxies, and may be responsible for central starbursts suggested in  observations (Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2008, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Nikolic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Scud-  der et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Scott & Kaviraj 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Knapen 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Thorp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The cold gas that is responsible for central starbursts may  also accrete onto a galaxy’s super-massive black hole (SMBH) and  ★ E-mail: rbickley@uvic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='ca  produce the observational signatures of an active galactic nucleus  (AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Satyapal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Lackner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Weston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2011, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Together,  increased star formation and energetic feedback from the AGN may  enrich the circum-galactic medium through ejective means (Johnson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The stellar and gas-phase kinematics  of merger progenitors are also disrupted in the process (Lynden-Bell  1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Toomre 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Negroponte & White 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hernquist 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Naab & Burkert 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Jesseit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Clauwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018), and star  formation may truncate rapidly due to either gas ejection or heating  after the epoch of intense star formation and AGN feedback is com-  plete (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Yesuf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Quai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Indeed, Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022) recently identified an  observational link between the post-merger phase and the signatures  of rapid quenching using the same post-merger sample studied in  Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021) and this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  While a merger-AGN connection is therefore well established in  pre-coalescence galaxy pairs, the precise role of post-coalescence  mergers in switching on AGN and feeding them is not well con-  strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Until recently, merger samples in the literature have been  either too small to perform precise statistics, or dubious in purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Without a large and highly pure merger sample, the quantitative role  of mergers in SMBH evolution cannot be studied effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='03681v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='GA]  9 Jan 2023  2  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Any effort to study the observed signatures of merger-induced  phenomena across the entire merger sequence requires a post-merger  sample that is both pure (containing as high a fraction as possible  of genuine mergers) and representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Merger identification is rel-  atively straightforward in spectroscopic galaxy pairs, which can be  identified by their visual appearances (Kampczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bundy  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Brinchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Darg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2010) or statistically,  by grouping galaxies together in angular position and line-of-sight  radial velocity in order to mitigate potential contamination by false  pairs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Thanks to the large number of spectroscopically-identified  galaxy pairs in redshift surveys, the statistical influence of the pair  phase has already been explored in great detail (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Barton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' De Propris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Lin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Small samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013), and detailed spatially re-  solved case studies of individual post-mergers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Thorp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Barrera-Ballesteros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019) offer a provisional  understanding of post-coalescence galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While these results have  hinted at changes in star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013, Elli-  son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020), chemical evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bustamante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020),  and intense SMBH activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Carpineti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2012) in the post-  merger epoch, fully coalesced galaxies are much more difficult to  identify since they are no longer spectroscopically distinct from their  companion(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Consequently, the specific quantitative contribution  of coalescence in producing these phenomena (especially SMBH  triggering and accretion) is still being evaluated in simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Sivasankaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022) and observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Because the characteristic features of the post-merger phase are  relatively faint, morphological merger identification methods require  imaging of adequate depth and resolution (as demonstrated by Bot-  trell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019a, Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019, Ćiprijanović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The Canada France Imaging Survey (CFIS), part of the Ul-  traviolet Near Infrared Optical Northern Survey (UNIONS) collab-  oration, offers a useful combination of imaging quality and volume,  with ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7 arcsecond seeing, and 𝑟-band imaging that will eventually  cover 5,000 square degrees of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The survey’s 5-𝜎 point-source  depth (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='85 mag in the 𝑟-band for the MegaCam wide-field optical  imager) is sufficient to capture the low-surface brightness features  necessary for merger identification in bright, low-redshift galaxies  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Sola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Estimates of the low-𝑧 merger rate suggest  that the UNIONS footprint will include thousands of post-mergers  (Lacey & Cole 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Lotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Casteels  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Convolutional neural networks (CNNs) have already been suc-  cessfully applied to a number of tasks in astronomy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Huertas-  Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Domínguez Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Jacobs  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Domínguez Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ntampaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hausen & Robertson 2020), and are  a natural candidate for merger identification in imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Acker-  mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Walmsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019, Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019a, Ferreira  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020) and in stellar velocity fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hung  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2016, McElroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022, Bottrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The specific task  of identifying a pure and complete post-merger sample in UNIONS  imaging with a simulation-trained CNN is discussed in principle in  Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021), and carried out in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' After  training and evaluation on realism-added mock CFIS observations of  1 Galaxies with small angular separations on the sky, but which are not  destined or likely to merge on account of large separations in radial distance  and/or velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  galaxies from the 100-1 run of the IllustrisTNG simulations (Mari-  nacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Naiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Pillepich  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019) the CNN was  used to classify all CFIS galaxies with available SDSS Data Release  7 (DR7) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021) noted that a CNN (or any automated merger  classification method) with an accuracy of < 100% (the CNN de-  ployed in this work has an accuracy of ~88% on test set galaxies)  would invariably fail to produce a pure post-merger sample on ac-  count of Bayesian statistics (Bayes & Price 1763) and the minuscule  prior probability that any given galaxy in the low-redshift universe  will be a post-merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' To address this issue, Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021)  suggest that a hybrid method, in which a subset of galaxies predicted  by the CNN to be post-mergers are subsequently inspected visually,  would offer a reasonable combination of efficiency and reproducibil-  ity without sacrificing the purity of the final post-merger sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The strengths of CNNs and human classifiers are complementary: a  simulation-trained CNN can classify galaxies quickly, rule out a large  number of galaxies unlikely to be mergers, and capture a breadth of  observed merger characteristics spanning the range of remnant stellar  masses, merger mass ratios, redshifts, orbital parameters, and obser-  vational conditions included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Conversely, human  classifiers are capable of meticulous classification with the goal of  sample purity in mind, and are capable of explaining and defending  their decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  As long as the CNN’s training data is observationally realistic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Bottrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ćiprijanović  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2021) and the simulated galaxies exhibit the same morpho-  logical characteristics as galaxies in the low-redshift universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Zanisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2021), visual classifiers  ought to inherit a sample with a high post-merger fraction from  the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Critically, this particular combination of classifiers also  improves over previous post-merger identification efforts in the di-  versity of post-mergers included in the final sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' If less disturbed  post-merger morphologies are included in the CNN’s training set  and given appropriate consideration in the visual classification phase  of the hybrid method, they can be preserved and studied alongside  more visually obvious major mergers as long as their disturbed mor-  phologies are sufficiently bright to be captured at the depth of CFIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Methods using shallower (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', SDSS) imaging, visual classifications  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013) or CNNs trained on visual classifications  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2020) do not reap the same  benefit — even when galaxies are inspected with great care, the most  dramatic merger morphologies always inspire more confidence in  visual classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In this work, we briefly review the simulation-trained CNN and  our method for its deployment in a hybrid classification scheme, as  well as three observational methods (optical spectroscopy, mid-IR  photometry, and radio classification) of AGN identification (Sec-  tion 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We next use the visually confirmed post-merger galaxies  from Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022) to investigate the link between the merger  sequence and the triggering of each AGN type (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1), and in-  vestigate the reasons for differences between the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013)  post-merger results and our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Finally, we select post-merger and  SDSS pair samples of optical AGN, and compare their [OIII] lu-  minosities with those of control AGN in order to approximate the  accretion rate enhancements produced by ongoing and completed  mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We assume cosmological parameters (Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3, ΩΛ0 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7, ℎ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7) when calculating luminosity distances, and for any other  cosmology-dependent quantities appearing in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2022)  AGN in UNIONS post-mergers  3  2 METHODS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 UNIONS data  Our census of AGN in post-mergers would not be possible without  high quality imaging resolution and depth over a large sky area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The  UNIONS collaboration is a new consortium of wide field imaging  surveys of the northern hemisphere and represents an excellent oppor-  tunity for merger searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' UNIONS consists of 1) CFIS conducted at  the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6-meter CFHT on Maunakea, 2) members of the Pan-STARRS  team, and 3) the Wide Imaging with Subaru HyperSuprimeCam of  the Euclid Sky (WISHES) team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021) used only the 𝑟-  band imaging in creating synthetic images, and the CNN predictions  used in this work are made exclusively on CFIS 𝑟-band imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The observing pattern employed by CFIS uses three single-  exposure visits with field-of-view (FOV) offsets in between for op-  timal astrometric and photometric calibration with respect to ob-  serving conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This also ensures that the entire survey footprint,  including areas in the "chip gaps" between MegaCam’s multiple  CCDs for a given exposure, will be visited for at least two exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  After raw images are collected by CFHT, they are detrended (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=',  the bias is removed and the images are flat-fielded using night sky  flats) with the software package MegaPipe (Gwyn 2008, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The  images are next astrometrically calibrated using Gaia data release  2 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2016, 2018) as a reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Pan-  STARRS 3𝜋 𝑟-band photometry (Chambers & Pan-STARRS Team  2016) is used to generate a run-by-run differential calibration across  the MegaCam mosaic, and an image-by-image absolute calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Finally, the individual images are stacked onto an evenly spaced grid  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5-degree-square tiles using Pan-STARRS PS1 stars as in-field  standards for photometric calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The resulting 𝑟-band images  have a typical 5-𝜎 point-source depth of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='85 mag, ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6-arcsecond  seeing, and a pixel size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='187 arcseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In order to prepare CFIS galaxy images for classification, we  cropped them to a physical scale of 100 kpc on a side based on  their redshifts, resized to 138 × 138 pixels, and normalized on a  linear scale with brightnesses of 0 and 1 assigned to the faintest and  brightest pixels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Since this method requires an accurate  spectroscopic redshift in order to crop and scale each galaxy image  appropriately, we restrict our post-merger search to galaxies with 𝑧 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 observed by both the CFIS data release 2 (DR2) and SDSS DR7,  and match the two catalogs with a 2-arcsecond tolerance, yielding  168,597 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Appearances of the same galaxies in the archival  Wide-field Infrared Survey Explorer (WISE) space telescope2 pho-  tometric catalog as well as the NRAO VLA Sky Survey (NVSS) and  Faint Images of the Radio Sky at Twenty-centimeters (FIRST) radio  catalogs will facilitate subsequent characterization of their SMBHs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' After processing, the galaxy images have  the same physical and pixel dimensions as the synthetic images used  for training in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The 𝑟-band imaging data is supplemented by derived measure-  ments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', redshifts) from the MPA-JHU catalog of derived proper-  ties3 from SDSS DR7, stellar mass estimates from Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  (2003a), and corrected optical emission line fluxes from Scudder  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2 we use non-parametric morpholog-  ical statistics from Pawlik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2016) to characterize degrees of  merger-induced disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2 wise2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='edu/docs/release/allsky/  3 wwwmpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='mpa-garching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='de/SDSS/DR7/  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2 The visually confirmed post-merger sample  In Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021), the CNN was trained on realism-added  images of IllustrisTNG post-merger remnants with mass ratios of  at least 1:10, stellar masses between 1010−12 M⊙, and which had  experienced coalescence between the previous and present simula-  tion snapshots (corresponding to a timescale of ≤160 Myr) every  snapshot since simulation 𝑧=1, and control galaxies that had not ex-  perienced a merger in at least 2 Gyr of simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  (2021) demonstrated that the network classified reserved simulated  post-mergers and control images with ~88% accuracy, but cautioned  that high accuracy would not directly lead to a pure predicted post-  merger sample on account of the rarity of post-mergers in the real  Universe (and simulated universes as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' As a result, Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  (2021) explored the utility of progressive cuts in CNN p(x), or the  floating point prediction between 0−1 returned by the network as a  prediction of merger status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In a sample with a realistic proportion  of post-mergers, using p(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 to select mergers would inevitably  lead to a sample of inadequate purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021) argued  that a hybrid approach, in which a more stringent p(x) cut is com-  bined with subsequent visual inspection, could be an efficient way  forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Our CNN was first used to classify real CFIS galaxies in Bickley  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022), and the same classifications are prerequisite to this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The CNN assigned p(x) predictions > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 to 6,778 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Because of the natural rarity of post-merger galaxies, visual inspec-  tion showed the galaxy sample with p(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 to be significantly  contaminated with non-interacting galaxies, a result consistent with  our expectations based on Bayesian statistics (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6 in  Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We therefore chose to visually inspect the 2,000  galaxies with classifications greater than p(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' There is no spe-  cial significance to p(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='75 other than the fact that it produces a  reasonable number of predicted post-mergers for subsequent inspec-  tion by a classification team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In order to identify a pure post-merger  sample, we used a conservative classification method that required a  consensus post-merger classification with mutual agreement by the  authors RWB, SLE, and DRP after careful inspection of the CFIS  galaxy image, the wider field, and spectroscopic companions via  SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Galaxies for which a consensus could not be reached were  discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Of the 2,000 galaxies inspected, 1,201 were rejected by  the classification team on account of doubtful merger status, and a  consensus was not reached for 100 additional galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' After visual  inspection, a sample of 699 galaxies with unambiguous post-merger  morphologies (described fully in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022) were confirmed  unanimously by all three authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The merger sample in Darg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2010) was selected based on  the merger vote fraction of citizen scientist volunteers on SDSS  imaging, and consisted of 3003 visually selected mergers (although  not all are post-coalescence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) inspected 370  galaxies from the Darg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2010) merger catalog which were  strongly perturbed but lacked a spectroscopic companion (suggesting  either a flyby or a completed merger) by eye and selected a smaller  sample of 97 post-mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Our post-merger catalog, first presented  in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022), offers an improvement in post-merger purity  over Darg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2010) and a substantial increase in size compared to  Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The post-mergers studied in this work were also  selected using deeper imaging, and as a result include fainter and  more subdued merger morphologies than could be detected using  SDSS imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  36 post-mergers from this visually confirmed sample are shown  with logarithmic brightness scaling in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' As with all con-  firmed post-mergers in the sample, the galaxies exhibit unambiguous  MNRAS 000, 1–13 (2022)  4  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  merger-induced morphologies: tidal tails, streams, or shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While  absolute knowledge of the status of each galaxy’s nucleus is limited  by the resolution and seeing of the imaging, each galaxy in the sample  appears to have only a single bright nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In addition, we ruled  out any galaxies whose merger-like features could plausibly have  been induced by any nearby object imaged by CFIS, or identified by  projected separation in SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' By eliminating companions both near  and far, we can confidently deduce that the merger features in each  of our 699 galaxies were produced by a partner that has been wholly  accreted at the time of imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The study of star formation in the visually confirmed post-merger  sample in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022) lent credence to the suggestion in  Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2021) that a hybrid approach to post-merger identifi-  cation, starting with a preliminary filtering by the CNN and ending  with rigorous human visual classifications, would be an efficient way  to identify a pure sample of galaxies belonging to the exceedingly  rare post-merger class in a large observational sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Even though  the visually confirmed sample is not free of biases, we maintain that  the Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022) visually confirmed post-merger sample is  likely representative of the remnants of major mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  We also compare the characteristics of the visually confirmed,  larger post-merger sample to those of the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-  mergers, which were visually selected from the larger Darg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  (2010) post-merger catalog by the author SLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Though lesser in  number (97 in the sample before any experiment-specific cuts are  applied), they provide useful context: they represent a highly pure  sample as well, albeit one that is likely biased towards mergers of  greater visual strength due to their selection from shallower SDSS  imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Figure 2 shows the stellar masses and redshifts of both post-merger  samples compared to the parent sample of galaxies from CFIS DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Both merger samples span the dynamic range of CFIS DR2 galaxy  stellar masses, but the new visually confirmed post-merger sample  identifies more distant post-merger galaxies out to 𝑧~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3, whereas  the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-mergers lie below 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The success  of the CNN at identifying galaxies over the full CFIS redshift range  likely stems from its training set, which was composed of simulated  galaxies inserted at redshifts drawn at random from the real CFIS  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The visually confirmed post-mergers also preferentially  lie at high stellar masses compared to the parent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This bias  is discussed at length in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022), and we posit two  candidates for the cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' First, the CNN was trained on galaxies with  masses M★ > 1010 M⊙ in order to ensure that their morphologies  were adequately resolved by the simulation, and it is natural for the  CNN to preferentially select galaxies similar to those on which it was  trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Second, the observability of merger-induced morphologies  depends on the brightness of a galaxy relative to that of the sky, and  more massive galaxies are likely to be brighter in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This  bias was affected during the CNN phase of the Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022)  hybrid merger identification effort, and therefore propagates through  to the visually confirmed post-merger sample as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3 Galaxy pairs  Though this work only addresses the AGN characteristics of ob-  served post-mergers, the characteristics of the post-merger epoch of  galaxy evolution is best contextualized within the merger sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  A sample of pre-coalescence galaxy pairs is therefore required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We  use the galaxy pair sample compiled by Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2016), and  select objects for comparison emulating the method employed in El-  lison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Whenever they are shown for context in this work,  SDSS pairs are required to have projected separations 𝑟 𝑝 ≤ 80ℎ−1  70  kpc, line of sight velocity differences ΔV ≤ 300 km s−1 (in order  to minimize accidental projections of a false pair), and stellar mass  ratios of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 ≤ M1/M2 ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Note that we are applying symmetrical mass ratio criteria to our  pair samples and the post-merger samples used to train our CNN (1:10  or more, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Although merger mass ratio may scale  with the degree of morphological disturbance of merger remnants,  we do not find that the inclusion / exclusion of minor galaxy pairs  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', changing the mass ratio criterion to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='25 ≤ M1/M2 ≤ 4)  has a significant influence on the excesses we calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We also  expect the median progenitor mass ratio of our post-merger sample  to increase following the visual inspection effort detailed in Bickley  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022), since merger candidates of unconvincing strength were  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We therefore present the physical characteristics of merger  remnants alongside galaxy pairs with the caveat that the pairs studied  may not be precise pre-merger analogues to the post-mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  A larger ΔV requirement, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', ≤ 1000 km s−1 could capture  a larger sample of galaxies that could conceivably be experiencing  interactions, but the risk of contamination by non-interacting galaxies  also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In order to restrict our analysis to galaxies that are  very likely to be pre-mergers, we therefore follow Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2016)  and use ΔV ≤ 300 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Regardless of the choice of ΔV, we expect  some contamination from interlopers to persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='4 Control pools  Wherever post-mergers are studied in this paper, we will compare  their physical characteristics to those of control galaxies that are sim-  ilar in mass and redshift, but which are not themselves post-mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  To this end, we again use the Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022) CNN classifica-  tions to our advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The vast majority (131,168 of 168,597) of  galaxies in the CFIS sample are assigned CNN p(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 by the  network, nominally indicating non-post-merger status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Even with an  accurate classifier, Bayesian statistics suggest that a small number  of genuine post-mergers will fall below this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Thanks to the  initial rarity of post-mergers, however, we can reasonably assume that  the extreme dilution of post-mergers below p(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 will effectively  eliminate their influence on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Wherever the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample is shown, their physical  characteristics are compared to those of their own mass- and 𝑧-  matched corresponding control sample with a Galaxy Zoo merger  vote fraction of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' As with the visually confirmed post-merger  sample, we expect that the signal from misclassified post-mergers  contaminating this control sample should be effectively zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The controls for galaxy pairs must have projected separations of  𝑟 𝑝 > 80ℎ−1  70 kpc, and Galaxy Zoo (Darg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2010) merger vote  fractions of zero in order to ensure that they do not belong to an  interacting pair, and have not merged recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The particular control  matching methodologies for each of our experiments are described  in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 Optical (Seyfert II) AGN  Since spectroscopic redshifts are required for appropriate treatment  of our CFIS galaxy images, all post-mergers identified in CFIS 𝑟-  band imaging by the CNN have available SDSS DR7 optical spectra  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Conveniently, we can use emission line measurements from  these same optical spectra to search for the luminous echoes of AGN  in our galaxies’ gaseous narrow-line regions (NLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Galaxies ex-  hibiting the NLR optical emission line characteristics of the Seyfert  II class will hereafter be referred to as "optical AGN".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2022)  AGN in UNIONS post-mergers  5  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' A mosaic of 36 post-mergers from our hybrid (CNN-identified and visually confirmed) post-merger sample, cropped to a physical size of 100 kpc  on a side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' All galaxies in the sample have distinctive merger-induced morphologies that could not be plausibly produced by any discernible object in the CFIS  imaging, or by any spectroscopic companions as identified using SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Furthermore, each of the post-mergers has only a single post-coalescence bright nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Galaxies are shown in log-scale with the contrast adjusted for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In order to quantify the incidence of optical AGN in post-mergers,  as well as estimate the accretion rates of their black holes, we use the  "maximum starburst line" on the Baldwin, Phillips, Terlevich (BPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 1981) diagram computed by Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The line represents the theoretical boundary separating AGN from  starbursts, determined using grids of models with varied metallicities  and ionization parameters, and is more stringent than the criterion of  Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2003b) which is often used to separate purely star-  forming galaxies from those with potential contributions from AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  We use Milky Way and host extinction-corrected optical emission  line measurements (from the SDSS MPA-JHU catalog and Scudder  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2012) to place our target and control galaxies on the BPT  diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In order to avoid contamination from shocks, which are  expected in gas-rich galaxy mergers, we do not count objects on the  AGN side of the Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2001) diagram as AGN if they fall  MNRAS 000, 1–13 (2022)  z: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
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+page_content='116  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The stellar masses and redshifts of the 699 visually confirmed post-  merger galaxies (magenta crosses) studied in the present paper and 97 Ellison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-mergers (black stars) superimposed over CFIS DR2 parent  sample (gradient histogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' CFIS 𝑟-band imaging was processed and used  as the input for CNN classification as well as for visual inspection in (Bickley  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  below either the [SII] or [OI] BPT diagram low-ionization nuclear  emission region (LINER) criteria described in Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  When quantifying AGN excesses (the ratio of the AGN fractions in  mergers and controls) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1, we require S/N ≥ 5 on all four  emission lines used in the BPT diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Consequently, the nebular  emission lines of galaxies that we count in our excess calculations are  dominated by the high-energy ionization associated with the AGN,  and our excess calculations capture the frequency with which mergers  induce strong, unambiguous optical evidence of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Later, when we approximate the SMBH accretion rate with the  luminosity of the [OIII] emission line in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2, we require an  ensemble of 5 or more controls (which must also be optical AGN) for  each merger in order to calculate a robust accretion rate enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The [OIII] emission line has been used many times to approximate  the accretion rates of optically identified AGN, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Brinchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2012),  but [OIII] flux can also be contaminated by star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The link  between [OIII] luminosity and star formation rate is complex, and its  dependence on several parameters is explored in Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Nonetheless, the Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2001) criterion we use is intended  to be a maximal starburst division, such that galaxies above it can  only be produced (in their models) by AGN contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We choose  galaxies for this experiment whose emission is unambiguously AGN-  dominated according to the same Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2001) criterion used  in our excess calculations, and compare the AGN found in mergers to  those found in non-mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The novel size of the visually confirmed  post-merger sample allows us to use the same S/N ≥ 5 cut as we  do when measuring optical AGN excesses, and additionally remove  LINERs using the [SII] and [OI] Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2006) criteria in order  to ensure a robust connection between observed [OIII] luminosity and  SMBH accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6 Infrared AGN  Though they are useful for identifying a particular subset of AGN,  optical emission lines do not represent a complete census of SMBH  activity in low-redshift galaxies (Hickox & Alexander 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' A num-  ber of studies have demonstrated the utility of mid-infrared (mid-IR)  observations to identify dust-obscured AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Lacy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Stern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Donley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hickox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Donley  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Eckart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Stern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Mateos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Many galaxies with ongoing SMBH accretion are obscured by dust  such that their optical emission line strengths cannot be reliably mea-  sured, while at the same time reddening their mid-IR colours (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Assef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In fact, mergers may preferentially produce dust-  obscured AGN (Satyapal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014, Weston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2017, Blecha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Models predict that when one or both companions involved in  the interaction is dusty, the kinematic disturbances induced by the fi-  nal stages of a merger may distribute the dust in the centre, obscuring  optical emission that might be observable in a dynamically settled  galaxy (Yutani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In order to identify dust-obscured AGN, we turn to legacy all-  sky mid-IR observations from the WISE space telescope, and follow  Satyapal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2014) and Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) in deploying a WISE  W1−W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 colour cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Because we are interested in studying  the proportion of galaxies with WISE photometry whose W1−W2  colours exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5, we also require that all galaxies (post-mergers as  identified by any method, galaxy pairs, and controls) included in our  WISE-related results have been detected by WISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' WISE sources  considered in our analysis have S/N of at least 2 for both W1 and  W2, but a more stringent S/N criterion does not change our results  since > 99% of our sources have S/N > 10 for both W1 and W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Mid-  IR observations can be affected by star formation, but even template  spectra for galaxies with extreme star formation have been shown  to fall below W1−W2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 at low 𝑧 (Assef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Satyapal  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Still, our visually-confirmed post-mergers are known to  be enhanced in star formation (Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022), so it remains  conceivable that a small contribution from star formation may affect  the signal in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Caution should therefore be exercised in  direct interpretation of the quantities reported therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7 LERGs  Optical and mid-IR AGN detections together provide a reasonably  complete census of the observational phenomena associated with ra-  diatively efficient, rapidly accreting SMBH with a luminous accretion  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Indeed, these AGN are most closely linked with the archetypal  understanding of the role of the merger sequence in galaxy evolution,  in which gas inflows simultaneously produce upticks in star forma-  tion and SMBH accretion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While useful,  this narrative largely excludes mergers between gas-poor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Although low-excitation AGN states are most often associated with  relatively isotropic accretion of hot gas from galactic halos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Best  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2005, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Hardcastle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Gaspari  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013), there has been some evidence of a role for mergers in  triggering radiatively inefficient accretion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Sabater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Garofalo 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In order to investigate the role of mergers in this context, we also  quantify the proportion of low-excitation radio galaxies (LERGs) in  our merger and pair samples relative to controls selected from the  same SDSS parent sample as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' These low-excitation  radio-selected AGN are taken from the compilation of Best & Heck-  man (2012), who match SDSS to a pair of radio catalogs: NVSS  and FIRST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' After classifying their detections as either star form-  MNRAS 000, 1–13 (2022)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  +  +  Stellar mass, log(Mo)  +  +  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  十  +  +  ++++  +  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  visually confirmed PMs  +  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013 PMs +★  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3  zAGN in UNIONS post-mergers  7  ing or AGN via a combination of 4000Å break strengths, radio-to-  emission-line luminosities, and BPT diagnostics, they use optical  emission lines and a second decision tree to distinguish the latter  into high-excitation radio galaxies (HERGs) and LERGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Because  the high-excitation AGN state is well described by our optical and  mid-IR observations, we consider only the LERGs in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='8 Overlap of AGN types  The visually confirmed post-merger sample, Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013)  post-merger sample, and SDSS galaxy pair sample contain 699,  96, and 17,566 galaxies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The visually confirmed post-  merger sample contains the following:  32 optical AGN meeting the Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2001) and Kewley  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2006) criteria (excluding star-forming, composite, and LIN-  ERs) with S/N > 5 for the emission lines used in BPT placement  66 mid-IR AGN with W1−W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  14 LERGs  17 galaxies that host AGN identifiable using both optical and  mid-IR criteria  No overlap between either the optical or mid-IR AGN and  LERGs  On average, the optical AGN have a WISE colour of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='35, and  the LERGs have a typical WISE colour of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Our choices of  criteria for optical and mid-IR AGN do not by definition preclude the  possibility of a galaxy being a LERG, and such dual-status objects  do appear in the SDSS DR7 and WISE catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' They are rare,  however, and as a result none appear in the visually confirmed post-  merger sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  3 RESULTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 AGN in post-mergers  After identifying our post-merger samples and establishing AGN cri-  teria, we can quantify the strength of the link between merger status  and SMBH incidence by calculating AGN excesses — the ratio of  the AGN fractions in matched samples of post-mergers (or pairs)  and controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' When controls are properly matched and statistics are  of adequate quality, any excess > 1 indicates that the differentiating  characteristic between the two samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', post-merger or pair sta-  tus) is responsible for (or correlated with a factor that is responsible  for) the increased AGN frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In calculating each excess, we match controls to the post-mergers  or pairs (hereafter referred to collectively as "target galaxies") on stel-  lar mass and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Controlling for environment using additional  parameters (halo mass, environment density) does not qualitatively  change our results, suggesting that the influence of the most rele-  vant interacting companion (in the case of galaxy pairs) or coalesced  companion (for the post-mergers) is largely responsible for the signal  uncovered in this Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In order to ensure we have the same num-  ber of controls per target galaxy, we identify the closest non-merger  match (without replacement) in 2-D parameter space for each tar-  get galaxy, assigning equal weight to each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' After finding  one control for each galaxy, a Kolmogorov–Smirnov (K-S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Smirnov  1948) statistic is calculated for the M★ and 𝑧 distributions of the  target galaxies and controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' If the K-S 𝑝 value is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='9, the con-  trol pool is finalized and we calculate the AGN fractions and excess  for the relevant observational type (optical, mid-IR, and LERG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' If  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Optical (top), mid-IR (centre), and LERG (bottom) AGN excesses  in SDSS spectroscopic pairs (blue), the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-merger  sample (teal), and the visually confirmed post-merger sample (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Vertical errors are calculated by adding the inverses of the binomial errors  on each fraction  √︁  𝑓 (1 − 𝑓 )/𝑁 , where f is the AGN fraction in the target  or control sample, and N is the size of that sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Horizontal errors are the  bin widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Our visually confirmed, more inclusive post-merger sample finds  a stronger optical AGN excess, and a weaker mid-IR AGN excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This is  consistent with the hypothesis that more minor mergers, which are present  in the visually confirmed sample and largely absent from the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  (2013) sample, are less likely to be dust-obscured after the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We find  LERG excesses for post-mergers and galaxy pairs consistent with unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Even  with a relatively simple control-matching scheme, the merger sequence does  not appear to be strongly connected to LERG status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  the 𝑝 value remains above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='9, we attempt to match additional equal-  sized batches of controls, checking the K-S 𝑝 value every time before  adding an additional batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We cap the number of controls per post-  merger at 10, as the counting statistics cease to improve significantly  by that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We use the same statistical control matching methodol-  ogy for each target sample (Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022 post-mergers, Ellison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013 post-mergers, and galaxy pairs), allowing for the control  pool (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='4) and number of controls per target galaxy to  vary (between 1−10 depending on the continued success of K-S tests)  between each subset of galaxy pairs (binned by projected separation)  and post-merger sample we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In the target samples and the matched control samples, the optical  AGN fraction is the number of BPT AGN above the Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  (2001) and Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2006) lines with S/N ≥ 5 divided by the total  number of target or control galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The WISE AGN fraction is the  number of galaxies with W1−W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 divided by the total number of  target or control galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' For the WISE excess, the target samples and  control pools are restricted to galaxies with WISE detections, even  though the matching parameters are derived from SDSS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The  MNRAS 000, 1–13 (2022)  4  Optical Excess  visually confirmed PMs  Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013 PMs  3  spectroscopic pairs  2  S  S  WISE Exces  10  5  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  ERG Excess  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='0  20  40  60  80  0  rp, kpc8  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  LERG fraction is the number of LERGs in each sample divided by  the total number of target or control galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' For the LERG excess  we require that the target and control pool galaxies are classified in  the Best & Heckman (2012) catalog as either star forming galaxies,  HERGs, or LERGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' All excesses are the ratio of the AGN fraction  in the target sample and the AGN fraction in the relevant matched  control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 Optical AGN excess  The top panel of Figure 3 shows the optical excesses for spectroscopic  pairs (blue) in 8 bins of projected separation between 0−80ℎ−1  70 kpc,  as well as for Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-mergers (teal) and the visually  confirmed post-merger sample (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In matching controls for  Figure 3, we reached the cap of 10 controls per target (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1  above) in both post-merger samples, and in each bin of projected sep-  aration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We uncover a decreasing trend of optical AGN excess with  increasing 𝑟 𝑝, peaking with an excess of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7 for galaxy pairs sepa-  rated by ≤ 10ℎ−1  70 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Using a more generous optical AGN criterion  (Stasińska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2006, which allows for the inclusion of composite  galaxies with significant contributions from both star formation and  AGN), and a different control matching algorithm (allowing for dif-  ferent numbers of controls to be selected for galaxies belonging to  the same target sample), Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2011) still reported quantita-  tively consistent pair-phase excesses, from ~0−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 for galaxy pairs  spanning separations of 80−0 ℎ−1  70 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We recalculate the post-  merger optical AGN excess in the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample in  order to compare using our updated control-matching method, and  find ~2 times as many optical AGN compared to controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We find  that optical AGN are even more common in the visually confirmed  post-merger sample, with an excess of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2 WISE AGN excess  There are 6,106 SDSS pairs, 364 visually confirmed post-mergers,  and 78 Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-mergers for which we can calculate a  WISE mid-IR colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The middle panel of Figure 3 shows the excess  of mid-IR AGN with W1−W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 in the same three main galaxy  samples as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 compared to matched control samples,  which are themselves required to have WISE detections so that their  WISE W1−W2 colours are calculable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Much like the optical AGN  excess, we find a decreasing excess of mid-IR AGN in bins of 𝑟 𝑝  with a peak value of ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6 for the closest spectroscopic pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' These  results are statistically consistent with the pair phase W1−W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5  results reported in Satyapal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2014), even though a different  control matching methodology was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Our visually confirmed  post-merger sample shows a mid-IR AGN excess consistent with the  closest pairs, but significantly below the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample,  which contains 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3 times as many WISE AGN per capita compared  to the control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Our pair phase and visually confirmed post-  merger sample excesses for both optical AGN and mid-IR AGN are in  reasonable concordance (spanning ~0−4 in the pair phase, and 3−4 in  the post-mergers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This agreement supports the view that the optical  and mid-IR AGN diagnostics are identifying different observational  phenomena associated with the same SMBH engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In order to investigate the discrepancies in optical and mid-IR  AGN excesses between the visually confirmed post-mergers and the  Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-mergers, we investigate differences in the  selection methods used for their identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Because the CNN,  Galaxy Zoo classifications, and expert visual classifications all rely  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Shape asymmetry statistics derived from SDSS imaging for SDSS  galaxies with 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='35 and M★ > 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 M⊙ (grey), the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013)  post-mergers (teal), and the visually confirmed post-mergers (magenta, top  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The vertical bars spanning both panels represent the median of each  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While both more disturbed than the typical SDSS galaxy, the  visually confirmed post-merger samples are less disturbed on average than  the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) mergers, suggesting that a greater number of minor  mergers are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This would be a natural consequence of the CNN’s  inclusive simulated training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The bottom panel shows the AGN fractions  in the SDSS sample as identified by optical (X-markers) and mid-IR (dia-  monds) criteria, see Section 2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The vertical errors are the binomial  errors on each fraction  √︁  𝑓 (1 − 𝑓 )/𝑁 , and the horizontal errors are the bin  widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Since the hybrid method typically selects galaxies with smaller shape  asymmetries, it also selects a proportionally lower fraction of dust-obscured  AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This accounts for the discrepancy between the two post-merger data  points in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  on morphology, we posit that a morphological bias may be respon-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In order to investigate differences in sample morphology, we  use shape asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Shape asymmetry is a non-parametric mor-  phological measurement that takes the asymmetry of a binary mask  that denotes a boundary between the galaxy and the background, de-  tailed in Pawlik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The binary mask used to measure shape  asymmetry is generated following the 8-connected structure detec-  tion method described in Pawlik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2016), in which a galaxy image  is smoothed using a 3×3 running average filter, and pixels above a  limiting surface brightness of approximately 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='7 mag arcsec −2,  equivalent to one standard deviation above the typical sky noise  level in SDSS imaging, are ascribed to the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' It was devel-  oped for the purpose of automated merger identification because  of its emphasis on the particular asymmetry of low-surface bright-  ness features whose importance would be overlooked by a traditional  asymmetry measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The merit of this metric for merger identi-  fication is explored in detail in Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' By deploying  shape asymmetry on galaxies whose post-merger status is already  confirmed, shape asymmetry instead measures the morphological  strength of the post-merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Tidally disturbed post-mergers who have  experienced dramatic, major mergers are more likely to have more  extended morphologies, and higher shape asymmetries compared to  those whose mergers have been relatively tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Shape asymmetry  has also been shown to fade in the several hundred Myr that fol-  MNRAS 000, 1–13 (2022)  5  all SDSS  Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013  p(As)  visually confirmed  WISE  optical  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='10  AGN fractior  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='00  AsAGN in UNIONS post-mergers  9  low coalescence (Pawlik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2016), but so too has it been shown  that the W1−W2 colour decreases after coalescence (Blecha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Shape asymmetry therefore also includes information about  the recency of the merger, along with the initial intensity of the  morphological disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Figure 4 investigates the shape asymmetry demographics of our  two post-merger samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The top panel shows the normalized dis-  tributions of shape asymmetry derived from SDSS imaging for the  entire SDSS DR7 galaxy population with spectroscopic redshifts <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='35, and masses > 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 M⊙ (grey, representing the area of M★−𝑧  parameter space encompassing both the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) and  visually-confirmed post-merger samples, see Figure 2), the Ellison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-merger sample (teal), and the new visually con-  firmed sample (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' It is important to note that shape asym-  metry does not trend strongly with either stellar mass or redshift,  and that the qualitative results of our shape asymmetry study do  not change when we compare our merger samples to their matched  control galaxies from Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2 instead of the SDSS  parent sample used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The median SDSS-derived shape asymme-  try of each sample is plotted over both panels as a dashed line of the  same colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Shape asymmetries derived from CFIS 𝑟-band imaging  are available for the visually confirmed post-mergers, but we present  only SDSS shape asymmetries in order to allow for direct compar-  ison to the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample, which does not appear in  full in CFIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Note that the shapes of the visually confirmed post-  merger sample, while of course more asymmetrical than SDSS in  general, are significantly less disturbed than the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013)  post-mergers, with ¯Δ𝐴𝑆 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We posit that the typical difference in  SDSS-derived shape asymmetry between the two post-merger sam-  ples is the result of the fact that the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-mergers  were identified by strictly visual means in shallow imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Con-  versely, the visually confirmed post-mergers were first identified by a  CNN trained on CFIS-depth simulated imaging of post-mergers with  mass ratios as small as 10:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' It is therefore plausible that a number  of relatively minor post-mergers were preserved by the CNN and  confirmed during visual classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The bottom panel of Figure 4 shows the local optical and WISE  AGN fractions of the SDSS parent sample (with 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='35 and M★ >  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 M⊙) in 10 bins of 𝐴𝑆 between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We find that the optical  AGN fraction is generally low and consistent with increasing shape  asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While the WISE AGN fraction does not trend monoton-  ically with 𝐴𝑆, the data show that more morphologically disturbed  galaxies in SDSS are indeed more likely to host an AGN that is iden-  tifiable by its mid-IR colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' These results indicate that the degree of  disturbance is unlikely to have a strong impact on the optical AGN  fraction, but this is not true for WISE AGN, since more disturbed  galaxies typically have higher mid-IR AGN fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Consequently,  the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample is more likely to contain highly dis-  turbed post-mergers, which host proportionally more dust-obscured  AGN and a consistent number of optical AGN, while the visually  confirmed sample is more inclusive of less-disturbed mergers, which  are less likely to host mid-IR AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  If the degree of morphological disturbance is responsible for the  difference in mid-IR AGN excess between the two post-merger sam-  ples, a subset of the visually confirmed post-mergers with the same  SDSS shape asymmetry demographics as the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013)  post-mergers ought to exhibit an excess that is in better agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In order to test this hypothesis, we match exactly one galaxy (with-  out replacement) from the visually confirmed post-merger sample to  each Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-merger on shape asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Where  multiple visually confirmed post-mergers have shape asymmetries  within ±5% of an Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-merger, we select the sin-  gle best match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Where there are no matches, we grow the tolerance  from 5% until a single match can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Of the 85 galaxies in the  Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample with shape asymmetries available, 82  have a match within 5% of their shape asymmetry in the visually con-  firmed sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The remaining 3 galaxies require 2, 3, and 5 growths,  respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' they are included for completeness but their exclusion  does not affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The shape asymmetry-matched visually  confirmed post-mergers have an optical AGN excess consistent with  the visually confirmed sample taken as a whole, but their mid-IR  AGN excess is increased from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6 up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While still not in perfect  agreement with the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample, this experiment con-  firms that the degree of morphological disturbance ( ¯Δ𝐴𝑆 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='13) and  sample selection are linked to an increased likelihood of a mid-IR  AGN detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This result is consistent with the physical narrative  presented in Yutani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022), wherein rapidly accreting AGN in  extremely recent (within ~4Myr) post-coalescence systems are more  likely to be observed as dust-obscured galaxies (DOGs) on account  of central and/or galaxy-scale dispersion of dust from the progenitor  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We posit that the longevity and intensity of the dust ob-  scuration may scale with the dynamic intensity of the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' As a  result, the specific method used to identify mergers based on their  morphology has a significant impact on the quantitative excesses we  calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3 LERGs in post-mergers  The bottom panel of Figure 3 investigates the role of mergers in  triggering LERGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In our mass- and redshift-matched study, we find  that LERGs are no more likely to exist in our post-merger or pair  samples than in controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This result is qualitatively discrepant with  the Pace & Salim (2014) finding that galaxies hosting radio AGN have  a 50% excess in the number of satellites, and the link between tidal  forces associated with pair phase interactions and LERG incidence  suggested by Sabater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2015) find a modest  pair phase LERG excess of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='4, and a small excess of ~4±2  (nearly consistent with unity as well) in the post-merger phase when  they match controls on stellar mass and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The lack of an  elevated LERG incidence rate in post-mergers or galaxy pairs in this  work is therefore in mild tension with the literature, even though the  conditions in a post-merger system are certainly not required for the  triggering of LERGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2 Optical AGN accretion rate enhancements  In addition to the initial triggering of AGN, we can use our post-  merger sample to determine typical merger-induced accretion rate  enhancements in optically-identified AGN, using [OIII] luminos-  ity as a proxy for accretion rate (see also Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Brinchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' As stated  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5, we again use a S/N criterion of at least 5 for the four  BPT emission lines, and explicitly disallow LINERS using the [SII]  and [OI] criteria of Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Because we are computing  a luminosity enhancement, we require an ensemble of at least 5 AGN  controls for each target galaxy in order to compare the luminosity of  each AGN post-merger or pair to a group of non-merger or non-pair  counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Rather than finding the nearest controls in parameter  space, we set initial tolerances of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 dex in M★ and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='05 in 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In practice, all of our target galaxies (pairs and post-mergers alike)  find at least 5 controls without any growths in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 31  post-mergers from the visually confirmed sample, 3 from the Elli-  son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample, and 263 SDSS pairs in bins of separation  MNRAS 000, 1–13 (2022)  10  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' [OIII] luminosities and luminosity enhancements in post-mergers  and pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The top panel shows log-scale [OIII] luminosity histograms for  optical AGN in the the galaxy pair sample described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='3 (blue), the  visually confirmed post-merger sample (magenta), the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013)  post-merger sample (teal), and the optical AGN control pool for the visually  confirmed post-merger sample (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The bottom panel shows Δlog(L[OIII])  for the same three target samples (galaxy pairs, visually confirmed post-  mergers, and Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013 post-mergers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Vertical error bars are the  statistical error on the median, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='253𝜎/  √  𝑁 , and horizontal error bars are  the bin widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We find enhancements approximately consistent with zero in  the pair phase, with some small local suppressions past 40ℎ−1  70 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In both  post-merger samples, we find significant positive excesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Post-mergers in  the visually confirmed sample are ~2 times as luminous in [OIII] as their  non-post-merger controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  between 0−80ℎ−1  70 kpc with mass ratios 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='1 ≤ M1/M2 ≤ 10 selected  from the Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2016) catalog with non-LINER optical AGN  and S/N of at least 5 on all emission lines used for placement on  the BPT diagram are ultimately included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The [OIII] luminosity en-  hancement, Δlog(L[OIII]), is calculated as the difference between the  logged [OIII] luminosity (in units of erg s−1) of the target galaxy and  the median logged luminosity of the control ensemble, and hence  captures the typical accretion rate difference between AGN triggered  my mergers and secular AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Figure 5, which shows L[OIII] (top panel) and Δlog(L[OIII]) (bot-  tom panel) for optical AGN in spectroscopic pairs, the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  (2013) post-mergers, and the visually confirmed post-merger sample,  suggests that accretion rates in AGN hosted by galaxies with a close  companion are consistent with isolated AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Conversely, both post-  merger samples are significantly enhanced compared to the matched  non-merger AGN control ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The visually confirmed post-  merger sample is in fact ~2 times as luminous in [OIII] on average  compared to non-post-merger controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The accretion rate enhance-  ments for optical AGN ushered in by coalescence therefore appear  to be significant, while those produced by pair-phase interactions  may just as likely be produced by secular or ambient processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' A  higher positive enhancement of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='4 dex is recovered for the three  post-mergers that remain after applying our quality control cuts to  the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) post-mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' While the sample size does  not invite extensive interpretation, it is possible that the increased  morphological disturbance that is typical of the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013)  sample is linked to more rapid gas infall and elevated accretion rates  in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The relationship between star formation rate and L[OIII] is of order  unity (with significant scatter) in BPT star forming galaxies beneath  the Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2003b) criterion with S/N>5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Therefore, in  order to achieve similar Δlog(L[OIII]), star forming galaxies would  need to have approximately doubled star formation rates (SFRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Still, since we select galaxies whose nebular emission could not  plausibly be produced by star formation alone, the [OIII] luminosity  enhancements uncovered in this Section are primarily indicative of  SMBH accretion-driven ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The result does not appear to be an effect of the SDSS fiber aper-  ture on the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) sample, which lies at low-𝑧 relative  to the visually confirmed sample (see Figure 2) as we find no corre-  lation of Δlog(L[OIII]) with 𝑧 for individual galaxies in the visually  confirmed sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' In the sample of Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2001) AGN with  S/N ≥ 5 for the four BPT emission lines and stellar masses be-  tween 1010−12 M⊙, the median measured [OIII] luminosity actually  increases from 1039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2 to 1041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='6 between 0 ≤ 𝑧 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The fiber  aperture effect therefore gives rise to higher [OIII] luminosities at  higher 𝑧, and moreover, our control-matching methodology accounts  for systematic changes in [OIII] luminosity with stellar mass and  redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Aperture effects (or more broadly, any redshift or stellar  mass effects) are not responsible for the difference in Δlog(L[OIII])  calculated between the Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013) and visually confirmed  post-mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  4 SUMMARY  In this work, we have used the CNN-identified and visually confirmed  post-merger sample introduced in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022) to study the  triggering and accretion of supermassive black holes in post-merger  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We also offer pair phase results in order to contextualize the  post-merger results within the merger sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We match control  (non-post-merger or non-pair) galaxies on M★ and 𝑧 to our target  (either post-merger or pair) galaxies in order to study the AGN excess  — that is, the ratio of the AGN fractions in the target sample and the  control sample — using optical narrow-line region (via SDSS, the  BPT diagram, and the Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2001 AGN selection), mid-IR  (via WISE and the colour criterion W1−W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='5 used by Satyapal  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2014 to select dust-obscured AGN), and LERG classifications  (capturing the low-excitation, isotropic SMBH accretion mode, as  determined by Best & Heckman 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We report the following:  We find that optical and mid-IR AGN excesses in the pair sample  increase from ~1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=', no excess) to ~2−4 as their projected separa-  tions decrease from 𝑟 𝑝~80 ℎ−1  70 kpc down to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We find that galaxy  pairs and mergers do not preferentially exhibit the characteristics of  LERGs, in mild tension with the literature (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  We find optical and mid-IR AGN excesses in a new sample of  visually confirmed post-mergers of ~3 − 4 over controls matched on  mass and redshift, suggesting that the nuclear conditions ushered in  by post-mergers increase the likelihood that an energetic AGN will  be triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  We find a significant connection between high shape asymme-  try derived from SDSS imaging (roughly analogous to the degree of  merger disturbance) and a galaxy’s likelihood to host a dust-obscured  MNRAS 000, 1–13 (2022)  PM control sample  galaxies  102  101  #  100  38  40  42  log(L[ol, erg/s)  Alog(L[ol), dex  visually confirmed PMs  Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2013 PMs  spectroscopic pairs  20  40  60  80  rp, kpcAGN in UNIONS post-mergers  11  AGN in an inclusive sample of SDSS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This is most likely  related to the tendency of mergers to disperse the central gas and dust  belonging to their participant galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This connection contributes  to the quantitative differences in the optical and mid-IR AGN ex-  cesses of our visually confirmed merger sample and that of Ellison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2013), which is composed of more visually dramatic merger  examples with higher shape asymmetries on average (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Following a number of efforts in the literature (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='2),  we use [OIII] luminosity as a proxy for SMBH accretion rate in a  sample of optical AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We find that optical AGN hosted by inter-  acting galaxy pairs are not preferentially enhanced in their accretion  rates (as measured by L[OIII]) compared with secularly driven AGN  in isolated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Conversely, we find our visually confirmed post-  merger sample to be ~2 times as bright in [OIII] than the AGN in  isolated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This suggests that the typical accretion rate en-  hancements produced during the pair phase of the merger sequence  are just as likely to be produced by secular or ambient processes  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' halo gas accretion, secular gas accretion from stellar winds or  supernovae), while the post-merger phase produces significant ac-  cretion rate enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This result, as well as our excess results,  support an important role for the post-merger epoch in triggering and  growing the SMBHs residing at the core of every galaxy (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  In addition to the above detailed census of AGN in post-  coalescence galaxies, we can also revisit the merit of a hybrid (CNN  plus human visual classification) post-merger identification frame-  work, which has allowed us to improve on the statistics of literature  studies of post-mergers, and propose revisions to other results whose  quantities were more heavily influenced by the selection functions of  their merger identification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Because the visually confirmed  CFIS merger sample is biased only by the training of the CNN and  the decisions of the visual classification team, we believe the galaxies  themselves (catalogued in Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 2022) will continue to pro-  vide value in the form of subsequent cross-survey characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Moreover, the hybrid classification framework itself shows promise  for future questions in astronomy surrounding rare and elusive ob-  servational phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The work detailed above was conducted at the University of Victoria  in Victoria, British Columbia, as well as in the Township of Esquimalt  in Greater Victoria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We acknowledge with respect the Lekwungen  peoples on whose unceded traditional territory the university stands,  and the Songhees, Esquimalt and ¯WS ´ANE ´C peoples who have stew-  arded the land for centuries and continue to do so today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  CFIS is conducted at the Canada-France-Hawaii Telescope on  Maunakea in Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' We also recognize and acknowledge with re-  spect the cultural importance of the summit of Maunakea to a broad  cross section of the Native Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  We thank Samir Salim, Christopher Agostino, and Connor Bottrell  for their indespensible feedback on this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  This work is based on data obtained as part of the Canada-  France Imaging Survey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' a CFHT large program of the National  Research Council of Canada and the French Centre National  de la Recherche Scientifique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' and on observations obtained with  MegaPrime/MegaCam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' a joint project of CFHT and CEA Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' at  the Canada-France-Hawaii Telescope (CFHT) which is operated by  the National Research Council (NRC) of Canada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the Institut Na-  tional des Science de l’Univers (INSU) of the Centre National de  la Recherche Scientifique (CNRS) of France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' and the University of  Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' This research used the facilities of the Canadian Astronomy  Data Centre operated by the National Research Council of Canada  with the support of the Canadian Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Data from the IllustrisTNG simulations are integral to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  We thank the Illustris Collaboration for making these data available  to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  Funding for the SDSS and SDSS-II has been provided by the Alfred  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Sloan Foundation, the Participating Institutions, the National Sci-  ence Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Department of Energy, the National Aero-  nautics and Space Administration, the Japanese Monbukagakusho,  the Max Planck Society, and the Higher Education Funding Council  for England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The SDSS Web Site is http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The SDSS  is managed by the Astrophysical Research Consortium for the Partic-  ipating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' The Participating Institutions are the American  Museum of Natural History,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Astrophysical Institute Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Uni-  versity of Basel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Case Western Reserve  University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Drexel University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Fermilab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the  Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the Japan Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Johns  Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the Joint Institute for Nuclear Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the  Kavli Institute for Particle Astrophysics and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the Korean  Scientist Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the Chinese Academy of Sciences (LAMOST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Los  Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the Max-Planck-Institute for Astron-  omy (MPIA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the Max-Planck-Institute for Astrophysics (MPA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' New  Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' University of Pitts-  burgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' the United  States Naval Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' and the University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  This research was enabled, in part, by the computing resources  provided by Compute Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  DATA AVAILABILITY  Simulation data from TNG100-1 used in the generation of training  images for this work are openly available on the IllustrisTNG website,  at tng-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='org/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Template versions of RealSim and RealSim-  CFIS, developed by Connor Bottrell with modifications by RWB are  publicly available via GitHub at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='com/cbottrell/RealSim and  github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='com/cbottrell/RealSim-CFIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Specific image training data  used to develop the findings of this study are available by request  from RWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  This publication makes use of data products from the Wide-field  Infrared Survey Explorer, which is a joint project of the University  of California, Los Angeles, and the Jet Propulsion Laboratory / Cal-  ifornia Institute of Technology, funded by the National Aeronautics  and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Data from the Sloan Digital Sky Survey  are available at sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The public visually confirmed post-merger catalog is available via  MNRAS as a digital resource along with Bickley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content='  The Canada France Imaging Survey is a legacy survey for the  Canadian and French communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' All data are proprietary within  the Canadian and French communities until Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' 1st, 2023 when the  individual frames will become available to the worldwide commu-  nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
+page_content=' Advanced curated CFIS data products will be made available to  the world as part of UNIONS public data releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE2T4oBgHgl3EQfIgZc/content/2301.03681v1.pdf'}
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+Automated Arrangements of Multi-Part Music for Sets of Monophonic
+Instruments
+MATTHEW MCCLOSKEY, GABRIELLE CURCIO, AMULYA BADINENI, KEVIN MCGRATH, and DIM-
+ITRIS PAPAMICHAIL, The College of New Jersey, USA
+Arranging music for a different set of instruments that it was originally written for is traditionally a tedious and time-consuming
+process, performed by experts with intricate knowledge of the specific instruments and involving significant experimentation. In this
+paper we study the problem of automating music arrangements for music pieces written for monophonic instruments or voices. We
+designed and implemented an algorithm that can always produce a music arrangement when feasible by transposing the music piece
+to a different scale, permuting the assigned parts to instruments/voices, and transposing individual parts by one or more octaves. We
+also published open source software written in Python that processes MusicXML files and allows musicians to experiment with music
+arrangements. It is our hope that our software can serve as a platform for future extensions that will include music reductions and
+inclusion of polyphonic instruments.
+CCS Concepts: • Applied computing → Sound and music computing.
+Additional Key Words and Phrases: music arrangement, music algorithms
+ACM Reference Format:
+Matthew McCloskey, Gabrielle Curcio, Amulya Badineni, Kevin McGrath, and Dimitris Papamichail. 2023. Automated Arrangements
+of Multi-Part Music for Sets of Monophonic Instruments. 1, 1 (January 2023), 7 pages. https://doi.org/XXXXXXX.XXXXXXX
+1
+INTRODUCTION
+Music arrangements involve the adaptation of a piece of music for different instruments or ensembles. This allows the
+music to be performed in a variety of settings, enhances the repertory of musicians, and can also help to bring new life
+to a piece that may have been composed for a specific instrument or ensemble [16]. Additionally, arrangements can
+help to showcase the unique strengths of different instruments or even create entirely new interpretations of a piece.
+The process of arranging a piece of music can be a creative endeavor in itself, giving the arranger the opportunity to
+put their own spin on a familiar work, greatly enhancing the listening experience for audiences [1, 5, 12].
+The computational complexity of arranging music written for a set of instruments toward a target single instrument,
+often employing reasonable reductive constraints, has been examined in the work of Moses and Demaine [4]. Complexi-
+ties of dealing with polyphonic instruments, such as piano and guitar, include the need of considering possible fingerings
+as well as reductions, the elimination of certain notes for playability of even feasibility. Most research in automating
+music arrangements has concentrated on the piano, primarily concerning orchestral pieces [2, 8, 10, 11, 13, 14]. Much
+of that work involves reductions to enable feasibility. Other work in the field has examined arrangements for the guitar
+[6, 7, 15], wind ensembles [9], and other orchestral instruments [3].
+Authors’ address: Matthew McCloskey; Gabrielle Curcio; Amulya Badineni; Kevin McGrath; Dimitris Papamichail, papamicd@tcnj.edu, The College of
+New Jersey, 2000 Pennington Road, Ewing, New Jersey, USA, 08618.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not
+made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components
+of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to
+redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2023 Association for Computing Machinery.
+Manuscript submitted to ACM
+Manuscript submitted to ACM
+1
+arXiv:2301.12084v1  [cs.SD]  28 Jan 2023
+
+2
+McCloskey and Curcio, et al.
+Fig. 1. Approximate sounding ranges of instruments and voices. Figure reproduced with permission from Dr. Brian Blood (dol-
+metch.com)
+Despite its obvious benefits, we are not aware of any published algorithm or widely available software that allows
+for the automated arrangement of a given music piece to a different set of instruments that it was originally written
+for in the general case. Working toward filling that need, we designed and implemented an algorithm that arranges
+music written for monophonic instruments and guarantees a successful outcome when an arrangement is possible
+without score reduction. Our recursive backtracking algorithm exhaustively examines all feasible assignments of parts
+to available instruments and all possible transpositions of the piece, including independent octave transpositions of
+individual parts, to determine a successful arrangement that minimally affects the musicality of the piece.
+2
+METHODS
+2.1
+Definitions
+For the purposes of our research, a music piece is written in a chromatic scale and notes are separated by the interval of
+a semitone. We will assume that all notes fall within a total range of 88 semitones, the notes of a traditional piano, from
+A0 to C8. We will assign an integer to each note in the range, such that all notes can be represented by an integer from
+1 to 88. For our discussion, a monophonic instrument is one that can only play one pitch at a time, such as the flute, the
+oboe, or a voice. Polyphonic instruments can play multiple notes simultaneously, such as the piano, guitar, or harp. A
+polyphonic instrument can always play a monophonic part within its range.
+For our study an input music piece will consist of 𝑛 parts, each being assigned to a single monophonic instrument or
+voice. Such parts are presented in the sheet music representation of the piece in an equal number of staves each. Our
+Manuscript submitted to ACM
+
+Haxp
+Accordian
+Guitar
+Xylophone
+Tinpani
+Chimes
+Piccolo
+Flute
+Sop sax-
+Alio sax 
+Tenor sax
+Baritone sax
+Bass sax
+Approximate
+Sop Clarinet
+Alte Clarinet
+Bass Clariet -
+Sounding
+Obee
+English Hern
+Ranges
+Bassoon
+Trurpet& Cornet
+FrenchHorn
+Trombone&Euphoiun
+Tuba
+Miolin-
+Viola
+Cello
+Soprao voice
+Altovoice
+Tenor
+Baritone
+Bass voice
+Piano KeyboardAutomated Arrangements of Multi-Part Music for Sets of Monophonic Instruments
+3
+clarinet = 1
+tenor-sax = 2
+alto-sax = 2
+(a) An example arrangement file
+[alto-sax]
+name = "AltoSaxophone"
+minimum = "Db3"
+maximum = "Bb5"
+key = "Eb"
+(b) An entry in the instrument metadata file
+Fig. 2. Examples of input instrument set and instrument information files
+algorithm preserves the rhythm, rhythmic values of notes and rests, as well as bar lines of the music piece. Clefs, key
+signatures and accidentals are adjusted based on the scale of the transposed music and the instruments/voices that
+parts are assigned to. Our algorithm does not control for instrument timbre that may be expected in any part of the
+music; similarly, the thickness of the piece is not being necessarily maintained.
+We will assume that an input music piece is originally written for 𝑛 instruments 𝐼1, 𝐼2, · · · , 𝐼𝑛, each assigned to play a
+part 𝑃𝑖 of the piece, with 1 ≤ 𝑖 ≤ 𝑛. We seek to arrange the music for 𝑛 output instruments 𝑂1,𝑂2, · · · ,𝑂𝑛. The range
+of each part 𝑖 is an integer interval 𝑅𝑖 = ⟦𝑎𝑖,𝑏𝑖⟧, where 𝑎𝑖 is the integer value corresponding to the lowest frequency
+note and 𝑏𝑖 to the highest frequency note played by instrument 𝐼𝑖 in part 𝑃𝑖, 1 ≤ 𝑖 ≤ 𝑛. Likewise, the playing range of
+each output instrument 𝑂𝑖 will be denoted by 𝑂𝑅𝑖, 1 ≤ 𝑖 ≤ 𝑛, indicating the integer interval of values corresponding to
+the notes the instrument is able to play. Approximate ranges for a set of instruments and voices can be seen in Figure 1.
+2.2
+Monophonic instrument set arrangement algorithm
+Our Monophonic Music Arrangement (MMS) algorithm performs a nearly comprehensive search of possible permuta-
+tions of parts. The music is transposed to all twelve keys, and the algorithm runs on each key, unless a solution has
+been found so far that results in fewer sharps/flats over all keys for each part. This is designed to prevent the "ideal"
+transposition from having a complex key signature if not necessary. Other than that, the search is fully comprehensive.
+For each part, the algorithm finds all possible transpositions of each part in the source piece that can be played by
+at least one available instrument. All permutations of these possible transpositions are then examined. If all parts
+can be played by at least one instrument, the algorithm then checks if there exists a set of part assignments that is
+valid. This is performed by a recursive function that is memoized to improve performance. If a transposed key yields
+valid permutations, the transposition with the least total deviation from the original composition is selected. Once all
+twelve keys have been checked, all permutations are tried using the selected transposition, unless there is no selected
+transposition, in which case the algorithm fails. All permutations are checked, and for those that are valid in the given
+transposition, the best arrangement is selected based on how closely the average pitch of each part matches the median
+pitch of the instrument’s range.
+The MMA algorithm implementation consists of four main function described in pseudocode below.
+2.3
+Implementation
+The MMA algorithm was implemented in Python utilizing the Music21 library and the MuseScore software. Our
+program requires two input files and produces a single output file with the music arrangement. The required input files
+consist of the original piece of music in MusicXML format and a TOML file listing the instrument set to arrange for,
+where an assigned value of 𝑘 to an instrument indicates 𝑘 parts should be arranged for that instrument. An example of
+a TOML file with an input instrument set consisting of one clarinet, two tenor saxophones, and two alto saxophones is
+Manuscript submitted to ACM
+
+4
+McCloskey and Curcio, et al.
+Algorithm 1 Find Transposed Options
+procedure FindTransposedOptions(𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑆𝑡𝑟𝑒𝑎𝑚,𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑃𝑎𝑟𝑡𝑠,𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠)
+𝑠𝑡𝑟𝑒𝑎𝑚 ←− 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑆𝑡𝑟𝑒𝑎𝑚 transposed by given 𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠
+𝑝𝑎𝑟𝑡𝑠 ←− new list
+for 𝑝𝑎𝑟𝑡 in 𝑠𝑡𝑟𝑒𝑎𝑚 do
+𝑐ℎ𝑜𝑖𝑐𝑒𝑠 ←− new list
+for each 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 do
+𝑠𝑒𝑡 ←− the subset of 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑃𝑎𝑟𝑡𝑠 that can play at this transposition
+add (𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠 + 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,𝑠𝑒𝑡) to 𝑐ℎ𝑜𝑖𝑐𝑒𝑠
+end for
+if 𝑐ℎ𝑜𝑖𝑐𝑒𝑠 is empty then
+return null
+end if
+add 𝑐ℎ𝑜𝑖𝑐𝑒𝑠 to 𝑝𝑎𝑟𝑡𝑠
+end for
+return 𝑝𝑎𝑟𝑡𝑠
+end procedure
+Algorithm 2 Run Transposed
+procedure RunTransposed(𝑠𝑡𝑟𝑒𝑎𝑚, 𝑝𝑎𝑟𝑡𝑠,𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠)
+𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 ←− new list
+for 𝑜𝑝𝑡𝑖𝑜𝑛 in all possible transpositions from FindTransposedOptions(𝑠𝑡𝑟𝑒𝑎𝑚, 𝑝𝑎𝑟𝑡𝑠,𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠) do
+𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑 ←− new list
+𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 ←− new list
+for 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 in 𝑜𝑝𝑡𝑖𝑜𝑛 do
+add set of parts covered to 𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑
+add deviation of transposition to 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛
+end for
+𝑎𝑙𝑙𝑃𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑 ←− the union of all sets in 𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑
+if 𝑎𝑙𝑙𝑃𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑 contains all parts and𝑉𝑎𝑙𝑖𝑑𝑎𝑡𝑒𝐴𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡(𝑝𝑎𝑟𝑡𝑠, 𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑, 𝑎𝑙𝑙𝑃𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑) then
+add 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 to 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠
+end if
+end for
+return 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠
+end procedure
+shown in Figure 2a. Metadata about each instrument, consisting of its key in notation and a reasonable note range,
+is defined in a separate TOML file which is loaded separately by the program and is populated with common music
+instruments. An example of an entry for the alto saxophone in the instrument metadata file is shown in Figure 2b.
+During execution our program checks whether the number of input instruments matches the number of parts in the
+piece, and then attempts to arrange for the given instruments as previously described. If arrangements are found, the
+best arrangement based on the criteria described in section 2.2 is output as a MusicXML file. If no feasible arrangement
+is found, or if the number of instruments does not match, then an error message is displayed and no output file is
+produced.
+Manuscript submitted to ACM
+
+Automated Arrangements of Multi-Part Music for Sets of Monophonic Instruments
+5
+Algorithm 3 Find Best Choice
+procedure FindBestChoice(𝑠𝑡𝑟𝑒𝑎𝑚, 𝑝𝑎𝑟𝑡𝑠)
+𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− null
+𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 ←− ∞
+for 𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠 from −6 through 5 do
+𝑠ℎ𝑎𝑟𝑝𝑠 ←− the total number of sharps/flats that would appear in the key signature for each part
+if 𝑠ℎ𝑎𝑟𝑝𝑠 ≤ 𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 then
+𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− element from RunTransposed(𝑠𝑡𝑟𝑒𝑎𝑚, 𝑝𝑎𝑟𝑡𝑠,𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠) with the least deviation
+if 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ≠ null and either 𝑠ℎ𝑎𝑟𝑝𝑠 < 𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 or deviation of 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 < deviation of
+𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 then
+𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒
+𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 ←− 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠
+end if
+end if
+end for
+return 𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒
+end procedure
+Algorithm 4 MMA Algorithm
+procedure MMA(𝑠𝑡𝑟𝑒𝑎𝑚, 𝑝𝑎𝑟𝑡𝑠)
+𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− FindBestChoice(𝑠𝑡𝑟𝑒𝑎𝑚, 𝑝𝑎𝑟𝑡𝑠)
+if 𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 = null then
+return null
+end if
+transpose each part by the resulting transposition
+𝑏𝑒𝑠𝑡𝐹𝑖𝑡 ←− ∞
+for each permutation of 𝑛𝑒𝑤𝑃𝑎𝑟𝑡𝑠 do
+if all parts are valid in the given permutation then
+𝑓 𝑖𝑡 ←− the total absolute difference between the average pitches and the median pitch of each part
+if 𝑓 𝑖𝑡 < 𝑏𝑒𝑠𝑡𝐹𝑖𝑡 then
+𝑏𝑒𝑠𝑡𝐹𝑖𝑡 ←− 𝑓 𝑖𝑡
+𝑏𝑒𝑠𝑡𝑃𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 ←− this permutation
+end if
+end if
+end for
+return 𝑏𝑒𝑠𝑡𝑃𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛
+end procedure
+3
+RESULTS
+We tested our software on a variety of music pieces written for monophonic instruments. In Figure 3 we show three
+measures, starting at measure 16, of the Puttin’ on the Ritz song by Irving Berlin. Part (a) shows the input score composed
+of four monophonic parts. Part (b) displays the arranged piece for saxophone quartet, consisting of a soprano, alto,
+tenor, and baritone saxophones. Similarly, in Figure 4 we display three measures of Carol of the Bells, as arranged and
+performed by the Pentatonix voice group, starting at measure 18 of the piece.
+Complete input/output files for three test cases of our software, including the Puttin’ on the Ritz and Carol of the Bells
+above, can be examined at: https://owd.tcnj.edu/∼papamicd/music/mma/examples/
+Manuscript submitted to ACM
+
+6
+McCloskey and Curcio, et al.
+The repository for this project can be found at: https://github.com/spazzylemons/music-arrangement/
+4
+CONCLUSIONS AND FUTURE WORK
+Our monophonic music arrangement algorithm and its software implementation create a platform for automating
+music arrangements with minimal user input. Although currently basic in its functionality, it can be readily extended in
+a number of different directions. For accommodating arrangements for a smaller sets of instruments than the number
+of parts in the music, score reduction techniques can be applied to eliminate certain parts or at least reduce the number
+of simultaneous notes that are played throughout the piece, while maintaining faithfulness to the original. To allow for
+the inclusion of polyphonic instruments in the arrangements, further work is required in analyzing and decomposing
+(a) Original Score
+(b) Arranged score
+Fig. 3. Three measures from an arrangement of ’Puttin’ on the Ritz’ from piano to saxophone quartet
+(a) Original Score
+(b) Arranged score
+Fig. 4. Three measures from an arrangement of ’Carol of the Bells’ from voices to saxophone quartet
+Manuscript submitted to ACM
+
+16
+112
+Pno.
+my
+112
+Pno.
+112
+Pno.
+0
+ff
+mf
+112
+0
+Pno.
+G16
+2
+SSax
+mf
+2
+ASax
+m
+2
+T Sax
+ff
+mf
+12
+Bar Sax
+418
+Pno.
+f
+Pno.
+Pno.
+Pno.18
+S Sax
+.ff
+A Sax
+mf
+T Sax
+Bar SaxAutomated Arrangements of Multi-Part Music for Sets of Monophonic Instruments
+7
+polyphonic parts into monophonic ones and inversely, while adhering to constraints related to fingerings and other
+instrument and player restrictions.
+ACKNOWLEDGMENTS
+The authors acknowledge use of the ELSA high performance computing cluster at The College of New Jersey for
+conducting the research reported in this paper. This cluster is funded in part by the National Science Foundation under
+grant numbers OAC-1826915 and OAC-1828163.
+REFERENCES
+[1] D. Baker. 1988. David Baker’s Arranging & Composing: For the Small Ensemble, Jazz, R & B, Jazz-rock. Alfred Publishing Company.
+https:
+//books.google.com/books?id=Le0EnwEACAAJ
+[2] Shih Chuan Chiu, Man Kwan Shan, and Jiun Long Huang. 2009. Automatic system for the arrangement of piano reductions. In ISM 2009 - 11th IEEE
+International Symposium on Multimedia. https://doi.org/10.1109/ISM.2009.105
+[3] Léopold Crestel and Philippe Esling. 2019. Live orchestral piano, a system for real-time orchestral music generation. In Proceedings of the 14th Sound
+and Music Computing Conference 2017, SMC 2017. arXiv:1609.01203
+[4] Erik D. Demaine and William S. Moses. 2017. 364Computational Complexity of Arranging Music. In The Mathematics of Various Entertaining Subjects:
+Research in Games, Graphs, Counting, and Complexity, Volume 2. Princeton University Press. https://doi.org/10.23943/princeton/9780691171920.003.
+0019
+[5] J.B. Elder. 2018. The Art of Arranging and Orchestration. Independently Published. https://books.google.com/books?id=mOQKugEACAAJ
+[6] Gen Hori, Hirokazu Kameoka, and Shigeki Sagayama. 2013. Input-output HMM applied to automatic arrangement for guitars. Journal of Information
+Processing (2013). https://doi.org/10.2197/ipsjjip.21.264
+[7] Gen Hori, Yuma Yoshinaga, Satoru Fukayama, Hirokazu Kameoka, and Shigeki Sagayama. 2012. Automatic arrangement for guitars using hidden
+Markov model. Proceedings of 9th Sound and Music Computing Conference (SMC2012) (7 2012), 450–455.
+[8] Jiun-Long Huang, Shih-Chuan Chiu, and Man-Kwan Shan. 2012. Towards an Automatic Music Arrangement Framework Using Score Reduction.
+ACM Trans. Multimedia Comput. Commun. Appl. 8, 1, Article 8 (feb 2012), 23 pages. https://doi.org/10.1145/2071396.2071404
+[9] Hiroshi Maekawa, Norio Emura, Masanobu Miura, and Masuzo Yanagida. 2006. On machine arrangement for smaller wind-orchestras based on
+scores for standard wind-orchestras. In International Conference on Music Perception and Cognition, ICMPC 2006. 268–273.
+[10] Eita Nakamura and Kazuyoshi Yoshii. 2018. Statistical piano reduction controlling performance difficulty. APSIPA Transactions on Signal and
+Information Processing (2018). https://doi.org/10.1017/ATSIP.2018.18 arXiv:1808.05006
+[11] Sho Onuma and Masatoshi Hamanaka. 2010. Piano arrangement system based on composers’ arrangement processes. In International Computer
+Music Conference, ICMC 2010.
+[12] T.H. Stefan Kostka, T.H. Dorothy Payne, and B. Almén. 2017. Tonal Harmony. McGraw-Hill Education.
+https://books.google.com/books?id=
+Cs2UAQAACAAJ
+[13] Hirofumi Takamori, Haruki Sato, Takayuki Nakatsuka, and Shigeo Morishima. 2019. Automatic arranging musical score for piano using important
+musical elements. In Proceedings of the 14th Sound and Music Computing Conference 2017, SMC 2017.
+[14] Moyu Terao, Yuki Hiramatsu, Ryoto Ishizuka, Yiming Wu, and Kazuyoshi Yoshii. 2022. Difficulty-Aware Neural Band-to-Piano Score Arrangement
+based on Note- and Statistic-Level Criteria. In ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
+196–200. https://doi.org/10.1109/ICASSP43922.2022.9747615
+[15] D.R. Tuohy and W.D. Potter. 2006. GA-based Music Arranging for Guitar. In 2006 IEEE International Conference on Evolutionary Computation.
+1065–1070. https://doi.org/10.1109/CEC.2006.1688427
+[16] G.C. White. 1992. Instrumental Arranging. McGraw-Hill. https://books.google.com/books?id=4c2KMwEACAAJ
+Manuscript submitted to ACM
+
diff --git a/kdFLT4oBgHgl3EQfcy-f/content/tmp_files/load_file.txt b/kdFLT4oBgHgl3EQfcy-f/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9207c376405888b656f8ac2b76cccb307705488f
--- /dev/null
+++ b/kdFLT4oBgHgl3EQfcy-f/content/tmp_files/load_file.txt
@@ -0,0 +1,330 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf,len=329
+page_content='Automated Arrangements of Multi-Part Music for Sets of Monophonic  Instruments  MATTHEW MCCLOSKEY, GABRIELLE CURCIO, AMULYA BADINENI, KEVIN MCGRATH, and DIM-  ITRIS PAPAMICHAIL, The College of New Jersey, USA  Arranging music for a different set of instruments that it was originally written for is traditionally a tedious and time-consuming  process, performed by experts with intricate knowledge of the specific instruments and involving significant experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' In this  paper we study the problem of automating music arrangements for music pieces written for monophonic instruments or voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' We  designed and implemented an algorithm that can always produce a music arrangement when feasible by transposing the music piece  to a different scale, permuting the assigned parts to instruments/voices, and transposing individual parts by one or more octaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' We  also published open source software written in Python that processes MusicXML files and allows musicians to experiment with music  arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' It is our hope that our software can serve as a platform for future extensions that will include music reductions and  inclusion of polyphonic instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  CCS Concepts: • Applied computing → Sound and music computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Additional Key Words and Phrases: music arrangement, music algorithms  ACM Reference Format:  Matthew McCloskey, Gabrielle Curcio, Amulya Badineni, Kevin McGrath, and Dimitris Papamichail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Automated Arrangements  of Multi-Part Music for Sets of Monophonic Instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 1, 1 (January 2023), 7 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='XXXXXXX  1  INTRODUCTION  Music arrangements involve the adaptation of a piece of music for different instruments or ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' This allows the  music to be performed in a variety of settings, enhances the repertory of musicians, and can also help to bring new life  to a piece that may have been composed for a specific instrument or ensemble [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Additionally, arrangements can  help to showcase the unique strengths of different instruments or even create entirely new interpretations of a piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  The process of arranging a piece of music can be a creative endeavor in itself, giving the arranger the opportunity to  put their own spin on a familiar work, greatly enhancing the listening experience for audiences [1, 5, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  The computational complexity of arranging music written for a set of instruments toward a target single instrument,  often employing reasonable reductive constraints, has been examined in the work of Moses and Demaine [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Complexi-  ties of dealing with polyphonic instruments, such as piano and guitar, include the need of considering possible fingerings  as well as reductions, the elimination of certain notes for playability of even feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Most research in automating  music arrangements has concentrated on the piano, primarily concerning orchestral pieces [2, 8, 10, 11, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Much  of that work involves reductions to enable feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Other work in the field has examined arrangements for the guitar  [6, 7, 15], wind ensembles [9], and other orchestral instruments [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Authors’ address: Matthew McCloskey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Gabrielle Curcio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Amulya Badineni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Kevin McGrath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Dimitris Papamichail, papamicd@tcnj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='edu, The College of  New Jersey, 2000 Pennington Road, Ewing, New Jersey, USA, 08618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not  made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Copyrights for components  of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' To copy otherwise, or republish, to post on servers or to  redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Manuscript submitted to ACM  Manuscript submitted to ACM  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='12084v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='SD]  28 Jan 2023  2  McCloskey and Curcio, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Approximate sounding ranges of instruments and voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Figure reproduced with permission from Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Brian Blood (dol-  metch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='com)  Despite its obvious benefits, we are not aware of any published algorithm or widely available software that allows  for the automated arrangement of a given music piece to a different set of instruments that it was originally written  for in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Working toward filling that need, we designed and implemented an algorithm that arranges  music written for monophonic instruments and guarantees a successful outcome when an arrangement is possible  without score reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Our recursive backtracking algorithm exhaustively examines all feasible assignments of parts  to available instruments and all possible transpositions of the piece, including independent octave transpositions of  individual parts, to determine a successful arrangement that minimally affects the musicality of the piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  2  METHODS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='1  Definitions  For the purposes of our research, a music piece is written in a chromatic scale and notes are separated by the interval of  a semitone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' We will assume that all notes fall within a total range of 88 semitones, the notes of a traditional piano, from  A0 to C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' We will assign an integer to each note in the range, such that all notes can be represented by an integer from  1 to 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' For our discussion, a monophonic instrument is one that can only play one pitch at a time, such as the flute, the  oboe, or a voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Polyphonic instruments can play multiple notes simultaneously, such as the piano, guitar, or harp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' A  polyphonic instrument can always play a monophonic part within its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  For our study an input music piece will consist of 𝑛 parts, each being assigned to a single monophonic instrument or  voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Such parts are presented in the sheet music representation of the piece in an equal number of staves each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Our  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Manuscript submitted to ACM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Haxp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Accordian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Guitar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Xylophone  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Tinpani  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Chimes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Piccolo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Flute  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Sop sax-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Alio sax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Tenor sax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Baritone sax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Bass sax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Approximate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Sop Clarinet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Alte Clarinet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Bass Clariet -  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Sounding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Obee  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='English Hern  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Ranges  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Bassoon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Trurpet& Cornet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='FrenchHorn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Trombone&Euphoiun  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Tuba  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Miolin-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Viola  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Cello  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Soprao voice  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Altovoice  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Tenor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Baritone  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Bass voice  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Piano KeyboardAutomated Arrangements of Multi-Part Music for Sets of Monophonic Instruments  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='clarinet = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='tenor-sax = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='alto-sax = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='(a) An example arrangement file  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='[alto-sax]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='name = "AltoSaxophone"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='minimum = "Db3"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='maximum = "Bb5"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='key = "Eb"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='(b) An entry in the instrument metadata file  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Examples of input instrument set and instrument information files  algorithm preserves the rhythm, rhythmic values of notes and rests, as well as bar lines of the music piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Clefs, key  signatures and accidentals are adjusted based on the scale of the transposed music and the instruments/voices that  parts are assigned to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Our algorithm does not control for instrument timbre that may be expected in any part of the  music;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' similarly, the thickness of the piece is not being necessarily maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  We will assume that an input music piece is originally written for 𝑛 instruments 𝐼1, 𝐼2, · · · , 𝐼𝑛, each assigned to play a  part 𝑃𝑖 of the piece, with 1 ≤ 𝑖 ≤ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' We seek to arrange the music for 𝑛 output instruments 𝑂1,𝑂2, · · · ,𝑂𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' The range  of each part 𝑖 is an integer interval 𝑅𝑖 = ⟦𝑎𝑖,𝑏𝑖⟧, where 𝑎𝑖 is the integer value corresponding to the lowest frequency  note and 𝑏𝑖 to the highest frequency note played by instrument 𝐼𝑖 in part 𝑃𝑖, 1 ≤ 𝑖 ≤ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Likewise, the playing range of  each output instrument 𝑂𝑖 will be denoted by 𝑂𝑅𝑖, 1 ≤ 𝑖 ≤ 𝑛, indicating the integer interval of values corresponding to  the notes the instrument is able to play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Approximate ranges for a set of instruments and voices can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='2  Monophonic instrument set arrangement algorithm  Our Monophonic Music Arrangement (MMS) algorithm performs a nearly comprehensive search of possible permuta-  tions of parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' The music is transposed to all twelve keys, and the algorithm runs on each key, unless a solution has  been found so far that results in fewer sharps/flats over all keys for each part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' This is designed to prevent the "ideal"  transposition from having a complex key signature if not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Other than that, the search is fully comprehensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  For each part, the algorithm finds all possible transpositions of each part in the source piece that can be played by  at least one available instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' All permutations of these possible transpositions are then examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' If all parts  can be played by at least one instrument, the algorithm then checks if there exists a set of part assignments that is  valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' This is performed by a recursive function that is memoized to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' If a transposed key yields  valid permutations, the transposition with the least total deviation from the original composition is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Once all  twelve keys have been checked, all permutations are tried using the selected transposition, unless there is no selected  transposition, in which case the algorithm fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' All permutations are checked, and for those that are valid in the given  transposition, the best arrangement is selected based on how closely the average pitch of each part matches the median  pitch of the instrument’s range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  The MMA algorithm implementation consists of four main function described in pseudocode below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='3  Implementation  The MMA algorithm was implemented in Python utilizing the Music21 library and the MuseScore software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Our  program requires two input files and produces a single output file with the music arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' The required input files  consist of the original piece of music in MusicXML format and a TOML file listing the instrument set to arrange for,  where an assigned value of 𝑘 to an instrument indicates 𝑘 parts should be arranged for that instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' An example of  a TOML file with an input instrument set consisting of one clarinet, two tenor saxophones, and two alto saxophones is  Manuscript submitted to ACM  4  McCloskey and Curcio, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Algorithm 1 Find Transposed Options  procedure FindTransposedOptions(𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑆𝑡𝑟𝑒𝑎𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑃𝑎𝑟𝑡𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠)  𝑠𝑡𝑟𝑒𝑎𝑚 ←− 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑆𝑡𝑟𝑒𝑎𝑚 transposed by given 𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠  𝑝𝑎𝑟𝑡𝑠 ←− new list  for 𝑝𝑎𝑟𝑡 in 𝑠𝑡𝑟𝑒𝑎𝑚 do  𝑐ℎ𝑜𝑖𝑐𝑒𝑠 ←− new list  for each 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 do  𝑠𝑒𝑡 ←− the subset of 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑃𝑎𝑟𝑡𝑠 that can play at this transposition  add (𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠 + 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑠𝑒𝑡) to 𝑐ℎ𝑜𝑖𝑐𝑒𝑠  end for  if 𝑐ℎ𝑜𝑖𝑐𝑒𝑠 is empty then  return null  end if  add 𝑐ℎ𝑜𝑖𝑐𝑒𝑠 to 𝑝𝑎𝑟𝑡𝑠  end for  return 𝑝𝑎𝑟𝑡𝑠  end procedure  Algorithm 2 Run Transposed  procedure RunTransposed(𝑠𝑡𝑟𝑒𝑎𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑝𝑎𝑟𝑡𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠)  𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 ←− new list  for 𝑜𝑝𝑡𝑖𝑜𝑛 in all possible transpositions from FindTransposedOptions(𝑠𝑡𝑟𝑒𝑎𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑝𝑎𝑟𝑡𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠) do  𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑 ←− new list  𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 ←− new list  for 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 in 𝑜𝑝𝑡𝑖𝑜𝑛 do  add set of parts covered to 𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑  add deviation of transposition to 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛  end for  𝑎𝑙𝑙𝑃𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑 ←− the union of all sets in 𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑  if 𝑎𝑙𝑙𝑃𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑 contains all parts and𝑉𝑎𝑙𝑖𝑑𝑎𝑡𝑒𝐴𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡(𝑝𝑎𝑟𝑡𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑝𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑎𝑙𝑙𝑃𝑎𝑟𝑡𝑠𝐶𝑜𝑣𝑒𝑟𝑒𝑑) then  add 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 to 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠  end if  end for  return 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠  end procedure  shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Metadata about each instrument, consisting of its key in notation and a reasonable note range,  is defined in a separate TOML file which is loaded separately by the program and is populated with common music  instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' An example of an entry for the alto saxophone in the instrument metadata file is shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  During execution our program checks whether the number of input instruments matches the number of parts in the  piece, and then attempts to arrange for the given instruments as previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' If arrangements are found, the  best arrangement based on the criteria described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='2 is output as a MusicXML file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' If no feasible arrangement  is found, or if the number of instruments does not match, then an error message is displayed and no output file is  produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Manuscript submitted to ACM  Automated Arrangements of Multi-Part Music for Sets of Monophonic Instruments  5  Algorithm 3 Find Best Choice  procedure FindBestChoice(𝑠𝑡𝑟𝑒𝑎𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑝𝑎𝑟𝑡𝑠)  𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− null  𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 ←− ∞  for 𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠 from −6 through 5 do  𝑠ℎ𝑎𝑟𝑝𝑠 ←− the total number of sharps/flats that would appear in the key signature for each part  if 𝑠ℎ𝑎𝑟𝑝𝑠 ≤ 𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 then  𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− element from RunTransposed(𝑠𝑡𝑟𝑒𝑎𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑝𝑎𝑟𝑡𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑠𝑒𝑚𝑖𝑡𝑜𝑛𝑒𝑠) with the least deviation  if 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ≠ null and either 𝑠ℎ𝑎𝑟𝑝𝑠 < 𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 or deviation of 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 < deviation of  𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 then  𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒  𝑏𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠 ←− 𝑡ℎ𝑖𝑠𝐵𝑒𝑠𝑡𝑆ℎ𝑎𝑟𝑝𝑠  end if  end if  end for  return 𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒  end procedure  Algorithm 4 MMA Algorithm  procedure MMA(𝑠𝑡𝑟𝑒𝑎𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑝𝑎𝑟𝑡𝑠)  𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 ←− FindBestChoice(𝑠𝑡𝑟𝑒𝑎𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 𝑝𝑎𝑟𝑡𝑠)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='if 𝑏𝑒𝑠𝑡𝐶ℎ𝑜𝑖𝑐𝑒 = null then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='return null  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='transpose each part by the resulting transposition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑏𝑒𝑠𝑡𝐹𝑖𝑡 ←− ∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='for each permutation of 𝑛𝑒𝑤𝑃𝑎𝑟𝑡𝑠 do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='if all parts are valid in the given permutation then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑓 𝑖𝑡 ←− the total absolute difference between the average pitches and the median pitch of each part  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='if 𝑓 𝑖𝑡 < 𝑏𝑒𝑠𝑡𝐹𝑖𝑡 then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑏𝑒𝑠𝑡𝐹𝑖𝑡 ←− 𝑓 𝑖𝑡  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='𝑏𝑒𝑠𝑡𝑃𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 ←− this permutation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='return 𝑏𝑒𝑠𝑡𝑃𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='end procedure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='RESULTS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='We tested our software on a variety of music pieces written for monophonic instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' In Figure 3 we show three  measures, starting at measure 16, of the Puttin’ on the Ritz song by Irving Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Part (a) shows the input score composed  of four monophonic parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Part (b) displays the arranged piece for saxophone quartet, consisting of a soprano, alto,  tenor, and baritone saxophones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Similarly, in Figure 4 we display three measures of Carol of the Bells, as arranged and  performed by the Pentatonix voice group, starting at measure 18 of the piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Complete input/output files for three test cases of our software, including the Puttin’ on the Ritz and Carol of the Bells  above, can be examined at: https://owd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='tcnj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='edu/∼papamicd/music/mma/examples/  Manuscript submitted to ACM  6  McCloskey and Curcio, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  The repository for this project can be found at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='com/spazzylemons/music-arrangement/  4  CONCLUSIONS AND FUTURE WORK  Our monophonic music arrangement algorithm and its software implementation create a platform for automating  music arrangements with minimal user input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Although currently basic in its functionality, it can be readily extended in  a number of different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' For accommodating arrangements for a smaller sets of instruments than the number  of parts in the music, score reduction techniques can be applied to eliminate certain parts or at least reduce the number  of simultaneous notes that are played throughout the piece, while maintaining faithfulness to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' To allow for  the inclusion of polyphonic instruments in the arrangements, further work is required in analyzing and decomposing  (a) Original Score  (b) Arranged score  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Three measures from an arrangement of ’Puttin’ on the Ritz’ from piano to saxophone quartet  (a) Original Score  (b) Arranged score  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
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+page_content=' Three measures from an arrangement of ’Carol of the Bells’ from voices to saxophone quartet  Manuscript submitted to ACM  16  112  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  my  112  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  112  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  0  ff  mf  112  0  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  G16  2  SSax  mf  2  ASax  m  2  T Sax  ff  mf  12  Bar Sax  418  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  f  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  Pno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='18  S Sax  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='ff  A Sax  mf  T Sax  Bar SaxAutomated Arrangements of Multi-Part Music for Sets of Monophonic Instruments  7  polyphonic parts into monophonic ones and inversely, while adhering to constraints related to fingerings and other  instrument and player restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors acknowledge use of the ELSA high performance computing cluster at The College of New Jersey for  conducting the research reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' This cluster is funded in part by the National Science Foundation under  grant numbers OAC-1826915 and OAC-1828163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
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+page_content=' Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Difficulty-Aware Neural Band-to-Piano Score Arrangement  based on Note- and Statistic-Level Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' In ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  196–200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='1109/ICASSP43922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='9747615  [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Tuohy and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Potter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' GA-based Music Arranging for Guitar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' In 2006 IEEE International Conference on Evolutionary Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='  1065–1070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='1109/CEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='1688427  [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' Instrumental Arranging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' McGraw-Hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content=' https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='com/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
+page_content='id=4c2KMwEACAAJ  Manuscript submitted to ACM' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFLT4oBgHgl3EQfcy-f/content/2301.12084v1.pdf'}
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+Visual Estimation of Fingertip Pressure on Diverse Surfaces
+using Easily Captured Data
+Patrick Grady1, Jeremy A. Collins1, Chengcheng Tang2, Christopher D. Twigg2,
+James Hays1, Charles C. Kemp1
+1Georgia Institute of Technology, 2Meta Reality Labs
+Input Image
+Estimated Pressure
+Input Image
+Prompt: Press index and middle, low force
+Prompt: Press thumb and index, high force
+Estimated Pressure
+Prompt: Press all fingers, high force
+Prompt: Press thumb and index, low force
+Prompt: Press thumb and index, high force
+Figure 1. By leveraging contact labels, ContactLabelNet estimates fingertip pressure when participants interact with diverse
+surfaces, including highly textured (left) and curved surfaces (right).
+Abstract
+Prior research has shown that deep models can estimate
+the pressure applied by a hand to a surface based on a
+single RGB image. Training these models requires high-
+resolution pressure measurements that are difficult to obtain
+with physical sensors. Additionally, even experts cannot re-
+liably annotate pressure from images. Thus, data collection
+is a critical barrier to generalization and improved perfor-
+mance. We present a novel approach that enables train-
+ing data to be efficiently captured from unmodified surfaces
+with only an RGB camera and a cooperative participant.
+Our key insight is that people can be prompted to perform
+actions that correspond with categorical labels (contact la-
+bels) describing contact pressure, such as using a specific
+fingertip to make low-force contact. We present Contact-
+LabelNet, which visually estimates pressure applied by fin-
+gertips. With the use of contact labels, ContactLabelNet
+achieves improved performance, generalizes to novel sur-
+faces, and outperforms models from prior work.
+1. Introduction
+People frequently interact with their surroundings by ap-
+plying pressure with their hands. Machine perception of
+hand contact pressure has been used for activity recogni-
+tion [51], ergonomics [45], user interfaces [48], and other
+applications. Most approaches use physical pressure sens-
+ing arrays, which can be expensive and difficult to mount.
+PressureVision [17] showed that computer vision can be
+used to estimate hand pressure, which could enable new ap-
+plications, such as touch interfaces for augmented reality.
+Their deep model, PressureVisionNet, estimates hand pres-
+sure from a single RGB image. The model performs well
+with diverse hands, but was only trained on two simple, flat,
+rigid surfaces, and was evaluated on the same surfaces.
+Collecting training data for diverse surfaces might en-
+able PressureVisionNet to perform well on varied surfaces,
+but data collection is challenging. Each RGB image in the
+training data requires ground truth from a high-resolution
+pressure sensing array registered with a camera. This re-
+quirement severely limits data collection for diverse sur-
+faces, since mounting a high-resolution pressure sensing ar-
+ray alters the appearance of the surface and the surface’s
+mechanical properties.
+We present a novel approach that enables training data
+for visual hand pressure estimation to be captured for unal-
+tered surfaces found in the wild. Instead of instrumenting
+the surface, our approach relies on people’s highly sensitive
+perception of touch and manual dexterity [26]. We prompt
+arXiv:2301.02310v1  [cs.CV]  5 Jan 2023
+
+SCHEDULEnoO人WORLDMAP
+WORLD GEOGRAPHY
+2979191
+Days
+Instructor
+Room特
+From-To
+Period
+CLASS SCHEDULEWORLD MAP
+WORLD OEOGRAPHY
+skeo
+Room#
+From-To
+Period
+CLASS SCHEDULEFigure 2. Instrumenting surfaces with pressure sensors without
+altering their properties is challenging. For example, pressure sen-
+sors must be transparent in order to instrument glass, or must be
+stretchable in order to instrument a deformable mat.
+participants to make contact with a surface in a specific way.
+This prompt serves as a categorical label, which we refer to
+as a contact label. A contact label consists of the regions of
+the hand that are in contact with a surface and the level of
+applied force.
+A contact label is a form of weak label [41] for an RGB
+image. For this paper, we use contact labels that define
+which fingertips are in contact and whether low or high
+force is being applied. We present ContactLabelNet, which
+estimates fingertip pressure on diverse surfaces, including
+surfaces with compliance, curvature, and complex textures.
+Training ContactLabelNet on RGB images paired with con-
+tact labels results in higher performance on diverse surfaces,
+outperforming prior work and generalizing to surfaces that
+are not represented in fully labeled training data.
+In summary, we make the following contributions:
+• Pressure estimation for diverse surfaces:
+We
+demonstrate visual estimation of pressure applied
+by multiple fingertips on diverse surfaces, including
+highly-textured, curved, and compliant surfaces.
+• Data collection using contact labels: We present a
+method suitable for collecting hand pressure data in the
+wild which involves prompting participants to make
+contact that adheres to a categorical label.
+• Deep model that benefits from contact labels: We
+present a deep model (ContactLabelNet) that achieves
+significantly higher performance when trained on RGB
+images with contact labels.
+• Open materials: We will release all trained models,
+data, and code associated with this paper.
+2. Related Work
+Our research relates to approaches that sense hand pres-
+sure, especially methods using computer vision. Our deep
+model, ContactLabelNet, uses methods inspired by research
+on unsupervised domain adaptation and learning from weak
+labels.
+Hand-Mounted Sensors:
+Hands are highly articulated
+and have compliant surfaces that make mounting pressure
+sensors difficult.
+Human factors also present challenges
+since mounting sensors can decrease comfort, interfere with
+tactile perception, and impact manual dexterity.
+Glove-
+based sensors have been developed by researchers [9, 51]
+and are commercially available [44, 52].
+However, the
+gloves occlude the surface of the hand, which interferes
+with data collection for visual models intended for bare
+hands.
+Environment-Mounted Sensors:
+Pressure sensors can
+be mounted on surfaces in the environment to sense hand
+pressure, including capacitive sensors [2, 13, 19], force-
+sensitive resistors [8,40,43], flexible sensors [4,28,52], and
+fabric-based sensors [36]. Sensors suitable for mounting on
+an in-the-wild surface tend to be custom or expensive. More
+importantly, mounting pressure sensors alters the visual ap-
+pearance and mechanical properties of the surface, which
+interferes with data collection for diverse in-the-wild sur-
+faces. To objectively assess the performance contact labels,
+in our study we mount textured overlays on a commercially
+produced pressure sensing array [40]. While useful, these
+surfaces lack compliance, curvature, and other properties
+associated with in-the-wild surfaces.
+Unconventional
+Imaging:
+Researchers
+have
+demon-
+strated methods that can detect where contact has occurred
+between a hand and an unmodified surface. Thermal imag-
+ing [6, 7] and laser speckle imaging [49] have been used
+to detect where hands have touched surfaces. These meth-
+ods require additional instrumentation, do not directly re-
+port pressure, and retrospectively report where contact oc-
+curred. They can also be sensitive to surface details, such
+as thermal properties and specularity. In contrast to these
+methods, our approach only requires an inexpensive off-the-
+shelf RGB camera and a surface that can be safely touched
+in accordance with the contact label, making our method
+suitable for crowd-sourcing and efficient in-the-wild data
+collection.
+Visual Hand Pressure Estimation:
+The force applied by
+a hand to a known object can be estimated by observing the
+object’s pose over time and finding the net force that would
+result in the object’s observed motion. If contact points are
+known, per-contact normal force can be estimated. Contact
+points can be estimated using neural networks [15, 31] or
+by combining markerless tracking of the hands with mesh-
+object intersection [42, 43, 47]. These methods can infer
+
+Thumb
+Index
+Middle
+Ring
+Pinky
+w0
+w2
+w1
+w3
+w4
+W = {thumb, index, low force}
+= [1 1 0 0 0 0]
+W = {all fingers, high force}
+= [1 1 1 1 1 1]
+W = {fingers, force level}
+= [w0 w1 w2 w3 w4 w5]
+Fully Labeled Data Capture
+Weakly Labeled Data Capture
+W = WØ
+W = WA
+W = WA
+W = WA
+W = WØ
+W = WA
+W = {index, middle, low force}
+= [0 1 1 0 0 0]
+Weakly Labeled Data
+Only Contact Labels
+Fully Labeled Data
+Pressure and Contact Labels
+Figure 3. We represent the contact labels as a six-dimensional vector. The first five elements are binary values indicating which fingers are
+prompted to be in contact, and the last element indicates the prompted force level: low (0), high (1), or unspecified (-1). Fully labeled data
+has both pressure and contact labels, while weakly labeled data has only contact labels. [46]
+contact that is occluded or out-of-view, but estimating net
+force based on an object requires that the object move, and
+thus fails for objects fixed to the environment like tabletops.
+Contact estimates based on mesh geometry are also sensi-
+tive to precision since contact depends on millimeter scale
+displacements [18].
+A number of approaches have demonstrated that vi-
+sual cues can be used to estimate hand pressure, includ-
+ing fingertip color changes [12, 37, 38], soft tissue defor-
+mation [25], and cast shadows [21, 22, 24]. In contrast to
+this prior work, our method uses an external camera to view
+the whole hand from a distance and deep learning to take
+advantage of multiple types of cues. As we describe else-
+where, our method builds on PressureVision [17].
+Deep Learning without Labels:
+Unsupervised domain
+adaptation addresses cases when an abundance of labeled
+data is available in a source domain, yet no labeled data is
+available in a target domain. To bridge this gap, a common
+technique is to learn domain-invariant features by minimiz-
+ing some measure of distance between source and target
+feature distributions [33, 53]. This can also be done by ad-
+versarially training a discriminator to classify the domain of
+the features [16] or class-conditioned features [27, 34, 55].
+Some methods instead deliberately learn domain-specific
+knowledge while leveraging domain-invariant information
+common to the source and target [5, 10]. In section 5.2
+we show that adversarial training [16] complements our
+method but is not an adequate substitute for contact labels.
+Deep Learning with Weak Labels:
+In cases where full
+labels are not available, approaches have been developed to
+still use partially or weakly labeled data. Prior work has
+used semantic segmentation as a motivating task, where
+generating per-pixel labels requires significant time from
+human annotators [14]. Techniques have been developed
+to leverage faster annotations, including image-level labels
+[1, 11, 30] and point labels [3]. Most similar to our paper
+is work that leverages image-level labels and an adversarial
+loss to transfer segmentation models to new domains [41].
+In contrast to weak labels applied by human annotators
+after data collection, our method prompts human behavior
+while data is being collected. People can often determine if
+an image shows a hand that is far from making contact or
+a hand that is holding a grasped object. In contrast, distin-
+guishing between near contact, low force contact, and high
+force contact for specific regions of the hand is challenging
+even for experts.
+3. Data Collection
+We leverage the human ability to achieve contact and
+force objectives with their hand at collection time. For each
+data capture sequence, we prompt a participant to make
+contact with a surface using a specific combination of fin-
+gertips to achieve a target force level. We assign a contact
+label associated with the prompted finger combination and
+force level to all images in our dataset.
+3.1. Collection Method
+We collect two types of data: fully labeled data and
+weakly labeled data. For both types of sequences, we col-
+lect contact labels by prompting the subject with specific
+contact cues.
+For the fully labeled sequences, we addi-
+tionally collect ground truth pressure labels using a high-
+resolution pressure sensing array [40].
+As shown in Figure 3, a contact label W is represented
+
+W = WP
+W = WA
+W = WØ
+W = WØ
+Fully Labeled Data Capture
+Weakly Labeled Data Capture
+W = WP
+Data Capture Setup
+W = WP
+Figure 4. Left: Data capture setup in one of three environments. Camera angles were varied between each participant. Center: Data
+involving dynamic contact was collected for participants interacting with a high-resolution pressure sensor. Frames with measured pressure
+are given the prompted contact label WP , and frames without measured pressure are given a no-contact label W∅. Right: Data involving
+static contact was collected for participants interacting with uninstrumented surfaces. All static contact frames are assigned the same
+contact label. Contact labels follow the format shown in Figure 3.
+Male
+70%
+Female
+30%
+Light
+25%
+Medium
+45%
+Dark
+30%
+Figure 5. Participants with a range of genders and skin tones were
+recruited for our study.
+as a vector with 6 elements. The first 5 elements indicate
+the presence or lack of contact at each of the 5 fingertips.
+The sixth element indicates if the participant was prompted
+to exert a low, high, or unspecified force.
+We represent a contact label W ∈ Z6 as follows:
+wi|0≤i≤4 ∈ {0, 1} ≡ {no contact, contact}
+wi|i=5 ∈ {−1, 0, 1} ≡ {unspecified, low, high force}
+For fully labeled data collection, both pressure labels and
+contact labels are collected. The participant is prompted to
+press a specified set of fingers onto a pressure sensor with a
+given force level. Data recording begins with the hand out
+of frame, then the participant presses and releases their hand
+multiple times on the sensor. The collection captures the
+onset and termination of contact as the participant performs
+multiple touches (Figure 4 middle). Frames where contact
+is detected by the pressure sensor are assigned the contact
+label associated with the prompted pressing action. Frames
+where no measurable force is detected are assigned a “no-
+contact” label.
+For weakly labeled data collection, only contact labels
+are collected. We use uninstrumented objects and ask par-
+ticipants to make contact for the entire duration of the
+recording. Before capture, the participant first makes con-
+tact with specified fingers on a surface.
+Once recording
+starts, the participant maintains contact for the duration of
+the video while varying the pose of the hand (Figure 4
+right). We rely on the participant to faithfully execute the
+prompted action.
+During all data capture sequences, the participant is
+prompted with a specific action, for example, “press ring
+finger, low force”. The participant performs the requested
+action while data is collected. This action corresponds with
+a contact label (Figure 3) used for training.
+3.2. Data Splits
+Our training data comes from fourteen participants. Our
+testing data comes from six participants who are not present
+in the training data. The details follow:
+• Fully labeled training set: Participants touched a
+solid-colored overlay (black, white, gray, or blue)
+placed on top of a pressure-sensing array.
+• Weakly labeled training set: Participants touched ten
+surfaces, including a textbook, a notebook, and glass.
+• Fully labeled test set: Participants touched an overlay
+(mirrored, granite, wood, or text) placed on top of a
+pressure-sensing array. The granite and wood overlays
+are not present in either training set.
+• Weakly labeled test set: Participants touched glass,
+foam, a wall, a football, a book, and a mirror. The
+book is not present in either training set. The other
+objects are present in the weakly labeled training set.
+
+Lw
+Ld
+Domain Label
+Lp
+Estimated Pressure
+Pressure Label
+Image Crop
+from Hand Detector
+Gradient Reversal
+Contact Label
+Contact Label 
+Classifier
+Domain 
+Discriminator
+Figure 6. ContactLabelNet architecture. An RGB input image is first cropped using the bounding box from an off-the-shelf hand detector.
+The cropped image is passed into an encoder-decoder network to estimate pressure for each pixel in the input image. Two classification
+heads are attached to the bottleneck of the network; one is trained to estimate the contact label, and the other uses an adversarial loss to
+discriminate between the source and target domains.
+We simultaneously collected video with up to six
+consumer-grade webcams manufactured by Logitech, Dell,
+and Elgato at 1080p resolution. We conducted data col-
+lection in three environments with different lighting condi-
+tions. We used a Sensel Morph [40] pressure sensing array.
+All data was synchronized and collected at 30 FPS. For each
+participant, we collected data with two objects – one object
+per hand.
+For all data collection procedures, we prompted partici-
+pants to press one of 8 combinations of fingertips onto a sur-
+face. For each combination, we prompted the participant to
+apply low force, high force, or slide with unspecified force.
+Additionally, for each combination, we prompted partici-
+pants to make “no contact” by hovering the specified fin-
+gertips just above the surface.
+3.3. Ethics
+Approval to conduct this study was obtained from a uni-
+versity Institutional Review Board (IRB). We recruited a di-
+verse set of 20 participants (Figure 5). All participants gave
+informed consent and were compensated for their time. We
+measured skin tone with a Pantone X-Rite RM200 spectro-
+colorimeter, and participants self-reported gender.
+4. Network Architecture
+We create a network (Figure 6) to take a single RGB im-
+age, I, as input and then output a pressure image, ˆP = f(I).
+For fully-labeled data, each RGB image is paired with a
+ground-truth pressure image obtained by projecting the out-
+put of a pressure sensing array into the image using a ho-
+mography transform. The output pressure ˆP is in image
+space, such that the input and output images are the same
+shape and can be superimposed (Figure 7).
+4.1. Pressure Estimation
+To estimate pressure, ContactLabelNet uses a binned
+representation and performs a classification across bins.
+The pressure range is split into NB = 9 bins divided across
+the pressure range, including one zero bin. Pressure estima-
+tion uses a structure-aware cross-entropy loss Lp [39, 50].
+Intuitively, the structure-aware loss penalizes large errors
+more than small errors. For each pressure pixel over the im-
+age x, y, the loss is computed over all bin indices b ∈ B
+using the ground truth index kb and the estimated probabil-
+ity for each bin ρx,y(b).
+Lp = −
+�
+x,y
+�
+b
+e−|b−kb|log(ρx,y(b))
+(1)
+Lp is only computed when fully labeled data is available.
+4.2. Contact Label Estimation
+In addition to estimating a pressure image, ContactLa-
+belNet performs the auxiliary task of estimating the contact
+label ˆW. The contact label classifier predicts ˆW using the
+features F at the network bottleneck (Figure 6). The addi-
+tion of the contact label classifier ensures that the features
+generated by the encoder are discriminative to the set of fin-
+gers in contact and the force level. The classifier pools fea-
+tures and uses a 2-layer MLP to estimate the contact label
+collected in Section 3.1. This classifier is trained with a bi-
+nary cross-entropy loss Lw. However, to account for cases
+when the force level is unspecified, loss is not calculated for
+negative values.
+4.3. Adversarial Domain Adaptation
+In addition to regularizing the network by applying the
+contact label loss, we apply an additional feature alignment
+
+Input RGB Image
+Baseline
+ContactLabelNet
+Ground Truth Pressure
+Surface/Prompt
+Press middle, high force
+W={all fingers, low force}
+Wood
+Press index, middle
+Force unspecified
+Granite
+Press all fingers
+High force
+Press index, slide
+W={all fingers, low force}
+Press index and thumb, slide
+W={all fingers, low force}
+Newspaper
+Press middle finger
+Low force
+Mirror
+Press pinky
+High force
+Figure 7. Results on the fully labeled test set. The baseline column is ContactLabelNet trained without either the domain loss or contact
+label loss. The full ContactLabelNet is shown to perform well on unseen surfaces, and benefits significantly from both losses.
+loss. This loss attempts to minimize the difference between
+the distributions of features generated from the source and
+target domains, following prior work in domain adapta-
+tion [16]. This loss is unsupervised, as it does not leverage
+contact label information.
+We implement this feature-alignment loss using a do-
+main discriminator D, which operates on features from the
+network bottleneck with a similar architecture to the weak
+label classifier. The discriminator attempts to identify if the
+image is from the fully labeled or weakly labeled domain.
+When backpropagating, gradients are reversed upstream of
+the domain discriminator [16]. For image features from the
+source domain Fs and target domain Ft, the domain loss
+function Ld is:
+Ld = −log(D(Fs)) − log(1 − D(Ft))
+(2)
+4.4. Training Details
+Due to the wide angle of the captured images, Contact-
+LabelNet operates on crops of the hand. We use Google
+MediaPipe [35] to produce hand detections, and use these
+bounding boxes to crop the image. Images are resized to
+448x448 pixels.
+ContactLabelNet is trained end-to-end using the follow-
+ing loss function:
+L = Lp + λ1Lw + λ2Ld
+(3)
+ContactLabelNet uses an SE-ResNeXt-50 encoder [20,
+23,54] and an FPN decoder [32,56]. ContactLabelNet was
+trained for 300k iterations, and optimized with the Adam
+optimizer [29]. We choose λ1 = 0.01 and λ2 = 0.001.
+Each batch contains an equal number of weakly and fully
+labeled samples.
+5. Evaluation
+We consider two types of evaluations, contact and pres-
+sure evaluations. Contact is a binary quantity indicating if
+the hand and object are touching, while pressure is a scalar
+indicating the magnitude of force. A binary contact image
+ˆC is generated by thresholding pressure ˆP at Pth = 1 kPa.
+These evaluations are inspired from prior work [17].
+• Contact Accuracy: the estimated contact image ˆC is
+used to determine if any contact is estimated. Accu-
+racy is calculated by counting the percentage of video
+frames for which ˆC corresponds with the contact label.
+• Contact IoU: intersection-over-union (IoU) is com-
+puted between the ground truth contact image C and
+estimated contact image ˆC. This is the upper bound on
+Volumetric IoU.
+• Volumetric IoU: as an extension of Contact IoU that
+considers the magnitude of pressure, 2D pressure im-
+
+Fully-Labeled Test Set
+Weakly-Labeled Test Set
+Method
+Contact Acc.
+Contact IoU
+Volumetric IoU
+Contact Acc.
+Zero Guesser
+49.0%
+0.0%
+0.0%
+23.4%
+PressureVisionNet [17] (original)
+52.8%
+1.5%
+1.2%
+40.8%
+PressureVisionNet [17] (retrained)
+70.2%
+16.3%
+13.1%
+45.6%
+ContactLabelNet (ours)
+90.4%
+44.3%
+30.6%
+87.4%
+Table 1. Performance compared to PressureVisionNet baseline [17]. Our method outperforms the baselines by a large margin.
+ages are viewed as 3D pressure volumes, where the
+height of the volume is proportional to the magnitude
+of pressure Px,y. Intersection-over-union is computed
+using these volumes.
+IoUvol =
+�
+x,y min(Px,y, ˆPx,y)
+�
+x,y max(Px,y, ˆPx,y)
+(4)
+For the same reasons that collecting fully-labeled train-
+ing data on diverse surfaces is difficult, collecting fully-
+labeled testing data also presents challenges. We evaluate
+both the fully labeled and weakly labeled test sets. How-
+ever, due to the lack of pressure measurements in the weakly
+labeled test set, only contact accuracy is computed.
+5.1. Performance Compared to Baselines
+We compare our method against two baselines:
+• Zero Guesser: The zero guesser always outputs a zero
+pressure image, and provides a simple baseline for
+Contact Accuracy due to the large number of frames
+with no contact.
+• PressureVisionNet: The network from [17], using
+either the original weights or retrained on our fully-
+labelled data. This method does not use contact labels.
+0
+20
+40
+60
+80
+100
+Percentage of Full Labels Enabled
+0
+5
+10
+15
+20
+Volumetric IoU
+Figure 8. To quantify the value of weakly-labeled images versus
+fully-labeled images, we remove a percentage of the pressure la-
+bels, but leave all images with contact labels. Volumetric IoU is
+evaluated on the fully-labeled test set.
+Table 1 shows the performance of baselines compared to
+ContactLabelNet on the test sets. ContactLabelNet signif-
+icantly outperforms prior work, improving on all metrics.
+Examples of pressure estimation on the fully-labeled test
+set are shown in Figure 7, and examples from the weakly-
+labeled test set are shown in Figure 9.
+ContactLabelNet estimates fingertip pressure on diverse
+surfaces, including textured, deformable, and curved sur-
+faces. Notably, none of the testing surfaces were included
+in the fully-labeled training set. Our method adapts to these
+surfaces using only weakly-labeled training data.
+Addi-
+tionally, two surface textures are completely unseen during
+training (wood and granite), indicating that our approach
+can successfully generalize to novel surfaces.
+When performance on the test set is separated by object
+type (full table in supplementary material), we observe that
+our approach performs most poorly on mirrors and glass,
+achieving a contact accuracy of 72% and 75%, respectively.
+We suspect that this is due to significant appearance dif-
+ferences from other surfaces in the training set (Figure 9),
+including a lack of shadows. As shown in prior work [17],
+performance is sensitive to shadows. We also observe that
+ContactLabelNet reports no contact for occluded fingertips.
+5.2. Ablating Domain Loss and Contact Label Loss
+Table 2 illustrates how the domain loss and contact la-
+bel loss impact ContactLabelNet’s performance. The table
+shows results from ablating the two auxilliary losses. We
+trained ContactLabelNet with each combination of the do-
+main loss and contact label loss enabled or disabled. Both
+losses significantly contribute to performance. Compared
+to the baseline with neither loss enabled, the domain loss
+improves the relative volumetric IoU by +27%, the contact
+label loss improves the metric by +51%, and both combined
+improve the metric by +69%.
+The performance improvement with contact labels
+demonstrates the value of weak supervision. The difficulty
+of weakly labeled data collection is comparable to that of
+unlabeled data collection, yet it provides a significant in-
+crease in performance over unlabeled data. Weakly-labeled
+data improves performance for diverse surfaces that can be
+difficult to instrument.
+
+Press all fingers
+High force
+Notebook
+Press all fingers
+Low force
+Notebook
+Press index, thumb
+Low force
+Glass
+Press index
+High force
+Glass
+Press ring
+Low force
+Football
+Press all fingers
+High force
+Press pinky
+High force
+Press index, middle
+Force unspecified
+Image
+ContactLabelNet
+Surface/Prompt
+Image
+ContactLabelNet
+Prompt
+Press all fingers
+Low force
+Press index, thumb
+Low force
+Press index
+High force
+No Contact
+Press all fingers
+High force
+No Contact
+Press all fingers
+High force
+Press middle
+Low force
+Press all fingers
+High force
+Press index
+Low force
+Image
+ContactLabelNet
+Prompt
+Image
+ContactLabelNet
+Surface/Prompt
+Foam mat
+Press all fingers
+High force
+Foam mat
+No contact
+Mirror
+Press all fingers
+High force
+Mirror
+Press middle
+Low force
+Wall
+Press index
+Low force
+Football
+Press all fingers
+High force
+Notebook page
+Press all fingers
+Low force
+Glass
+No Contact
+Figure 9. Results on the surfaces in weakly labeled test set, none of which are included in the fully labeled training set. ContactLabelNet
+produces qualitatively accurate results on highly-textured, curved, and compliant surfaces, but may not perform as well on surfaces that are
+transparent and reflective (see bottom row). Note that we did not obtain full pressure labels for these objects.
+Fully-Labeled Test Set
+Weakly-Labeled Test Set
+Domain Loss
+Contact Label Loss
+Contact Acc.
+Contact IoU
+Volumetric IoU
+Contact Acc.
+73.6%
+21.5%
+18.1%
+60.0%
+✓
+82.6%
+32.5%
+23.1%
+64.8%
+✓
+88.8%
+40.2%
+27.3%
+84.0%
+✓
+✓
+90.4%
+44.3%
+30.6%
+87.4%
+Table 2. We evaluate ContactLabelNet when the network losses are ablated. The domain loss and the contact label loss enable leveraging
+weakly-labeled training data, and both improve performance significantly.
+5.3. Performance with Full versus Weak Labels
+We perform an additional evaluation to quantify the
+value of weakly-labeled images versus fully-labeled im-
+ages. We train the network on only the fully-labeled train-
+ing set, and artificially remove the full labels, providing the
+network with 1% to 100% fully-labeled data. As shown in
+Figure 8, performance quickly improves as the percentage
+of fully-labeled data is increased, indicating that weakly-
+labeled data may be a strong substitute for fully labeled
+data.
+6. Conclusion
+Training deep models to visually estimate the pressure
+applied by fingertips relies on ground-truth pressure mea-
+surements that are difficult to obtain. We presented Con-
+tactLabelNet which uses more easily obtained contact la-
+bels collected by prompting participants to achieve specific
+types of contact. Leveraging this weakly supervised data is
+shown to improve pressure estimation on diverse surfaces
+and outperforms prior methods.
+
+WilsonWilson2272WORLDM
+WORLD GWORLD M
+WORLDG655550WORLDM
+WORLDOWORLD M
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diff --git a/ktE0T4oBgHgl3EQfYwAL/content/tmp_files/load_file.txt b/ktE0T4oBgHgl3EQfYwAL/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf,len=549
+page_content='Visual Estimation of Fingertip Pressure on Diverse Surfaces  using Easily Captured Data  Patrick Grady1, Jeremy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Collins1, Chengcheng Tang2, Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Twigg2,  James Hays1, Charles C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Kemp1  1Georgia Institute of Technology, 2Meta Reality Labs  Input Image  Estimated Pressure  Input Image  Prompt: Press index and middle, low force  Prompt: Press thumb and index, high force  Estimated Pressure  Prompt: Press all fingers, high force  Prompt: Press thumb and index, low force  Prompt: Press thumb and index, high force  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' By leveraging contact labels, ContactLabelNet estimates fingertip pressure when participants interact with diverse  surfaces, including highly textured (left) and curved surfaces (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Abstract  Prior research has shown that deep models can estimate  the pressure applied by a hand to a surface based on a  single RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Training these models requires high-  resolution pressure measurements that are difficult to obtain  with physical sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Additionally, even experts cannot re-  liably annotate pressure from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Thus, data collection  is a critical barrier to generalization and improved perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We present a novel approach that enables train-  ing data to be efficiently captured from unmodified surfaces  with only an RGB camera and a cooperative participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Our key insight is that people can be prompted to perform  actions that correspond with categorical labels (contact la-  bels) describing contact pressure, such as using a specific  fingertip to make low-force contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We present Contact-  LabelNet, which visually estimates pressure applied by fin-  gertips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' With the use of contact labels, ContactLabelNet  achieves improved performance, generalizes to novel sur-  faces, and outperforms models from prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Introduction  People frequently interact with their surroundings by ap-  plying pressure with their hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Machine perception of  hand contact pressure has been used for activity recogni-  tion [51], ergonomics [45], user interfaces [48], and other  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Most approaches use physical pressure sens-  ing arrays, which can be expensive and difficult to mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  PressureVision [17] showed that computer vision can be  used to estimate hand pressure, which could enable new ap-  plications, such as touch interfaces for augmented reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Their deep model, PressureVisionNet, estimates hand pres-  sure from a single RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The model performs well  with diverse hands, but was only trained on two simple, flat,  rigid surfaces, and was evaluated on the same surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Collecting training data for diverse surfaces might en-  able PressureVisionNet to perform well on varied surfaces,  but data collection is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Each RGB image in the  training data requires ground truth from a high-resolution  pressure sensing array registered with a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This re-  quirement severely limits data collection for diverse sur-  faces, since mounting a high-resolution pressure sensing ar-  ray alters the appearance of the surface and the surface’s  mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  We present a novel approach that enables training data  for visual hand pressure estimation to be captured for unal-  tered surfaces found in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Instead of instrumenting  the surface, our approach relies on people’s highly sensitive  perception of touch and manual dexterity [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We prompt  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='02310v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='CV]  5 Jan 2023  SCHEDULEnoO人WORLDMAP  WORLD GEOGRAPHY  2979191  Days  Instructor  Room特  From-To  Period  CLASS SCHEDULEWORLD MAP  WORLD OEOGRAPHY  skeo  Room#  From-To  Period  CLASS SCHEDULEFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Instrumenting surfaces with pressure sensors without  altering their properties is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For example, pressure sen-  sors must be transparent in order to instrument glass, or must be  stretchable in order to instrument a deformable mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  participants to make contact with a surface in a specific way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  This prompt serves as a categorical label, which we refer to  as a contact label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' A contact label consists of the regions of  the hand that are in contact with a surface and the level of  applied force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  A contact label is a form of weak label [41] for an RGB  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For this paper, we use contact labels that define  which fingertips are in contact and whether low or high  force is being applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We present ContactLabelNet, which  estimates fingertip pressure on diverse surfaces, including  surfaces with compliance, curvature, and complex textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Training ContactLabelNet on RGB images paired with con-  tact labels results in higher performance on diverse surfaces,  outperforming prior work and generalizing to surfaces that  are not represented in fully labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  In summary, we make the following contributions:  Pressure estimation for diverse surfaces:  We  demonstrate visual estimation of pressure applied  by multiple fingertips on diverse surfaces, including  highly-textured, curved, and compliant surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Data collection using contact labels: We present a  method suitable for collecting hand pressure data in the  wild which involves prompting participants to make  contact that adheres to a categorical label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Deep model that benefits from contact labels: We  present a deep model (ContactLabelNet) that achieves  significantly higher performance when trained on RGB  images with contact labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Open materials: We will release all trained models,  data, and code associated with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Related Work  Our research relates to approaches that sense hand pres-  sure, especially methods using computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Our deep  model, ContactLabelNet, uses methods inspired by research  on unsupervised domain adaptation and learning from weak  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Hand-Mounted Sensors:  Hands are highly articulated  and have compliant surfaces that make mounting pressure  sensors difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Human factors also present challenges  since mounting sensors can decrease comfort, interfere with  tactile perception, and impact manual dexterity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Glove-  based sensors have been developed by researchers [9, 51]  and are commercially available [44, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  However, the  gloves occlude the surface of the hand, which interferes  with data collection for visual models intended for bare  hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Environment-Mounted Sensors:  Pressure sensors can  be mounted on surfaces in the environment to sense hand  pressure, including capacitive sensors [2, 13, 19], force-  sensitive resistors [8,40,43], flexible sensors [4,28,52], and  fabric-based sensors [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Sensors suitable for mounting on  an in-the-wild surface tend to be custom or expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' More  importantly, mounting pressure sensors alters the visual ap-  pearance and mechanical properties of the surface, which  interferes with data collection for diverse in-the-wild sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' To objectively assess the performance contact labels,  in our study we mount textured overlays on a commercially  produced pressure sensing array [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' While useful, these  surfaces lack compliance, curvature, and other properties  associated with in-the-wild surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Unconventional  Imaging:  Researchers  have  demon-  strated methods that can detect where contact has occurred  between a hand and an unmodified surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Thermal imag-  ing [6, 7] and laser speckle imaging [49] have been used  to detect where hands have touched surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' These meth-  ods require additional instrumentation, do not directly re-  port pressure, and retrospectively report where contact oc-  curred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' They can also be sensitive to surface details, such  as thermal properties and specularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' In contrast to these  methods, our approach only requires an inexpensive off-the-  shelf RGB camera and a surface that can be safely touched  in accordance with the contact label, making our method  suitable for crowd-sourcing and efficient in-the-wild data  collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Visual Hand Pressure Estimation:  The force applied by  a hand to a known object can be estimated by observing the  object’s pose over time and finding the net force that would  result in the object’s observed motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' If contact points are  known, per-contact normal force can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Contact  points can be estimated using neural networks [15, 31] or  by combining markerless tracking of the hands with mesh-  object intersection [42, 43, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' These methods can infer  Thumb  Index  Middle  Ring  Pinky  w0  w2  w1  w3  w4  W = {thumb, index, low force}  = [1 1 0 0 0 0]  W = {all fingers, high force}  = [1 1 1 1 1 1]  W = {fingers, force level}  = [w0 w1 w2 w3 w4 w5]  Fully Labeled Data Capture  Weakly Labeled Data Capture  W = WØ  W = WA  W = WA  W = WA  W = WØ  W = WA  W = {index, middle, low force}  = [0 1 1 0 0 0]  Weakly Labeled Data  Only Contact Labels  Fully Labeled Data  Pressure and Contact Labels  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We represent the contact labels as a six-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The first five elements are binary values indicating which fingers are  prompted to be in contact, and the last element indicates the prompted force level: low (0), high (1), or unspecified (-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Fully labeled data  has both pressure and contact labels, while weakly labeled data has only contact labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' [46]  contact that is occluded or out-of-view, but estimating net  force based on an object requires that the object move, and  thus fails for objects fixed to the environment like tabletops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Contact estimates based on mesh geometry are also sensi-  tive to precision since contact depends on millimeter scale  displacements [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  A number of approaches have demonstrated that vi-  sual cues can be used to estimate hand pressure, includ-  ing fingertip color changes [12, 37, 38], soft tissue defor-  mation [25], and cast shadows [21, 22, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' In contrast to  this prior work, our method uses an external camera to view  the whole hand from a distance and deep learning to take  advantage of multiple types of cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' As we describe else-  where, our method builds on PressureVision [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Deep Learning without Labels:  Unsupervised domain  adaptation addresses cases when an abundance of labeled  data is available in a source domain, yet no labeled data is  available in a target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' To bridge this gap, a common  technique is to learn domain-invariant features by minimiz-  ing some measure of distance between source and target  feature distributions [33, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This can also be done by ad-  versarially training a discriminator to classify the domain of  the features [16] or class-conditioned features [27, 34, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Some methods instead deliberately learn domain-specific  knowledge while leveraging domain-invariant information  common to the source and target [5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='2  we show that adversarial training [16] complements our  method but is not an adequate substitute for contact labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Deep Learning with Weak Labels:  In cases where full  labels are not available, approaches have been developed to  still use partially or weakly labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Prior work has  used semantic segmentation as a motivating task, where  generating per-pixel labels requires significant time from  human annotators [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Techniques have been developed  to leverage faster annotations, including image-level labels  [1, 11, 30] and point labels [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Most similar to our paper  is work that leverages image-level labels and an adversarial  loss to transfer segmentation models to new domains [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  In contrast to weak labels applied by human annotators  after data collection, our method prompts human behavior  while data is being collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' People can often determine if  an image shows a hand that is far from making contact or  a hand that is holding a grasped object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' In contrast, distin-  guishing between near contact, low force contact, and high  force contact for specific regions of the hand is challenging  even for experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Data Collection  We leverage the human ability to achieve contact and  force objectives with their hand at collection time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For each  data capture sequence, we prompt a participant to make  contact with a surface using a specific combination of fin-  gertips to achieve a target force level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We assign a contact  label associated with the prompted finger combination and  force level to all images in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Collection Method  We collect two types of data: fully labeled data and  weakly labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For both types of sequences, we col-  lect contact labels by prompting the subject with specific  contact cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  For the fully labeled sequences, we addi-  tionally collect ground truth pressure labels using a high-  resolution pressure sensing array [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  As shown in Figure 3, a contact label W is represented  W = WP  W = WA  W = WØ  W = WØ  Fully Labeled Data Capture  Weakly Labeled Data Capture  W = WP  Data Capture Setup  W = WP  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Left: Data capture setup in one of three environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Camera angles were varied between each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Center: Data  involving dynamic contact was collected for participants interacting with a high-resolution pressure sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Frames with measured pressure  are given the prompted contact label WP , and frames without measured pressure are given a no-contact label W∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Right: Data involving  static contact was collected for participants interacting with uninstrumented surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' All static contact frames are assigned the same  contact label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Contact labels follow the format shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Male  70%  Female  30%  Light  25%  Medium  45%  Dark  30%  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Participants with a range of genders and skin tones were  recruited for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  as a vector with 6 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The first 5 elements indicate  the presence or lack of contact at each of the 5 fingertips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  The sixth element indicates if the participant was prompted  to exert a low, high, or unspecified force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  We represent a contact label W ∈ Z6 as follows:  wi|0≤i≤4 ∈ {0, 1} ≡ {no contact, contact}  wi|i=5 ∈ {−1, 0, 1} ≡ {unspecified, low, high force}  For fully labeled data collection, both pressure labels and  contact labels are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The participant is prompted to  press a specified set of fingers onto a pressure sensor with a  given force level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Data recording begins with the hand out  of frame, then the participant presses and releases their hand  multiple times on the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The collection captures the  onset and termination of contact as the participant performs  multiple touches (Figure 4 middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Frames where contact  is detected by the pressure sensor are assigned the contact  label associated with the prompted pressing action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Frames  where no measurable force is detected are assigned a “no-  contact” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  For weakly labeled data collection, only contact labels  are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We use uninstrumented objects and ask par-  ticipants to make contact for the entire duration of the  recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Before capture, the participant first makes con-  tact with specified fingers on a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Once recording  starts, the participant maintains contact for the duration of  the video while varying the pose of the hand (Figure 4  right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We rely on the participant to faithfully execute the  prompted action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  During all data capture sequences, the participant is  prompted with a specific action, for example, “press ring  finger, low force”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The participant performs the requested  action while data is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This action corresponds with  a contact label (Figure 3) used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Data Splits  Our training data comes from fourteen participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Our  testing data comes from six participants who are not present  in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The details follow:  Fully labeled training set: Participants touched a  solid-colored overlay (black, white, gray, or blue)  placed on top of a pressure-sensing array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Weakly labeled training set: Participants touched ten  surfaces, including a textbook, a notebook, and glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Fully labeled test set: Participants touched an overlay  (mirrored, granite, wood, or text) placed on top of a  pressure-sensing array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The granite and wood overlays  are not present in either training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Weakly labeled test set: Participants touched glass,  foam, a wall, a football, a book, and a mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The  book is not present in either training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The other  objects are present in the weakly labeled training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Lw  Ld  Domain Label  Lp  Estimated Pressure  Pressure Label  Image Crop  from Hand Detector  Gradient Reversal  Contact Label  Contact Label  Classifier  Domain  Discriminator  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' ContactLabelNet architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' An RGB input image is first cropped using the bounding box from an off-the-shelf hand detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  The cropped image is passed into an encoder-decoder network to estimate pressure for each pixel in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Two classification  heads are attached to the bottleneck of the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' one is trained to estimate the contact label, and the other uses an adversarial loss to  discriminate between the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  We simultaneously collected video with up to six  consumer-grade webcams manufactured by Logitech, Dell,  and Elgato at 1080p resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We conducted data col-  lection in three environments with different lighting condi-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We used a Sensel Morph [40] pressure sensing array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  All data was synchronized and collected at 30 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For each  participant, we collected data with two objects – one object  per hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  For all data collection procedures, we prompted partici-  pants to press one of 8 combinations of fingertips onto a sur-  face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For each combination, we prompted the participant to  apply low force, high force, or slide with unspecified force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Additionally, for each combination, we prompted partici-  pants to make “no contact” by hovering the specified fin-  gertips just above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Ethics  Approval to conduct this study was obtained from a uni-  versity Institutional Review Board (IRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We recruited a di-  verse set of 20 participants (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' All participants gave  informed consent and were compensated for their time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We  measured skin tone with a Pantone X-Rite RM200 spectro-  colorimeter, and participants self-reported gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Network Architecture  We create a network (Figure 6) to take a single RGB im-  age, I, as input and then output a pressure image, ˆP = f(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  For fully-labeled data, each RGB image is paired with a  ground-truth pressure image obtained by projecting the out-  put of a pressure sensing array into the image using a ho-  mography transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The output pressure ˆP is in image  space, such that the input and output images are the same  shape and can be superimposed (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Pressure Estimation  To estimate pressure, ContactLabelNet uses a binned  representation and performs a classification across bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  The pressure range is split into NB = 9 bins divided across  the pressure range, including one zero bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Pressure estima-  tion uses a structure-aware cross-entropy loss Lp [39, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Intuitively, the structure-aware loss penalizes large errors  more than small errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For each pressure pixel over the im-  age x, y, the loss is computed over all bin indices b ∈ B  using the ground truth index kb and the estimated probabil-  ity for each bin ρx,y(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Lp = −  �  x,y  �  b  e−|b−kb|log(ρx,y(b))  (1)  Lp is only computed when fully labeled data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Contact Label Estimation  In addition to estimating a pressure image, ContactLa-  belNet performs the auxiliary task of estimating the contact  label ˆW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The contact label classifier predicts ˆW using the  features F at the network bottleneck (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The addi-  tion of the contact label classifier ensures that the features  generated by the encoder are discriminative to the set of fin-  gers in contact and the force level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The classifier pools fea-  tures and uses a 2-layer MLP to estimate the contact label  collected in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This classifier is trained with a bi-  nary cross-entropy loss Lw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' However, to account for cases  when the force level is unspecified, loss is not calculated for  negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Adversarial Domain Adaptation  In addition to regularizing the network by applying the  contact label loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' we apply an additional feature alignment  Input RGB Image  Baseline  ContactLabelNet  Ground Truth Pressure  Surface/Prompt  Press middle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' high force  W={all fingers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' low force}  Wood  Press index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' middle  Force unspecified  Granite  Press all fingers  High force  Press index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' slide  W={all fingers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' low force}  Press index and thumb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' slide  W={all fingers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' low force}  Newspaper  Press middle finger  Low force  Mirror  Press pinky  High force  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Results on the fully labeled test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The baseline column is ContactLabelNet trained without either the domain loss or contact  label loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The full ContactLabelNet is shown to perform well on unseen surfaces, and benefits significantly from both losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This loss attempts to minimize the difference between  the distributions of features generated from the source and  target domains, following prior work in domain adapta-  tion [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This loss is unsupervised, as it does not leverage  contact label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  We implement this feature-alignment loss using a do-  main discriminator D, which operates on features from the  network bottleneck with a similar architecture to the weak  label classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The discriminator attempts to identify if the  image is from the fully labeled or weakly labeled domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  When backpropagating, gradients are reversed upstream of  the domain discriminator [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' For image features from the  source domain Fs and target domain Ft, the domain loss  function Ld is:  Ld = −log(D(Fs)) − log(1 − D(Ft))  (2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Training Details  Due to the wide angle of the captured images, Contact-  LabelNet operates on crops of the hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We use Google  MediaPipe [35] to produce hand detections, and use these  bounding boxes to crop the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Images are resized to  448x448 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  ContactLabelNet is trained end-to-end using the follow-  ing loss function:  L = Lp + λ1Lw + λ2Ld  (3)  ContactLabelNet uses an SE-ResNeXt-50 encoder [20,  23,54] and an FPN decoder [32,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' ContactLabelNet was  trained for 300k iterations, and optimized with the Adam  optimizer [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We choose λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='01 and λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Each batch contains an equal number of weakly and fully  labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Evaluation  We consider two types of evaluations, contact and pres-  sure evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Contact is a binary quantity indicating if  the hand and object are touching, while pressure is a scalar  indicating the magnitude of force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' A binary contact image  ˆC is generated by thresholding pressure ˆP at Pth = 1 kPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  These evaluations are inspired from prior work [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Contact Accuracy: the estimated contact image ˆC is  used to determine if any contact is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Accu-  racy is calculated by counting the percentage of video  frames for which ˆC corresponds with the contact label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Contact IoU: intersection-over-union (IoU) is com-  puted between the ground truth contact image C and  estimated contact image ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This is the upper bound on  Volumetric IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Volumetric IoU: as an extension of Contact IoU that  considers the magnitude of pressure, 2D pressure im-  Fully-Labeled Test Set  Weakly-Labeled Test Set  Method  Contact Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Contact IoU  Volumetric IoU  Contact Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Zero Guesser  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='0%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='4%  PressureVisionNet [17] (original)  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='8%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='5%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='2%  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='8%  PressureVisionNet [17] (retrained)  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='2%  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='3%  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='1%  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='6%  ContactLabelNet (ours)  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='4%  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='3%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='6%  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='4%  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Performance compared to PressureVisionNet baseline [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Our method outperforms the baselines by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  ages are viewed as 3D pressure volumes, where the  height of the volume is proportional to the magnitude  of pressure Px,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Intersection-over-union is computed  using these volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  IoUvol =  �  x,y min(Px,y, ˆPx,y)  �  x,y max(Px,y, ˆPx,y)  (4)  For the same reasons that collecting fully-labeled train-  ing data on diverse surfaces is difficult, collecting fully-  labeled testing data also presents challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We evaluate  both the fully labeled and weakly labeled test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' How-  ever, due to the lack of pressure measurements in the weakly  labeled test set, only contact accuracy is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Performance Compared to Baselines  We compare our method against two baselines:  Zero Guesser: The zero guesser always outputs a zero  pressure image, and provides a simple baseline for  Contact Accuracy due to the large number of frames  with no contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  PressureVisionNet: The network from [17], using  either the original weights or retrained on our fully-  labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' This method does not use contact labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  0  20  40  60  80  100  Percentage of Full Labels Enabled  0  5  10  15  20  Volumetric IoU  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' To quantify the value of weakly-labeled images versus  fully-labeled images, we remove a percentage of the pressure la-  bels, but leave all images with contact labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Volumetric IoU is  evaluated on the fully-labeled test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Table 1 shows the performance of baselines compared to  ContactLabelNet on the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' ContactLabelNet signif-  icantly outperforms prior work, improving on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Examples of pressure estimation on the fully-labeled test  set are shown in Figure 7, and examples from the weakly-  labeled test set are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  ContactLabelNet estimates fingertip pressure on diverse  surfaces, including textured, deformable, and curved sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Notably, none of the testing surfaces were included  in the fully-labeled training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Our method adapts to these  surfaces using only weakly-labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Addi-  tionally, two surface textures are completely unseen during  training (wood and granite), indicating that our approach  can successfully generalize to novel surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  When performance on the test set is separated by object  type (full table in supplementary material), we observe that  our approach performs most poorly on mirrors and glass,  achieving a contact accuracy of 72% and 75%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  We suspect that this is due to significant appearance dif-  ferences from other surfaces in the training set (Figure 9),  including a lack of shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' As shown in prior work [17],  performance is sensitive to shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We also observe that  ContactLabelNet reports no contact for occluded fingertips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Ablating Domain Loss and Contact Label Loss  Table 2 illustrates how the domain loss and contact la-  bel loss impact ContactLabelNet’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The table  shows results from ablating the two auxilliary losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We  trained ContactLabelNet with each combination of the do-  main loss and contact label loss enabled or disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Both  losses significantly contribute to performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Compared  to the baseline with neither loss enabled, the domain loss  improves the relative volumetric IoU by +27%, the contact  label loss improves the metric by +51%, and both combined  improve the metric by +69%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  The performance improvement with contact labels  demonstrates the value of weak supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The difficulty  of weakly labeled data collection is comparable to that of  unlabeled data collection, yet it provides a significant in-  crease in performance over unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Weakly-labeled  data improves performance for diverse surfaces that can be  difficult to instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Press all fingers  High force  Notebook  Press all fingers  Low force  Notebook  Press index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' thumb  Low force  Glass  Press index  High force  Glass  Press ring  Low force  Football  Press all fingers  High force  Press pinky  High force  Press index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' middle  Force unspecified  Image  ContactLabelNet  Surface/Prompt  Image  ContactLabelNet  Prompt  Press all fingers  Low force  Press index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' thumb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Low force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press index  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='High force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='No Contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press all fingers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='High force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='No Contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press all fingers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='High force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press middle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Low force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press all fingers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='High force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press index  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Low force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='ContactLabelNet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Prompt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='ContactLabelNet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Surface/Prompt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Foam mat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press all fingers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='High force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Foam mat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='No contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Mirror  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press all fingers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='High force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Mirror  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press middle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Low force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Wall  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press index  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Low force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Football  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Press all fingers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='High force  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
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+page_content='No Contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Results on the surfaces in weakly labeled test set, none of which are included in the fully labeled training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' ContactLabelNet  produces qualitatively accurate results on highly-textured, curved, and compliant surfaces, but may not perform as well on surfaces that are  transparent and reflective (see bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Note that we did not obtain full pressure labels for these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Fully-Labeled Test Set  Weakly-Labeled Test Set  Domain Loss  Contact Label Loss  Contact Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  Contact IoU  Volumetric IoU  Contact Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
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+page_content='0%  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
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+page_content='4%  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We evaluate ContactLabelNet when the network losses are ablated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' The domain loss and the contact label loss enable leveraging  weakly-labeled training data, and both improve performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Performance with Full versus Weak Labels  We perform an additional evaluation to quantify the  value of weakly-labeled images versus fully-labeled im-  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We train the network on only the fully-labeled train-  ing set, and artificially remove the full labels, providing the  network with 1% to 100% fully-labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' As shown in  Figure 8, performance quickly improves as the percentage  of fully-labeled data is increased, indicating that weakly-  labeled data may be a strong substitute for fully labeled  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Conclusion  Training deep models to visually estimate the pressure  applied by fingertips relies on ground-truth pressure mea-  surements that are difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' We presented Con-  tactLabelNet which uses more easily obtained contact la-  bels collected by prompting participants to achieve specific  types of contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Leveraging this weakly supervised data is  shown to improve pressure estimation on diverse surfaces  and outperforms prior methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  WilsonWilson2272WORLDM  WORLD GWORLD M  WORLDG655550WORLDM  WORLDOWORLD M  WORLDGReferences  [1] Jiwoon Ahn and Suha Kwak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Learning pixel-level semantic  affinity with image-level supervision for weakly supervised  semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  In Proceedings of the IEEE con-  ference on computer vision and pattern recognition, pages  4981–4990, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' 3  [2] Karan Ahuja, Paul Streli, and Christian Holz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' TouchPose:  Hand pose prediction, depth estimation, and touch classifica-  tion from capacitive images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' In The 34th Annual ACM Sym-  posium on User Interface Software and Technology, pages  997–1009, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' 2  [3] Amy Bearman, Olga Russakovsky, Vittorio Ferrari, and Li  Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' What’s the point: Semantic segmentation with point  supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content='  In European conference on computer vision,  pages 549–565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
+page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf'}
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+arXiv:2301.03915v1  [cs.DC]  10 Jan 2023
+Learning nonlinear hybrid automata from
+input–output time-series data
+Amit Gurung
+, Masaki Waga
+, and Kohei Suenaga
+Kyoto University, Kyoto, Japan
+amit.gurung@cyphai.io, {mwaga, ksuenaga}@fos.kuis.kyoto-u.ac.jp
+Abstract. Learning an automaton that approximates the behavior of a
+black-box system is a long-studied problem. Besides its theoretical signif-
+icance, its application to search-based testing and model understanding
+is recently recognized. We present an algorithm to learn a nonlinear hy-
+brid automaton (HA) that approximates a black-box hybrid system (HS)
+from a set of input–output traces generated by the HS. Our method is
+novel in handling (1) both exogenous and endogenous HS and (2) HA
+with reset associated with each transition. To our knowledge, ours is the
+first method that achieves both features. We applied our algorithm to
+various benchmarks and confirmed its effectiveness.
+Keywords: Automata Learning · Inferring Hybrid Systems · Learning
+Cyber-Physical Systems.
+1
+Introduction
+Mathematical modeling of the behavior of a system is one of the main tasks in
+science and engineering. If a system exhibits only continuous dynamics, it is well
+modeled by ordinary differential equations (ODE). However, many systems ex-
+hibit continuous and discrete dynamics, being instances of hybrid systems (HS).
+For instance, in modeling an automotive engine, the ODE must be switched fol-
+lowing the status of the gear. A similar combination of continuous and discrete
+dynamics also appears in many other systems, e.g., biological systems [5].
+˙v = −g
+˙x = v
+x ≤ 0
+v := −cv
+Fig. 1: A bounc-
+ing ball model
+Hybrid automata (HAs) [3] is a formalism for HS. Fig. 1 il-
+lustrates an HA modeling a bouncing ball. In an HA, a set
+of locations (represented by a circle in Fig. 1) and transitions
+between them (represented by an arrow) expresses its discrete
+dynamics. An ODE associated with each location expresses
+continuous dynamics. In the HA in Fig. 1, the ODEs at the
+location show the free-fall behavior of the system, and the
+transition shows the discrete jump caused by bouncing on
+the floor (i.e., a change in the ball’s velocity.)
+It is a natural research direction to automatically identify an
+HA given system’s behavior. Not only is it interesting as re-
+search, but it is also of a practical impact since learning a
+
+2
+A. Gurung et al.
+Table 1: Comparison of hybrid automata learning methods
+Non-linear ODEs? Exo- and Endogenous? Infer Resets? Support Inputs?
+Ours
+Polynomial
+Yes
+Yes
+Yes
+[16]
+Polynomial
+Only Exogenous
+No
+Yes
+[25]
+Linear
+Yes
+No
+Yes
+[20]
+Linear
+Only Endogenous
+No
+No
+model of a black-box system is recently being applied to au-
+tomated testing (e.g., black-box checking [15, 19, 23].) There
+have been various techniques to infer an HA from a set of input–output sys-
+tem trajectories. However, as Table 1 shows, all the existing methods have some
+limitations in the inferred HA. To the best of our knowledge, there is no ex-
+isting work that achieves all of the following features: (1) Learned HAs may
+involve nonlinear ODE as a flow; (2) Learned HAs may be exogenous (i.e., mode
+changes caused by external events) and endogenous (i.e., mode changes caused
+by internal events); and (3) Learned HAs may involve resetting of variables at a
+transition.
+This paper proposes an HA-learning algorithm that achieves the three features
+above. Namely, our algorithm learns an HA that may be exogenous, endogenous
+or both. A learned HA can reset variables at transitions. These two features make
+it possible to infer the bouncing ball example in Fig. 1, which is not possible
+in some of the previous work [16] despite its simplicity. Furthermore, an HA
+learned by our algorithm may involve ODEs with polynomial flow functions,
+whereas existing work like [20, 25] can infer only HAs with linear ODEs.
+Fig. 2 shows the overview of our HA learning algorithm. Our algorithm consists
+of a location identification step and a transition identification step. We explain
+each step below.
+Location identification step. To identify the locations, our algorithm first splits
+trajectories into segments so that each segment consists only of continuous dy-
+namics. To this end, the algorithm estimates the derivative of each point on a
+trajectory with the linear multistep method (LMM) [12] and detects the points
+where the derivative changes discontinuously.
+Then, the segments are grouped into clusters so that the segments in each clus-
+ter have similar continuous dynamics. For this, we conduct clustering based on
+the distance determined by dynamic time warping (DTW) [4], which takes two
+segments and computes their similarity in terms of the “shape.” We treat each
+cluster as a location of an HA in the following steps.
+After the clustering, our algorithm synthesizes an ODE that best describes the
+continuous dynamics of the segments in each cluster. Our ODE inference is by
+a template-based approach. For each location, we fix a polynomial template—a
+
+Learning nonlinear hybrid automata from input–output time-series data
+3
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+Fig. 2: Overview of our HA learning algorithm. In the below center figure, the
+circle and the star points stand for each segment’s first and last points.
+polynomial whose coefficients are symbols for unknowns—for the flow function
+of the ODE. Then, we obtain coefficients of the polynomial via linear regression
+of the values in a trajectory and the derivative estimated by LMM.
+Transition identification step. Once locations are identified, our algorithm next
+synthesizes the transition relation. It first identifies the pairs of locations (or
+clusters) between which there is a transition. Concretely, the algorithm identifies
+a transition from location c1 to location c2 if (1) there is a segment s1 in c1 and
+s2 in c2 and (2) s1 immediately precedes s2 in a trajectory.
+The algorithm then synthesizes the guard and the reset on each transition. We
+synthesize the guard and the reset on each transition in a data-driven man-
+ner. Moreover, we introduce type annotation to improve the inference of resets
+utilizing domain knowledge; we explain the method in detail in Section 3.2.
+Contributions
+Our contributions are summarized as follows.
+– We propose an algorithm inferring a general subclass of HAs from a set of
+input and output trajectories.
+– We introduce type annotations to improve the inference of the resets.
+– We experimentally show that our algorithm infers HAs fairly close to the
+original system under learning.
+
+4
+A. Gurung et al.
+Related Work Despite the maturity of switched-system identification [9, 12],
+only a few algorithms have been proposed to infer HAs. This scarcity of work in
+HA learning may be attributed to the additional information that needs to be
+inferred for HAs (e.g., variable assignments.)
+Table 1 summarizes algorithms inferring an HA from a set of trajectories. In [20],
+an HA is learned from a set of trajectories; however, it does not support systems
+with inputs. Moreover, only linear ODEs can be learned. In [25], an HA with
+inputs and outputs is learned from trajectories, but the learned ODEs are still
+limited to linear functions. In [16], an HA with polynomial ODEs is learned from
+inputs and outputs trajectories. However, the guards in the transitions consist
+only of the input variables and timing constraints. Due to this limitation, their
+method cannot infer an endogenous HA such as the bouncing ball model in
+Fig. 1. Compared with these methods, our algorithm supports the most general
+class of HAs, to our knowledge.
+We remark that most of the technical ingredients used in our algorithm are
+already presented in the previous papers. For example, using LMM for segmen-
+tation and inference of polynomial ODEs is also used by Jin et al. [12]. The use
+of DTW for clustering is common in [16]. We argue that our significant technical
+contribution is the achievement of learning the general class of HAs by an appro-
+priate adaptation and combination of these techniques, e.g., by projecting the
+input dimensions during segmentation. The use of type annotation to improve
+the inference of variable assignments is also our novelty, up to our knowledge.
+Organization After reviewing the preliminaries in Section 2, we present our
+HA learning algorithm and an experimental evaluation of it in Section 3 and
+Section 4, respectively. Finally, we conclude in Section 5.
+2
+Preliminaries
+For a set X, we denote its powerset by P(X). For a pair p := (a, b), we write
+pr 1(p) for a and pr 2(p) for b.
+We denote naturals and reals by N and R,
+respectively. For vectors u, and v with the same dimension, the relative difference
+between them is rd(u, v) :=
+∥u−v∥
+∥u∥+∥v∥ where ∥u∥ is the Euclidean norm of u. We
+write [a, b] for the inclusive interval between a and b.
+2.1
+Trajectories and Hybrid automata
+For a time domain [0, T ] ⊆ R and n ∈ N, an n-dimensional (continuous) signal
+σ is a function assigning an n-dimensional vector σ(t) ∈ RN to each timepoint
+t ∈ [0, T ]. Execution of a system with n1 dimensional inputs and n2 dimensional
+outputs can be modeled by an (n1 + n2)-dimensional signal.
+A (discrete) trajectory is a sequence of vectors with timestamps. Concretely,
+an n-dimensional trajectory (t1, x1), (t2, x2), . . . , (tN, xN) is a finite sequence
+
+Learning nonlinear hybrid automata from input–output time-series data
+5
+of pairs of timestamp ti ∈ R and the corresponding value xi ∈ Rn sat-
+isfying t1 < t2 < · · · < tN. For a signal σ: [0, T ] → Rn, a trajectory
+(t1, x1), (t2, x2), . . . , (tN, xN) is a discretization of σ if for any i ∈ {1, 2, . . ., N},
+we have xi = σ(ti). We call each vector (ti, xi) in a trajectory as a (sampling)
+point.
+Hybrid automata (HAs) [3, 14] is a formalism to model a system exhibiting
+an interplay between continuous and discrete dynamics. Since we aim to learn
+an HA from a set of trajectories with inputs and outputs, we employ HAs with
+input and output variables. To define HAs, we fix a finite set of (continuous state)
+variables X, input variables I, and output variables O such that X = I ⊎ O. A
+valuation is a mapping δ ∈ RX that represents the value of each variable.
+Definition 1 (Hybrid automaton). A hybrid automaton (HA) H, is a tuple
+(L, Inv, Init, Flow, Trans) where:
+- L is a finite set of locations;
+- Inv : L → P(RX ) is a function mapping each location ℓ to the invariant at
+ℓ;
+- Init, the initial condition, is a pair (ℓ0, δ0) such that ℓ0 ∈ L and δ0 ∈ Inv(ℓ0);
+- Flow is a flow function mapping each location ℓ ∈ L to ODEs of the form
+˙x = f(x, u), called flow equation, where x is the vector of all the variables
+in O and u is the vector of all the variables in I;
+- Trans is the set of discrete transitions denoted by a tuple e = (ℓ, G, M, ℓ′),
+where ℓ, ℓ′ ∈ L are the source and target locations, G ⊆ P(RX ) is the guard,
+and M : RX → RO is the assignment function.
+For a transition e ∈ Trans, we write G(e) and M(e) for the guard and the
+assignment function of e, respectively.
+Intuitively, a guard G(e) of a transition e is the condition that enables the tran-
+sition: A transition e can be fired if a valuation δ for the variables satisfies
+δ ∈ G(e). An assignment M(e) specifies how a valuation is updated if the tran-
+sition e fired: A valuation is updated from δ to δ′ such that for each x ∈ O and
+u ∈ I, we have δ′(x) = M(e)(δ)(x) and δ′(u) = δ(u) if e is fired.
+The semantics of an HA is formalized by the notion of a run. A state of an HA
+H is a pair (ℓ, δ), where ℓ is a location of H and δ ∈ RX is a valuation.
+Definition 2 (Run). A run of an HA (L, Inv, Init, Flow, Trans) is a sequence
+(ℓ0, δ0)
+τ0
+−→ (ℓ0, δ′
+0)
+e0
+−→ (ℓ1, δ1)
+τ1
+−→ (ℓ1, δ′
+1)
+e1
+−→ . . .
+eN−1
+−−−→ (ℓN, δN)
+τN
+−−→ (ℓN, δ′
+N)
+satisfying (ℓ0, δ0) ∈ Init and for each i ∈ {0, 1, . . ., N}, there are signals
+σx
+i : [0, τi] → RO and σu
+i : [0, τi] → RI such that (i) for any x ∈ O and
+u ∈ I, we have σx
+i (0)(x) = δi(x) and σu
+i (0)(u) = δi(u), σx
+i (τi)(x) = δ′
+i(x), and
+
+6
+A. Gurung et al.
+σx
+i (τi)(u) = δ′
+i(u), (ii) for any t ∈ [0, τi], we have (σx
+i (t), σu
+i (t)) ∈ Inv(ℓi) and
+˙σx
+i (t) = Flow(ℓi)(σx
+i (t), σu
+i (t)), and (iii) we have δ′
+i ∈ G(ei) and δi+1 is such that
+for each x ∈ O and u ∈ I, we have δi+1(x) = M(ei)(δ′
+i)(x) and δi+1(u) = δ′
+i(u).
+For such a run ρ, a signal σ: [0, TN] → RX is the signal over ρ if σ is such that
+σ(t)(x) = σx
+i (t−Ti−1) and σ(t)(u) = σu
+i (t−Ti−1)(u) for each x ∈ O, u ∈ I, and
+i ∈ {0, 1, . . ., N} such that Ti ≤ t < Ti+1, where Ti = �i
+j∈0 τj and TN+1 = ∞.
+2.2
+Linear Multistep Method
+The linear multistep method (LMM) [6] is a technique to numerically solve
+an initial value problem of an ODE
+˙
+x(t) = f(x, t). Concretely, it approxi-
+mates the value of x(tn+M) by using the values of x(tn), . . . , x(tn+M−1) and
+f(xn, tn), . . . , f(xn+M−1, tn+M−1)—namely, M previous discretized values of x
+and f(x, t)—where tn+i = tn + ih for some h > 0. For this purpose, LMM
+assumes the following approximation parameterized over (αi)i and (βi)i:
+M
+�
+i=0
+αix(tn−i) ≈ h
+M
+�
+i=0
+βif(x(tn−i), tn−i).
+Then, LMM determines the values of (αi)i and (βi)i so that the error of the above
+approximation, quantified with Taylor’s theorem, is minimum; see [6, 12, 22] for
+more detail. The approximation with the determined values of (αi)i and (βi)i is
+used to successively determine the values of x(t) from its initial value.
+In the context of our work, we estimate the derivative of a trajectory at
+each point without knowing the ODE. To this end, we use backwards differ-
+entiation formula (BDF) [13, 22] derived from LMM. The idea is to com-
+pute the polynomial passing all the points (tn, x(tn)), . . . , (tn+M−1, x(tn+M−1))
+using Lagrange interpolation [6, 13] and derive the formula to approx-
+imate
+the
+derivative
+at
+(tn+M, x(tn+M−1))
+from
+the
+polynomial
+using
+LMM. Concretely, Lagrange interpolation results in the following poly-
+nomial: x(t)
+≈
+�M
+m=0 x(tn−m) �
+i̸=m
+t−tn−m
+tn−i−tn−m . By taking the derivative
+of both sides and setting t to tn, we obtain
+˙x(tn)
+=
+f(x(tn), tn)
+≈
+�M
+m=0 x(tn−m) �
+i̸=m( d
+dt
+t−tn−m
+tn−i−tn−m )
+���
+t=tn. We use this formula to estimate the
+derivative at each point in a trajectory. For instance, the formula to estimate
+the derivatives with M = 2 is: f(x(tn)) = 1
+h( 3
+2x(tn)− 4
+2x(tn−1)+ 1
+2x(tn−2)) [22].
+The above formula estimates the derivative at x(tn) using M previous points—
+hence called backward BDF. Dually, we can derive a formula that estimates the
+derivative at x(tn) using M following points called forward BDF. We use both
+in our algorithm.
+2.3
+Dynamic Time Warping (DTW)
+Our algorithm introduced in Section 3 first splits given trajectories so that each
+segment includes only continuous dynamics. Then, it classifies the generated
+
+Learning nonlinear hybrid automata from input–output time-series data
+7
+segments based on the “similarity” of the ODE behind. For the classification
+purpose, we use dynamic time warping (DTW) [4]—one of the methods for
+quantifying the similarity between time-series data in their shapes—as the mea-
+sure of the similarity inspired by [16]. The previous work [16] applies DTW for
+HA learning and confirms its effectiveness.
+The DTW distance between two time-series data X := (x1, x2, . . . , xM) and
+Y
+:= (y1, y2, . . . , yN), where M, N ∈ N, is defined as follows. The align-
+ment path between X and Y is a finite sequence P := (p1, . . . , pl) where
+pi ∈ {1, 2, . . ., M}×{1, 2, . . ., N} and P is an alignment between {1, . . . , M} and
+{1, . . . , N}. Concretely, P should satisfy the following conditions: (1) p1 = (1, 1);
+(2) pl = (M, N); (3) (ai+1 − ai, bi+1 − bi) is either (1, 0), (0, 1), or (1, 1) for any
+(pi, pi+1) = ((ai, bi), (ai+1, bi+1)). For example, ((1, 1), (1, 2), (2, 3), (3, 3), (3, 4))
+is an alignment path between (x1, . . . , x3) and (y1, . . . , y4).
+An alignment path P = (p1, . . . , pl) between X := (x1, x2, . . . , xM) and Y :=
+(y1, y2, . . . , yN) determines the sum dP := �l
+i=1 ||x(pr1(pi)) − y(pr2(pi))|| of the
+distances between corresponding points in X and Y . Then, the DTW distance
+DTWdist(X, Y ) between X and Y is defined by minP dP , where P moves all the
+alignment paths between X and Y . There is an efficient algorithm computing
+DTWdist(X, Y ) in O(MN) based on dynamic programming [18].
+For X and Y , let P be the alignment that gives the optimal sum of dis-
+tances between X and Y . We write DTWcorrel(X, Y ) for correl(P1, P2), where
+P1 := (pr 1(p1), . . . , pr 1(pl)), P2 := (pr 2(p1), . . . , pr 2(pl)), and correl(P1, P2) is
+the Pearson product-moment correlation coefficients between P1 and P2. This
+value becomes larger if P1 and P2 increase evenly. Thus, the higher this value is,
+the more X is similar to Y . The effectiveness of this value in classifying segments
+is also shown in [16].
+3
+HA Learning from Input–Output Trajectories
+Here, we present our HA learning algorithm from given trajectories. Our problem
+setting is formalized as follows.
+Passive HA learning problem:
+Input: trajectories {(ti
+1, xi
+1), (ti
+2, xi
+2), . . . , (ti
+Ni, xi
+Ni) | i ∈ {1, 2, . . ., M}} that
+are discretizations of signals over runs of an HA H
+Output: an HA H approximating H
+Our current algorithm learns an HA such that (i) the invariant of each location
+is true, (ii) each guard is expressed as a polynomial inequality, and (iii) each
+assignment function is a linear function.
+We assume that (i) each location of
+H has different ODEs and (ii) for each pair (ℓ, ℓ′) of locations of H, there is at
+most one transition from ℓ to ℓ′.
+Fig. 2 outlines our HA learning algorithm. We first present the identification of
+the locations and then present that of the transitions.
+
+8
+A. Gurung et al.
+Algorithm 1 Outline of our segmentation algorithm
+Input: A trajectory τ = (t1, x1), (t2, x2), . . . , (tN, xN), the step size M in BDF, and
+the thresholds εFwdBwd and εBwd
+Output: cp ⊆ {1, 2, . . . , N} is the set of change points
+1: candidates ← ∅; C ← ∅
+2: for all i ∈ {M + 1, M + 2, . . . , N − M} do
+3:
+fwdi ← fF(τ|O, i, M)
+⊲ Compute the forward BDF
+4:
+bwd i ← fB(τ|O, i, M)
+⊲ Compute the backward BDF
+5:
+if rd(fwdi, bwd i) > εFwdBwd then
+6:
+add i to candidates
+7: while candidates ̸= ∅ do
+8:
+i ← min(candidates); remove i from candidates
+9:
+if i + 1 ̸∈ candidates or rd(bwdi, bwd i+1) ≥ εBwd then
+10:
+add i to cp
+11:
+while i + 1 ∈ candidates do
+12:
+remove i + 1 from candidates; i ← i + 1
+(a) fwd i ≈ bwd i
+(b) rd(fwdi, bwd i) ≫ 0 and
+bwd i ≈ bwd i+1
+(c) rd(fwdi, bwd i) ≫ 0 and
+rd(bwdi, bwd i+1) ≫ 0
+Fig. 3: Illustration of our segmentation algorithm near a boundary of a segment.
+The red circle in the right figure is the change point because it is the first point
+satisfying rd(fwd i, bwd i) > εFwdBwd and rd(bwd i, bwdi+1) > εBwd.
+3.1
+Identification of Locations
+We identify the locations of an HA by the following three steps: (i) segmentation
+of the given trajectories, (ii) clustering of the segments, and (iii) inference of
+ODEs and initial locations.
+Segmentation of the Trajectories The first step in our HA learning algo-
+rithm is segmentation. Each trajectory is divided into segments so that the dy-
+namics in each segment are jump-free. We perform segmentation by identifying
+the change points—the points where the derivative discontinuously changes—
+along a trajectory; this step is an enhancement of the idea by Jin et al. [12].
+Algorithm 1 outlines our segmentation algorithm. For simplicity, we present an
+algorithm for a single trajectory; this algorithm is applied to each trajectory
+obtained from the system. First, for each point in the trajectory, we estimate
+the derivative using forward and backward BDF (fwd i and bwdi, respectively)
+and deem the point as a candidate of change points if rd(fwd i, bwdi) exceeds
+
+T
+B
+FCT
+CT
+B
+FCT
+F
+BLearning nonlinear hybrid automata from input–output time-series data
+9
+Algorithm 2 Outline of the clustering of the segmented trajectories
+Input: Set Sg of segments and thresholds εdst and εcor for distance and diagonality
+Output: C = {C1, C2, . . . , Cn} is a set of set of segments such that each Ci is a cluster
+1: C ← ∅
+2: while Sg ̸= ∅ do
+3:
+pick sg from Sg
+⊲ We still have sg ∈ Sg after picking it.
+4:
+C ′ ← {sg ′ ∈ Sg | DTWdist(sg|O, sg′|O) < εdst ∧ DTWcorrel(sg|O, sg′|O) > εcor}
+5:
+Sg ← Sg \ C ′
+⊲ C ′ always includes sg, and sg is removed from Sg.
+6:
+add C ′ to C
+the threshold. For example, among the three red circles in Fig. 3, we have
+fwd i ≈ bwdi for the one in Fig. 3a and rd(fwd i, bwdi) ≫ 0 for the others.
+Thus, the red circles in Figs. 3b and 3c are the candidates of the change point.
+We remark that fwdi and bwdi are computed with the trajectory τ|O projected
+to the output variables, and our segmentation is not sensitive to the change in
+the input variables I.
+When there are consecutive candidates of the change points, we take the first
+one satisfying rd(bwd i, bwdi+1) ≥ εBwd to precisely estimate the change point.
+Such an optimization is justified under the assumption that there are at least
+2M − 1 points between two consecutive mode changes. For example, in the
+example shown in Fig. 3, the red circle in Fig. 3c is deemed to be the change
+point because this is the first candidate satisfying rd(bwd i, bwdi+1) ≥ εBwd.
+The algorithm splits the trajectories at the identified change points into seg-
+ments; the change points are not included in the segments. Notice that our
+twofold identification of change points is an enhancement to Jin et al. [12],
+which takes all our candidates as change points and drops them from the re-
+sulting segments. Our enhancement allows more points to be included in the
+resulting segments, which makes the subsequent phases more precise.
+Clustering of the Segments Then, we cluster the segmented trajectories so
+that the segments with similar continuous behaviors are included in the same
+cluster. For instance, in Fig. 2, the continuous behaviors in S1a, S2a, S2d, and S3a
+are similar and hence included in a single cluster. We use the identified clusters
+as the set of locations in the resulting HA. This construction is justified when
+each location has a different ODE.
+Algorithm 2 outlines our clustering algorithm. The overall idea is, the algorithm
+picks one segment (line 3) and creates a cluster by merging similar segments
+(line 4). We use both DTWdist and DTWcorrel to determine the similarity be-
+tween segments. We remark that we compare the segments sg|O and sg′|O pro-
+jected to the output variables to ignore the similarity in the input variables.
+Inference of ODEs and Initial Locations Our ODE inference is by a
+template-based linear regression. First, we fix a template Φ(x; θ) = θ1f1(x) +
+
+10
+A. Gurung et al.
+θ2f2(x)+ · · ·+ θNfN(x) of the ODE. In our current implementation, each fi is a
+monomial whose degree is less than a value specified by a user, but an arbitrary
+template can be used. Then, for each cluster Ci and for each output variable
+o ∈ O, we construct the set Pi,o of points in Ci, and the derivative of o at this
+point. Formally, Pi,o = {(x, ˙x(o)) | ∃sg ∈ Ci. x ∈ sg}. The derivative ˙x(o) is,
+for example, computed by BDF. Moreover, we can reuse the derivative used in
+Algorithm 1. Finally, we use linear regression to compute the coefficients θ such
+that for each (x, ˙x(o)) ∈ Pi,o, we have ˙x(o) ≈ Φ(x; θ).
+In the resulting HA, the initial locations are the locations such that the cor-
+responding cluster contains the first segment for some trajectories. Therefore,
+we have multiple initial locations if there are trajectories such that their first
+segments do not satisfy the similarity condition during clustering.
+3.2
+Identification of Transitions
+�
+�
+��
+��
+��
+��
+��
+��
+���������
+����������
+Fig. 4: Illustration of the points
+connecting clusters Ci and Cj
+After identifying the locations of the result-
+ing HA by clustering the segments, we con-
+struct transitions. Let sg1, sg2, . . . , sgm be the
+segments obtained from a single trajectory by
+Algorithm 1 and ordered in chronological or-
+der; segment sgi immediately precedes sgi+1
+in the original trajectory. For each segment
+sgg, we denote its initial point, the second last
+point, and the last point by sgη1
+g , sgη2
+g , and
+sgη3
+g , respectively.
+The
+idea
+of
+the
+transition
+identification
+is to make one transition for each triple
+(sgη2
+g , sgη3
+g , sgη1
+g+1)—called
+a
+connection
+triple—and use these points in a triple to infer its guard and assignment; see
+Fig. 4 for an illustration. We note that such a triple is always defined since each
+segment has at least three points.
+Formally, for clusters Ci and Cj and a segmented trajectory sg1, sg2, . . . , sgm,
+the set Ti,j of connection triples from Ci to Cj is as follows:
+Ti,j = {(sgη2
+g , sgη3
+g , sgη1
+g+1) | g ∈ {1, 2, . . ., m − 1}, sgg ∈ Ci, sgg+1 ∈ Cj}
+If there are multiple trajectories in HA learning, we construct Ti,j for each tra-
+jectory and take their union.
+We infer guards and assignments using Ti,j. For each cluster pair (Ci, Cj),
+the guard of the transition from Ci to Cj is obtained using a support vector
+machine (SVM) to classify the second last points and the last points. More
+precisely, for T ⊥
+i,j = {sgη2
+g
+| ∃(sgη2
+g , sgη3
+g , sgη1
+g+1) ∈ Ti,j} and T ⊤
+i,j = {sgη3
+g
+|
+∃(sgη2
+g , sgη3
+g , sgη1
+g+1) ∈ Ti,j}, we compute an equation of hyperplane separating
+
+Learning nonlinear hybrid automata from input–output time-series data
+11
+T ⊥
+i,j and T ⊤
+i,j using SVM and construct an inequality constraint G that is satisfied
+by the points in T ⊤
+i,j but not by that in T ⊥
+i,j.
+For each cluster pair (Ci, Cj), the assignment in the transition from Ci to Cj
+is obtained using linear regression to approximate the relationship between the
+valuation before and after the transition. More precisely, we use linear regression
+to compute an equation M such that for each (sgη2
+g , sgη3
+g , sgη1
+g+1) ∈ Ti,j and for
+each x ∈ O, sgη1
+g+1(x) is close to M(sgη3
+g )(x). Such M is used as the assignment.
+Improving Assignments Inference with Type Annotation If we have no
+prior knowledge of the system under learning, we infer assignments using linear
+regression, as mentioned above. However, even if the exact system dynamics are
+unknown, we often know how each variable behaves at jumps. For instance, it
+is reasonable to believe that a variable representing temperature is continuous;
+hence, it does not change its value at jumps. Such domain knowledge is helpful
+in inferring precise assignments rather than one using linear regression.
+To easily enforce the constraints from domain knowledge on variables, we extend
+our assignment inference to allow users to annotate each variable with types
+expressing how a variable is assumed to behave at jumps. We currently support
+the following types.
+No assignments If a variable is continuous at a jump (e.g., the variable rep-
+resenting temperature mentioned above), one annotates the variable with “no
+assignments”. For a variable x with this annotation, the procedure above infers
+an assignment that does not change the value of x. For example, in the HA in
+the bouncing ball HA in Fig. 1, x is a continuous variable while v is not.
+Constant pool If the value assigned to a variable at a jump is chosen from a finite
+set, one annotates the variable with “Constant pool” accompanied with the finite
+set {v1, . . . , vn}. An example of such a variable is one representing the gear in
+a model of an automotive. For a variable with this annotation, our algorithm
+infers the assignment at a jump by majority poll: For a transition from cluster
+Ci to Cj, it chooses the value most frequently occurring in Ti,j as sgη1
+g+1.
+4
+Experiments
+We implemented our proposed algorithm using a combination of C++, Python,
+and MATLAB/Simulink/Stateflow: The HA learning algorithm is written in
+Python; The learned model is translated into a Simulink/Stateflow model by a
+C++ program; We use MATLAB to simulate the learned model. We optimized
+the ODE inference by using only a part of the trajectories when they were
+sufficiently many. We take M = 5 as the step size for BDF.
+We conducted experiments (i) to compare the performance of our algorithm
+against a state-of-the-art method and (ii) to evaluate how the type annotation
+
+12
+A. Gurung et al.
+helps our learning algorithm. For the former evaluation, we compared our al-
+gorithm against one of the latest HA learning methods called POSEHAD [16].
+We compared our algorithm with and without a type annotation for the latter
+evaluation, denoted as “Type” and “W/o Type,” respectively.
+Each benchmark consists of a Simulink/Stateflow model, which we call an origi-
+nal model, and two sets of trajectories generated from the original model, which
+we call training and test sets. We generated trajectories by feeding random in-
+put trajectories and random initial values of the state variables to the original
+model. The training set is used to learn an HA, which we call a learned model,
+and the test set is used to evaluate the accuracy of the learned model. For each
+benchmark, the size of the training and test sets are 64 and 32, respectively.
+To evaluate the accuracy of the learned model, we feed the same input trajec-
+tories and the same initial values to the original and the learned models and
+compared their output trajectories. The comparison is based on the DTW dis-
+tance DTWdist. A low DTW distance indicates higher accuracy of the learned
+model. We denote as δO1 and δO2 the DTW distances between trajectories gener-
+ated from the original and the learned model on the output variable, O1, and O2,
+respectively. We note that, in POSEHAD, the DTW distance is not computed
+with the entire trajectories but with the segmented trajectories. All the experi-
+ments reported in this paper are conducted on a machine with an Intel Core i9
+CPU, 2.40GHz, and 32 GiB RAM. We used εBwd = 0.01 in all our experiments.
+4.1
+Benchmark Description
+We briefly describe the benchmarks used in our experiments.
+Ball This is a benchmark modeling a bouncing ball taken from the demo ex-
+ample of Simulink [2]. Fig. 1 shows the HA. The acceleration due to the gravity g
+is taken as input. The range of g is [−9.9, −9.5]. We modify the original Simulink
+model to parameterize the initial values of x and v. We let x ∈ [10.2, 10.5] and
+v = 15. The reset factor c in Fig. 1 is c = −0.8. We execute the model for a time
+horizon of 13 units with a sampling time of 0.001, i.e., each trajectory consists
+of 13,000 points. We use εFwdBwd = 0.1, εdst = 9.0, and εcor = 0.8.
+Tanks This benchmark models a two tanks system [11]. Fig. 7a shows the HA.
+The system consists of two tanks with liquid levels x1 and x2. The first tank
+has in/out flow controlled by a valve v1, whereas, the second tank has outflow
+controlled by the other valve v2. Both tanks have external in/out flow controlled
+by the input signal u. There is also a flow from the first tank to the second tank.
+In summary, the system has four locations for on and off of v1 and v2. The range
+of the input is u ∈ [−0.1, 0.1], the initial liquid level of the two tanks are x1 = 1.2
+and x2 = 1, and the initial location is off_off. We execute the model for a time
+horizon of 9.3 units with a sampling time of 0.001, i.e., each trajectory consists
+of 9,300 points. We use εFwdBwd = 0.01, εdst = 1.5 and εcor = 0.7.
+
+Learning nonlinear hybrid automata from input–output time-series data
+13
+y + 0.714286x ≥ 0
+x ≤ 0
+˙y = −y − 0.7
+˙x = −2x + 1.4
+loc4
+x ≤ 0
+y + 0.714286x = 0
+y + 0.714286x ≤ 0
+loc3
+x ≤ 0
+0.714286x + y ≥ 0
+˙x = −2x − 1.4
+˙y = −y + 0.7
+x ≥ 0
+x ≥ 0
+y ≤ −0.714286x
+0.714286x + y ≤ 0
+x = 0
+x = 0
+˙x = −2x − 1.4
+˙y = −y + 0.7
+loc1
+˙x = −2x + 1.4
+˙y = −y − 0.7
+y ≥ −0.714286x
+x ≥ 0
+y < −0.714286x
+loc2
+(a) An HA model of Osci
+Early_Repolarization
+Upstroke
+Plateau
+Final_Repolarization
+˙x = −1.52
+34 ≤ x ≤ 46
+−76 ≤ x ≤ 46
+˙x = 130.02
+44 ≤ x ≤ 46
+−76 ≤ x ≤ −74
+−6 ≤ x ≤ −4
+˙x = −0.76
+34 ≤ x ≤ 36
+−76 ≤ x ≤ 36
+˙x = −2.13
+−6 ≤ x ≤ 36
+(b) An HA model of Cells
+Fig. 5: HA models for Osci and Cells benchmarks
+Osci This is a benchmark modeling a switched oscillator without filters [8].
+Osci is an affine system with two variables, x and y oscillating between two
+equilibria to maintain a stable oscillation. The HA is shown in Fig. 5a. All the
+transitions have constant assignments. This system has no inputs. The initial
+values are x, y ∈ [0.01, 0.09], and the initial location is loc1. We execute the model
+for a time horizon of 10 units with a sampling time of 0.01, i.e., each trajectory
+consists of 1,000 points. We use εFwdBwd = 0.1, εdst = 1.0 and εcor = 0.89.
+Cells This is a benchmark modeling excitable cells [10, 26], which is a biological
+system exhibiting hybrid behavior. We use a variant of the excitable cell used
+in [21]. Our HA model is shown in Fig. 5b. This model has no inputs. We take the
+initial values for the voltage x ∈ [−76, −74]. The Upstroke is the initial location.
+We execute the model for a time horizon of 500 units with a sampling time
+of 0.01, i.e., each trajectory consists of 50,000 points. We use εFwdBwd = 0.01,
+εdst = 1.0, and εcor = 0.92.
+Engine This benchmark models an engine timing system taken from the demo
+examples in the Simulink automotive category [1]. The model is a complex non-
+linear system with two inputs and one output signal. The inputs are the desired
+speed of the system and the load torque, while the output signal is the engine’s
+speed. We simulate the model for a time horizon of 10 units with a sampling
+time of 0.01, i.e., each trajectory consists of 1,000 points. We use εFwdBwd = 0.99,
+εdst = 560 and εcor = 0.89.
+4.2
+Results and Discussion
+Overall Discussion Table 2 shows the summary of the results. In columns
+δO1 and δO2, we observe that for all the benchmarks, the HAs learned by our
+algorithm (both “W/o Type” and “Type”) achieved higher accuracy in terms of
+
+14
+A. Gurung et al.
+Table 2: Summary of the results. The columns δO1 and δO2 show the mini-
+mum (Min), maximum(Max), average (Avg), and standard deviation (Std) of
+the DTW distance between trajectories generated by the original model and the
+learned model feeding the test set. The columns Time show the total running
+time in seconds for learning an HA. Cells with the best results are highlighted.
+Model Measure
+W/o Type
+Type
+POSEHAD
+δO1
+δO2
+Time
+δO1
+δO2
+Time
+δO1
+δO2
+Time
+Ball
+Min(δ)
+54.8
+15.9
+332.3
+134.9
+61.1
+332.4
+125.1
+2.1e+6
+56719.7
+Max (δ)
+66.9
+42.0
+195.0
+82.0
+4.0e+4
+1.6e+9
+Avg (δ)
+60.6
+27.1
+169.2
+70.8
+1.1e+4
+3.3e+8
+Std (δ)
+2.8
+6.5
+15.2
+4.6
+1.3e+4
+3.6e+8
+Tanks
+Min(δ)
+4.2
+2.6
+356.5
+2.5
+1.7
+332.7
+37.8
+7.8e+12
+13771.5
+Max (δ)
+6.8
+5.0
+5.1
+4.1
+2.3e+4
+2.0e+14
+Avg (δ)
+5.6
+3.8
+3.9
+2.8
+8.1e+3
+9.5e+13
+Std (δ)
+0.7
+0.6
+0.6
+0.5
+7.6e+3
+5.9e+13
+Osci
+Min(δ)
+2.2
+2.2
+24.6
+0.2
+0.2
+23.5
+15.8
+8.8
+404.2
+Max (δ)
+2.9
+3.0
+0.3
+0.6
+1.5e+3
+933.9
+Avg (δ)
+2.4
+2.4
+0.2
+0.2
+1.2e+3
+716.0
+Std (δ)
+0.2
+0.2
+0.03
+0.1
+404.0
+313.4
+Cells
+Min(δ)
+152.1
+–
+2404.2
+1.3
+–
+2358.5
+2.5e+9
+–
+191050.0
+Max (δ) 410.9
+–
+150.5
+–
+5.1e+9
+–
+Avg (δ)
+205.2
+–
+58.1
+–
+3.1e+9
+–
+Std (δ)
+55.6
+–
+57.3
+–
+8.3e+8
+–
+Engine
+Min(δ) 7.9e+4
+–
+44.6
+1.2e+4
+–
+44.5
+2.8e+3
+–
+197.6
+Max (δ) 2.9e+5
+–
+3.8e+4
+–
+4.2e+14
+–
+Avg (δ) 1.9e+5
+–
+1.9e+4
+–
+1.3e+13
+–
+Std (δ) 8.0e+4
+–
+5.8e+3
+–
+7.4e+8
+–
+Avg(δ). This is because of the adequate handling of the input variables and the
+inference of the resets at transitions. We also observe that using type annotation
+usually improves the accuracy of the learned model. For instance, in benchmarks
+Tanks, Osci, and Cells, the results obtained with type annotation performed
+better than without type annotation.
+We also observe that for the HAs learned by our learning algorithm, the maxi-
+mum DTW distance Max(δ) tends to be close to the minimum DTW distance
+Min(δ). This indicates that trajectories generated by our learned model do not
+have a high deviation from the trajectories generated by the original model. We
+discuss the detail later in this section. In contrast, in the POSEHAD algorithm,
+they tend to have a high difference between Min(δ) and Max(δ). We also observe
+that for the HAs learned by the POSEHAD algorithm, the standard deviation
+Std(δ) is much larger than that learned from ours. This suggests that our learn-
+ing algorithm is better at generalization. Moreover, our algorithm takes much
+less time than POSEHAD. For instance, in the Cells benchmark, our algorithm
+takes less than one hour, whereas POSEHAD takes more than 53 hours.
+
+Learning nonlinear hybrid automata from input–output time-series data
+15
+off_off
+on_off
+off_on
+on_on
+x1 ≥ −1
+˙x2 = x1 + u
+˙x1 = −x1 − 2 + u
+x1 = −1
+˙x2 = x1 + u
+˙x1 = −x1 + 3 + u
+x2 = 0
+˙x2 = x1 − x2 − 5 + u
+x1 = 1
+˙x1 = −x1 + 3 + u
+˙x2 = x1 − x2 − 5 + u
+x2 = 1
+x2 = 0
+˙x1 = −x1 − 2 + u
+x1 = −1
+x2 = 1
+x2 ≥ 0
+x1 ≤ 1
+x2 ≥ 0
+x1 ≥ −1
+x2 ≤ 1
+x2 ≤ 1
+(a) The original HA
+˙x2 = 1.0u + 1.0x1 + 0.0x2 + 0.0
+˙x1 = 1.0u − 1.0x1 + 0.0x2 − 2.0
+˙x2 = 1.0u + 1.0x1 − 1.0x2 − 5.0
+˙x1 = 1.0u − 1.0x1 + 0.0x2 − 2.0
+˙x2 = 1.0u + 1.0x1 + 0.0x2 + 0.0
+˙x1 = 1.0u − 1.0x1 + 0.0x2 + 3.0
+loc2
+loc1
+loc3
+−16.8u + 2.1x1 − 4825.6x2 + 4821.1 ≤ 0
+1.0u + 146.4x1 − 0.004x2 + 145.0 ≤ 0
+1.0u − 0.1x1 + 159.0x2 − 1.6 ≤ 0
+(b) The HA learned by our algorithm with type
+annotation
+Fig. 7: HAs on Tanks benchmark
+v := 0.0093g + 0.4026x − 0.7997v + 0.036
+˙x = 0.0g + 0.0x + 1.0v
+˙v = 1.0g + 0.0x + 0.0v
+1.0g + 17.7709x + 0.0092v + 9.474 ≤ 0
+loc1
+x := 1.0x
+Fig. 6: The HA learned by our algorithm
+with type annotation on Ball
+Discussion for each benchmark
+Fig. 6 shows the learned HA for Ball
+produced by our algorithm with a
+type annotation. We observe that the
+ODE is precisely learned. Although
+the guard is far from the expected con-
+dition x ≤ 0, it is close to the expected
+condition given the range of the state
+variables; when we have v ≈ −20.55
+and g ≈ −9.8, the condition is about
+x ≤ 0.028, which is reasonably close to
+x ≤ 0. Similarly, the assignment of v is
+reasonably close to v ::= −0.8v when
+we have x ≈ 0 and g ≈ −9.8. In Figs. 8a and 8b, we show plots of the trajec-
+tories obtained from the HAs learned by our algorithm (with and without type
+annotation), the output trajectory predicted by POSEHAD, and the trajectory
+obtained from the original model. In Fig. 8b, we did not include the predicted
+trajectory by POSEHAD due to its high error. We observe that the trajectories
+obtained from our learned models coincide with the original benchmark trajec-
+tory, while the trajectory predicted by POSEHAD does not.
+Fig. 7b shows the HA learned by our algorithm with type annotation on the
+Tanks benchmark. Since the initial value, x2 = 1, is satisfied by the guard at
+the initial location, the system takes an instant transition to location off_on (see
+Fig. 7a). Therefore, all trajectories contain data starting from this location, and
+our algorithm identifies this to be the initial location. Moreover, the trajectories
+given to the learning algorithm do not include data visiting the location on_on,
+and this mode is not present in the learned model. We observe that the ODEs
+are exactly learned, and the guards are close to the original model. In Fig. 8c,
+
+16
+A. Gurung et al.
+we show a plot of the trajectories obtained from the HAs learned by our algo-
+rithm (with and without type annotation), the output trajectory predicted by
+POSEHAD, and the trajectory obtained from the original model. The models
+learned by our algorithm produced trajectories close to the original model, while
+several parts predicted by POSEHAD are far from the original one.
+0
+5
+10
+15
+time
+0
+5
+10
+15
+20
+x
+W/o Type
+Type
+POSEHAD
+Original
+(a) Position of the ball
+0
+5
+10
+15
+time
+-20
+-10
+0
+10
+20
+v
+W/o Type
+Type
+Original
+(b) Velocity of the ball
+0
+5
+10
+time
+-2
+0
+2
+4
+6
+8
+10
+x1
+W/o Type
+Type
+POSEHAD
+Original
+(c) Liquid level x1 of tank 1
+0
+5
+10
+time
+-0.5
+0
+0.5
+1
+1.5
+2
+x
+W/o Type
+Type
+POSEHAD
+Original
+(d) Position of x
+(e) Voltage of the cell
+0
+5
+10
+time
+500
+1000
+1500
+2000
+2500
+3000
+3500
+speed
+W/o Type
+Type
+POSEHAD
+Original
+(f) Engine speed
+Fig. 8: Trajectories on (a-b) Ball (c) Tanks (d) Osci (e) Cells and (f) Engine
+For the Engine model, due to the system’s complexity, our algorithm produced
+HAs at most with 20 locations and 130 transitions. In Fig. 8f, we show a plot
+of the trajectories obtained from the HAs learned by our algorithm (with and
+without type annotation), the output trajectory predicted by POSEHAD, and
+the trajectory obtained from the original model. The models learned by our
+algorithm produced trajectories uniformly close to the original model, while
+several parts predicted by POSEHAD are far from the original one. Similar
+observations on accuracy can be drawn from Figs. 8d and 8e on Osci and Cells
+benchmarks, respectively.
+5
+Conclusion
+This paper presents an algorithm to learn an HA with polynomial ODEs from
+input–output trajectories. We identify the locations by segmenting the given
+trajectories, clustering the segments, and inferring ODEs. We learn transition
+guards using SVM with a polynomial kernel and assignment functions using
+linear regression. Our experimental evaluation suggests that our algorithm pro-
+duces more accurate HAs than one of the state-of-the-art algorithms. Moreover,
+we extended the inference of assignments with type annotations to utilize prior
+knowledge of a user. In future work, we plan to utilize our learned HA model to
+perform black-box checking [15, 19, 23] for falsification, model-bounded moni-
+toring of hybrid systems [24], and controller synthesis [7, 17].
+
+100
+W/o Type
+Type
+50
+- Original400
+500x
+0
+-50
+0
+100
+200
+300
+timeLearning nonlinear hybrid automata from input–output time-series data
+17
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+A
+Additional Detailed
+A.1
+Experiment settings
+For our experiment in Table 2, the maximum number of trajectory segments we
+take for each location in the ODE inference helps our algorithm improve perfor-
+mance compared to POSEHAD. For the benchmarks Ball, Tanks, and Osci,
+we take 50 trajectory segments, 100 for Engine, and 3 for Cells, respectively.
+
+20
+A. Gurung et al.
+loc1
+˙x = −2.0047674x − 0.0044129y + 1.40154
+˙y = 0.0026296x − 0.9976326y − 0.700885
+loc2
+1.0x + 1.41332y + 0.00552 ≤ 0
+˙x = −2.0000007x − 0.0y − 1.4000005
+˙y = 0.0x − 1.0y + 0.7
+1.0x − 0.76904y + 0.99291 ≤ 0
+(a) Learned HA model of Osci
+˙x = 0.0x − 1.52
+loc2
+−3.137x + 137.974 ≤ 0
+loc3
+˙x = 0.0x − 0.76
+1.0x − 36.004 ≤ 0
+˙x = 0.0x + 130.02
+loc1
+˙x = 0.0x − 2.13
+loc4
+1.0x + 74.009 ≤ 0
+1.0x + 4.003 ≤ 0
+(b) Learned HA model of Cells
+Fig. 9: Our learned HA models using type annotation
+We thank the authors for providing us with the source code of the POSEHAD
+algorithm. POSEHAD also uses the DTW algorithm for clustering similar seg-
+mented trajectories. However, for segmentation, they use a different off-the-shelf
+Python library named Rupture to detect change points in trajectories. There-
+fore, the threshold parameters that we use for our algorithm may not be the best
+for POSEHAD. So, as recommended in their paper, we perform a simple manual
+grid search of parameters, including the thresholds used in our approach. We
+fix a parameter that performs the best and keeps the implementation running
+without returning errors during the entire search. In the original POSEHAD
+implementation, pre-processing is applied to the input-output data by scaling
+the data values to 0 and 1. To perform a fair comparison, we skip this pre-
+processing in the experiment. POSEHAD learns an HA model for each output
+variable independently.
+A.2
+Learned HA models
+A.3
+Additional experimental results
+Detailed discussion for each benchmark Fig. 9a shows our learned HA
+model for an Osci model generated using the Type annotation approach. Ob-
+serve that in the original model, locations loc1 and loc4 have the same ODE,
+and there is no assignment logic (cause for a change point) for a discrete transi-
+tion. Therefore our segmentation process considers these locations to be a single
+mode. Similarly, locations loc2 and loc3 have the same dynamics, so our approach
+also learns a single location for this. The ODE and the transition guards in the
+learned model are relatively close to the original model. In Figs. 8d and 10, we
+compare output trajectories obtained by our learned models (with and without
+Type annotation), POSEHAD prediction, and the original benchmark. The tra-
+jectory obtained by our learned model using Type annotation overlaps precisely
+with the original benchmark trajectory. Without a Type annotation, the trajec-
+tory either overlaps or passes close to the original trajectory. On the other hand,
+
+Learning nonlinear hybrid automata from input–output time-series data
+21
+several sections of the predicted trajectory by POSEHAD are either incorrectly
+predicted or do not overlap with the original trajectory.
+0
+5
+10
+time
+-0.5
+0
+0.5
+1
+1.5
+y
+W/o Type
+Type
+POSEHAD
+Original
+Fig. 10: Trajectories of Osci model on
+variable y
+Fig. 9b shows our learned HA model
+for the Cells model produced us-
+ing the Type annotation approach. We
+learned a four-location HA with de-
+terministic guards for each transition.
+Note that the learned ODE is exact
+to the original model for the associ-
+ated locations, and the guard condi-
+tions are close to the actual guards.
+In Fig. 8e, we show the accuracy of
+our learned model where trajectories
+generated by our learned models (with
+and without Type annotation) coin-
+cide with the trajectory obtained by
+the original benchmark. Due to high errors, we could not show the predicted
+trajectory by POSEHAD here in a single figure.
+
diff --git a/mdE2T4oBgHgl3EQfeQfB/content/tmp_files/load_file.txt b/mdE2T4oBgHgl3EQfeQfB/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='DC]  10 Jan 2023  Learning nonlinear hybrid automata from  input–output time-series data  Amit Gurung  , Masaki Waga  , and Kohei Suenaga  Kyoto University, Kyoto, Japan  amit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='gurung@cyphai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='io, {mwaga, ksuenaga}@fos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='kuis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='jp  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Learning an automaton that approximates the behavior of a  black-box system is a long-studied problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Besides its theoretical signif-  icance, its application to search-based testing and model understanding  is recently recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We present an algorithm to learn a nonlinear hy-  brid automaton (HA) that approximates a black-box hybrid system (HS)  from a set of input–output traces generated by the HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Our method is  novel in handling (1) both exogenous and endogenous HS and (2) HA  with reset associated with each transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' To our knowledge, ours is the  first method that achieves both features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We applied our algorithm to  various benchmarks and confirmed its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Keywords: Automata Learning · Inferring Hybrid Systems · Learning  Cyber-Physical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  1  Introduction  Mathematical modeling of the behavior of a system is one of the main tasks in  science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' If a system exhibits only continuous dynamics, it is well  modeled by ordinary differential equations (ODE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' However, many systems ex-  hibit continuous and discrete dynamics, being instances of hybrid systems (HS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  For instance, in modeling an automotive engine, the ODE must be switched fol-  lowing the status of the gear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' A similar combination of continuous and discrete  dynamics also appears in many other systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', biological systems [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  ˙v = −g  ˙x = v  x ≤ 0  v := −cv  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1: A bounc-  ing ball model  Hybrid automata (HAs) [3] is a formalism for HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1 il-  lustrates an HA modeling a bouncing ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In an HA, a set  of locations (represented by a circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1) and transitions  between them (represented by an arrow) expresses its discrete  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' An ODE associated with each location expresses  continuous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In the HA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1, the ODEs at the  location show the free-fall behavior of the system, and the  transition shows the discrete jump caused by bouncing on  the floor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', a change in the ball’s velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=')  It is a natural research direction to automatically identify an  HA given system’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Not only is it interesting as re-  search, but it is also of a practical impact since learning a  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Table 1: Comparison of hybrid automata learning methods  Non-linear ODEs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Exo- and Endogenous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Infer Resets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Support Inputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Ours  Polynomial  Yes  Yes  Yes  [16]  Polynomial  Only Exogenous  No  Yes  [25]  Linear  Yes  No  Yes  [20]  Linear  Only Endogenous  No  No  model of a black-box system is recently being applied to au-  tomated testing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', black-box checking [15, 19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=') There  have been various techniques to infer an HA from a set of input–output sys-  tem trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' However, as Table 1 shows, all the existing methods have some  limitations in the inferred HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' To the best of our knowledge, there is no ex-  isting work that achieves all of the following features: (1) Learned HAs may  involve nonlinear ODE as a flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' (2) Learned HAs may be exogenous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', mode  changes caused by external events) and endogenous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', mode changes caused  by internal events);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' and (3) Learned HAs may involve resetting of variables at a  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  This paper proposes an HA-learning algorithm that achieves the three features  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Namely, our algorithm learns an HA that may be exogenous, endogenous  or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' A learned HA can reset variables at transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' These two features make  it possible to infer the bouncing ball example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1, which is not possible  in some of the previous work [16] despite its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Furthermore, an HA  learned by our algorithm may involve ODEs with polynomial flow functions,  whereas existing work like [20, 25] can infer only HAs with linear ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 2 shows the overview of our HA learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Our algorithm consists  of a location identification step and a transition identification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We explain  each step below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Location identification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' To identify the locations, our algorithm first splits  trajectories into segments so that each segment consists only of continuous dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' To this end, the algorithm estimates the derivative of each point on a  trajectory with the linear multistep method (LMM) [12] and detects the points  where the derivative changes discontinuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Then, the segments are grouped into clusters so that the segments in each clus-  ter have similar continuous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For this, we conduct clustering based on  the distance determined by dynamic time warping (DTW) [4], which takes two  segments and computes their similarity in terms of the “shape.” We treat each  cluster as a location of an HA in the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  After the clustering, our algorithm synthesizes an ODE that best describes the  continuous dynamics of the segments in each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Our ODE inference is by  a template-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For each location, we fix a polynomial template—a  Learning nonlinear hybrid automata from input–output time-series data  3  �  �  ���  ������  ������������  ��������  ������������  ����������  ��������  ���  ���  ���  ���  ���  ���  ���  ���  ���  ��� ���  ���  ���  ���  ���  ���  ���  ����  ����  ����  ����  ����  ��  ��  ��  ��  ��  ��  ���  ��� ���  ���  ���  ���  ��� ���  ���  ��  ��  ��  �����������������  ��  ��  ��  ��  ��  ��  ��������  ���������  ����������  ��  ��  ��  �������  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='��������  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 2: Overview of our HA learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In the below center figure, the  circle and the star points stand for each segment’s first and last points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  polynomial whose coefficients are symbols for unknowns—for the flow function  of the ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Then, we obtain coefficients of the polynomial via linear regression  of the values in a trajectory and the derivative estimated by LMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Transition identification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Once locations are identified, our algorithm next  synthesizes the transition relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' It first identifies the pairs of locations (or  clusters) between which there is a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Concretely, the algorithm identifies  a transition from location c1 to location c2 if (1) there is a segment s1 in c1 and  s2 in c2 and (2) s1 immediately precedes s2 in a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The algorithm then synthesizes the guard and the reset on each transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We  synthesize the guard and the reset on each transition in a data-driven man-  ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Moreover, we introduce type annotation to improve the inference of resets  utilizing domain knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' we explain the method in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Contributions  Our contributions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  – We propose an algorithm inferring a general subclass of HAs from a set of  input and output trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  – We introduce type annotations to improve the inference of the resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  – We experimentally show that our algorithm infers HAs fairly close to the  original system under learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Related Work Despite the maturity of switched-system identification [9, 12],  only a few algorithms have been proposed to infer HAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This scarcity of work in  HA learning may be attributed to the additional information that needs to be  inferred for HAs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', variable assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=')  Table 1 summarizes algorithms inferring an HA from a set of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In [20],  an HA is learned from a set of trajectories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' however, it does not support systems  with inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Moreover, only linear ODEs can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In [25], an HA with  inputs and outputs is learned from trajectories, but the learned ODEs are still  limited to linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In [16], an HA with polynomial ODEs is learned from  inputs and outputs trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' However, the guards in the transitions consist  only of the input variables and timing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Due to this limitation, their  method cannot infer an endogenous HA such as the bouncing ball model in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Compared with these methods, our algorithm supports the most general  class of HAs, to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We remark that most of the technical ingredients used in our algorithm are  already presented in the previous papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For example, using LMM for segmen-  tation and inference of polynomial ODEs is also used by Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The use  of DTW for clustering is common in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We argue that our significant technical  contribution is the achievement of learning the general class of HAs by an appro-  priate adaptation and combination of these techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', by projecting the  input dimensions during segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The use of type annotation to improve  the inference of variable assignments is also our novelty, up to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Organization After reviewing the preliminaries in Section 2, we present our  HA learning algorithm and an experimental evaluation of it in Section 3 and  Section 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Finally, we conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  2  Preliminaries  For a set X, we denote its powerset by P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For a pair p := (a, b), we write  pr 1(p) for a and pr 2(p) for b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We denote naturals and reals by N and R,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For vectors u, and v with the same dimension, the relative difference  between them is rd(u, v) :=  ∥u−v∥  ∥u∥+∥v∥ where ∥u∥ is the Euclidean norm of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We  write [a, b] for the inclusive interval between a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  Trajectories and Hybrid automata  For a time domain [0, T ] ⊆ R and n ∈ N, an n-dimensional (continuous) signal  σ is a function assigning an n-dimensional vector σ(t) ∈ RN to each timepoint  t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Execution of a system with n1 dimensional inputs and n2 dimensional  outputs can be modeled by an (n1 + n2)-dimensional signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  A (discrete) trajectory is a sequence of vectors with timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Concretely,  an n-dimensional trajectory (t1, x1), (t2, x2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , (tN, xN) is a finite sequence  Learning nonlinear hybrid automata from input–output time-series data  5  of pairs of timestamp ti ∈ R and the corresponding value xi ∈ Rn sat-  isfying t1 < t2 < · · · < tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For a signal σ: [0, T ] → Rn, a trajectory  (t1, x1), (t2, x2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , (tN, xN) is a discretization of σ if for any i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', N},  we have xi = σ(ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We call each vector (ti, xi) in a trajectory as a (sampling)  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Hybrid automata (HAs) [3, 14] is a formalism to model a system exhibiting  an interplay between continuous and discrete dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Since we aim to learn  an HA from a set of trajectories with inputs and outputs, we employ HAs with  input and output variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' To define HAs, we fix a finite set of (continuous state)  variables X, input variables I, and output variables O such that X = I ⊎ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' A  valuation is a mapping δ ∈ RX that represents the value of each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Definition 1 (Hybrid automaton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' A hybrid automaton (HA) H, is a tuple  (L, Inv, Init, Flow, Trans) where:  L is a finite set of locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Inv : L → P(RX ) is a function mapping each location ℓ to the invariant at  ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Init, the initial condition, is a pair (ℓ0, δ0) such that ℓ0 ∈ L and δ0 ∈ Inv(ℓ0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Flow is a flow function mapping each location ℓ ∈ L to ODEs of the form  ˙x = f(x, u), called flow equation, where x is the vector of all the variables  in O and u is the vector of all the variables in I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Trans is the set of discrete transitions denoted by a tuple e = (ℓ, G, M, ℓ′),  where ℓ, ℓ′ ∈ L are the source and target locations, G ⊆ P(RX ) is the guard,  and M : RX → RO is the assignment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  For a transition e ∈ Trans, we write G(e) and M(e) for the guard and the  assignment function of e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Intuitively, a guard G(e) of a transition e is the condition that enables the tran-  sition: A transition e can be fired if a valuation δ for the variables satisfies  δ ∈ G(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' An assignment M(e) specifies how a valuation is updated if the tran-  sition e fired: A valuation is updated from δ to δ′ such that for each x ∈ O and  u ∈ I, we have δ′(x) = M(e)(δ)(x) and δ′(u) = δ(u) if e is fired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The semantics of an HA is formalized by the notion of a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' A state of an HA  H is a pair (ℓ, δ), where ℓ is a location of H and δ ∈ RX is a valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Definition 2 (Run).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' A run of an HA (L, Inv, Init, Flow, Trans) is a sequence  (ℓ0, δ0)  τ0  −→ (ℓ0, δ′  0)  e0  −→ (ℓ1, δ1)  τ1  −→ (ℓ1, δ′  1)  e1  −→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  eN−1  −−−→ (ℓN, δN)  τN  −−→ (ℓN, δ′  N)  satisfying (ℓ0, δ0) ∈ Init and for each i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', N}, there are signals  σx  i : [0, τi] → RO and σu  i : [0, τi] → RI such that (i) for any x ∈ O and  u ∈ I, we have σx  i (0)(x) = δi(x) and σu  i (0)(u) = δi(u), σx  i (τi)(x) = δ′  i(x), and  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  σx  i (τi)(u) = δ′  i(u), (ii) for any t ∈ [0, τi], we have (σx  i (t), σu  i (t)) ∈ Inv(ℓi) and  ˙σx  i (t) = Flow(ℓi)(σx  i (t), σu  i (t)), and (iii) we have δ′  i ∈ G(ei) and δi+1 is such that  for each x ∈ O and u ∈ I, we have δi+1(x) = M(ei)(δ′  i)(x) and δi+1(u) = δ′  i(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  For such a run ρ, a signal σ: [0, TN] → RX is the signal over ρ if σ is such that  σ(t)(x) = σx  i (t−Ti−1) and σ(t)(u) = σu  i (t−Ti−1)(u) for each x ∈ O, u ∈ I, and  i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', N} such that Ti ≤ t < Ti+1, where Ti = �i  j∈0 τj and TN+1 = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  Linear Multistep Method  The linear multistep method (LMM) [6] is a technique to numerically solve  an initial value problem of an ODE  ˙  x(t) = f(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Concretely, it approxi-  mates the value of x(tn+M) by using the values of x(tn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , x(tn+M−1) and  f(xn, tn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , f(xn+M−1, tn+M−1)—namely, M previous discretized values of x  and f(x, t)—where tn+i = tn + ih for some h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For this purpose, LMM  assumes the following approximation parameterized over (αi)i and (βi)i:  M  �  i=0  αix(tn−i) ≈ h  M  �  i=0  βif(x(tn−i), tn−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Then, LMM determines the values of (αi)i and (βi)i so that the error of the above  approximation, quantified with Taylor’s theorem, is minimum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' see [6, 12, 22] for  more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The approximation with the determined values of (αi)i and (βi)i is  used to successively determine the values of x(t) from its initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  In the context of our work, we estimate the derivative of a trajectory at  each point without knowing the ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' To this end, we use backwards differ-  entiation formula (BDF) [13, 22] derived from LMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The idea is to com-  pute the polynomial passing all the points (tn, x(tn)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , (tn+M−1, x(tn+M−1))  using Lagrange interpolation [6, 13] and derive the formula to approx-  imate  the  derivative  at  (tn+M, x(tn+M−1))  from  the  polynomial  using  LMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Concretely, Lagrange interpolation results in the following poly-  nomial: x(t)  ≈  �M  m=0 x(tn−m) �  i̸=m  t−tn−m  tn−i−tn−m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' By taking the derivative  of both sides and setting t to tn, we obtain  ˙x(tn)  =  f(x(tn), tn)  ≈  �M  m=0 x(tn−m) �  i̸=m( d  dt  t−tn−m  tn−i−tn−m )  ���  t=tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use this formula to estimate the  derivative at each point in a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For instance, the formula to estimate  the derivatives with M = 2 is: f(x(tn)) = 1  h( 3  2x(tn)− 4  2x(tn−1)+ 1  2x(tn−2)) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The above formula estimates the derivative at x(tn) using M previous points—  hence called backward BDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Dually, we can derive a formula that estimates the  derivative at x(tn) using M following points called forward BDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use both  in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3  Dynamic Time Warping (DTW)  Our algorithm introduced in Section 3 first splits given trajectories so that each  segment includes only continuous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Then, it classifies the generated  Learning nonlinear hybrid automata from input–output time-series data  7  segments based on the “similarity” of the ODE behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For the classification  purpose, we use dynamic time warping (DTW) [4]—one of the methods for  quantifying the similarity between time-series data in their shapes—as the mea-  sure of the similarity inspired by [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The previous work [16] applies DTW for  HA learning and confirms its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The DTW distance between two time-series data X := (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , xM) and  Y  := (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , yN), where M, N ∈ N, is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The align-  ment path between X and Y is a finite sequence P := (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , pl) where  pi ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', M}×{1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', N} and P is an alignment between {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , M} and  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Concretely, P should satisfy the following conditions: (1) p1 = (1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  (2) pl = (M, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' (3) (ai+1 − ai, bi+1 − bi) is either (1, 0), (0, 1), or (1, 1) for any  (pi, pi+1) = ((ai, bi), (ai+1, bi+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For example, ((1, 1), (1, 2), (2, 3), (3, 3), (3, 4))  is an alignment path between (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , x3) and (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , y4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  An alignment path P = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , pl) between X := (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , xM) and Y :=  (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , yN) determines the sum dP := �l  i=1 ||x(pr1(pi)) − y(pr2(pi))|| of the  distances between corresponding points in X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Then, the DTW distance  DTWdist(X, Y ) between X and Y is defined by minP dP , where P moves all the  alignment paths between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' There is an efficient algorithm computing  DTWdist(X, Y ) in O(MN) based on dynamic programming [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  For X and Y , let P be the alignment that gives the optimal sum of dis-  tances between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We write DTWcorrel(X, Y ) for correl(P1, P2), where  P1 := (pr 1(p1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , pr 1(pl)), P2 := (pr 2(p1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , pr 2(pl)), and correl(P1, P2) is  the Pearson product-moment correlation coefficients between P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This  value becomes larger if P1 and P2 increase evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Thus, the higher this value is,  the more X is similar to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The effectiveness of this value in classifying segments  is also shown in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  3  HA Learning from Input–Output Trajectories  Here, we present our HA learning algorithm from given trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Our problem  setting is formalized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Passive HA learning problem:  Input: trajectories {(ti  1, xi  1), (ti  2, xi  2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , (ti  Ni, xi  Ni) | i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', M}} that  are discretizations of signals over runs of an HA H  Output: an HA H approximating H  Our current algorithm learns an HA such that (i) the invariant of each location  is true, (ii) each guard is expressed as a polynomial inequality, and (iii) each  assignment function is a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We assume that (i) each location of  H has different ODEs and (ii) for each pair (ℓ, ℓ′) of locations of H, there is at  most one transition from ℓ to ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 2 outlines our HA learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We first present the identification of  the locations and then present that of the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Algorithm 1 Outline of our segmentation algorithm  Input: A trajectory τ = (t1, x1), (t2, x2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , (tN, xN), the step size M in BDF, and  the thresholds εFwdBwd and εBwd  Output: cp ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , N} is the set of change points  1: candidates ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' C ← ∅  2: for all i ∈ {M + 1, M + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , N − M} do  3:  fwdi ← fF(τ|O, i, M)  ⊲ Compute the forward BDF  4:  bwd i ← fB(τ|O, i, M)  ⊲ Compute the backward BDF  5:  if rd(fwdi, bwd i) > εFwdBwd then  6:  add i to candidates  7: while candidates ̸= ∅ do  8:  i ← min(candidates);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' remove i from candidates  9:  if i + 1 ̸∈ candidates or rd(bwdi, bwd i+1) ≥ εBwd then  10:  add i to cp  11:  while i + 1 ∈ candidates do  12:  remove i + 1 from candidates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' i ← i + 1  (a) fwd i ≈ bwd i  (b) rd(fwdi, bwd i) ≫ 0 and  bwd i ≈ bwd i+1  (c) rd(fwdi, bwd i) ≫ 0 and  rd(bwdi, bwd i+1) ≫ 0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 3: Illustration of our segmentation algorithm near a boundary of a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The red circle in the right figure is the change point because it is the first point  satisfying rd(fwd i, bwd i) > εFwdBwd and rd(bwd i, bwdi+1) > εBwd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  Identification of Locations  We identify the locations of an HA by the following three steps: (i) segmentation  of the given trajectories, (ii) clustering of the segments, and (iii) inference of  ODEs and initial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Segmentation of the Trajectories The first step in our HA learning algo-  rithm is segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Each trajectory is divided into segments so that the dy-  namics in each segment are jump-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We perform segmentation by identifying  the change points—the points where the derivative discontinuously changes—  along a trajectory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' this step is an enhancement of the idea by Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Algorithm 1 outlines our segmentation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For simplicity, we present an  algorithm for a single trajectory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' this algorithm is applied to each trajectory  obtained from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' First, for each point in the trajectory, we estimate  the derivative using forward and backward BDF (fwd i and bwdi, respectively)  and deem the point as a candidate of change points if rd(fwd i, bwdi) exceeds  T  B  FCT  CT  B  FCT  F  BLearning nonlinear hybrid automata from input–output time-series data  9  Algorithm 2 Outline of the clustering of the segmented trajectories  Input: Set Sg of segments and thresholds εdst and εcor for distance and diagonality  Output: C = {C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , Cn} is a set of set of segments such that each Ci is a cluster  1: C ← ∅  2: while Sg ̸= ∅ do  3:  pick sg from Sg  ⊲ We still have sg ∈ Sg after picking it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  4:  C ′ ← {sg ′ ∈ Sg | DTWdist(sg|O, sg′|O) < εdst ∧ DTWcorrel(sg|O, sg′|O) > εcor}  5:  Sg ← Sg \\ C ′  ⊲ C ′ always includes sg, and sg is removed from Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  6:  add C ′ to C  the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For example, among the three red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 3, we have  fwd i ≈ bwdi for the one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 3a and rd(fwd i, bwdi) ≫ 0 for the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Thus, the red circles in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 3b and 3c are the candidates of the change point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We remark that fwdi and bwdi are computed with the trajectory τ|O projected  to the output variables, and our segmentation is not sensitive to the change in  the input variables I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  When there are consecutive candidates of the change points, we take the first  one satisfying rd(bwd i, bwdi+1) ≥ εBwd to precisely estimate the change point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Such an optimization is justified under the assumption that there are at least  2M − 1 points between two consecutive mode changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For example, in the  example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 3, the red circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 3c is deemed to be the change  point because this is the first candidate satisfying rd(bwd i, bwdi+1) ≥ εBwd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The algorithm splits the trajectories at the identified change points into seg-  ments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' the change points are not included in the segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Notice that our  twofold identification of change points is an enhancement to Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' [12],  which takes all our candidates as change points and drops them from the re-  sulting segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Our enhancement allows more points to be included in the  resulting segments, which makes the subsequent phases more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Clustering of the Segments Then, we cluster the segmented trajectories so  that the segments with similar continuous behaviors are included in the same  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 2, the continuous behaviors in S1a, S2a, S2d, and S3a  are similar and hence included in a single cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use the identified clusters  as the set of locations in the resulting HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This construction is justified when  each location has a different ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Algorithm 2 outlines our clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The overall idea is, the algorithm  picks one segment (line 3) and creates a cluster by merging similar segments  (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use both DTWdist and DTWcorrel to determine the similarity be-  tween segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We remark that we compare the segments sg|O and sg′|O pro-  jected to the output variables to ignore the similarity in the input variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Inference of ODEs and Initial Locations Our ODE inference is by a  template-based linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' First, we fix a template Φ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' θ) = θ1f1(x) +  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  θ2f2(x)+ · · ·+ θNfN(x) of the ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In our current implementation, each fi is a  monomial whose degree is less than a value specified by a user, but an arbitrary  template can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Then, for each cluster Ci and for each output variable  o ∈ O, we construct the set Pi,o of points in Ci, and the derivative of o at this  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Formally, Pi,o = {(x, ˙x(o)) | ∃sg ∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' x ∈ sg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The derivative ˙x(o) is,  for example, computed by BDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Moreover, we can reuse the derivative used in  Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Finally, we use linear regression to compute the coefficients θ such  that for each (x, ˙x(o)) ∈ Pi,o, we have ˙x(o) ≈ Φ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  In the resulting HA, the initial locations are the locations such that the cor-  responding cluster contains the first segment for some trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Therefore,  we have multiple initial locations if there are trajectories such that their first  segments do not satisfy the similarity condition during clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  Identification of Transitions  �  �  ��  ��  ��  ��  ��  ��  ���������  ����������  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 4: Illustration of the points  connecting clusters Ci and Cj  After identifying the locations of the result-  ing HA by clustering the segments, we con-  struct transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Let sg1, sg2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , sgm be the  segments obtained from a single trajectory by  Algorithm 1 and ordered in chronological or-  der;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' segment sgi immediately precedes sgi+1  in the original trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For each segment  sgg, we denote its initial point, the second last  point, and the last point by sgη1  g , sgη2  g , and  sgη3  g , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The  idea  of  the  transition  identification  is to make one transition for each triple  (sgη2  g , sgη3  g , sgη1  g+1)—called  a  connection  triple—and use these points in a triple to infer its guard and assignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We note that such a triple is always defined since each  segment has at least three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Formally, for clusters Ci and Cj and a segmented trajectory sg1, sg2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , sgm,  the set Ti,j of connection triples from Ci to Cj is as follows:  Ti,j = {(sgη2  g , sgη3  g , sgη1  g+1) | g ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', m − 1}, sgg ∈ Ci, sgg+1 ∈ Cj}  If there are multiple trajectories in HA learning, we construct Ti,j for each tra-  jectory and take their union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We infer guards and assignments using Ti,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For each cluster pair (Ci, Cj),  the guard of the transition from Ci to Cj is obtained using a support vector  machine (SVM) to classify the second last points and the last points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' More  precisely, for T ⊥  i,j = {sgη2  g  | ∃(sgη2  g , sgη3  g , sgη1  g+1) ∈ Ti,j} and T ⊤  i,j = {sgη3  g  |  ∃(sgη2  g , sgη3  g , sgη1  g+1) ∈ Ti,j}, we compute an equation of hyperplane separating  Learning nonlinear hybrid automata from input–output time-series data  11  T ⊥  i,j and T ⊤  i,j using SVM and construct an inequality constraint G that is satisfied  by the points in T ⊤  i,j but not by that in T ⊥  i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  For each cluster pair (Ci, Cj), the assignment in the transition from Ci to Cj  is obtained using linear regression to approximate the relationship between the  valuation before and after the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' More precisely, we use linear regression  to compute an equation M such that for each (sgη2  g , sgη3  g , sgη1  g+1) ∈ Ti,j and for  each x ∈ O, sgη1  g+1(x) is close to M(sgη3  g )(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Such M is used as the assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Improving Assignments Inference with Type Annotation If we have no  prior knowledge of the system under learning, we infer assignments using linear  regression, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' However, even if the exact system dynamics are  unknown, we often know how each variable behaves at jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For instance, it  is reasonable to believe that a variable representing temperature is continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  hence, it does not change its value at jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Such domain knowledge is helpful  in inferring precise assignments rather than one using linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  To easily enforce the constraints from domain knowledge on variables, we extend  our assignment inference to allow users to annotate each variable with types  expressing how a variable is assumed to behave at jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We currently support  the following types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  No assignments If a variable is continuous at a jump (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', the variable rep-  resenting temperature mentioned above), one annotates the variable with “no  assignments”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For a variable x with this annotation, the procedure above infers  an assignment that does not change the value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For example, in the HA in  the bouncing ball HA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1, x is a continuous variable while v is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Constant pool If the value assigned to a variable at a jump is chosen from a finite  set, one annotates the variable with “Constant pool” accompanied with the finite  set {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' , vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' An example of such a variable is one representing the gear in  a model of an automotive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For a variable with this annotation, our algorithm  infers the assignment at a jump by majority poll: For a transition from cluster  Ci to Cj, it chooses the value most frequently occurring in Ti,j as sgη1  g+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  4  Experiments  We implemented our proposed algorithm using a combination of C++, Python,  and MATLAB/Simulink/Stateflow: The HA learning algorithm is written in  Python;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The learned model is translated into a Simulink/Stateflow model by a  C++ program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use MATLAB to simulate the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We optimized  the ODE inference by using only a part of the trajectories when they were  sufficiently many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We take M = 5 as the step size for BDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We conducted experiments (i) to compare the performance of our algorithm  against a state-of-the-art method and (ii) to evaluate how the type annotation  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  helps our learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For the former evaluation, we compared our al-  gorithm against one of the latest HA learning methods called POSEHAD [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We compared our algorithm with and without a type annotation for the latter  evaluation, denoted as “Type” and “W/o Type,” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Each benchmark consists of a Simulink/Stateflow model, which we call an origi-  nal model, and two sets of trajectories generated from the original model, which  we call training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We generated trajectories by feeding random in-  put trajectories and random initial values of the state variables to the original  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The training set is used to learn an HA, which we call a learned model,  and the test set is used to evaluate the accuracy of the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For each  benchmark, the size of the training and test sets are 64 and 32, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  To evaluate the accuracy of the learned model, we feed the same input trajec-  tories and the same initial values to the original and the learned models and  compared their output trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The comparison is based on the DTW dis-  tance DTWdist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' A low DTW distance indicates higher accuracy of the learned  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We denote as δO1 and δO2 the DTW distances between trajectories gener-  ated from the original and the learned model on the output variable, O1, and O2,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We note that, in POSEHAD, the DTW distance is not computed  with the entire trajectories but with the segmented trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' All the experi-  ments reported in this paper are conducted on a machine with an Intel Core i9  CPU, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='40GHz, and 32 GiB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We used εBwd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='01 in all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  Benchmark Description  We briefly describe the benchmarks used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Ball This is a benchmark modeling a bouncing ball taken from the demo ex-  ample of Simulink [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1 shows the HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The acceleration due to the gravity g  is taken as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The range of g is [−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9, −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We modify the original Simulink  model to parameterize the initial values of x and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We let x ∈ [10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5] and  v = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The reset factor c in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 1 is c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We execute the model for a time  horizon of 13 units with a sampling time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='001, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', each trajectory consists  of 13,000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use εFwdBwd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1, εdst = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0, and εcor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Tanks This benchmark models a two tanks system [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 7a shows the HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  The system consists of two tanks with liquid levels x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The first tank  has in/out flow controlled by a valve v1, whereas, the second tank has outflow  controlled by the other valve v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Both tanks have external in/out flow controlled  by the input signal u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' There is also a flow from the first tank to the second tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  In summary, the system has four locations for on and off of v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The range  of the input is u ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1], the initial liquid level of the two tanks are x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  and x2 = 1, and the initial location is off_off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We execute the model for a time  horizon of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3 units with a sampling time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='001, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', each trajectory consists  of 9,300 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use εFwdBwd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='01, εdst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5 and εcor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Learning nonlinear hybrid automata from input–output time-series data  13  y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x ≥ 0  x ≤ 0  ˙y = −y − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  ˙x = −2x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  loc4  x ≤ 0  y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x = 0  y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x ≤ 0  loc3  x ≤ 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x + y ≥ 0  ˙x = −2x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  ˙y = −y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  x ≥ 0  x ≥ 0  y ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x + y ≤ 0  x = 0  x = 0  ˙x = −2x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  ˙y = −y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  loc1  ˙x = −2x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  ˙y = −y − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  y ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x  x ≥ 0  y < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='714286x  loc2  (a) An HA model of Osci  Early_Repolarization  Upstroke  Plateau  Final_Repolarization  ˙x = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='52  34 ≤ x ≤ 46  −76 ≤ x ≤ 46  ˙x = 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='02  44 ≤ x ≤ 46  −76 ≤ x ≤ −74  −6 ≤ x ≤ −4  ˙x = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='76  34 ≤ x ≤ 36  −76 ≤ x ≤ 36  ˙x = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='13  −6 ≤ x ≤ 36  (b) An HA model of Cells  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 5: HA models for Osci and Cells benchmarks  Osci This is a benchmark modeling a switched oscillator without filters [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Osci is an affine system with two variables, x and y oscillating between two  equilibria to maintain a stable oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The HA is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' All the  transitions have constant assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This system has no inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The initial  values are x, y ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='09], and the initial location is loc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We execute the model  for a time horizon of 10 units with a sampling time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='01, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', each trajectory  consists of 1,000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use εFwdBwd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1, εdst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0 and εcor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Cells This is a benchmark modeling excitable cells [10, 26], which is a biological  system exhibiting hybrid behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use a variant of the excitable cell used  in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Our HA model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This model has no inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We take the  initial values for the voltage x ∈ [−76, −74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The Upstroke is the initial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We execute the model for a time horizon of 500 units with a sampling time  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='01, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', each trajectory consists of 50,000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use εFwdBwd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='01,  εdst = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0, and εcor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Engine This benchmark models an engine timing system taken from the demo  examples in the Simulink automotive category [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The model is a complex non-  linear system with two inputs and one output signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The inputs are the desired  speed of the system and the load torque, while the output signal is the engine’s  speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We simulate the model for a time horizon of 10 units with a sampling  time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='01, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', each trajectory consists of 1,000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We use εFwdBwd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='99,  εdst = 560 and εcor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  Results and Discussion  Overall Discussion Table 2 shows the summary of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In columns  δO1 and δO2, we observe that for all the benchmarks, the HAs learned by our  algorithm (both “W/o Type” and “Type”) achieved higher accuracy in terms of  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Table 2: Summary of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The columns δO1 and δO2 show the mini-  mum (Min), maximum(Max), average (Avg), and standard deviation (Std) of  the DTW distance between trajectories generated by the original model and the  learned model feeding the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The columns Time show the total running  time in seconds for learning an HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Cells with the best results are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Model Measure  W/o Type  Type  POSEHAD  δO1  δO2  Time  δO1  δO2  Time  δO1  δO2  Time  Ball  Min(δ)  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9  332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3  134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1e+6  56719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  Max (δ)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0e+4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6e+9  Avg (δ)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1e+4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3e+8  Std (δ)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3e+4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6e+8  Tanks  Min(δ)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8e+12  13771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  Max (δ)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3e+4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0e+14  Avg (δ)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1e+3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5e+13  Std (δ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6e+3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9e+13  Osci  Min(δ)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8  404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  Max (δ)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5e+3  933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9  Avg (δ)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2e+3  716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  Std (δ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4  Cells  Min(δ)  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  –  2404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3  –  2358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5e+9  –  191050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  Max (δ) 410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9  –  150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  –  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1e+9  –  Avg (δ)  205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  –  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  –  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1e+9  –  Std (δ)  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  –  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3  –  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3e+8  –  Engine  Min(δ) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9e+4  –  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2e+4  –  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8e+3  –  197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6  Max (δ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9e+5  –  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8e+4  –  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2e+14  –  Avg (δ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9e+5  –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='9e+4  –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3e+13  –  Std (δ) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0e+4  –  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8e+3  –  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4e+8  –  Avg(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This is because of the adequate handling of the input variables and the  inference of the resets at transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We also observe that using type annotation  usually improves the accuracy of the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For instance, in benchmarks  Tanks, Osci, and Cells, the results obtained with type annotation performed  better than without type annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  We also observe that for the HAs learned by our learning algorithm, the maxi-  mum DTW distance Max(δ) tends to be close to the minimum DTW distance  Min(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This indicates that trajectories generated by our learned model do not  have a high deviation from the trajectories generated by the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We  discuss the detail later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In contrast, in the POSEHAD algorithm,  they tend to have a high difference between Min(δ) and Max(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We also observe  that for the HAs learned by the POSEHAD algorithm, the standard deviation  Std(δ) is much larger than that learned from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' This suggests that our learn-  ing algorithm is better at generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Moreover, our algorithm takes much  less time than POSEHAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For instance, in the Cells benchmark, our algorithm  takes less than one hour, whereas POSEHAD takes more than 53 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Learning nonlinear hybrid automata from input–output time-series data  15  off_off  on_off  off_on  on_on  x1 ≥ −1  ˙x2 = x1 + u  ˙x1 = −x1 − 2 + u  x1 = −1  ˙x2 = x1 + u  ˙x1 = −x1 + 3 + u  x2 = 0  ˙x2 = x1 − x2 − 5 + u  x1 = 1  ˙x1 = −x1 + 3 + u  ˙x2 = x1 − x2 − 5 + u  x2 = 1  x2 = 0  ˙x1 = −x1 − 2 + u  x1 = −1  x2 = 1  x2 ≥ 0  x1 ≤ 1  x2 ≥ 0  x1 ≥ −1  x2 ≤ 1  x2 ≤ 1  (a) The original HA  ˙x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  ˙x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  ˙x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x2 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  ˙x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  ˙x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  ˙x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0  loc2  loc1  loc3  −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8u + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1x1 − 4825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6x2 + 4821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1 ≤ 0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u + 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4x1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='004x2 + 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0 ≤ 0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0u − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1x1 + 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='6 ≤ 0  (b) The HA learned by our algorithm with type  annotation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 7: HAs on Tanks benchmark  v := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0093g + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4026x − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7997v + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='036  ˙x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0g + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0v  ˙v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0g + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0v  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0g + 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7709x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0092v + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='474 ≤ 0  loc1  x := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 6: The HA learned by our algorithm  with type annotation on Ball  Discussion for each benchmark  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 6 shows the learned HA for Ball  produced by our algorithm with a  type annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We observe that the  ODE is precisely learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Although  the guard is far from the expected con-  dition x ≤ 0, it is close to the expected  condition given the range of the state  variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' when we have v ≈ −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='55  and g ≈ −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8, the condition is about  x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='028, which is reasonably close to  x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Similarly, the assignment of v is  reasonably close to v ::= −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8v when  we have x ≈ 0 and g ≈ −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8a and 8b, we show plots of the trajec-  tories obtained from the HAs learned by our algorithm (with and without type  annotation), the output trajectory predicted by POSEHAD, and the trajectory  obtained from the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8b, we did not include the predicted  trajectory by POSEHAD due to its high error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We observe that the trajectories  obtained from our learned models coincide with the original benchmark trajec-  tory, while the trajectory predicted by POSEHAD does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 7b shows the HA learned by our algorithm with type annotation on the  Tanks benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Since the initial value, x2 = 1, is satisfied by the guard at  the initial location, the system takes an instant transition to location off_on (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Therefore, all trajectories contain data starting from this location, and  our algorithm identifies this to be the initial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Moreover, the trajectories  given to the learning algorithm do not include data visiting the location on_on,  and this mode is not present in the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We observe that the ODEs  are exactly learned, and the guards are close to the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8c,  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Gurung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  we show a plot of the trajectories obtained from the HAs learned by our algo-  rithm (with and without type annotation), the output trajectory predicted by  POSEHAD, and the trajectory obtained from the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The models  learned by our algorithm produced trajectories close to the original model, while  several parts predicted by POSEHAD are far from the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  0  5  10  15  time  0  5  10  15  20  x  W/o Type  Type  POSEHAD  Original  (a) Position of the ball  0  5  10  15  time  20  10  0  10  20  v  W/o Type  Type  Original  (b) Velocity of the ball  0  5  10  time  2  0  2  4  6  8  10  x1  W/o Type  Type  POSEHAD  Original  (c) Liquid level x1 of tank 1  0  5  10  time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  2  x  W/o Type  Type  POSEHAD  Original  (d) Position of x  (e) Voltage of the cell  0  5  10  time  500  1000  1500  2000  2500  3000  3500  speed  W/o Type  Type  POSEHAD  Original  (f) Engine speed  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8: Trajectories on (a-b) Ball (c) Tanks (d) Osci (e) Cells and (f) Engine  For the Engine model, due to the system’s complexity, our algorithm produced  HAs at most with 20 locations and 130 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8f, we show a plot  of the trajectories obtained from the HAs learned by our algorithm (with and  without type annotation), the output trajectory predicted by POSEHAD, and  the trajectory obtained from the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The models learned by our  algorithm produced trajectories uniformly close to the original model, while  several parts predicted by POSEHAD are far from the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Similar  observations on accuracy can be drawn from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8d and 8e on Osci and Cells  benchmarks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  5  Conclusion  This paper presents an algorithm to learn an HA with polynomial ODEs from  input–output trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We identify the locations by segmenting the given  trajectories, clustering the segments, and inferring ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We learn transition  guards using SVM with a polynomial kernel and assignment functions using  linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Our experimental evaluation suggests that our algorithm pro-  duces more accurate HAs than one of the state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Moreover,  we extended the inference of assignments with type annotations to utilize prior  knowledge of a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In future work, we plan to utilize our learned HA model to  perform black-box checking [15, 19, 23] for falsification, model-bounded moni-  toring of hybrid systems [24], and controller synthesis [7, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='1007/978-3-030-88494-9_6  Learning nonlinear hybrid automata from input–output time-series data  19  [20] Soto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=' In: Bogomolov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Jungers,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Schilling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=': Membership-  based synthesis of linear hybrid automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In: Dillig, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Tasiran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=') Computer Aided Verification - 31st International Conference,  CAV 2019, New York City, NY, USA, July 15-18, 2019, Proceed-  ings, Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Lecture Notes in Computer Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 11561, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=' Springer (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='1007/978-3-030-25540-4_16  [22] Süli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Mayers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' : An introduction to numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Cambridge  university press (2003)  [23] Waga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=' In: Ames, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=') HSCC  ’20: 23rd ACM International Conference on Hybrid Systems: Computation  and Control, Sydney, New South Wales, Australia, April 21-24, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='  11:1–11:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' ACM (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=', André, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Hasuo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=': Model-bounded monitoring of hy-  brid  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='1145/  3529095  [25] Yang,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=',  Beg,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=',  Kenigsberg,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=',  Johnson,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content=' :  A  frame-  work for identification and validation of affine hybrid automata from  input-output traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='1145/  3470455  [26] Ye, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Entcheva, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Grosu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=', Smolka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=': Efficient modeling of excitable  cells using hybrid automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' of CMSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 216–227 (2005)  A  Additional Detailed  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='1  Experiment settings  For our experiment in Table 2, the maximum number of trajectory segments we  take for each location in the ODE inference helps our algorithm improve perfor-  mance compared to POSEHAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' For the benchmarks Ball, Tanks, and Osci,  we take 50 trajectory segments, 100 for Engine, and 3 for Cells, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='  loc1  ˙x = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0047674x − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+page_content='41332y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='00552 ≤ 0  ˙x = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0000007x − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0y − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='4000005  ˙y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='76904y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='99291 ≤ 0  (a) Learned HA model of Osci  ˙x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='52  loc2  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='137x + 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='974 ≤ 0  loc3  ˙x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x − 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='004 ≤ 0  ˙x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x + 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='02  loc1  ˙x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='13  loc4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x + 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='009 ≤ 0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='0x + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='003 ≤ 0  (b) Learned HA model of Cells  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 9: Our learned HA models using type annotation  We thank the authors for providing us with the source code of the POSEHAD  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' POSEHAD also uses the DTW algorithm for clustering similar seg-  mented trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' However, for segmentation, they use a different off-the-shelf  Python library named Rupture to detect change points in trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' There-  fore, the threshold parameters that we use for our algorithm may not be the best  for POSEHAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' So, as recommended in their paper, we perform a simple manual  grid search of parameters, including the thresholds used in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We  fix a parameter that performs the best and keeps the implementation running  without returning errors during the entire search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In the original POSEHAD  implementation, pre-processing is applied to the input-output data by scaling  the data values to 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' To perform a fair comparison, we skip this pre-  processing in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' POSEHAD learns an HA model for each output  variable independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='2  Learned HA models  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='3  Additional experimental results  Detailed discussion for each benchmark Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 9a shows our learned HA  model for an Osci model generated using the Type annotation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Ob-  serve that in the original model, locations loc1 and loc4 have the same ODE,  and there is no assignment logic (cause for a change point) for a discrete transi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Therefore our segmentation process considers these locations to be a single  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Similarly, locations loc2 and loc3 have the same dynamics, so our approach  also learns a single location for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The ODE and the transition guards in the  learned model are relatively close to the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8d and 10, we  compare output trajectories obtained by our learned models (with and without  Type annotation), POSEHAD prediction, and the original benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' The tra-  jectory obtained by our learned model using Type annotation overlaps precisely  with the original benchmark trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Without a Type annotation, the trajec-  tory either overlaps or passes close to the original trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' On the other hand,  Learning nonlinear hybrid automata from input–output time-series data  21  several sections of the predicted trajectory by POSEHAD are either incorrectly  predicted or do not overlap with the original trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  0  5  10  time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='5  y  W/o Type  Type  POSEHAD  Original  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 10: Trajectories of Osci model on  variable y  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 9b shows our learned HA model  for the Cells model produced us-  ing the Type annotation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' We  learned a four-location HA with de-  terministic guards for each transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  Note that the learned ODE is exact  to the original model for the associ-  ated locations, and the guard condi-  tions are close to the actual guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' 8e, we show the accuracy of  our learned model where trajectories  generated by our learned models (with  and without Type annotation) coin-  cide with the trajectory obtained by  the original benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
+page_content=' Due to high errors, we could not show the predicted  trajectory by POSEHAD here in a single figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfeQfB/content/2301.03915v1.pdf'}
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+Altermagnetism in MnTe: origin, predicted manifestations, and routes to detwinning
+I. I. Mazin
+Department of Physics and Astronomy, George Mason University, Fairfax, VA 22030, USA and
+Quantum Science and Engineering Center, George Mason University, Fairfax, VA 22030, USA
+MnTe has recently attracted attention as an altermagnetic candidate. Experimentally it has an
+altermagnetic order of ferromagnetic ab planes, stacked antiferromagnetically along c. We show that
+this magnetic order (by itself non-trivial, since the in-plane exchange in antiferromagnetic) opens
+intriguing possibility of manufacturing altermagnetically-detwinned samples and generate observable
+magnetooptical response (which we calculate from first principles) as a signature of altermagnetism.
+The recently discovered phenomenon of spin-split
+bands in collinear symmetry-compensated antiferromag-
+nets, dubbed “altermagnetism” (AM)[2, 3, 7], has at-
+tracted considerable attention.
+While a number of al-
+termagnets have been theoretically identified, there is a
+big experimental challenges in assessing this, for a num-
+ber of reasons: First, most of them are not metals, so
+anomalous Hall conductivity cannot be measured. Sec-
+ond, many have the easy magnetization direction not
+compatible with anomalous response. Third, statistically
+these materials form chiral domains, so that the anoma-
+lous response of opposite signs largely cancels.
+There are ways to overcome these difficulties. First,
+since the nondiagonal optical conductivity, accessible
+through magnetooptical effects, is governed by the same
+selection rules as the anomalous Hall conductivity, it can
+be used in its place to detect the AM response. An ad-
+ditional advantage is that, as discussed later in the pa-
+per, calculations of the finite-frequency response from the
+first principles is mush easier and more reliable that in
+the static (Hall) limit. Finally, while the chiral domains
+necessarily form statistically, as the magnetic phase is
+nucleating upon cooling simultaneously in different parts
+of the sample, it does not carry, as opposed to ferromag-
+netics, any energetic advantage, only the energy cost of
+forming domain walls.
+This suggests that carefull an-
+nealing through the Neel temperature, preferably with a
+temperature gradient, in order to suppress independent
+nucleation in different parts of the sample, or on a ferro-
+magnetic substrate, in order to encourage a single domain
+on the interface, may result in a single domain sample,
+or domains large enough to be probed by polarized light
+independently. However, before urging experimentalists
+to pursue this path, a better and more quantitative un-
+derstanding of this material is imperative.
+Specifically, two main issues need to be understood:
+(i) magnetic interactions in MnTe, as they eventually de-
+terming the domain wall dynamics, and (ii) frequencies at
+which the strongest magnetoptical response is expected,
+and an estimate of the latter. In this paper we will pro-
+vide both.
+MnTe crystallizes in the NiAs crystal structure, as is
+known since 1956[4], which can be viewed as the hexag-
+onal analog of the metastable cubic MnTe (crystallized
+in the NaCl structure)[1]. In the latter, both Mn and O
+form triangular layers stacked along 111 as AbCaBc (the
+uppercase letters correspond to the Mn layers). In the
+former, the stacking sequence is AbAc, and the structure
+is expanded in the direction perpendicular to the trian-
+gular planes, and squeezed in the planes (Fig. 1).
+As a result, while the Mn-Mn interlayer distance is
+2.60 ˚A in the cubuc MnTe, it is 3.37 ˚A in the hexagonal
+one, which is also the shortes Mn-Mn bond. The next
+bond connects two Mn in the ab plane, and is 4.15 ˚A
+long; both are shorter than the corresponding bonds in
+the cubic material, which is 4.23 ˚A. The corresponding
+Mn-Te-Mn angles (Fig. 2) are 70.3◦ and 90.1◦. The third
+neighbors correspond to the second neighbors in the cubic
+structure, where they are bridged by Te along the straight
+line (a 180◦ angles) and the distance is 5.98 ˚A; in the
+hexagonal structure it is 5.35 ˚A and the angle is 131.7◦.
+MnTe has been studied a lot, both experimentally and
+theoretically.
+The latest and the most comprehensive
+study was probably Ref.
+[6] (see also the references
+therein).
+Experimentally, there is full consensus that
+MnTe forms an A-type antiferromagnetic structure with
+q = (0, 0, 0), and the magnetic moments are collinear
+and aligned with the (210) direction (i.e., perpendicular
+to the Mn-Mn bond). The in-plane magnetic anisotropy
+energy K was found to be too small to be measured by
+neutrons in Ref. [11], and too small to be calculated re-
+liably in Ref.
+[6].
+The in-plane spin-flop field in Ref.
+[6] was between 2 and 6 T, which, using the leading ex-
+change coupling of J ∼ 40 meV (see below), corresponds
+to K ≈
+√
+2KJ ≈ 0.2 − 1.4 µeV.
+Spin-wave dispersion was fitted with three nearest
+neighbor Heisenberg exchange coupling, defined via the
+Hamiltonian
+H =
+�
+i=1−3
+Ji ˆm · ˆm′,
+(1)
+where the summation is over all different bonds of a
+given length, and ˆm, ˆm′ are the unit vectors of spins
+forming the bond. The resulting parameters are listed
+in Table 1, together with those calculated in Ref.
+[8]
+and our own calculations. Note that both DFT calcu-
+lations, while performed by different methods (VASP[5]
+in Ref.
+[8], LAPW[9] here), give the nearest-neighbor
+in-plane exchange J2 antiferromagnetic, while Ref. [11]
+reports a very small ferromagnetic value.
+We believe
+that this is an experimental artifact, maybe due to ne-
+glect of the longer interactions in the spin-wave analy-
+arXiv:2301.08573v1  [cond-mat.mtrl-sci]  20 Jan 2023
+
+2
+TABLE I. Calculated and experimental Heisenberg exchange
+parameters, in meV.
+J1
+J2
+J3 J4
+TCW (K)
+Expt. ([11])
+46.2 -1.44 6.2
+-
+612a, 581b
+Calc. ([8])
+38.4 0.34 5.0 2.0
+552
+Calc. (this work) 42.1 0.91 5.3
+-
+592
+a calculated from the exchange parameters in Ref. [11].
+b measured[10].
+sis. Indeed, for Mn2+ there is no superexchange mecha-
+nism that could generate a ferromagnetic coupling, and
+no itinerant electrons to promote ferromagnetism. Since
+the bond angle in this case is nearly exactly 90◦, only
+pdσ × pdπ superexchange processes are allowed, but,
+since both t2g and eg states are occupied, their contri-
+bution is antiferromagnetic (as opposed to, for instance,
+Cr3+), and proportional to t2
+pdσt2
+pdπ/U∆2, where U is
+the Hubbard repulsion and ∆ is the Mn(d)−Te(π) en-
+ergy separation. The Goodenough-Kanamori ferromag-
+netic exchange is of course present, but proportional to
+JH(O)(t4
+pdσ + t4
+pdπ)/∆2, which is much smaller.
+With this in mind, one way wonder what drives the fer-
+romagnetic order in plane. The answer is that this is J3,
+which is sizable and has high degeneracy of 12, and tries
+to make the nearest neighbors in the plane antiparallel
+to the once-removed Mn in the neigboring plane, that is,
+parallel to each other. It can thus easily overcome the
+antiferromagnetic J2.
+These findings suggest that the ab domain walls, that
+is to say, walls perperdicular to the ab plane, should form
+more easily that those parallel to ab (see Fig. 1 We have
+verified that through direct density functional (DFT) cal-
+culations, using the standard VASP package[5], with the
+following settings: a 20 formula units supercell, the k-
+point mesh parallel to the domain boundary 12x12, per-
+pendicular 3, pseudopotentials PAW PBE Te and Mn pv,
+energy cutoff 400 eV, and applying U − J = 4 eV, which
+gives a reasonable direct optical gap of 1.7 eV and in-
+direct gap of 0.8 eV. The results are shown in TableII,
+where we also show the effect of lattice optimization (po-
+sitions only).
+TABLE II. Calculated energy of the domain walls, in meV
+per Mn at boundary.
+ab domain
+c domain
+not optmized optimized not optmized optimized
+19.1
+19.0
+65.2
+55.4
+As expected, the c wall has a much higher energy and
+is much less likely to form.
+On the other hand, since
+individual ab planes are ferromagnetic, growing MnTe
+on a single-domain ferromagnetic substrate (with can be
+FIG. 1. Supercells used for the domain wall energy calcula-
+tions for an ab domain (left) and a c domain (right).
+easily achieved by applying an in-plane magnetic field)
+should prevent the ab domains from forming. Numerous
+antiferromagnets and ferromagnets with stacked ferro-
+magnetic layer with an in-layer easy axis are know, and
+many have transition temperature above that of MnTe
+(∼ 310 K), such as NaOsO3 (610 K), (Sc,Ga)FeO3 (up
+to 408 K), Fe2O3 (960 K), Mn3(Cu,Ge) (380 K), FeBO3
+(348 K), CuMnAs (480 K), but especially promising is
+LiMn6Sn6, which in naturally layered, has TC ≈ 380 K,
+and, in addition, has a nearly perfect epitaxial match
+with MnTe (assuming a
+√
+5×
+√
+5 superlattice, ˜a =10.977
+˚A for the latter and 2×2, ˜a =10.982 ˚A for the former, a
+0.05% match). While epitaxial coherence is not required,
+it would serve to reduce the distance from the substrate
+and enhance coupling.
+Thus, MnTe is a prime candidate to singe-domain al-
+termagnetism. Unfortunately, it is an insulator, so direct
+measurement of the anomalous Hall effect is not possi-
+ble. Fortunately, the altermagnetism there can be probed
+by magnetooptical tools, such as MOKE (magnetoopti-
+cal Kerr effect). Also fortunately, the nondiagonal part
+of the optical conductivity σxy(ω) can be reliably calcu-
+lated by modern DFT codes, such as VASP — as op-
+posed to the Hall conductivity, the zero-frequency limit
+of σxy(ω), which is impossible to converge in existing cal-
+culations, and all current first principle calculations rely
+upon Wannier-based interpolation, which adds consid-
+erable ambiguity.
+In order to inform the experiments,
+which, we hope, will be encouraged by this paper, we
+have calculated the non-diagonal part of the optical con-
+ductivity, for the experimental easy magnetization axis
+of 210, that is, at α = 30◦ to the Mn-Mn bond. As ex-
+pected, only σxy is nonzero. We show the convergence of
+σxy(ω) in Fig. 2. Note that the results are reasonably
+well converged already at the k-mesh of 20 × 20 × 20; for
+the Hall conductivity σxy(0) in similar materials an order
+of magnitude larger linear density is required. Consistent
+with the symmetry analysis[2], only σxy(ω) is nonzero,
+and only for α ̸= 0. In Fig. 3 we show the angular de-
+pendence of σxy(ω)/ sin 3α as a function of α. One can see
+that lowest-order linear dependence of σxy(ω) on sin 3α
+holds with a good accuracy.
+In summary, we (a) explained the microscopic origin
+of the ferromagnetic ordering in the ab plane of MnTe,
+as driven not by a ferromagnetic in-plane exchange in-
+teraction (which has in fact the antiferromagnetic sign),
+but by the second-interlayer-neighbors antiferromagnetic
+counplig, (b) computed the energy of the antiferromag-
+netic domain walls in MnTe, and showed it to be substan-
+tial, encouraging growing single-domaun samples, where
+the predicted magnetooptica response can be measured,
+
+88888888888888888
+8.8888.88.88.88.88.88.88.88.3
+-0.06
+-0.04
+-0.02
+ 0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+ 0.1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+σ, 1/mΩ.cm-1
+ω, eV
+20x20x20
+20x20x19
+20x20x18
+-0.08
+-0.06
+-0.04
+-0.02
+ 0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+ 0.1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+σ, 1/mΩ.cm-1
+ω, eV
+20x20x20
+21x21x20
+22x22x20
+23x23x20
+24x24x20
+FIG. 2.
+Calculated nondiagonal optical conductivity σxy.
+The two panels show convergence with the respect to the in-
+plane and out-of-plane k-point mesh, respectively.
+and (c) calculated the said response and foud it to be
+sizeable, with a symmetry following the theoretical pre-
+diction. We hope that this work will encourage experiem-
+natl studies of altermagnetism in this compound.
+-0.2
+-0.15
+-0.1
+-0.05
+ 0
+ 0.05
+ 0.1
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+σ, 1/mΩ.cm-1
+ω, eV
+30o
+25o
+20o
+15o
+10o
+5o
+FIG. 3. Dependence of σxy on the angle that Mn spins form
+with Mn-Mn-bond direction α (see the inset), divided by
+sin(3α).
+ACKNOWLEDGMENTS
+The author acknowledges support from the Army Re-
+search Office
+[1] C. H. Griffiths. Cubic manganous telluride. Journal of
+Materials Science, 13(3):513–518, Mar 1978.
+[2] Libor ˇSmejkal, Jairo Sinova, and Tomas Jungwirth.
+Beyond conventional ferromagnetism and antiferromag-
+netism: A phase with nonrelativistic spin and crystal
+rotation symmetry. Phys. Rev. X, 12:031042, Sep 2022.
+[3] Libor ˇSmejkal, Jairo Sinova, and Tomas Jungwirth.
+Emerging research landscape of altermagnetism. Phys.
+Rev. X, 12:040501, Dec 2022.
+[4] Robert Juza, Albrecht Rabenau, and Gertrud Pascher.
+Uber feste losungen in den systemen zns/mns, znse/mnse
+und znte/mnte. Zeitschrift f¨ur anorganische und allge-
+meine Chemie, 285(1-2):61–69, 1956.
+[5] G. Kresse and D. Joubert. From ultrasoft pseudopoten-
+tials to the projector augmented-wave method.
+Phys.
+Rev. B, 59(3):1758–1775, Jan 1999.
+[6] D. Kriegner, H. Reichlova, J. Grenzer, W. Schmidt,
+E. Ressouche, J. Godinho, T. Wagner, S. Y. Martin,
+A. B. Shick, V. V. Volobuev, G. Springholz, V. Hol´y,
+J. Wunderlich, T. Jungwirth, and K. V´yborn´y. Magnetic
+anisotropy in antiferromagnetic hexagonal mnte. Phys.
+Rev. B, 96:214418, Dec 2017.
+[7] Igor Mazin.
+Editorial: Altermagnetism—a new punch
+line of fundamental magnetism. Phys. Rev. X, 12:040002,
+Dec 2022.
+[8] Sai
+Mu,
+Rapha¨el
+P.
+Hermann,
+St´ephane
+Gorsse,
+Huaizhou Zhao, Michael E. Manley, Randy S. Fishman,
+and L. Lindsay.
+Phonons, magnons, and lattice ther-
+mal transport in antiferromagnetic semiconductor mnte.
+Phys. Rev. Mater., 3:025403, Feb 2019.
+[9] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,
+and J. Luitz. Wien2k, 2002. ISBN 3-9501031-1-2.
+[10] I. Sergueev, H.-C. Wille, R. P. Hermann, D. Bessas,
+Yu. V. Shvyd’ko, M. Zajac, and R. R¨uffer.
+Milli-
+electronvolt monochromatization of hard X-rays with a
+sapphire backscattering monochromator. Journal of Syn-
+chrotron Radiation, 18(5):802–810, Sep 2011.
+[11] W. Szuszkiewicz, E. Dynowska, B. Witkowska, and
+B. Hennion. Spin-wave measurements on hexagonal mnte
+of nias-type structure by inelastic neutron scattering.
+Phys. Rev. B, 73:104403, Mar 2006.
+
diff --git a/o9FAT4oBgHgl3EQfeB1V/content/tmp_files/load_file.txt b/o9FAT4oBgHgl3EQfeB1V/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cdb314f41d5bb6a50c9da4814470187d3c106ffa
--- /dev/null
+++ b/o9FAT4oBgHgl3EQfeB1V/content/tmp_files/load_file.txt
@@ -0,0 +1,297 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf,len=296
+page_content='Altermagnetism in MnTe: origin, predicted manifestations, and routes to detwinning  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Mazin  Department of Physics and Astronomy, George Mason University, Fairfax, VA 22030, USA and  Quantum Science and Engineering Center, George Mason University, Fairfax, VA 22030, USA  MnTe has recently attracted attention as an altermagnetic candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Experimentally it has an  altermagnetic order of ferromagnetic ab planes, stacked antiferromagnetically along c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' We show that  this magnetic order (by itself non-trivial, since the in-plane exchange in antiferromagnetic) opens  intriguing possibility of manufacturing altermagnetically-detwinned samples and generate observable  magnetooptical response (which we calculate from first principles) as a signature of altermagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  The recently discovered phenomenon of spin-split  bands in collinear symmetry-compensated antiferromag-  nets, dubbed “altermagnetism” (AM)[2, 3, 7], has at-  tracted considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  While a number of al-  termagnets have been theoretically identified, there is a  big experimental challenges in assessing this, for a num-  ber of reasons: First, most of them are not metals, so  anomalous Hall conductivity cannot be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Sec-  ond, many have the easy magnetization direction not  compatible with anomalous response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Third, statistically  these materials form chiral domains, so that the anoma-  lous response of opposite signs largely cancels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  There are ways to overcome these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' First,  since the nondiagonal optical conductivity, accessible  through magnetooptical effects, is governed by the same  selection rules as the anomalous Hall conductivity, it can  be used in its place to detect the AM response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' An ad-  ditional advantage is that, as discussed later in the pa-  per, calculations of the finite-frequency response from the  first principles is mush easier and more reliable that in  the static (Hall) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Finally, while the chiral domains  necessarily form statistically, as the magnetic phase is  nucleating upon cooling simultaneously in different parts  of the sample, it does not carry, as opposed to ferromag-  netics, any energetic advantage, only the energy cost of  forming domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  This suggests that carefull an-  nealing through the Neel temperature, preferably with a  temperature gradient, in order to suppress independent  nucleation in different parts of the sample, or on a ferro-  magnetic substrate, in order to encourage a single domain  on the interface, may result in a single domain sample,  or domains large enough to be probed by polarized light  independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' However, before urging experimentalists  to pursue this path, a better and more quantitative un-  derstanding of this material is imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Specifically, two main issues need to be understood:  (i) magnetic interactions in MnTe, as they eventually de-  terming the domain wall dynamics, and (ii) frequencies at  which the strongest magnetoptical response is expected,  and an estimate of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' In this paper we will pro-  vide both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  MnTe crystallizes in the NiAs crystal structure, as is  known since 1956[4], which can be viewed as the hexag-  onal analog of the metastable cubic MnTe (crystallized  in the NaCl structure)[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' In the latter, both Mn and O  form triangular layers stacked along 111 as AbCaBc (the  uppercase letters correspond to the Mn layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' In the  former, the stacking sequence is AbAc, and the structure  is expanded in the direction perpendicular to the trian-  gular planes, and squeezed in the planes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  As a result, while the Mn-Mn interlayer distance is  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='60 ˚A in the cubuc MnTe, it is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='37 ˚A in the hexagonal  one, which is also the shortes Mn-Mn bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The next  bond connects two Mn in the ab plane, and is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='15 ˚A  long;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' both are shorter than the corresponding bonds in  the cubic material, which is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='23 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The corresponding  Mn-Te-Mn angles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 2) are 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='3◦ and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The third  neighbors correspond to the second neighbors in the cubic  structure, where they are bridged by Te along the straight  line (a 180◦ angles) and the distance is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='98 ˚A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' in the  hexagonal structure it is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='35 ˚A and the angle is 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  MnTe has been studied a lot, both experimentally and  theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  The latest and the most comprehensive  study was probably Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [6] (see also the references  therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Experimentally, there is full consensus that  MnTe forms an A-type antiferromagnetic structure with  q = (0, 0, 0), and the magnetic moments are collinear  and aligned with the (210) direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=', perpendicular  to the Mn-Mn bond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The in-plane magnetic anisotropy  energy K was found to be too small to be measured by  neutrons in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' [11], and too small to be calculated re-  liably in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  The in-plane spin-flop field in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [6] was between 2 and 6 T, which, using the leading ex-  change coupling of J ∼ 40 meV (see below), corresponds  to K ≈  √  2KJ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='4 µeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Spin-wave dispersion was fitted with three nearest  neighbor Heisenberg exchange coupling, defined via the  Hamiltonian  H =  �  i=1−3  Ji ˆm · ˆm′,  (1)  where the summation is over all different bonds of a  given length, and ˆm, ˆm′ are the unit vectors of spins  forming the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The resulting parameters are listed  in Table 1, together with those calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [8]  and our own calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Note that both DFT calcu-  lations, while performed by different methods (VASP[5]  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [8], LAPW[9] here), give the nearest-neighbor  in-plane exchange J2 antiferromagnetic, while Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' [11]  reports a very small ferromagnetic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  We believe  that this is an experimental artifact, maybe due to ne-  glect of the longer interactions in the spin-wave analy-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='08573v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='mtrl-sci]  20 Jan 2023  2  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Calculated and experimental Heisenberg exchange  parameters, in meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  J1  J2  J3 J4  TCW (K)  Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' ([11])  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='44 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='2 612a, 581b  Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' ([8])  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='0  552  Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' (this work) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='3 592  a calculated from the exchange parameters in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  b measured[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Indeed, for Mn2+ there is no superexchange mecha-  nism that could generate a ferromagnetic coupling, and  no itinerant electrons to promote ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Since  the bond angle in this case is nearly exactly 90◦, only  pdσ × pdπ superexchange processes are allowed, but,  since both t2g and eg states are occupied, their contri-  bution is antiferromagnetic (as opposed to, for instance,  Cr3+), and proportional to t2  pdσt2  pdπ/U∆2, where U is  the Hubbard repulsion and ∆ is the Mn(d)−Te(π) en-  ergy separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The Goodenough-Kanamori ferromag-  netic exchange is of course present, but proportional to  JH(O)(t4  pdσ + t4  pdπ)/∆2, which is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  With this in mind, one way wonder what drives the fer-  romagnetic order in plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The answer is that this is J3,  which is sizable and has high degeneracy of 12, and tries  to make the nearest neighbors in the plane antiparallel  to the once-removed Mn in the neigboring plane, that is,  parallel to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' It can thus easily overcome the  antiferromagnetic J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  These findings suggest that the ab domain walls, that  is to say, walls perperdicular to the ab plane, should form  more easily that those parallel to ab (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 1 We have  verified that through direct density functional (DFT) cal-  culations, using the standard VASP package[5], with the  following settings: a 20 formula units supercell, the k-  point mesh parallel to the domain boundary 12x12, per-  pendicular 3, pseudopotentials PAW PBE Te and Mn pv,  energy cutoff 400 eV, and applying U − J = 4 eV, which  gives a reasonable direct optical gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='7 eV and in-  direct gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' The results are shown in TableII,  where we also show the effect of lattice optimization (po-  sitions only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Calculated energy of the domain walls, in meV  per Mn at boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  ab domain  c domain  not optmized optimized not optmized optimized  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='1  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='0  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='2  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='4  As expected, the c wall has a much higher energy and  is much less likely to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  On the other hand, since  individual ab planes are ferromagnetic, growing MnTe  on a single-domain ferromagnetic substrate (with can be  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Supercells used for the domain wall energy calcula-  tions for an ab domain (left) and a c domain (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  easily achieved by applying an in-plane magnetic field)  should prevent the ab domains from forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Numerous  antiferromagnets and ferromagnets with stacked ferro-  magnetic layer with an in-layer easy axis are know,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' and  many have transition temperature above that of MnTe  (∼ 310 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' such as NaOsO3 (610 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' (Sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='Ga)FeO3 (up  to 408 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Fe2O3 (960 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Mn3(Cu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='Ge) (380 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' FeBO3  (348 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' CuMnAs (480 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' but especially promising is  LiMn6Sn6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' which in naturally layered,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' has TC ≈ 380 K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' has a nearly perfect epitaxial match  with MnTe (assuming a  √  5×  √  5 superlattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' ˜a =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='977  ˚A for the latter and 2×2, ˜a =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='982 ˚A for the former, a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='05% match).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' While epitaxial coherence is not required,  it would serve to reduce the distance from the substrate  and enhance coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Thus, MnTe is a prime candidate to singe-domain al-  termagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Unfortunately, it is an insulator, so direct  measurement of the anomalous Hall effect is not possi-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Fortunately, the altermagnetism there can be probed  by magnetooptical tools, such as MOKE (magnetoopti-  cal Kerr effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Also fortunately, the nondiagonal part  of the optical conductivity σxy(ω) can be reliably calcu-  lated by modern DFT codes, such as VASP — as op-  posed to the Hall conductivity, the zero-frequency limit  of σxy(ω), which is impossible to converge in existing cal-  culations, and all current first principle calculations rely  upon Wannier-based interpolation, which adds consid-  erable ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  In order to inform the experiments,  which, we hope, will be encouraged by this paper, we  have calculated the non-diagonal part of the optical con-  ductivity, for the experimental easy magnetization axis  of 210, that is, at α = 30◦ to the Mn-Mn bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' As ex-  pected, only σxy is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' We show the convergence of  σxy(ω) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Note that the results are reasonably  well converged already at the k-mesh of 20 × 20 × 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' for  the Hall conductivity σxy(0) in similar materials an order  of magnitude larger linear density is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Consistent  with the symmetry analysis[2], only σxy(ω) is nonzero,  and only for α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 3 we show the angular de-  pendence of σxy(ω)/ sin 3α as a function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' One can see  that lowest-order linear dependence of σxy(ω) on sin 3α  holds with a good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' we (a) explained the microscopic origin  of the ferromagnetic ordering in the ab plane of MnTe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  as driven not by a ferromagnetic in-plane exchange in-  teraction (which has in fact the antiferromagnetic sign),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  but by the second-interlayer-neighbors antiferromagnetic  counplig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' (b) computed the energy of the antiferromag-  netic domain walls in MnTe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' and showed it to be substan-  tial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' encouraging growing single-domaun samples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' where  the predicted magnetooptica response can be measured,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  88888888888888888  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='8888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  5  σ, 1/mΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='cm-1  ω, eV  20x20x20  20x20x19  20x20x18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='5  5  σ, 1/mΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='cm-1  ω, eV  20x20x20  21x21x20  22x22x20  23x23x20  24x24x20  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Calculated nondiagonal optical conductivity σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  The two panels show convergence with the respect to the in-  plane and out-of-plane k-point mesh, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  and (c) calculated the said response and foud it to be  sizeable, with a symmetry following the theoretical pre-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' We hope that this work will encourage experiem-  natl studies of altermagnetism in this compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content='5  5  σ, 1/mΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='cm-1  ω, eV  30o  25o  20o  15o  10o  5o  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Dependence of σxy on the angle that Mn spins form  with Mn-Mn-bond direction α (see the inset), divided by  sin(3α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The author acknowledges support from the Army Re-  search Office  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Griffiths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Cubic manganous telluride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Journal of  Materials Science, 13(3):513–518, Mar 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [2] Libor ˇSmejkal, Jairo Sinova, and Tomas Jungwirth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Beyond conventional ferromagnetism and antiferromag-  netism: A phase with nonrelativistic spin and crystal  rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' X, 12:031042, Sep 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [3] Libor ˇSmejkal, Jairo Sinova, and Tomas Jungwirth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Emerging research landscape of altermagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' X, 12:040501, Dec 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [4] Robert Juza, Albrecht Rabenau, and Gertrud Pascher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Uber feste losungen in den systemen zns/mns, znse/mnse  und znte/mnte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Zeitschrift f¨ur anorganische und allge-  meine Chemie, 285(1-2):61–69, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Joubert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' From ultrasoft pseudopoten-  tials to the projector augmented-wave method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content='  [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Volobuev, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' B, 96:214418, Dec 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content='  [8] Sai  Mu,  Rapha¨el  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Manley, Randy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Fishman,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Lindsay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Phonons, magnons, and lattice ther-  mal transport in antiferromagnetic semiconductor mnte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=', 3:025403, Feb 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Schwarz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Madsen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Kvasnicka,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Luitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Wien2k, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' ISBN 3-9501031-1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [10] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Sergueev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Wille, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Hermann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Bessas,  Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Shvyd’ko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Zajac, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' R¨uffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  Milli-  electronvolt monochromatization of hard X-rays with a  sapphire backscattering monochromator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Journal of Syn-  chrotron Radiation, 18(5):802–810, Sep 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content='  [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Szuszkiewicz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Dynowska, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Witkowska, and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Hennion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' Spin-wave measurements on hexagonal mnte  of nias-type structure by inelastic neutron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
+page_content=' B, 73:104403, Mar 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FAT4oBgHgl3EQfeB1V/content/2301.08573v1.pdf'}
diff --git a/odE3T4oBgHgl3EQfLQlL/vector_store/index.faiss b/odE3T4oBgHgl3EQfLQlL/vector_store/index.faiss
new file mode 100644
index 0000000000000000000000000000000000000000..f2efbd0c570758a919dc209f75fad1bf5f970fda
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+Draft version January 23, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+The formation of hard VHE spectra from GRB afterglow via Two-Zone Synchrotron Self-Compton
+Emission
+Dmitry Khangulyan,1 Andrew M. Taylor,2 and Felix Aharonian3, 4
+1Graduate School of Artificial Intelligence and Science, Rikkyo University, Nishi-Ikebukuro 3-34-1, Toshima-ku, Tokyo 171-8501, Japan
+2DESY, D-15738 Zeuthen, Germany
+3Dublin Institute for Advanced Studies, School of Cosmic Physics, 31 Fitzwilliam Place, Dublin 2, Ireland
+4Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
+ABSTRACT
+Electron Compton scattering of target photons into the gamma-ray energy band (inverse Compton
+scattering –IC–) is commonly expected to dominate the very high energy spectra in gamma-ray bursts
+especially during the afterglow phase. For sufficiently large center-of-mass energies in these collisions,
+the effect of the electron recoil starts reducing the scattering cross section (the Klein-Nishina regime).
+The IC spectra generated in the Klein-Nishina regime is softer and has a smaller flux level compared
+to the synchrotron spectra produced by the same electrons.
+The detection of afterglow emission
+from nearby GRB190829A in the very high energy (VHE) domain with H.E.S.S. has revealed an
+unexpected feature: the slope of the VHE spectrum matches well the slope of the X-ray spectra, despite
+expectations that for the IC production process, the impact of the Klein-Nishina effect should be strong.
+The multi-wavelength spectral energy distribution appears to be inconsistent with predictions of one-
+zone synchrotron-self-Compton models. We study the possible impact of two-zone configuration on the
+properties of IC emission when the magnetic field strength differs considerably between the two zones.
+Synchrotron photons from the strong magnetic field zone provide the dominant target for cooling of the
+electrons in the weak magnetic field zone, which results in a formation of hard electron distribution and
+consequently of a hard IC emission. We show that the two-zone model can provide a good description
+of the X-ray XRT and VHE H.E.S.S. data.
+Keywords: Non-thermal radiation sources(1119) — Gamma-ray transient sources(1853) — Gamma-
+ray bursts(629) — Gamma-ray astronomy(628) — Particle astrophysics(96) — X-ray
+sources(1822)
+1. INTRODUCTION
+The very high energy (VHE; > 100 GeV) emission de-
+tected from gamma-ray burst (GRB) afterglows with
+H.E.S.S. and MAGIC (Abdalla et al. 2019; MAGIC Col-
+laboration et al. 2019a,b; H. E. S. S. Collaboration et al.
+2021) is considered by many to have inverse Compton
+(IC) origin (see, e.g, Zhang 2019). The emission com-
+ponent produced by relativistic protons is expected to
+have a significantly lower flux, due to the very low ra-
+diative efficiency of hadronic interactions (see, e.g., Ab-
+dalla et al. 2019).
+If the VHE emission is produced
+by relativistic electrons, then because of the so-called
+Corresponding author: Dmitry Khangulyan
+d.khangulyan@rikkyo.ac.jp
+synchrotron burn-off limit (Guilbert et al. 1983) the
+synchrotron component is expected to reach the VHE
+regime only if the bulk Lorentz factor is very high,
+Γ ≥ 103. Such high bulk Lorentz factors are excluded
+during the afterglow phase by energy conservation argu-
+ments (e.g., related to self-similar solution for relativistic
+blast wave obtained by Blandford & McKee 1976) mak-
+ing IC scattering the most feasible radiation mechanism
+for the VHE GRB emission during the afterglow period.
+However, the hard intrinsic spectral slope inferred from
+observations by H.E.S.S. of GRB190829A afterglow can-
+not be easily reproduced with standard IC models (see,
+e.g., H. E. S. S. Collaboration et al. 2021). This leaves
+one of two possibilities: (i) invoke alternative radiation
+mechanisms, or (ii) develop a more sophisticated IC sce-
+arXiv:2301.08578v1  [astro-ph.HE]  20 Jan 2023
+
+2
+Khangulyan et al.
+nario to provide a better description of the observational
+data.
+Synchrotron radiation is a very efficient radiative
+emission mechanism of electrons during the afterglow
+phase of GRBs. If the synchrotron component extends
+into the VHE domain, it can reproduce the flux level and
+spectral slope revealed with H.E.S.S. from GRB190829A
+afterglow (H. E. S. S. Collaboration et al. 2021). While
+the conservation of energy, used to constrain the bulk
+Lorentz factor, is a robust argument, the burn-off en-
+ergy limit can be avoided in certain non-standard sce-
+narios. For example, if the strength of the accelerating
+electric field, E, exceeds the strength of the magnetic
+field, B (in a plasma such configurations require non-
+ideal magnetohydrodynamics) then synchrotron emis-
+sion can extend beyond the burn-off limit by the fac-
+tor of E/B. Alternatively, in highly turbulent magnetic
+fields magnetobremsstrahlung radiation can extend be-
+yond the burn-off limit (Kelner et al. 2013). If the cor-
+relation length of the magnetic field is large compared
+to the photon formation length, mec2/e ¯B (here me and
+e are electron mass and charge, respectively; c and ¯B
+are light speed and averaged magnetic field), then the
+radiation is generated in the synchrotron regime, result-
+ing in the burn-off limit for the synchrotron maximum
+energy (for a detailed consideration, see, e.g., in Kel-
+ner et al. 2013; Derishev & Aharonian 2019). However,
+if the correlation length is short compared to the pho-
+ton formation length, then the electrons instead emit
+in the jitter regime, and the emission peaks at higher
+energy compared to the synchrotron case, alleviating
+the limit from the burn-off limit (Kelner et al. 2013).
+Finally, the electron synchrotron spectrum can extend
+beyond the burn-off limit in two-zone systems, where
+the physical conditions at the acceleration site and in
+the radiation production region differ substantially (Ku-
+mar et al. 2012; Khangulyan et al. 2021). In conclusion,
+there are several ways of expanding the energy spec-
+trum of magneto-bremsstrahlung to high or even very
+high energies. However, the feasibility of these scenar-
+ios depends on the implementation of many factors and
+requires extreme assumptions.
+In contrast, IC scattering is a natural and very effec-
+tive channel of VHE gamma-ray production. Although
+the recent observations of VHE gamma rays during the
+GRB afterglows challenge the simple one-zone IC model,
+more sophisticated scenarios cannot be excluded. In this
+paper, we study the spectral properties of gamma rays in
+the two-zone IC model in which the production region of
+the target (synchrotron) photons and the IC gamma-ray
+emitter are separated. One can propose several possi-
+ble realizations for such a two-zone setup. For example,
+one may expect quite different conditions at the for-
+ward and reverse shocks, which propagate through the
+circumburst medium (CBM) and the jet, respectively. If
+the emission from the reverse shock appears to be impor-
+tant at certain frequencies then a two-zone description
+for GRB afterglow emission should be considered (see,
+e.g., Dichiara et al. 2022; Salafia et al. 2022). Alterna-
+tively, the shock region itself can be quite complex po-
+tentially providing quite different physical conditions for
+particle acceleration and radiation. Indeed, simulations
+suggest that downstream shock material, the dominant
+emission site during the afterglow phase, is expected to
+be highly inhomogeneous, an aspect usually neglected in
+GRB afterglow emission modelling. Below we consider
+the impact of a strongly inhomogeneous magnetic field
+on the properties of IC emission. We show that under
+reasonable assumptions, even a two-zone synchrotron
+self-Compton (SSC) scenario can provide a considerably
+improved description of the broadband spectra reported
+from GRB190829A.
+2. STANDARD ONE-ZONE SSC SCENARIO
+The standard GRB afterglow emission framework pos-
+tulates that this emission is generated via the syn-
+chrotron and IC channels, with synchrotron radiation
+providing the dominant target for IC scattering – the
+so called SSC scenario.
+The analysis of the spectral
+energy distribution (SED) in SSC models is straightfor-
+ward if the IC emission is generated in the Thomson
+regime (see, e.g. Sari & Esin 2001), as in this case the
+energy loss rate,
+˙E, has a simple form ˙E ∝ E2 (here
+E is electron energy). In this regime, a power-law in-
+jection of non-thermal electrons, q ∝ E−α (here α is
+the injection index, for conventional acceleration mech-
+anisms one typically assumes α ≈ 2), leads to the forma-
+tion of a broken-power-law distribution of radiating elec-
+trons. The synchrotron and IC (Thomson) components
+generated by these electrons also reflect this broken-
+power-law shape, with the IC component dominating at
+higher energies. The subsequent broadband SED pro-
+duced is double-humped, with the relative emissivity of
+the synchrotron and IC components being determined
+by phenomenological parameters (typically, by the radi-
+ation efficiency, i.e., by the fraction of energy radiated
+away). The photon index of the synchrotron spectrum,
+produced by electrons with energies above the cooling
+break, is γs = (α + 2)/2, provided that α > 1. In the
+single zone SSC scenario, the corresponding IC spectrum
+has the same photon index, if generated in the Thomson
+regime.
+Typically,
+during the afterglow phase the (syn-
+chrotron) X-ray spectrum is observed to be hard, with
+
+Hard VHE emission from SSC sources
+3
+E
+dN
+dE
+Thomson cooling
+E−(α+1)
+Tr
+a
+n
+si
+ti
+o
+n
+co
+oli
+ng
+Asymptotic K-N cooling
+E−(α−1)
+Synchrotron cooling
+E−(α+1)
+Synchrotron cooling
+E−(α+1)
+Figure 1.
+A sketch that illustrates the formation of the
+particle spectrum in the case of dominant synchrotron losses
+and dominant IC losses. The part of the spectrum formed in
+the fast cooling regime is shown.
+ε
+εFε
+Synchrotron dominated case
+Synchrotron
+Inverse Compton
+ε− (α+2)
+2
+Thomson
+ε− (α+2)
+2
+Transition
+Klein-Nishina
+ε−(α+2)
+ε
+εFε
+IC dominated case
+Synchrotron
+Inverse Compton
+ε− (α+2)
+2
+ε− α
+2
+ε− (α+2)
+2
+Thomson
+ε− (α+2)
+2
+Transition
+Klein-Nishina
+ε−(α+2)
+Figure 2. A sketch that illustrates the formation of the SED
+in the case of dominant synchrotron losses and dominant IC
+losses.
+a photon index ∼ 2. Thus the photons detected in the
+X-ray band provide a non-negligible target for IC scat-
+tering. In the plasma co-moving frame, the energy of
+the electron, E, generating the VHE emission, detected
+at energy1 ε′
+vhe, satisfy the condition: E > ε′
+vhe/Γ. If
+electrons of this energy up-scatter photons from a com-
+ponent detected by the observer at energy ε′
+x, then the
+typical product of the target photon and electron ener-
+gies, which determines the scattering regime, is
+Eεx
+m2ec4 > ε′
+vheε′
+x
+m2ec4Γ2 ≈ 4
+� Γ
+10
+�−2 �
+ε′
+vhe
+0.1 TeV
+� �
+ε′
+x
+1 keV
+�
+.
+(1)
+1 Note that we prime the quantities in the progenitor frame, and
+we neglect the cosmological redshift effect.
+Here me and c are the electron mass and speed of light,
+respectively.
+Unless the bulk Lorentz factor is high,
+Γ ≥ 102, the electrons that produce the VHE emission
+up-scatter a considerable part of the photon targets in
+the Klein-Nishina regime. The study of the VHE prop-
+erties of GRB afterglows should therefore be conducted
+with models that account for the change of the IC cross-
+section in the relativistic regime.
+The influence of the Klein-Nishina regime on the SED
+is two-fold, as one must account for both the change
+of the emission and energy loss rates (see, e.g., Derishev
+et al. 2003; Nakar et al. 2009). In the fast cooling regime,
+the particle spectrum, dN = n dE, is determined by
+the injection spectrum, q, and by the cooling time τ =
+E/| ˙E|:
+n(E) = τ(E)
+E
+∞
+�
+E
+d ˜E q
+�
+˜E
+�
+.
+(2)
+If the injection is a power-law q ∝ E−α, then the particle
+spectrum is
+n(E) ∝ τ(E)E−α .
+(3)
+(Note that here we assume that the injection spectrum
+is sufficiently steep so as to ensure the integral is domi-
+nated by the low energy limit).
+If the synchrotron losses dominate over the Compton
+losses (more specifically if the energy density of the mag-
+netic field is larger than the energy density of the target
+photons) then τ(E) ∝ E−1, and a power-law spectral
+injection also yields a power-law distribution of parti-
+cles: n(E) ∝ E−(α+1) (see Fig. 1 for a sketch of the
+cooled particle spectrum).
+Subsequently, a power-law
+synchrotron component is produced with photon index
+γs.
+The inverse Compton of radiation has the same power-
+law photon index as long as the scattering takes place in
+the Thompson regime. In the Klein-Nishina regime, the
+IC slope should (asymptotically, i.e., ignoring the log-
+arithmic term) approach γkn ≈ (α + 2) (provided that
+the emitting electrons obey a power-law energy distribu-
+tion with index α+1, Blumenthal & Gould 1970). Thus,
+since the slope of IC component generated in the Thom-
+son regime matches that of the synchrotron radiation,
+γs, the Klein-Nishina effect causes a spectral softening
+by ∆γ ≈ γkn − γs ≈ (α + 2)/2. For example, if α ≈ 2
+then the spectral slope changes from γs ≈ 2 to γkn ≈ 4,
+and the spectral softening is ∆γ ≈ 2. A schematic of the
+SED is shown in Fig. 2. One should note that for a broad
+target photon distribution, the transition to the Klein-
+Nishina regime is spread over a broad energy range and
+can have a rather complex character.
+The situation changes dramatically when the energy
+density of target photons is larger than the energy den-
+
+4
+Khangulyan et al.
+sity of the magnetic field. In this case, the impact of
+the Klein-Nishina effect on the formation of the elec-
+tron spectrum becomes a dominant factor. The radia-
+tive cooling time τ(E) can be approximated by a broken
+power-law function: for sufficiently low electron ener-
+gies, the IC interaction proceeds in the Thomson regime,
+thus τ(E) ∝ E−1. At higher energies, the IC interac-
+tions occur in the Klein-Nishina regime where the en-
+ergy loss rate is energy-independent, thus τ(E) ∝ E.
+Finally at even higher energies, denoted E∗, the syn-
+chrotron losses (as their rate increases with particle
+energy) begin to dominate over the IC energy losses,
+and the original energy dependence of the cooling time
+is recovered: τ(E) ∝ E−1.
+As follows from Eq. (3),
+for a power-law injection spectrum, the particle spec-
+trum formed in the fast cooling regime should also be
+a double-broken-power-law (with the power-law index
+changing as α + 1 → α − 1 → α + 1: see Fig. 1). The
+E−(α+1) part of the spectrum formed under dominant
+(Thomson regime) IC losses changes to, ∝ E−(α−1),
+formed under the dominant IC (Klein-Nishina regime)
+losses.
+Finally, above E∗, the spectrum softens back
+to E−(α+1). We note, however, that the transition to
+the Klein-Nishina regime proceeds smoothly, therefore
+the spectrum does not follow precisely the schematic
+shape explained above.
+For example, as can be seen
+from Fig. 4, the IC cooling time in the transition regime
+is better approximated as a constant, τ ≈ const. There-
+fore, the corresponding transformation of the electron
+spectrum is better approximated as α + 1 → α → α + 1
+(note that this power law index is indicated in the bot-
+tom panel of Fig. 4 with a black guide line).
+As for the synchrotron radiation, electrons cooled by
+IC in the Thomson regime produce a spectrum with pho-
+ton index γs; at higher energies the hardening of the elec-
+tron spectrum due to the dominant Klein-Nishina en-
+ergy losses results in a hard synchrotron spectrum with
+photon index in the range between γs and γs,kn ≈ α/2
+(γs,kn is the limiting value achieved under IC cooling in
+the deep Klein-Nishina regime: see Fig. 2). In the tran-
+sition region with an approximately constant IC cooling
+time, the slope of the synchrotron spectrum is approxi-
+mately (α + 1)/2, as indicated by the black guide lines
+in Figs. 5 and 6.
+Finally, the emission produced by
+electrons with energies exceeding E∗ has the standard
+synchrotron slope γs. As the synchrotron and IC energy
+loss rates for particles with E∗ are equal, the narrow-
+band luminosity of the synchrotron and IC components
+produced by particles with E∗ are (almost) equal.
+The spectral shape of the IC component is different to
+that of the synchrotron spectrum. The component gen-
+erated in the Thomson regime has a spectral index of
+γs. At higher energies the impact of the Klein-Nishina
+effect on the particle spectrum is partially compensated
+by the reduction of the cross section. For example, in
+the limiting regime, a spectrum ∝ E−(α−1) generates
+in the Klein-Nishina regime a E−α IC spectrum. For
+α ≈ 2 a Thomson spectrum with photon index (α+2)/2
+transits smoothly into the Klein-Nishina spectrum with
+photon index α. However, in the region of transition
+to the Klein-Nishina regime, this asymptotic photon
+index might be quite a coarse approximation.
+More-
+over, above E∗ the synchrotron losses dominate, thus
+the Klein-Nishina spectrum eventually softens to α + 2
+above E∗. Note that in the Klein-Nishina regime almost
+all the electron energy is transferred to the up-scattered
+photon, so the photon energy in the co-moving frame is
+equal to that of the incident electron energy, εic ≈ E∗.
+Observations of GRB190829A with H.E.S.S. revealed
+that VHE component, corrected for the extragalactic
+background light (EBL) attenuation, is best described as
+a single power-law spectrum extending up to 3 TeV with
+a hard photon index of γvhe = 2.07 ± 0.09 (H. E. S. S.
+Collaboration et al. 2021). Strikingly, this slope matches
+well the slope of the X-ray spectrum measured with
+Swift-XRT (e.g., γxrt = 2.03±0.06 during the first night
+H. E. S. S. Collaboration et al. 2021). Also, the Swift-
+XRT and H.E.S.S. observations revealed that the fluxes
+in the X-ray and VHE bands appeared to be similar (po-
+tentially a natural feature of pair loading feedback, see
+Derishev & Piran 2016, 2019, for detail).
+In the VHE band the influence of the Klein-Nishina
+effect should be noticeable. However, this spectral ef-
+fect was not observed in the H.E.S.S. measurements.
+In the framework of the simple one-zone analysis intro-
+duced above, the slope and flux level match implies that
+the cooling of TeV emitting electrons proceeds in the
+Klein-Nishina regime, and that the X-ray synchrotron
+is produced by particles with energy exceeding E∗. As
+the hard VHE spectrum extends up to 3 TeV, then
+E∗ > 0.3(Γ/10)−1 TeV. The synchrotron emission pro-
+duced by the high-energy electrons is detected by the
+observer at
+ε′
+syn > 60
+� Γ
+10
+�−1 B
+1 G keV .
+(4)
+This estimate shows that a very low magnetic field of
+∼ mG level is required by the VHE measurements.
+Such a low magnetic field, however, is incompatible with
+the required radiation efficiency of the production region
+given the adiabatic cooling time is τad ∼ t′
+trΓ, where t′
+tr
+is time since the GRB trigger (as measured by a distant
+observer at rest in the progenitor reference frame). The
+broad-band SED obtained with Swift-XRT and H.E.S.S.
+
+Hard VHE emission from SSC sources
+5
+therefore cannot be reproduced in the framework of the
+standard one-zone SSC scenario (see also Huang et al.
+2022). To resolve the spectral issue in SSC scenario one
+needs either: (1) assume that there is an important low-
+energy target photon field, probably of external origin;
+or (2) consider a two-zone scenario.
+The former scenario requires the presence of an exter-
+nal target that provides a target of an energy density
+comparable to that of the magnetic field in the plasma
+co-moving frame:
+wext ∼ 4 × 10−2
+� B
+1 G
+�2
+erg cm−3 .
+(5)
+If the photons are isotropic in the progenitor frame, then
+we obtain w′
+ext ∼ 4 × 10−4(10B/(Γ G))2 erg cm−3. The
+VHE emission detected from GRB190829A lasted for
+almost ∆t = 50 h (H. E. S. S. Collaboration et al.
+2021), and the forward shock covered a distance of
+∆R′ ∼ Γ2∆tc ∼ 1017(Γ/10)2 cm.
+The luminosity of
+the photon field should therefore be
+L′
+ext ∼ 4π∆R′2w′
+extc ∼ 1042
+� B
+1 G
+�2
+erg s−1 .
+(6)
+If the magnetic field is weak, B ≪ 1 G, then an external
+photon field of reasonable luminosity can provide a suffi-
+ciently dense external photon field (see, e.g., Zhang et al.
+2021), however external IC scenarios with an equivalent
+Gauss-strength magnetic field cannot be realized.
+3. TWO-ZONE SSC EMISSION SCENARIO
+3.1. Physical justification
+We consider the emission region consisting of two
+zones: the first zone with a strong magnetic field, B1,
+and the second zone with a weak magnetic field, B2,
+with B1 ≫ B2. Should particles themselves also easily
+mix between the two zones, then one would not expect a
+significant difference between the energy distributions of
+particles in these zones. We here, however, assume that
+the particle exchange between the zones is inefficient,
+and thus two distinct particle distributions, n1 and n2,
+are formed in the two zones.
+The target photons, however, travel freely between the
+two zones. The specific realization of the scenario, in
+particular the shapes and relative location of the zones,
+determines the actual distribution of target photons in
+the zones. Let us qualitatively consider several possible
+realizations of the two-zone scenario: (i) two distinct
+regions with typical sizes of r1 and r2 separated by a
+distance r0; (ii) two converging shells of radius r1 and r0;
+(iii) N compact regions (of typical size r1) with strong
+magnetic field embedded within a larger zone of size r0.
+r1
+r2
+r0
+Scenario (i)
+Strong B-field region
+Weak B-field region
+Photons from the
+strong B-field region
+Photons from the
+weak B-field region
+r1
+r0
+Scenario (ii)
+r1
+r0
+Scenario (iii)
+Figure 3. Examples of three different geometries that allow
+the scenario realization. Scenario (i): two distinct regions
+with typical sizes of r1 and r2 separated by a distance r0;
+scenario (ii): two converging shells of radius r1 and r0; sce-
+nario (iii): a large number of compact regions (of typical size
+r1) with strong magnetic field embedded within a larger zone
+of size r0.
+These three possibilities are shown in Fig. 3. Although
+less apparent, scenario (iii) is two-zone in the sense that
+the physical conditions and processes are the same in
+the compact regions, and differ substantially from those
+in the larger zone.
+The synchrotron luminosity of each of the zones is L1
+and L2, respectively. In scenario (iii) we define L1 as the
+total luminosity of N regions of enhanced B-field. We
+consider a situation L1 ≫ L2. Thus, when considering
+the processes in the first zone, we can ignore the photons
+supplied by the second zone. The energy density of the
+
+6
+Khangulyan et al.
+locally generated photons in the first zone is
+w1→1 ∼
+L1
+r2
+1Nc ,
+(7)
+where N = 1 for scenarios (i) and (ii). Equation (7)
+ignores a numerical factor, which depends on the pro-
+duction region geometry and the distribution of emitting
+particles. For example, in the case of a spherical homo-
+geneous production region, the volume average energy
+density of target photons is given by Eq. 7 with a factor
+9/(16π) (for detail, see in Atoyan & Aharonian 1996).
+We note that such factors do not affect our conclusions,
+we therefore safely ignore them.
+In the second zone one needs to account for the con-
+tribution of locally generated photons:
+w(i)
+2→2 ∼ L2
+r2
+2c
+and
+w(ii)/(iii)
+2→2
+∼ L2
+r2
+0c
+(8)
+and the photons supplied from the first zone, w1→2. For
+the each of the above defined geometries one obtains
+w1→2 ∼ L1
+r2
+0c .
+(9)
+The suggested scenario is realized if the photon field
+produced in the first zone (being locally a subdominant)
+provides the dominant target for the particle cooling in
+the second zone:
+w1→1 ≪ B2
+1
+8π
+and
+w1→2 ≫ B2
+2
+8π .
+(10)
+The photon field in the second zone is diluted com-
+pared to the first zone: w1→1 > w1→2, thus the scenario
+requires that B1 ≫ B2.
+The difference of the mag-
+netic fields determines the dilution of the photon field,
+κ = w1→2/w1→1, that allows the scenario realization
+(i.e, the conditions given by Eq. 10).
+The possible ratio of the magnetic fields should be de-
+termined by the physical arguments unique to each spe-
+cific realization of the scenario. However, from the gen-
+eral point of view, it is obvious that if the photon field
+is significantly diluted in the second zone, κ ≪ 1, the
+required difference between the magnetic field strength
+becomes larger, making the realization of the scenario
+less feasible (although not excluded).
+For example, a
+strong dilution might be expected in scenario (i) pro-
+vided that r0 ≫ r1.
+In contrast, in scenario (ii) the
+dilution of the photon field in the second zone is small,
+by a factor of ∼ 2, provided that two shells are of com-
+parable radius, r1 ≈ r0. Similarly, in scenario (iii) one
+obtains
+w1→2
+w1→1
+∼ r2
+1N
+r2
+0
+= fr0
+r1
+,
+(11)
+where f is filling factor. If the above ratio is not small
+(i.e., w1→2/w1→1 ≳ 1) then the photon field is nearly
+homogeneous in the entire production region, i.e., κ ≈ 1.
+For the sake of simplicity we will consider a single
+common photon target being present in the two zones.
+In the first place, this seems to be a perfectly suitable
+choice for scenarios (ii) and (iii) if r1 ≈ r0 and fr0/r1 ≳
+1, respectively. Even if these conditions are not fulfilled,
+the model calculations should reproduce correctly the
+part of SED formed in the fast cooling regime (provided
+that IC losses dominate over the synchrotron cooling in
+the second zone: w1→2 ≫ B2
+2/(8π)).
+Although the scenario can be realized also in scenario
+(i), if the magnetic field in the second zone is sufficiently
+weak to remain subdominant compared to the signifi-
+cantly diluted photon field provided from the first zone,
+scenarios (ii) and (iii) seem to be less demanding. In
+particular, these geometries can be formed during the
+afterglow phase of GRBs. The shells assumed in sce-
+nario (ii) may correspond to the reverse and forward
+shocks. Also an onion-like structure may be formed in
+the inner part of the forward shock downstream region,
+where the competing processes of magnetic field ampli-
+fication and decay may lead to the formation of a layer
+with an enhanced magnetic field. If the magnetic field
+amplification in the downstream proceeds in a highly
+non-homogeneous manner, then instead of a shell-like
+structure one should expect rather a large number of
+magnetized blobs in the production region, i.e., scenario
+(iii).
+Although scenarios (ii) and (iii) are character-
+ized by quite similar geometries, the angular distribu-
+tion of the target photons in the second zone may be
+quite different in these two cases.
+While in scenario
+(iii), the target photons are nearly isotropic, scenario
+(ii) features a substantial anisotropy of the target pho-
+tons in the second zone (as depicted in Fig. 3). As the
+emitting particles are isotropized in the plasma frame,
+this photon anisotropy should not have any impact on
+the cooling process. However, one may need to account
+for anisotropic IC cross-section (see, e.g., Aharonian &
+Atoyan 1981) for accurate computation of the IC spec-
+tra. For example, if the emission generated in the direc-
+tion of the observer is predominately produced by scat-
+tering target photons at small scattering angles, then
+the IC spectra appear to be harder compared to the
+spectra computed with angle-averaged IC cross-section
+(see, e.g., Khangulyan et al. 2008).
+Because of the Doppler boosting effect, the observer
+can detect the emission coming from a patch of the shell
+with a typical size of R′/Γ, where R′ ∼ t′
+trΓ2c. Thus,
+one obtains the patch size as t′
+trΓc ≫ 1015 cm (provided
+that t′
+tr > 1 h for the afterglow period). The realiza-
+
+Hard VHE emission from SSC sources
+7
+tion of scenario (iii) requires that the size of the blobs is
+small, r1 ≪ t′
+trΓc. Verification of this condition from the
+first principles may require detailed plasma simulations,
+which are beyond the scope of this study.
+As in the
+case of GRB afterglow, the GeV emission seems to be-
+long to the same component as the synchrotron, we may
+therefore speculate that the acceleration in the blobs are
+limited by the synchrotron cooling and the acceleration
+process is efficient, ηacc ∼ 1 (here ηacc is the accelera-
+tion efficiency). Thus, the size of the blobs should be
+sufficiently large to confine particles with energy
+E ≈ 60
+� B
+1 G
+�−1/2
+TeV .
+(12)
+The corresponding gyro radius,
+RG ≈ 2 × 1011
+� B
+1 G
+�−3/2
+cm ,
+(13)
+is significantly smaller than the patch size. This likely
+implies that there are no fundamental constrains from
+the plasma physics forbidding the scenario realization.
+3.2. Mathematical setup
+For each of the zones we consider the injection-cooling
+equation:
+∂n1
+∂t + ∂ ˙E1n1
+∂E
+= q1(E) −
+n1
+τ1→2
++
+n2
+τ2→1
+,
+(14)
+∂n2
+∂t + ∂ ˙E2n2
+∂E
+= q2(E) +
+n1
+τ1→2
+−
+n2
+τ2→1
+.
+(15)
+Here qi is the injection term; ˙Ei is the energy loss rate in
+each zone; and τi→j is the probability of particle escape
+from zone i to zone j. To illustrate the possible impact
+of the two-zone setup on the IC spectrum we assume
+that in the energy range of interest τi→j ≫ τ1, τ2, where
+τi = E/| ˙Ei| is radiative cooling time in zone i. Since we
+are interested in the high-energy part of the spectrum,
+which is formed in the fast cooling regime, we consider
+the following simplified equations:
+∂ ˙Eini
+∂E
+= qi(E) .
+(16)
+The magnetic field strengths differ significantly in each
+zone, thus we do not adopt a universal cutoff energy in
+the injection spectrum, but instead find different injec-
+tion rates, Ai, and cutoff energies, Ecut,i, within the two
+zones. We, however, assume that within both zones the
+injection function has a common power-law spectral in-
+dex:
+qi = AiE−α exp
+�
+−
+E
+Ecut,i
+�
+,
+(17)
+where the cutoff energy is found through the balance of
+acceleration and loss time-scales:
+ηaccEcut,i
+eBic
+= −
+Ecut,i
+˙Ei(Ecut,i)
+.
+(18)
+Here e is electron charge. Whilst the acceleration pa-
+rameter ηacc is assumed to be the same in both zones,
+the energy losses are computed independently for each
+zone.
+We assume that dimensionless parameters κ1 and κ2
+(κ1 + κ2 = 1) define the fraction of the total energy
+injected into zone one and two, respectively:
+1
+κi
+∞
+�
+Emin
+dE Eqi(E) = L0 ,
+(19)
+where L0 is the total power injected in the production
+region. The minimum energy, Emin, we set to a value of
+Γmec2 (here me is electron mass and c is the speed of
+light). While Eq. (18) defines the cutoff energy, Eq. (19)
+determines the normalization coefficients in each zone,
+Ai.
+As in the high-energy regime, the synchrotron and IC
+losses are expected to provide the dominant energy loss
+channels, we therefore only take account of these two en-
+ergy loss mechanisms. Since the photon field is common
+between the zones, the difference between the energy
+loss rate in the zones is due to the different synchrotron
+losses within each zone:
+˙Ei = ˙Esyn,i + ˙Eic .
+(20)
+The synchrotron energy losses in zone i are determined
+by the magnetic field strength
+˙Esyn,i = −16π
+3
+e4E2
+m4ec7
+�2
+3
+B2
+i
+8π
+�
+.
+(21)
+Note that the equation above is averaged over pitch an-
+gle. The synchrotron cooling time is
+τsyn,i =
+E
+| ˙Esyn,i|
+≈ 400
+� Bi
+1 G
+�−2�
+E
+1 TeV
+�−1
+s .
+(22)
+The IC losses are determined by the energy distribu-
+tion and number density of target (synchrotron) pho-
+tons. As photons can freely cross the zone boundaries
+we assume that the photon distribution is the same
+throughout the entire production region, i.e., it includes
+the contributions from both zones. We compute the syn-
+chrotron emission using the particle distribution in each
+zone and the corresponding magnetic field:
+dNsyn,i
+dε dt
+≈
+∞
+�
+mec2
+dE niKsyn,ε(E, Bi) ,
+(23)
+
+8
+Khangulyan et al.
+where ε is the target (synchrotron) photon energy. For
+the synchrotron integral kernel, Ksyn,ε, we use a sim-
+ple analytic approximation for the pitch angle averaged
+synchrotron spectrum (for detail see in Aharonian et al.
+2010). Finally, we compute the energy distribution of
+the target photons as
+dNsyn
+dε dV ≈
+1
+R2c
+�dNsyn,1
+dε dt
++ dNsyn,2
+dε dt
+�
+.
+(24)
+Here R is size of the production region.
+The rate of IC scattering is determined by the angle
+averaged scattering cross section (for detail see in Jones
+1968):
+dνic
+dεγ dε = 8πcr2
+0
+bE
+dNsyn
+dV dε×
+�
+1 +
+z2
+2(1 − z) +
+z
+b(1 − z) −
+2z2
+b2(1 − z)2 −
+z3
+2b(1 − z)2 −
+2z
+b(1 − z) ln b(1 − z)
+z
+�
+.
+(25)
+Here r0 = e2/mec2 is the electron classical radius; the
+Klein-Nishina parameter is given by b = 4εE/(m2
+ec4);
+and z is the ratio of the up-scattered photon to electron
+energy, z = εγ/E. The IC energy loss rate depends on
+the energy distribution of target photons as
+˙Eic ≈
+∞
+�
+0
+dε
+εmax,γ
+�
+εmin,γ
+dεγ (ε − εγ) dνic
+dεγ dε ,
+(26)
+where the maximum/minimum energy of up-scattered
+gamma-ray, εmax/min,γ, is determined by kinematic con-
+straints. If electrons up-scatter low-energy target pho-
+tons (i.e., the Klein-Nishina parameter is small, b ≪ 1),
+then the IC energy loss rate depends only on the energy
+density of the target photons, wph:
+˙ET,i = −32π
+9
+e4E2
+m4ec7 wph ,
+(27)
+analogous to the corresponding angle averaged energy
+loss rate in a magnetic field given in Eq. (21).
+3.3. Model calculations
+For the model calculations, magnetic field values of
+B1 = 1 G and B2 = 10−3 G are assumed. The injection
+power is set to ∼ 1039 erg s−1, and for the size of the
+production region we consider a value close to 1016 cm.
+If one considers this size in the context of a GRB after-
+glow, one should compare it to the forward shock radius,
+which depends on the time passed since the trigger, t′
+tr:
+R ∼ Γ2t′
+trc ∼ 3 × 1016
+� Γ
+10
+�2 t′
+tr
+3 h cm .
+(28)
+The typical energy density of the target photons in the
+production region is
+wph ∼ 4×10−5κ1ηrad
+�
+R
+3 × 1016 cm
+�−2
+erg cm−3 , (29)
+where ηrad is the radiation efficiency in zone 1 (in what
+follows we ignore this factor, setting ηrad = 1, for the
+sake of simplicity). This energy density corresponds to
+an equivalent magnetic field strength of
+Beq ∼ 3 × 10−2κ
+1/2
+1
+�
+R
+3 × 1016 cm
+�−1
+G .
+(30)
+This photon field is the dominant target in zone 2,
+whereas it is negligible in zone 1.
+The correspond-
+ing cooling time scales are shown in Fig. 4 (top
+panel). Whilst at high energies (approaching 1 PeV), the
+Klein-Nishina losses approach their asymptotic energy-
+dependence, τkn ∝ E, for the parameter set consid-
+ered, the particles cool in the transition regime with
+τ ∝ const. Thus the spectrum formed is not as hard as
+expected from our earlier qualitative analysis.
+The effect of the onset of Klein Nishina cooling on the
+electron spectrum is shown in Fig. 4 (bottom panel),
+where the energy distribution of electrons in both zones
+are shown.
+For the calculations here we adopted the
+following parameters: linear size R = 1016 cm; total
+power of acceleration of non-thermal particles L0 =
+1039erg s−1, which is distributed between the zones with
+κ1 = 0.90 and κ2 = 0.10; the injection index α = 2.2
+(the “main case”). Finally, the acceleration efficiency
+was set to ηacc = 102, for which the cutoff energy in
+zone 1 is determined to be:
+Ecut,1 ≈ 6
+�ηacc
+102
+�−1/2 � B1
+1 G
+�−1/2
+TeV .
+(31)
+For this acceleration efficiency the cutoff energy in zone
+2 is at ≈ 200 TeV, which is close to the energy at which
+the synchrotron losses dominate over the IC losses,
+E∗ ≈ 20 TeV, thus the influence of the high energy cut-
+off becomes prominent at energies just above the Klein-
+Nishina hardening energy scale.
+The energy dependence of the electron distribution
+is directly reflected in the synchrotron spectrum from
+zone 2.
+As can be seen from Fig. 5, this component
+is subdominant to the luminous synchrotron component
+from zone 1. The photon index of the hardest part of the
+spectrum is (α+1)/2 ≈ 1.5, which is considerably softer
+than the limiting photon index of γs,kn(= α/2). This
+is caused by the smooth broad transition to the Klein-
+Nishina regime.
+While the broad transition from the
+Thomson to Klein-Nishina regimes causes the electron
+
+Hard VHE emission from SSC sources
+9
+distribution to be not as hard as naively expected, the
+IC component appears to be somewhat harder than in
+the limiting case. As can be seen in Fig. 5, a power-law
+component extends from a few GeV to beyond 10 TeV
+with a photon index of ≈ (α + 1)/2. Note that for our
+calculations we set α = 2.2, and the production region
+bulk Lorentz factor was assumed to be Γ = 10.
+Figure 4. Top panel: Synchrotron, IC cooling time together
+with the acceleration time. Bottom panel: Electron distri-
+bution in two zones. Black guide lines indicate power-law
+approximations.
+To illustrate the influence of the model parameters, we
+performed calculations for a range of different parame-
+ter sets. The results of these calculations are shown in
+Fig. 6. For the “case A” we adopted a different value for
+the injection index: α = 2 instead of α = 2.2 used in the
+“main case”. For the “case B” we adopted a different
+value for the acceleration efficiency: ηacc = 104 instead
+of ηacc = 102 used in the “main case”.
+The adopted
+model parameter values are summarized in Table 2.
+Low-energy target photons can be an important role
+in the formation of a hard VHE spectrum in the case
+of the conventional one-zone SSC models. To demon-
+strate the relatively small influence of low-energy target
+Figure 5. Spectral energy distribution of synchrotron and
+IC emission from two zones.
+Black guide lines show the
+power-law approximations.
+Figure 6. Spectral energy distribution of synchrotron and
+IC emission from two zones.
+Black guide lines show the
+power-law approximations.
+Top panel:
+Case A; Bottom
+panel: Case B. Thin lines in the top panel correspond to a
+case when the electron distribution in the first zone features
+a cooling break at E ≈ 10 GeV.
+photons in the framework of our considered two-zone
+approach, in the top panel of Fig. 6 we also plot the
+
+Synch. Zone 1
+三
+Synch. Zone 2
+IC
+1010
+ITT
+108
+106
+s
+104
+L
+102
+WT
+100
+Acceleration Zone1
+Acceleration Zone2
+10-2
+102
+10-1
+100
+101
+103
+104
+105
+106
+E, GeVZone 1: particles (x103)
+Zone 2: particles
+56
+-α
+10
+TTTTT
+1055
+xE'α-1
+1054
+10-1
+100
+101
+102
+103
+104
+105
+E, GeV46
+0
+L
+Zone 1
+-(α + 2)/2
+Zone 2
+45
+10
+, arbitrary units
+44
+10
+43
+-(α+1)/2
+3
+42
+C
+Syn
+41
+10
+l
+100
+10-6
+10-2
+102
+104
+10~8
+, GeV46
+T
+(α + 2)/2
+Zone 1
+Zone 2
+45
+0
+-(α+2)/2
+x
++1)/2
+-(α+
+43
+10°
+3
+42
+TTT
+C
+7
+sVn
+1
+41
+10-6
+100
+102
+104
+10-2
+, GeV46
+0
+Zone 1
+α + 2)/2
+Zone 2
+45
+10
+, arbitrary units
+44
+10
+43
+10
+-(α+1)/2
+3
+42
+10
+IC
+41
+yn
+1
+E
+Ll
+100
+10~2
+102
+104
+10~8
+10-6
+10-4
+, GeV10
+Khangulyan et al.
+SED obtained under the same conditions assuming that
+the particle spectrum in the first zone features a cool-
+ing break at E ≈ 10 GeV. The corresponding spectra
+are shown with thin lines. Under this assumption, the
+synchrotron spectrum from the first zone features the
+cooling breaks, as expected. The IC spectrum is more
+strongly suppressed: one sees here the impact of both
+the cooling break and reduced target photon density.
+The reduction of the IC loss rate leads to a consider-
+able enhancement of the synchrotron emission from the
+second zone (note that this component still remains sub-
+dominant). The IC spectra from the second zone shows,
+however, only minor changes, noticeably only close to
+the high- and low-energy cutoffs regions.
+This quite
+weak influence of the target photon spectrum on the
+spectral properties of the IC component from the second
+zone is caused by the fact that the IC losses determine
+the particle spectrum, as we assume that the emission
+is generated in the fast cooling regime. Therefore, the
+electron spectrum adjusts to the rate of the dominant
+losses, and the spectral properties of the IC component
+are largely determined by the injection spectrum.
+4. DISCUSSION AND CONCLUSION
+The need for studying energy losses in the inhomoge-
+neous emission region downstream can be easily realized
+by considering the evolution of the magnetic field from
+the upstream to downstream regions. Based on the hy-
+drodynamics of the forward shock propagating through
+the CBM, one can obtain the following estimate for the
+downstream magnetic field strength:
+B ∼ 3 × 102 Γ
+10
+Bcbm
+10 µG µG .
+(32)
+This estimate depends on the typical strength of the
+CBM magnetic field, Bcbm, and accounts for the trans-
+formation of this field to the forward shock rest frame,
+and for the increase of the field strength at a weakly
+magnetized relativistic shock due to the shock compres-
+sion.
+The magnetic field given by Eq. (32) appears signifi-
+cantly below the Gauss-level required for the afterglow
+radiation. Therefore, one needs to assume an efficient
+magnetic field amplification process, which can increase
+the energy density of the magnetic field to the level com-
+parable to the plasma energy density in the downstream:
+w ∼ ncbmmpΓ2 ≈ 0.15
+� ncbm
+1 cm3
+�� Γ
+10
+�2
+erg cm−3 , (33)
+where ncmb is CBM density. This estimates shows that
+the magnetic field in the downstream can be amplified
+up to a strength of
+Beq =
+√
+8πw ∼ 2
+� ncbm
+1 cm3
+�1/2� Γ
+10
+�
+G .
+(34)
+Gauss-strength magnetic fields in the afterglow produc-
+tion region are also favored on theoretical grounds by
+afterglow emission modeling.
+If the magnetic field is
+indeed amplified by a factor of ∼ 103, it is natural to
+further assume that this amplification is inhomogeneous
+throughout the volume resulting in a magnetic field con-
+figuration with strong spatial fluctuations. For example,
+magnetic field amplification by turbulent dynamo shows
+that the magnetic energy is predominantly localized in
+small blobs (Zhang et al. 2009). Moreover, this may be
+a general effect: the field amplification predominately
+operates on small scale fields (Kazantsev 1968).
+The highly inhomogeneous structure of the down-
+stream region can have important implications for the
+properties of the non-thermal emission generated.
+In
+particular, such a structure in the production region
+can significantly alter the synchrotron radiation emis-
+sion, with clumps of highly amplified magnetic field
+leading to the synchrotron emission extending signifi-
+cantly beyond the one-zone synchrotron burn-off limit
+(Khangulyan et al. 2021). This scenario requires that
+particles are accelerated in a region of weak magnetic
+field, and subsequently penetrate into a second zone of
+amplified magnetic field, where they rapidly cool pro-
+ducing VHE synchrotron radiation.
+The requirement
+of effective particle exchange between the two zones of
+strong and weak magnetic field is an important element
+of this scenario.
+It should be noted, however, that efficient particle ex-
+change between the zones is a significant assumption.
+Processes exist, which can hinder particle exchange be-
+tween the two zones. For example, if the change of the
+magnetic field strength is relatively smooth, the mag-
+netic adiabatic invariant prevents particles from the zone
+of weak magnetic field reaching a strong magnetic field
+zone (see the discussion in Khangulyan et al. 2021). The
+particle escape from the zone of strong to weak mag-
+netic field is not forbidden by the magnetic adiabatic
+invariant, but it seems feasible that one can neglect this
+process. Because of the much higher rate of synchrotron
+losses in the strong magnetic field zone, the total number
+of particles in this zone is naturally significantly reduced
+to that in the weak magnetic field zone, particularly for
+the highest energy particles with energies close to the
+maximum energy.
+On the other hand, synchrotron photons can freely
+travel between the two zones.
+The photon exchange
+between the zones have two major effects: (i) altering
+
+Hard VHE emission from SSC sources
+11
+Figure 7.
+Spectral energy distribution of synchrotron
+and IC emission from two zones. Filled zones show X-ray
+(XRT) and VHE intrinsic (H.E.S.S.) spectra (for detail see
+in H. E. S. S. Collaboration et al. 2021).
+particle energy losses, and (ii) change the properties of
+IC emission.
+We suggest a simple model that allows
+one to study these two effects. We find that for feasi-
+ble model parameters, IC scattering dominates the cool-
+ing process in the zone of weak magnetic field. Due to
+the Klein-Nishina effect, the particle spectrum formed
+in the fast cooling regime appears to be significantly
+harder than the spectrum formed for the case when syn-
+chrotron losses dominate. While the synchrotron emis-
+sion from this zone may appear completely sub-luminous
+with respect to the synchrotron emission generated in
+the strong magnetic field zone, the IC component from
+the weak magnetic field zone would be expected to dom-
+inate.
+The second signature of the hard particle spectrum is
+expected in the IC component generated by these parti-
+cles. This spectrum appears to be hard, with a photon
+index coinciding with the value expected for the syn-
+chrotron/Thomson spectra, (α + 2)/2 (where α is the
+injection index).
+The IC spectrum therefore appears
+to have the same slope as the dominant synchrotron
+emission. The relative flux through these two channels
+is determined by the phenomenological parameters, κi,
+which determine the ratio of the acceleration powers in
+the two zones. Our simulations presented in Fig. 7 show
+that for an acceleration spectrum with a spectral slope
+of α = 2.1, which allows the slope of the X-ray spec-
+tra for GRB190829A to be reproduced, the X-ray and
+VHE flux ratio seen in GRB190829A implies κ1 = 0.7
+and κ2 = 0.3 (see column “GRB190829A” in Table 2).
+This suggests that acceleration processes of comparable
+power operate in the both zones. However, the acceler-
+ation in the zone of stronger magnetic field is somewhat
+more efficient. The obtained hard VHE IC spectrum ex-
+tends beyond 10 TeV for a modest bulk Lorentz factor
+of Γ = 10. This implies that a hard multi-TeV IC com-
+ponent can be generated also during the late afterglow
+phases, when the forward shock transits into the mildly
+relativistic regime. During the prompt or early afterglow
+phases, when the bulk Lorentz factor can be significantly
+larger, Γ ≥ 100, the intrinsically hard IC component can
+extend up to the ultra high energy domain (≥ 100 TeV).
+However, we note that the extragalactic EBL attenua-
+tion is severe already in the VHE domain for even the
+most local GRB redshift values.
+DK acknowledges support by the RSF grant No. 21-12-
+00416.
+1
+2
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+Parameter
+Notation
+Comment
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+R
+Similar to the forward shock radius
+Magnetic field
+B
+B1,2 in zone 1 or 2; Bcbm in the shock upstream
+Bulk Lorentz factor
+Γ
+We do not distinguish the bulk Lorentz factors of emitting plasma or forward shock
+Injection power
+L0
+Eq. (19) is in the co-moving frame, but note that L0 is a Lorentz invariant
+Normalization factors
+κ1,2
+Power distribution between two zone; κ1 + κ2 = 1
+Energy density
+w
+e.g., wext is energy density of external photon fields
+Electron energy
+E
+–
+Electron energy density
+n
+i.e., dN = n dE
+Injection index
+α
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+Injection rate
+q
+i.e., dN = q dE dt
+Acceleration efficiency
+ηacc
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+τ
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+ncmb
+i.e. dN cmb = ncmb dV
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+Parameter
+Notation
+units
+Main case
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+Case B
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+–
+10
+10
+10
+10
+Luminosity
+L0
+erg s−1
+1039
+1039
+1039
+2.2 × 1038
+Strong magnetic field
+B1
+G
+1
+1
+1
+1
+Weak magnetic field
+B2
+G
+10−3
+10−3
+10−3
+10−3
+Zone 1 power fraction
+κ1
+–
+0.9
+0.9
+0.9
+0.7
+Zone 2 power fraction
+κ2
+–
+0.1
+0.1
+0.1
+0.3
+Injection slope
+α
+–
+2.2
+2
+2.2
+2.1
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+–
+102
+102
+104
+102
+Radiation efficiency
+ηrad
+–
+1
+1
+1
+1
+Distance
+D
+Mpc
+–
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+page_content=' Rikkyo University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Nishi-Ikebukuro 3-34-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
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+page_content=' Germany  3Dublin Institute for Advanced Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' School of Cosmic Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 31 Fitzwilliam Place,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Dublin 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Ireland  4Max-Planck-Institut f¨ur Kernphysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Saupfercheckweg 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 69117 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Germany  ABSTRACT  Electron Compton scattering of target photons into the gamma-ray energy band (inverse Compton  scattering –IC–) is commonly expected to dominate the very high energy spectra in gamma-ray bursts  especially during the afterglow phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For sufficiently large center-of-mass energies in these collisions,  the effect of the electron recoil starts reducing the scattering cross section (the Klein-Nishina regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The IC spectra generated in the Klein-Nishina regime is softer and has a smaller flux level compared  to the synchrotron spectra produced by the same electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The detection of afterglow emission  from nearby GRB190829A in the very high energy (VHE) domain with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' has revealed an  unexpected feature: the slope of the VHE spectrum matches well the slope of the X-ray spectra, despite  expectations that for the IC production process, the impact of the Klein-Nishina effect should be strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The multi-wavelength spectral energy distribution appears to be inconsistent with predictions of one-  zone synchrotron-self-Compton models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We study the possible impact of two-zone configuration on the  properties of IC emission when the magnetic field strength differs considerably between the two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Synchrotron photons from the strong magnetic field zone provide the dominant target for cooling of the  electrons in the weak magnetic field zone, which results in a formation of hard electron distribution and  consequently of a hard IC emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We show that the two-zone model can provide a good description  of the X-ray XRT and VHE H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Keywords: Non-thermal radiation sources(1119) — Gamma-ray transient sources(1853) — Gamma-  ray bursts(629) — Gamma-ray astronomy(628) — Particle astrophysics(96) — X-ray  sources(1822)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' INTRODUCTION  The very high energy (VHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' > 100 GeV) emission de-  tected from gamma-ray burst (GRB) afterglows with  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' and MAGIC (Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' MAGIC Col-  laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2019a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  2021) is considered by many to have inverse Compton  (IC) origin (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g, Zhang 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The emission com-  ponent produced by relativistic protons is expected to  have a significantly lower flux, due to the very low ra-  diative efficiency of hadronic interactions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Ab-  dalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  If the VHE emission is produced  by relativistic electrons, then because of the so-called  Corresponding author: Dmitry Khangulyan  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='khangulyan@rikkyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='jp  synchrotron burn-off limit (Guilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 1983) the  synchrotron component is expected to reach the VHE  regime only if the bulk Lorentz factor is very high,  Γ ≥ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Such high bulk Lorentz factors are excluded  during the afterglow phase by energy conservation argu-  ments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', related to self-similar solution for relativistic  blast wave obtained by Blandford & McKee 1976) mak-  ing IC scattering the most feasible radiation mechanism  for the VHE GRB emission during the afterglow period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  However, the hard intrinsic spectral slope inferred from  observations by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' of GRB190829A afterglow can-  not be easily reproduced with standard IC models (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This leaves  one of two possibilities: (i) invoke alternative radiation  mechanisms, or (ii) develop a more sophisticated IC sce-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='08578v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='HE]  20 Jan 2023  2  Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  nario to provide a better description of the observational  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Synchrotron radiation is a very efficient radiative  emission mechanism of electrons during the afterglow  phase of GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' If the synchrotron component extends  into the VHE domain, it can reproduce the flux level and  spectral slope revealed with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' from GRB190829A  afterglow (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' While  the conservation of energy, used to constrain the bulk  Lorentz factor, is a robust argument, the burn-off en-  ergy limit can be avoided in certain non-standard sce-  narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example, if the strength of the accelerating  electric field, E, exceeds the strength of the magnetic  field, B (in a plasma such configurations require non-  ideal magnetohydrodynamics) then synchrotron emis-  sion can extend beyond the burn-off limit by the fac-  tor of E/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Alternatively, in highly turbulent magnetic  fields magnetobremsstrahlung radiation can extend be-  yond the burn-off limit (Kelner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' If the cor-  relation length of the magnetic field is large compared  to the photon formation length, mec2/e ¯B (here me and  e are electron mass and charge, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' c and ¯B  are light speed and averaged magnetic field), then the  radiation is generated in the synchrotron regime, result-  ing in the burn-off limit for the synchrotron maximum  energy (for a detailed consideration, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', in Kel-  ner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Derishev & Aharonian 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' However,  if the correlation length is short compared to the pho-  ton formation length, then the electrons instead emit  in the jitter regime, and the emission peaks at higher  energy compared to the synchrotron case, alleviating  the limit from the burn-off limit (Kelner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Finally, the electron synchrotron spectrum can extend  beyond the burn-off limit in two-zone systems, where  the physical conditions at the acceleration site and in  the radiation production region differ substantially (Ku-  mar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In conclusion,  there are several ways of expanding the energy spec-  trum of magneto-bremsstrahlung to high or even very  high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' However, the feasibility of these scenar-  ios depends on the implementation of many factors and  requires extreme assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  In contrast, IC scattering is a natural and very effec-  tive channel of VHE gamma-ray production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Although  the recent observations of VHE gamma rays during the  GRB afterglows challenge the simple one-zone IC model,  more sophisticated scenarios cannot be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In this  paper, we study the spectral properties of gamma rays in  the two-zone IC model in which the production region of  the target (synchrotron) photons and the IC gamma-ray  emitter are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' One can propose several possi-  ble realizations for such a two-zone setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example,  one may expect quite different conditions at the for-  ward and reverse shocks, which propagate through the  circumburst medium (CBM) and the jet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' If  the emission from the reverse shock appears to be impor-  tant at certain frequencies then a two-zone description  for GRB afterglow emission should be considered (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Dichiara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Salafia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Alterna-  tively, the shock region itself can be quite complex po-  tentially providing quite different physical conditions for  particle acceleration and radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Indeed, simulations  suggest that downstream shock material, the dominant  emission site during the afterglow phase, is expected to  be highly inhomogeneous, an aspect usually neglected in  GRB afterglow emission modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Below we consider  the impact of a strongly inhomogeneous magnetic field  on the properties of IC emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We show that under  reasonable assumptions, even a two-zone synchrotron  self-Compton (SSC) scenario can provide a considerably  improved description of the broadband spectra reported  from GRB190829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' STANDARD ONE-ZONE SSC SCENARIO  The standard GRB afterglow emission framework pos-  tulates that this emission is generated via the syn-  chrotron and IC channels, with synchrotron radiation  providing the dominant target for IC scattering – the  so called SSC scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The analysis of the spectral  energy distribution (SED) in SSC models is straightfor-  ward if the IC emission is generated in the Thomson  regime (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Sari & Esin 2001), as in this case the  energy loss rate,  ˙E, has a simple form ˙E ∝ E2 (here  E is electron energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In this regime, a power-law in-  jection of non-thermal electrons, q ∝ E−α (here α is  the injection index, for conventional acceleration mech-  anisms one typically assumes α ≈ 2), leads to the forma-  tion of a broken-power-law distribution of radiating elec-  trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The synchrotron and IC (Thomson) components  generated by these electrons also reflect this broken-  power-law shape, with the IC component dominating at  higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The subsequent broadband SED pro-  duced is double-humped, with the relative emissivity of  the synchrotron and IC components being determined  by phenomenological parameters (typically, by the radi-  ation efficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', by the fraction of energy radiated  away).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The photon index of the synchrotron spectrum,  produced by electrons with energies above the cooling  break, is γs = (α + 2)/2, provided that α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In the  single zone SSC scenario, the corresponding IC spectrum  has the same photon index, if generated in the Thomson  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Typically,  during the afterglow phase the (syn-  chrotron) X-ray spectrum is observed to be hard, with  Hard VHE emission from SSC sources  3  E  dN  dE  Thomson cooling  E−(α+1)  Tr  a  n  si  ti  o  n  co  oli  ng  Asymptotic K-N cooling  E−(α−1)  Synchrotron cooling  E−(α+1)  Synchrotron cooling  E−(α+1)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  A sketch that illustrates the formation of the  particle spectrum in the case of dominant synchrotron losses  and dominant IC losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The part of the spectrum formed in  the fast cooling regime is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  ε  εFε  Synchrotron dominated case  Synchrotron  Inverse Compton  ε− (α+2)  2  Thomson  ε− (α+2)  2  Transition  Klein-Nishina  ε−(α+2)  ε  εFε  IC dominated case  Synchrotron  Inverse Compton  ε− (α+2)  2  ε− α  2  ε− (α+2)  2  Thomson  ε− (α+2)  2  Transition  Klein-Nishina  ε−(α+2)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' A sketch that illustrates the formation of the SED  in the case of dominant synchrotron losses and dominant IC  losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  a photon index ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Thus the photons detected in the  X-ray band provide a non-negligible target for IC scat-  tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In the plasma co-moving frame, the energy of  the electron, E, generating the VHE emission, detected  at energy1 ε′  vhe, satisfy the condition: E > ε′  vhe/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' If  electrons of this energy up-scatter photons from a com-  ponent detected by the observer at energy ε′  x, then the  typical product of the target photon and electron ener-  gies, which determines the scattering regime, is  Eεx  m2ec4 > ε′  vheε′  x  m2ec4Γ2 ≈ 4  � Γ  10  �−2 �  ε′  vhe  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='1 TeV  � �  ε′  x  1 keV  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (1)  1 Note that we prime the quantities in the progenitor frame, and  we neglect the cosmological redshift effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Here me and c are the electron mass and speed of light,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Unless the bulk Lorentz factor is high,  Γ ≥ 102, the electrons that produce the VHE emission  up-scatter a considerable part of the photon targets in  the Klein-Nishina regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The study of the VHE prop-  erties of GRB afterglows should therefore be conducted  with models that account for the change of the IC cross-  section in the relativistic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The influence of the Klein-Nishina regime on the SED  is two-fold, as one must account for both the change  of the emission and energy loss rates (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Derishev  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Nakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In the fast cooling regime,  the particle spectrum, dN = n dE, is determined by  the injection spectrum, q, and by the cooling time τ =  E/| ˙E|:  n(E) = τ(E)  E  ∞  �  E  d ˜E q  �  ˜E  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (2)  If the injection is a power-law q ∝ E−α, then the particle  spectrum is  n(E) ∝ τ(E)E−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (3)  (Note that here we assume that the injection spectrum  is sufficiently steep so as to ensure the integral is domi-  nated by the low energy limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  If the synchrotron losses dominate over the Compton  losses (more specifically if the energy density of the mag-  netic field is larger than the energy density of the target  photons) then τ(E) ∝ E−1, and a power-law spectral  injection also yields a power-law distribution of parti-  cles: n(E) ∝ E−(α+1) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 1 for a sketch of the  cooled particle spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Subsequently, a power-law  synchrotron component is produced with photon index  γs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The inverse Compton of radiation has the same power-  law photon index as long as the scattering takes place in  the Thompson regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In the Klein-Nishina regime, the  IC slope should (asymptotically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', ignoring the log-  arithmic term) approach γkn ≈ (α + 2) (provided that  the emitting electrons obey a power-law energy distribu-  tion with index α+1, Blumenthal & Gould 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Thus,  since the slope of IC component generated in the Thom-  son regime matches that of the synchrotron radiation,  γs, the Klein-Nishina effect causes a spectral softening  by ∆γ ≈ γkn − γs ≈ (α + 2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example, if α ≈ 2  then the spectral slope changes from γs ≈ 2 to γkn ≈ 4,  and the spectral softening is ∆γ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' A schematic of the  SED is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' One should note that for a broad  target photon distribution, the transition to the Klein-  Nishina regime is spread over a broad energy range and  can have a rather complex character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The situation changes dramatically when the energy  density of target photons is larger than the energy den-  4  Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  sity of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In this case, the impact of  the Klein-Nishina effect on the formation of the elec-  tron spectrum becomes a dominant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The radia-  tive cooling time τ(E) can be approximated by a broken  power-law function: for sufficiently low electron ener-  gies, the IC interaction proceeds in the Thomson regime,  thus τ(E) ∝ E−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' At higher energies, the IC interac-  tions occur in the Klein-Nishina regime where the en-  ergy loss rate is energy-independent, thus τ(E) ∝ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Finally at even higher energies, denoted E∗, the syn-  chrotron losses (as their rate increases with particle  energy) begin to dominate over the IC energy losses,  and the original energy dependence of the cooling time  is recovered: τ(E) ∝ E−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  As follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (3),  for a power-law injection spectrum, the particle spec-  trum formed in the fast cooling regime should also be  a double-broken-power-law (with the power-law index  changing as α + 1 → α − 1 → α + 1: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The  E−(α+1) part of the spectrum formed under dominant  (Thomson regime) IC losses changes to, ∝ E−(α−1),  formed under the dominant IC (Klein-Nishina regime)  losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Finally, above E∗, the spectrum softens back  to E−(α+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We note, however, that the transition to  the Klein-Nishina regime proceeds smoothly, therefore  the spectrum does not follow precisely the schematic  shape explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  For example, as can be seen  from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 4, the IC cooling time in the transition regime  is better approximated as a constant, τ ≈ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' There-  fore, the corresponding transformation of the electron  spectrum is better approximated as α + 1 → α → α + 1  (note that this power law index is indicated in the bot-  tom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 4 with a black guide line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  As for the synchrotron radiation, electrons cooled by  IC in the Thomson regime produce a spectrum with pho-  ton index γs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' at higher energies the hardening of the elec-  tron spectrum due to the dominant Klein-Nishina en-  ergy losses results in a hard synchrotron spectrum with  photon index in the range between γs and γs,kn ≈ α/2  (γs,kn is the limiting value achieved under IC cooling in  the deep Klein-Nishina regime: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In the tran-  sition region with an approximately constant IC cooling  time, the slope of the synchrotron spectrum is approxi-  mately (α + 1)/2, as indicated by the black guide lines  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Finally, the emission produced by  electrons with energies exceeding E∗ has the standard  synchrotron slope γs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' As the synchrotron and IC energy  loss rates for particles with E∗ are equal, the narrow-  band luminosity of the synchrotron and IC components  produced by particles with E∗ are (almost) equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The spectral shape of the IC component is different to  that of the synchrotron spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The component gen-  erated in the Thomson regime has a spectral index of  γs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' At higher energies the impact of the Klein-Nishina  effect on the particle spectrum is partially compensated  by the reduction of the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example, in  the limiting regime, a spectrum ∝ E−(α−1) generates  in the Klein-Nishina regime a E−α IC spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For  α ≈ 2 a Thomson spectrum with photon index (α+2)/2  transits smoothly into the Klein-Nishina spectrum with  photon index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' However, in the region of transition  to the Klein-Nishina regime, this asymptotic photon  index might be quite a coarse approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  More-  over, above E∗ the synchrotron losses dominate, thus  the Klein-Nishina spectrum eventually softens to α + 2  above E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Note that in the Klein-Nishina regime almost  all the electron energy is transferred to the up-scattered  photon, so the photon energy in the co-moving frame is  equal to that of the incident electron energy, εic ≈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Observations of GRB190829A with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' revealed  that VHE component, corrected for the extragalactic  background light (EBL) attenuation, is best described as  a single power-law spectrum extending up to 3 TeV with  a hard photon index of γvhe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='09 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Strikingly, this slope matches  well the slope of the X-ray spectrum measured with  Swift-XRT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', γxrt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='06 during the first night  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Also, the Swift-  XRT and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' observations revealed that the fluxes  in the X-ray and VHE bands appeared to be similar (po-  tentially a natural feature of pair loading feedback, see  Derishev & Piran 2016, 2019, for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  In the VHE band the influence of the Klein-Nishina  effect should be noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' However, this spectral ef-  fect was not observed in the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  In the framework of the simple one-zone analysis intro-  duced above, the slope and flux level match implies that  the cooling of TeV emitting electrons proceeds in the  Klein-Nishina regime, and that the X-ray synchrotron  is produced by particles with energy exceeding E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' As  the hard VHE spectrum extends up to 3 TeV, then  E∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='3(Γ/10)−1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The synchrotron emission pro-  duced by the high-energy electrons is detected by the  observer at  ε′  syn > 60  � Γ  10  �−1 B  1 G keV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (4)  This estimate shows that a very low magnetic field of  ∼ mG level is required by the VHE measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Such a low magnetic field, however, is incompatible with  the required radiation efficiency of the production region  given the adiabatic cooling time is τad ∼ t′  trΓ, where t′  tr  is time since the GRB trigger (as measured by a distant  observer at rest in the progenitor reference frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The  broad-band SED obtained with Swift-XRT and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Hard VHE emission from SSC sources  5  therefore cannot be reproduced in the framework of the  standard one-zone SSC scenario (see also Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' To resolve the spectral issue in SSC scenario one  needs either: (1) assume that there is an important low-  energy target photon field, probably of external origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  or (2) consider a two-zone scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The former scenario requires the presence of an exter-  nal target that provides a target of an energy density  comparable to that of the magnetic field in the plasma  co-moving frame:  wext ∼ 4 × 10−2  � B  1 G  �2  erg cm−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (5)  If the photons are isotropic in the progenitor frame, then  we obtain w′  ext ∼ 4 × 10−4(10B/(Γ G))2 erg cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The  VHE emission detected from GRB190829A lasted for  almost ∆t = 50 h (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  2021), and the forward shock covered a distance of  ∆R′ ∼ Γ2∆tc ∼ 1017(Γ/10)2 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The luminosity of  the photon field should therefore be  L′  ext ∼ 4π∆R′2w′  extc ∼ 1042  � B  1 G  �2  erg s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (6)  If the magnetic field is weak, B ≪ 1 G, then an external  photon field of reasonable luminosity can provide a suffi-  ciently dense external photon field (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  2021), however external IC scenarios with an equivalent  Gauss-strength magnetic field cannot be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' TWO-ZONE SSC EMISSION SCENARIO  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Physical justification  We consider the emission region consisting of two  zones: the first zone with a strong magnetic field, B1,  and the second zone with a weak magnetic field, B2,  with B1 ≫ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Should particles themselves also easily  mix between the two zones, then one would not expect a  significant difference between the energy distributions of  particles in these zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We here, however, assume that  the particle exchange between the zones is inefficient,  and thus two distinct particle distributions, n1 and n2,  are formed in the two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The target photons, however, travel freely between the  two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The specific realization of the scenario, in  particular the shapes and relative location of the zones,  determines the actual distribution of target photons in  the zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Let us qualitatively consider several possible  realizations of the two-zone scenario: (i) two distinct  regions with typical sizes of r1 and r2 separated by a  distance r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (ii) two converging shells of radius r1 and r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (iii) N compact regions (of typical size r1) with strong  magnetic field embedded within a larger zone of size r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  r1  r2  r0  Scenario (i)  Strong B-field region  Weak B-field region  Photons from the  strong B-field region  Photons from the  weak B-field region  r1  r0  Scenario (ii)  r1  r0  Scenario (iii)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Examples of three different geometries that allow  the scenario realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Scenario (i): two distinct regions  with typical sizes of r1 and r2 separated by a distance r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  scenario (ii): two converging shells of radius r1 and r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' sce-  nario (iii): a large number of compact regions (of typical size  r1) with strong magnetic field embedded within a larger zone  of size r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  These three possibilities are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Although  less apparent, scenario (iii) is two-zone in the sense that  the physical conditions and processes are the same in  the compact regions, and differ substantially from those  in the larger zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The synchrotron luminosity of each of the zones is L1  and L2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In scenario (iii) we define L1 as the  total luminosity of N regions of enhanced B-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We  consider a situation L1 ≫ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Thus, when considering  the processes in the first zone, we can ignore the photons  supplied by the second zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The energy density of the  6  Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  locally generated photons in the first zone is  w1→1 ∼  L1  r2  1Nc ,  (7)  where N = 1 for scenarios (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Equation (7)  ignores a numerical factor, which depends on the pro-  duction region geometry and the distribution of emitting  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example, in the case of a spherical homo-  geneous production region, the volume average energy  density of target photons is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 7 with a factor  9/(16π) (for detail, see in Atoyan & Aharonian 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  We note that such factors do not affect our conclusions,  we therefore safely ignore them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  In the second zone one needs to account for the con-  tribution of locally generated photons:  w(i)  2→2 ∼ L2  r2  2c  and  w(ii)/(iii)  2→2  ∼ L2  r2  0c  (8)  and the photons supplied from the first zone, w1→2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For  the each of the above defined geometries one obtains  w1→2 ∼ L1  r2  0c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (9)  The suggested scenario is realized if the photon field  produced in the first zone (being locally a subdominant)  provides the dominant target for the particle cooling in  the second zone:  w1→1 ≪ B2  1  8π  and  w1→2 ≫ B2  2  8π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (10)  The photon field in the second zone is diluted com-  pared to the first zone: w1→1 > w1→2, thus the scenario  requires that B1 ≫ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The difference of the mag-  netic fields determines the dilution of the photon field,  κ = w1→2/w1→1, that allows the scenario realization  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e, the conditions given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The possible ratio of the magnetic fields should be de-  termined by the physical arguments unique to each spe-  cific realization of the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' However, from the gen-  eral point of view, it is obvious that if the photon field  is significantly diluted in the second zone, κ ≪ 1, the  required difference between the magnetic field strength  becomes larger, making the realization of the scenario  less feasible (although not excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  For example, a  strong dilution might be expected in scenario (i) pro-  vided that r0 ≫ r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  In contrast, in scenario (ii) the  dilution of the photon field in the second zone is small,  by a factor of ∼ 2, provided that two shells are of com-  parable radius, r1 ≈ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Similarly, in scenario (iii) one  obtains  w1→2  w1→1  ∼ r2  1N  r2  0  = fr0  r1  ,  (11)  where f is filling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' If the above ratio is not small  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', w1→2/w1→1 ≳ 1) then the photon field is nearly  homogeneous in the entire production region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', κ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  For the sake of simplicity we will consider a single  common photon target being present in the two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  In the first place, this seems to be a perfectly suitable  choice for scenarios (ii) and (iii) if r1 ≈ r0 and fr0/r1 ≳  1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Even if these conditions are not fulfilled,  the model calculations should reproduce correctly the  part of SED formed in the fast cooling regime (provided  that IC losses dominate over the synchrotron cooling in  the second zone: w1→2 ≫ B2  2/(8π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Although the scenario can be realized also in scenario  (i), if the magnetic field in the second zone is sufficiently  weak to remain subdominant compared to the signifi-  cantly diluted photon field provided from the first zone,  scenarios (ii) and (iii) seem to be less demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' In  particular, these geometries can be formed during the  afterglow phase of GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The shells assumed in sce-  nario (ii) may correspond to the reverse and forward  shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Also an onion-like structure may be formed in  the inner part of the forward shock downstream region,  where the competing processes of magnetic field ampli-  fication and decay may lead to the formation of a layer  with an enhanced magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' If the magnetic field  amplification in the downstream proceeds in a highly  non-homogeneous manner, then instead of a shell-like  structure one should expect rather a large number of  magnetized blobs in the production region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', scenario  (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Although scenarios (ii) and (iii) are character-  ized by quite similar geometries, the angular distribu-  tion of the target photons in the second zone may be  quite different in these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  While in scenario  (iii), the target photons are nearly isotropic, scenario  (ii) features a substantial anisotropy of the target pho-  tons in the second zone (as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' As the  emitting particles are isotropized in the plasma frame,  this photon anisotropy should not have any impact on  the cooling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' However, one may need to account  for anisotropic IC cross-section (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Aharonian &  Atoyan 1981) for accurate computation of the IC spec-  tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example, if the emission generated in the direc-  tion of the observer is predominately produced by scat-  tering target photons at small scattering angles, then  the IC spectra appear to be harder compared to the  spectra computed with angle-averaged IC cross-section  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Because of the Doppler boosting effect, the observer  can detect the emission coming from a patch of the shell  with a typical size of R′/Γ, where R′ ∼ t′  trΓ2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Thus,  one obtains the patch size as t′  trΓc ≫ 1015 cm (provided  that t′  tr > 1 h for the afterglow period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The realiza-  Hard VHE emission from SSC sources  7  tion of scenario (iii) requires that the size of the blobs is  small, r1 ≪ t′  trΓc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Verification of this condition from the  first principles may require detailed plasma simulations,  which are beyond the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  As in the  case of GRB afterglow, the GeV emission seems to be-  long to the same component as the synchrotron, we may  therefore speculate that the acceleration in the blobs are  limited by the synchrotron cooling and the acceleration  process is efficient, ηacc ∼ 1 (here ηacc is the accelera-  tion efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Thus, the size of the blobs should be  sufficiently large to confine particles with energy  E ≈ 60  � B  1 G  �−1/2  TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (12)  The corresponding gyro radius,  RG ≈ 2 × 1011  � B  1 G  �−3/2  cm ,  (13)  is significantly smaller than the patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This likely  implies that there are no fundamental constrains from  the plasma physics forbidding the scenario realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Mathematical setup  For each of the zones we consider the injection-cooling  equation:  ∂n1  ∂t + ∂ ˙E1n1  ∂E  = q1(E) −  n1  τ1→2  +  n2  τ2→1  ,  (14)  ∂n2  ∂t + ∂ ˙E2n2  ∂E  = q2(E) +  n1  τ1→2  −  n2  τ2→1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (15)  Here qi is the injection term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' ˙Ei is the energy loss rate in  each zone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' and τi→j is the probability of particle escape  from zone i to zone j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' To illustrate the possible impact  of the two-zone setup on the IC spectrum we assume  that in the energy range of interest τi→j ≫ τ1, τ2, where  τi = E/| ˙Ei| is radiative cooling time in zone i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Since we  are interested in the high-energy part of the spectrum,  which is formed in the fast cooling regime, we consider  the following simplified equations:  ∂ ˙Eini  ∂E  = qi(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (16)  The magnetic field strengths differ significantly in each  zone, thus we do not adopt a universal cutoff energy in  the injection spectrum, but instead find different injec-  tion rates, Ai, and cutoff energies, Ecut,i, within the two  zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We, however, assume that within both zones the  injection function has a common power-law spectral in-  dex:  qi = AiE−α exp  �  −  E  Ecut,i  �  ,  (17)  where the cutoff energy is found through the balance of  acceleration and loss time-scales:  ηaccEcut,i  eBic  = −  Ecut,i  ˙Ei(Ecut,i)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (18)  Here e is electron charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Whilst the acceleration pa-  rameter ηacc is assumed to be the same in both zones,  the energy losses are computed independently for each  zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  We assume that dimensionless parameters κ1 and κ2  (κ1 + κ2 = 1) define the fraction of the total energy  injected into zone one and two, respectively:  1  κi  ∞  �  Emin  dE Eqi(E) = L0 ,  (19)  where L0 is the total power injected in the production  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The minimum energy, Emin, we set to a value of  Γmec2 (here me is electron mass and c is the speed of  light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' While Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (18) defines the cutoff energy, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (19)  determines the normalization coefficients in each zone,  Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  As in the high-energy regime, the synchrotron and IC  losses are expected to provide the dominant energy loss  channels, we therefore only take account of these two en-  ergy loss mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Since the photon field is common  between the zones, the difference between the energy  loss rate in the zones is due to the different synchrotron  losses within each zone:  ˙Ei = ˙Esyn,i + ˙Eic .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (20)  The synchrotron energy losses in zone i are determined  by the magnetic field strength  ˙Esyn,i = −16π  3  e4E2  m4ec7  �2  3  B2  i  8π  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (21)  Note that the equation above is averaged over pitch an-  gle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The synchrotron cooling time is  τsyn,i =  E  | ˙Esyn,i|  ≈ 400  � Bi  1 G  �−2�  E  1 TeV  �−1  s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (22)  The IC losses are determined by the energy distribu-  tion and number density of target (synchrotron) pho-  tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' As photons can freely cross the zone boundaries  we assume that the photon distribution is the same  throughout the entire production region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', it includes  the contributions from both zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We compute the syn-  chrotron emission using the particle distribution in each  zone and the corresponding magnetic field:  dNsyn,i  dε dt  ≈  ∞  �  mec2  dE niKsyn,ε(E, Bi) ,  (23)  8  Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  where ε is the target (synchrotron) photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For  the synchrotron integral kernel, Ksyn,ε, we use a sim-  ple analytic approximation for the pitch angle averaged  synchrotron spectrum (for detail see in Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Finally, we compute the energy distribution of  the target photons as  dNsyn  dε dV ≈  1  R2c  �dNsyn,1  dε dt  + dNsyn,2  dε dt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (24)  Here R is size of the production region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The rate of IC scattering is determined by the angle  averaged scattering cross section (for detail see in Jones  1968):  dνic  dεγ dε = 8πcr2  0  bE  dNsyn  dV dε×  �  1 +  z2  2(1 − z) +  z  b(1 − z) −  2z2  b2(1 − z)2 −  z3  2b(1 − z)2 −  2z  b(1 − z) ln b(1 − z)  z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (25)  Here r0 = e2/mec2 is the electron classical radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' the  Klein-Nishina parameter is given by b = 4εE/(m2  ec4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  and z is the ratio of the up-scattered photon to electron  energy, z = εγ/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The IC energy loss rate depends on  the energy distribution of target photons as  ˙Eic ≈  ∞  �  0  dε  εmax,γ  �  εmin,γ  dεγ (ε − εγ) dνic  dεγ dε ,  (26)  where the maximum/minimum energy of up-scattered  gamma-ray, εmax/min,γ, is determined by kinematic con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' If electrons up-scatter low-energy target pho-  tons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', the Klein-Nishina parameter is small, b ≪ 1),  then the IC energy loss rate depends only on the energy  density of the target photons, wph:  ˙ET,i = −32π  9  e4E2  m4ec7 wph ,  (27)  analogous to the corresponding angle averaged energy  loss rate in a magnetic field given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Model calculations  For the model calculations, magnetic field values of  B1 = 1 G and B2 = 10−3 G are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The injection  power is set to ∼ 1039 erg s−1, and for the size of the  production region we consider a value close to 1016 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  If one considers this size in the context of a GRB after-  glow, one should compare it to the forward shock radius,  which depends on the time passed since the trigger, t′  tr:  R ∼ Γ2t′  trc ∼ 3 × 1016  � Γ  10  �2 t′  tr  3 h cm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (28)  The typical energy density of the target photons in the  production region is  wph ∼ 4×10−5κ1ηrad  �  R  3 × 1016 cm  �−2  erg cm−3 , (29)  where ηrad is the radiation efficiency in zone 1 (in what  follows we ignore this factor, setting ηrad = 1, for the  sake of simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This energy density corresponds to  an equivalent magnetic field strength of  Beq ∼ 3 × 10−2κ  1/2  1  �  R  3 × 1016 cm  �−1  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (30)  This photon field is the dominant target in zone 2,  whereas it is negligible in zone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The correspond-  ing cooling time scales are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 4 (top  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Whilst at high energies (approaching 1 PeV), the  Klein-Nishina losses approach their asymptotic energy-  dependence, τkn ∝ E, for the parameter set consid-  ered, the particles cool in the transition regime with  τ ∝ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Thus the spectrum formed is not as hard as  expected from our earlier qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The effect of the onset of Klein Nishina cooling on the  electron spectrum is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 4 (bottom panel),  where the energy distribution of electrons in both zones  are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  For the calculations here we adopted the  following parameters: linear size R = 1016 cm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' total  power of acceleration of non-thermal particles L0 =  1039erg s−1, which is distributed between the zones with  κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='90 and κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' the injection index α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='2  (the “main case”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Finally, the acceleration efficiency  was set to ηacc = 102, for which the cutoff energy in  zone 1 is determined to be:  Ecut,1 ≈ 6  �ηacc  102  �−1/2 � B1  1 G  �−1/2  TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (31)  For this acceleration efficiency the cutoff energy in zone  2 is at ≈ 200 TeV, which is close to the energy at which  the synchrotron losses dominate over the IC losses,  E∗ ≈ 20 TeV, thus the influence of the high energy cut-  off becomes prominent at energies just above the Klein-  Nishina hardening energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The energy dependence of the electron distribution  is directly reflected in the synchrotron spectrum from  zone 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 5, this component  is subdominant to the luminous synchrotron component  from zone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The photon index of the hardest part of the  spectrum is (α+1)/2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='5, which is considerably softer  than the limiting photon index of γs,kn(= α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This  is caused by the smooth broad transition to the Klein-  Nishina regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  While the broad transition from the  Thomson to Klein-Nishina regimes causes the electron  Hard VHE emission from SSC sources  9  distribution to be not as hard as naively expected, the  IC component appears to be somewhat harder than in  the limiting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 5, a power-law  component extends from a few GeV to beyond 10 TeV  with a photon index of ≈ (α + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Note that for our  calculations we set α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='2, and the production region  bulk Lorentz factor was assumed to be Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Top panel: Synchrotron, IC cooling time together  with the acceleration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Bottom panel: Electron distri-  bution in two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Black guide lines indicate power-law  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  To illustrate the influence of the model parameters, we  performed calculations for a range of different parame-  ter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The results of these calculations are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For the “case A” we adopted a different value for  the injection index: α = 2 instead of α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='2 used in the  “main case”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For the “case B” we adopted a different  value for the acceleration efficiency: ηacc = 104 instead  of ηacc = 102 used in the “main case”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The adopted  model parameter values are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Low-energy target photons can be an important role  in the formation of a hard VHE spectrum in the case  of the conventional one-zone SSC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' To demon-  strate the relatively small influence of low-energy target  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Spectral energy distribution of synchrotron and  IC emission from two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Black guide lines show the  power-law approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Spectral energy distribution of synchrotron and  IC emission from two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Black guide lines show the  power-law approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Top panel:  Case A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Bottom  panel: Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Thin lines in the top panel correspond to a  case when the electron distribution in the first zone features  a cooling break at E ≈ 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  photons in the framework of our considered two-zone  approach, in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 6 we also plot the  Synch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Zone 1  三  Synch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Zone 2  IC  1010  ITT  108  106  s  104  L  102  WT  100  Acceleration Zone1  Acceleration Zone2  10-2  102  10-1  100  101  103  104  105  106  E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=" GeVZone 1: particles (x103)  Zone 2: particles  56  α  10  TTTTT  1055  xE'α-1  1054  10-1  100  101  102  103  104  105  E," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' GeV46  0  L  Zone 1  (α + 2)/2  Zone 2  45  10  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' arbitrary units  44  10  43  (α+1)/2  3  42  C  Syn  41  10  l  100  10-6  10-2  102  104  10~8  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' GeV46  T  (α + 2)/2  Zone 1  Zone 2  45  0  (α+2)/2  x  +1)/2  (α+  43  10°  3  42  TTT  C  7  sVn  1  41  10-6  100  102  104  10-2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' GeV46  0  Zone 1  α + 2)/2  Zone 2  45  10  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' arbitrary units  44  10  43  10  (α+1)/2  3  42  10  IC  41  yn  1  E  Ll  100  10~2  102  104  10~8  10-6  10-4  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' GeV10  Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  SED obtained under the same conditions assuming that  the particle spectrum in the first zone features a cool-  ing break at E ≈ 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The corresponding spectra  are shown with thin lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Under this assumption, the  synchrotron spectrum from the first zone features the  cooling breaks, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The IC spectrum is more  strongly suppressed: one sees here the impact of both  the cooling break and reduced target photon density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The reduction of the IC loss rate leads to a consider-  able enhancement of the synchrotron emission from the  second zone (note that this component still remains sub-  dominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The IC spectra from the second zone shows,  however, only minor changes, noticeably only close to  the high- and low-energy cutoffs regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  This quite  weak influence of the target photon spectrum on the  spectral properties of the IC component from the second  zone is caused by the fact that the IC losses determine  the particle spectrum, as we assume that the emission  is generated in the fast cooling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Therefore, the  electron spectrum adjusts to the rate of the dominant  losses, and the spectral properties of the IC component  are largely determined by the injection spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSION  The need for studying energy losses in the inhomoge-  neous emission region downstream can be easily realized  by considering the evolution of the magnetic field from  the upstream to downstream regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Based on the hy-  drodynamics of the forward shock propagating through  the CBM, one can obtain the following estimate for the  downstream magnetic field strength:  B ∼ 3 × 102 Γ  10  Bcbm  10 µG µG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (32)  This estimate depends on the typical strength of the  CBM magnetic field, Bcbm, and accounts for the trans-  formation of this field to the forward shock rest frame,  and for the increase of the field strength at a weakly  magnetized relativistic shock due to the shock compres-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The magnetic field given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (32) appears signifi-  cantly below the Gauss-level required for the afterglow  radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Therefore, one needs to assume an efficient  magnetic field amplification process, which can increase  the energy density of the magnetic field to the level com-  parable to the plasma energy density in the downstream:  w ∼ ncbmmpΓ2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='15  � ncbm  1 cm3  �� Γ  10  �2  erg cm−3 , (33)  where ncmb is CBM density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This estimates shows that  the magnetic field in the downstream can be amplified  up to a strength of  Beq =  √  8πw ∼ 2  � ncbm  1 cm3  �1/2� Γ  10  �  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  (34)  Gauss-strength magnetic fields in the afterglow produc-  tion region are also favored on theoretical grounds by  afterglow emission modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  If the magnetic field is  indeed amplified by a factor of ∼ 103, it is natural to  further assume that this amplification is inhomogeneous  throughout the volume resulting in a magnetic field con-  figuration with strong spatial fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example,  magnetic field amplification by turbulent dynamo shows  that the magnetic energy is predominantly localized in  small blobs (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Moreover, this may be  a general effect: the field amplification predominately  operates on small scale fields (Kazantsev 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The highly inhomogeneous structure of the down-  stream region can have important implications for the  properties of the non-thermal emission generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  In  particular, such a structure in the production region  can significantly alter the synchrotron radiation emis-  sion, with clumps of highly amplified magnetic field  leading to the synchrotron emission extending signifi-  cantly beyond the one-zone synchrotron burn-off limit  (Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This scenario requires that  particles are accelerated in a region of weak magnetic  field, and subsequently penetrate into a second zone of  amplified magnetic field, where they rapidly cool pro-  ducing VHE synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The requirement  of effective particle exchange between the two zones of  strong and weak magnetic field is an important element  of this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  It should be noted, however, that efficient particle ex-  change between the zones is a significant assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Processes exist, which can hinder particle exchange be-  tween the two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' For example, if the change of the  magnetic field strength is relatively smooth, the mag-  netic adiabatic invariant prevents particles from the zone  of weak magnetic field reaching a strong magnetic field  zone (see the discussion in Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The  particle escape from the zone of strong to weak mag-  netic field is not forbidden by the magnetic adiabatic  invariant, but it seems feasible that one can neglect this  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Because of the much higher rate of synchrotron  losses in the strong magnetic field zone, the total number  of particles in this zone is naturally significantly reduced  to that in the weak magnetic field zone, particularly for  the highest energy particles with energies close to the  maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  On the other hand, synchrotron photons can freely  travel between the two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The photon exchange  between the zones have two major effects: (i) altering  Hard VHE emission from SSC sources  11  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Spectral energy distribution of synchrotron  and IC emission from two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Filled zones show X-ray  (XRT) and VHE intrinsic (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=') spectra (for detail see  in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  particle energy losses, and (ii) change the properties of  IC emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  We suggest a simple model that allows  one to study these two effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' We find that for feasi-  ble model parameters, IC scattering dominates the cool-  ing process in the zone of weak magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Due to  the Klein-Nishina effect, the particle spectrum formed  in the fast cooling regime appears to be significantly  harder than the spectrum formed for the case when syn-  chrotron losses dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' While the synchrotron emis-  sion from this zone may appear completely sub-luminous  with respect to the synchrotron emission generated in  the strong magnetic field zone, the IC component from  the weak magnetic field zone would be expected to dom-  inate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The second signature of the hard particle spectrum is  expected in the IC component generated by these parti-  cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This spectrum appears to be hard, with a photon  index coinciding with the value expected for the syn-  chrotron/Thomson spectra, (α + 2)/2 (where α is the  injection index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  The IC spectrum therefore appears  to have the same slope as the dominant synchrotron  emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The relative flux through these two channels  is determined by the phenomenological parameters, κi,  which determine the ratio of the acceleration powers in  the two zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Our simulations presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 7 show  that for an acceleration spectrum with a spectral slope  of α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='1, which allows the slope of the X-ray spec-  tra for GRB190829A to be reproduced, the X-ray and  VHE flux ratio seen in GRB190829A implies κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='7  and κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='3 (see column “GRB190829A” in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  This suggests that acceleration processes of comparable  power operate in the both zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' However, the acceler-  ation in the zone of stronger magnetic field is somewhat  more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' The obtained hard VHE IC spectrum ex-  tends beyond 10 TeV for a modest bulk Lorentz factor  of Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' This implies that a hard multi-TeV IC com-  ponent can be generated also during the late afterglow  phases, when the forward shock transits into the mildly  relativistic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' During the prompt or early afterglow  phases, when the bulk Lorentz factor can be significantly  larger, Γ ≥ 100, the intrinsically hard IC component can  extend up to the ultra high energy domain (≥ 100 TeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  However, we note that the extragalactic EBL attenua-  tion is severe already in the VHE domain for even the  most local GRB redshift values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  DK acknowledges support by the RSF grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 21-12-  00416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
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+page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
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+page_content='237  Derishev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
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+page_content='3847/1538-4357/ab536a  Derishev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', & Piran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 2019, ApJL, 880, L27,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='3847/2041-8213/ab2d8a  10-10  S  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='12  2  10  Total  Zone 1  Zone 2  10-13  3  GRB190829A  IC  Syn  10-14  l  10-8  10-6  10-2  100  102  104  , GeV12  Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Used notations  Parameter  Notation  Comment  Production region size  R  Similar to the forward shock radius  Magnetic field  B  B1,2 in zone 1 or 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Bcbm in the shock upstream  Bulk Lorentz factor  Γ  We do not distinguish the bulk Lorentz factors of emitting plasma or forward shock  Injection power  L0  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (19) is in the co-moving frame, but note that L0 is a Lorentz invariant  Normalization factors  κ1,2  Power distribution between two zone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' κ1 + κ2 = 1  Energy density  w  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', wext is energy density of external photon fields  Electron energy  E  –  Electron energy density  n  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', dN = n dE  Injection index  α  see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 17  Injection rate  q  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', dN = q dE dt  Acceleration efficiency  ηacc  see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' (18)  Cooling time  τ  Synchrotron, IC, and adiabatic cooling or escape  Energy loss rate  ˙E  Synchrotron, IC, and adiabatic  Photon energy  ε  For photons produced through the synchrotron or IC channels  Photon index  γ  In particular, γs, γkn  Radiation efficiency  ηrad  We assume ηrad = 1  Circumburst medium density  ncmb  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' dN cmb = ncmb dV  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' Parameter values used for model calculations  Parameter  Notation  units  Main case  Case A  Case B  GRB190829A  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 4&5  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 6  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 6  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content=' 7  Size  R  cm  1016  1016  1016  1016  Bulk Lorentz factor  Γ  –  10  10  10  10  Luminosity  L0  erg s−1  1039  1039  1039  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='2 × 1038  Strong magnetic field  B1  G  1  1  1  1  Weak magnetic field  B2  G  10−3  10−3  10−3  10−3  Zone 1 power fraction  κ1  –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='7  Zone 2 power fraction  κ2  –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='3  Injection slope  α  –  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
+page_content='2  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFAT4oBgHgl3EQfex2-/content/2301.08578v1.pdf'}
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+ARXIV VERSION UPLOADED: JANUARY 09, 2023
+1
+Non-contact Respiratory Anomaly Detection using
+Infrared Light Wave Sensing
+Md Zobaer Islam, Brenden Martin, Carly Gotcher, Tyler Martinez, John F. O’Hara, Senior Member, IEEE,
+and Sabit Ekin, Senior Member, IEEE
+Abstract—Human respiratory rate and its pattern convey
+important information about the physical and psychological
+states of the subject. Abnormal breathing can be a sign of fatal
+health issues which may lead to further diagnosis and treatment.
+Wireless light wave sensing (LWS) using incoherent infrared light
+turns out to be promising in human breathing monitoring in
+a safe, discreet, efficient and non-invasive way without raising
+any privacy concerns. The regular breathing patterns of each
+individual are unique, hence the respiration monitoring system
+needs to learn the subject’s usual pattern in order to raise flags for
+breathing anomalies. Additionally, the system needs to be capable
+of validating that the collected data is a breathing waveform,
+since any faulty data generated due to external interruption or
+system malfunction should be discarded. In order to serve both of
+these needs, breathing data of normal and abnormal breathing
+were collected using infrared light wave sensing technology in
+this study. Two machine learning algorithms, decision tree and
+random forest, were applied to detect breathing anomalies and
+faulty data. Finally, model performance was evaluated using
+average classification accuracies found through cross-validation.
+The highest classification accuracy of 96.6% was achieved with
+the data collected at 0.5 m distance using decision tree model.
+Ensemble models like random forest were found to perform better
+than a single model in classifying the data that were collected at
+multiple distances from the light wave sensing setup.
+Index Terms—Non-contact vitals monitoring, respiration mon-
+itoring, light wave sensing, machine learning, anomaly detection,
+data classification.
+I. INTRODUCTION
+P
+HYSICAL and psychological states of human beings are
+reflected in their respiratory patterns which can be moni-
+tored to raise general health-consciousness and generate alarm
+for anomalous breathing before more serious health issues
+occur. Anomalous breathing can occur due to various respi-
+ratory illnesses like asthma, obstructive sleep apnea, chronic
+bronchitis, chronic obstructive pulmonary disease (COPD),
+emphysema and COVID-19. Also, it can be a symptom of
+unstable mental conditions including stress, panic, anxiety,
+fatigue, anger and so on. Doctors evaluate the breathing quality
+This work was supported by the National Science Foundation under Grant
+2008556. (Corresponding author: Md Zobaer Islam, Sabit Ekin.)
+Md Zobaer Islam, Brenden Martin, Carly Gotcher, Tyler Martinez, and John
+F. O’Hara are with the School of Electrical and Computer Engineering, Okla-
+homa State University, Oklahoma, USA (e-mail: zobaer.islam, brenden.martin,
+carly.gotcher, tyler.martinez, oharaj {@okstate.edu})
+Sabit Ekin is with the Department of Engineering Technology and Industrial
+Distribution, Texas A&M University, College Station, Texas, USA (e-mail:
+sabitekin@tamu.edu)
+This work has been submitted to the IEEE for possible publication.
+Copyright may be transferred without notice, after which this version may
+no longer be accessible.
+of patients most often by manual observation which may
+generate wrong perception due to its subjective nature. This
+approach is not suitable for continuous monitoring either.
+Automated breathing monitoring solutions are rarely utilized
+and they mostly use contact-based or wearable sensors. When
+a person is aware that their breathing is being monitored, they
+may undergo a shift in breathing pattern from their typical
+signature. Moreover, the subject can be too young (infants at
+neonatal intensive care unit or NICU) or too ill (burn unit
+patients) to apply contact-based approaches [1]. For patients
+with contagious diseases like COVID-19, any contact-based
+respiration monitoring can spread the disease further. Hence,
+there is a need for non-contact, discreet respiration monitoring.
+Research and development on non-contact vitals monitoring
+using electromagnetic (EM) signals have been ongoing for
+many years. Received signal strength (RSS) and channel
+state information (CSI) of WiFi networks were utilized for
+human breathing detection, estimation and monitoring [2]–
+[5]. Remote monitoring of breathing has also been studied
+using different types of radar including microwave bands [6],
+[7], mmWave [8], ultra-wideband (UWB) [9], [10], frequency-
+modulated continuous wave (FMCW) [11], [12], step fre-
+quency continuous wave (SFCW) [13] etc. These radio-
+frequency (RF) based respiration monitoring approaches are
+prone to electromagnetic interference with EM signals from
+nearby devices [14], [15]. Also, continuous exposure to RF
+signals can be detrimental to the human body [16], [17]. The
+WiFi-based approaches usually require a receiver device held
+at chest, hence they are not fully contactless methods [2].
+Besides RF signal based approaches, researchers also exploited
+recorded videos or images from RGB cameras [18]–[25]
+and thermal infrared cameras [26]–[32] to extract breathing
+information. These approaches may raise privacy concerns
+among users because the subjects’ image data is discreetly
+captured. Also, post-processing of video and image data is
+computationally more expensive than one dimensional time
+series data [33].
+Non-contact vitals monitoring using light wave sensing may
+prove superior to existing technologies because of the safe,
+ubiquitous and harmless nature of light as well as the absence
+of privacy issues as there are with camera-based approaches.
+Non-contact vitals monitoring using visible light has already
+been performed as a proof of concept that showed >94%
+accuracy in breathing rate measurement in comparison to FDA
+approved contact-based counterpart [33], [34]. But visible light
+can be troublesome to subjects in dark environments, espe-
+cially during sleep, hence more subtle means are preferred. In
+arXiv:2301.03713v1  [eess.SP]  9 Jan 2023
+
+ARXIV VERSION UPLOADED: JANUARY 09, 2023
+2
+the current study, respiration has been monitored using widely
+available incoherent infrared (IR) light in a discreet way (since
+IR light is not visible to the naked human eye) and machine
+learning algorithms were applied on the collected data to detect
+breathing anomalies. The main contributions of this project are
+summarized below:
+1) Development of a novel IR based light wave sensing sys-
+tem model for human respiration monitoring and anomaly
+detection.
+2) Breathing data collection using the developed system in a
+controlled environment, with precision and repeatability.
+3) Handcrafted feature extraction from the collected breath-
+ing data for classifying them.
+4) Smart detection of anomalous breathing and faulty data
+from the collected breathing data using machine learning
+models.
+The remainder of this manuscript is organized as follows.
+Section II includes an overview of related works on the
+application of machine learning to breathing data collected
+by different technologies. Section III describes various human
+breathing patterns from the literature to be used as breathing
+classes for anomaly detection. Section IV presents the system
+model, relevant theory, hardware setup and the overall process
+to detect breathing anomalies. Section V presents the exper-
+imental evaluation of the approach described in Section IV.
+That includes details of data collection and processing, feature
+extraction, data classification and the results. Finally, Sec-
+tion VI presents the conclusions drawn from the whole effort
+and forecasts future research directions.
+II. RELATED WORKS
+Researchers have been applying machine learning and arti-
+ficial intelligence on human breathing data collected through
+various technologies, both contact-based and non-contact,
+for multiple applications like posture detection [35], [36],
+identity authentication [37]–[39], activity classification [40],
+stress classification [41], exercise detection [42], breathing
+anomaly detection [12], [43] etc. Most of these efforts made
+use of handcrafted features which were mostly dependent on
+their system model and hardware setup [44]. Some of the
+common categories of features that were used for breathing
+data classification in the literature were statistical features from
+the data (mean, standard deviation, skewness, kurtosis, root
+mean-square value, range etc.), signal-processing based fea-
+tures (Fourier co-efficients, autoregressive integrated moving
+average co-efficients, wavelet decomposition, mel-frequency
+cepstral coefficients, linear predictive coding etc.), and respi-
+ration related features (breathing rate, amplitude, inspiratory
+time, expiratory time etc.) [12], [26], [35], [41]–[47]. In some
+research efforts, convolutional neural networks were trained to
+recognize subtle features from breathing data (2-dimensional
+image or 1-dimensional time series data) thus making manual
+feature extraction redundant [48]–[52].
+Some past classification efforts involved one-class classifi-
+cation or outlier detection, as in [42] where the model was
+trained using human breathing data in resting condition to
+predict if the person was exercising in new examples. Binary
+classification between normal breathing and apnea were per-
+formed in [43] to detect obstructive sleep apnea. Multiclass
+breathing classification efforts considered different types of
+breathing anomalies like tachypnea, bradypnea, hyperpnea,
+hypopnea etc. and sometimes more complicated anomalies like
+Cheyne-Stoke’s, Biot’s and Apneustic breathing as separate
+classes [26], [44], [49], [50], [53]. These breathing patterns are
+explained in Section III. Data for these efforts were collected
+from human subjects who are generally unable to breathe
+using precise frequency, amplitude and pattern. Sometimes,
+breathing data from patients with breathing disorder were
+used which had limitations too because even the patients
+might not breathe in consistent abnormal pattern all the time.
+In the current study, more reliable data were generated by
+using a programmable robot with precise human-like breath-
+ing capability. The machine learning techniques used in the
+literature for classifying breathing data were decision tree,
+random forest, support vector machine, XGBoost, K-nearest
+neighbours, feedforward neural network, logistic regression,
+ensemble learning etc. Model performances have been anal-
+ysed in these works by using confusion matrices, K-fold cross-
+validation, accuracy, precision, sensitivity or recall, specificity,
+F1-score and so on [12], [26], [35], [37]–[41], [43], [44], [46],
+[48].
+III. HUMAN BREATHING PATTERNS
+Various types of human breathing patterns (both normal and
+abnormal) have been identified from the literature. The major
+7 types of human breathing are described as follows:
+1) Eupnea: This is regular human breathing with a uniform
+depth, rate and pattern. The depths and rates that are
+considered as normal vary according to the age and
+activity level of human being. For adult people, the
+regular breathing rate is 12-20 BPM (breaths per minute)
+at resting conditions [12], [44], [54], [55]. Breathing
+depth is measured from rib cage movement and it is
+expressed as a percentage of the maximum movement
+of the rib cage. For healthy adults of 20-39 years, the
+breathing depth was experimentally found to be 44±14%
+or 30-58% of the maximum rib cage movement in [56].
+2) Apnea: Temporary cessation of breathing is known as
+apnea [54], [57]. It often occurs during sleep which is
+known as sleep apnea.
+3) Tachypnea: This is an anomalous breathing condition
+where the human breaths faster than usual [26], [45], [54],
+[57]–[59]. When the breathing rate becomes higher than
+20 BPM, then it can be considered as Tachypnea [44],
+[53]. Sometimes, the threshold frequency for Tachypnea
+is taken to be 25 BPM [60] or 30 BPM [57] too.
+4) Bradypnea: Bradypnea is defined as slow breathing [12],
+[26], [53], [54], [57]. So, any breathing rate less than
+12 BPM can be considered as Bradypnea [44].
+5) Hyperpnea: Hyperpnea is a breathing pattern with in-
+creased depth of breathing at normal rate [12], [58].
+6) Hypopnea: In hypopnea, the breathing becomes shallow
+with at least 50% decrease in the regular air flow volume
+for ≥10 seconds [43], [61], [62].
+
+ARXIV VERSION UPLOADED: JANUARY 09, 2023
+3
+7) Kussmaul’s breathing: When tachypnea and hyperpnea
+occur together and thus the breathing becomes rapid, deep
+and labored, then it is known as Kussmaul’s breathing.
+So, in Kussmaul’s breathing, both of breathing depth
+and rate will be higher than those of Eupnea or normal
+breathing [45], [53]–[55], [58].
+The above 7 types of breathing will be addressed in this
+study. There are a few other more complicated breathing pat-
+terns too like Cheyne-Stoke’s, Biot’s and apneustic breathing
+which are composed of periods of these 7 base classes [12],
+[44], [45], [53], [55], [63], [64].
+Fig. 1: Proposed system model for respiration monitoring.
+IV. SYSTEM DESIGN AND IMPLEMENTATION
+A. Theory of Operation
+The proposed system model for non-contact respiration
+monitoring using IR sensing and anomaly detection is depicted
+in Fig. 1. In this model, modulated IR light is sent towards the
+chest of a human being. Light propagation from light emitting
+diodes (LED) follows Lambertian propagation model [65],
+[66]. According to this model, if Pt is the optical power
+transmitted from a point source, then power at distance d,
+is given by
+Pd = (n + 1)APt
+2πdγ
+cosn(φ) cos(θ), ∀ θ < φ1/2,
+(1)
+where A is the area intercepted, γ is the empirical path-loss
+exponent, and φ and θ are irradiance and incident angles,
+respectively [33]. φ1/2 is the half-power angle in the field
+of view of the light source. n is the order of the Lambertian
+model and is given by n =
+− ln(2)
+ln{cos(φ 1
+2 )}.
+Propagated optical power intercepts the subject’s chest, part
+of which is absorbed by the clothing material while the rest is
+further modulated by the chest movement due to inhaling and
+exhaling and is scattered back to the photodetector. If Pr is the
+scattered optical power received by the photodetector, Rpd is
+the responsivity and id is the dark current of the photodetector,
+then the generated photocurrent ipd = id + RpdPr [67]. The
+transimpedance amplifier in the photodetector converts this
+current into voltage
+Vsig = gpd (id + RpdPr) ,
+(2)
+where gpd is the transimpedance gain.
+This signal is contaminated by noise from the photodetector
+and the environment. When the signal of interest is buried
+in noise, lock-in detection technique can be used to detect
+the signal accurately. Hence, the output voltage from the
+photodetector (Vsig) is fed to the lock-in amplifier [68] as
+input. If Vsig is a sinusoidal wave with only one frequency ωs
+and phase φs i.e. Vsig = Vs sin(ωst + φs) and the reference
+input to the lock-in amplifier is a periodic signal of the same
+frequency ωs and amplitude Vr, then the output magnitude of
+the lock-in amplifier can be shown to be R = 1
+2VsVr, which
+is proportional to Vs [69]. R can be scaled to the desired level
+by varying the sensitivity S = Vfs
+G
+of the lock-in amplifier,
+where Vfs is the full-scale voltage (generally 10V) and G is
+the overall gain of the lock-in amplifier, to produce
+Rscaled = VfsVr
+2S
+Vs.
+(3)
+In practice, the signal Vsig is distorted by high frequency
+noises which will be attenuated by the low pass filters inside
+the lock-in amplifier1. The output signal of the lock-in ampli-
+fier is collected, stored and processed for feature extraction.
+The extracted features are fed to machine learning models for
+breathing anomaly detection.
+B. Experimental Setup
+To collect respiration data in a controlled, precise and
+repeatable way, a robot was developed to emulate human
+respiration using 3-D printed structures and two servo motors.
+The breathing rate, pattern, and depth could be set using a
+Raspberry Pi. External and internal views of the robot are
+shown in Fig. 2a and 2b. The robot could be controlled by one
+of two programs written in Python. The first program allowed
+the robot to breathe following a list of predefined waveforms
+including |sin|, sin2, sin4 and sin6 patterns. The breathing
+rate could be varied from 0 to 50 BPM with maximum
+depth ranging from 0 to 30 mm (denoted by 0% to 100%
+in the program). Also, an offset could be provided to set the
+initial position of the chest. Fig. 2c presents the graphical
+user interface of this software. A secondary control software
+could actuate arbitrary waveforms from a text file generated
+by MATLAB code.
+A light wave sensing system was developed for collecting
+respiration data using infrared light. An IRLED matrix source
+(FY-48 940 nm IR lamp board [70]) consisting of 48 IRLEDs
+was used as the light source. This source gives the highest
+intensity of light when it is connected to ≈12 V. The light
+source was modulated by a 1 kHz sinusoidal voltage wave
+of 3.8 V peak-to-peak amplitude and 8.1 V DC offset from
+a Keysight 33500B function generator [71]. The voltage am-
+plitude and DC offset were chosen as such to keep the light
+intensity high enough (by applying 11.9 V peak voltage), while
+1The noise frequencies that are very close to the reference frequency (ωs)
+may exist in the filtered signal depending on the filter bandwidth and roll-off.
+Filter bandwidth can be adjusted by varying the time constant, τ =
+1
+2πfc of
+the lock-in amplifier, where fc is the 3-dB cut-off frequency of the low pass
+filter. A narrowband filter will suppress most of the noises while keeping the
+signal to be detected intact as a true DC component
+
+Distance
+Warning if
+anomaly
+source
+detected
+Inhale
+Classification
+Exhale
+Photodetector
+using ML
+Enlarged view of the
+respiration monitoring device
+Lock-in
+Data
+Data
+Feature
+detection
+collection
+processing
+extractionARXIV VERSION UPLOADED: JANUARY 09, 2023
+4
+(a) External view.
+(b) Internal view.
+(c) User interface
+Fig. 2: External and internal views and the graphical user
+interface of the robot.
+maintaining linear LED operation. For collecting the reflected
+light from the chest of the breathing robot, Thorlabs PDA100A
+photodetector [72] with a converging lens of 25.4 mm focal
+length was used. The photodetector comprised of a p-i-n
+photodiode and transimpedance amplifier with adjustable gain.
+For lock-in detection, SR830DSP frequency lock-in ampli-
+fier [73] was used. Finally, for data collection and storage,
+a second Raspberry Pi preceded by analog to digital converter
+(DAQC2Piplate ADC [74]) was used. A diagram of the setup
+used to generate and collect breathing data is shown in Fig. 3.
+Also, a picture of the overall setup is shown in Fig. 4.
+C. Anomaly Detection Process
+Breathing anomaly detection was performed using machine
+learning based classification of labeled breathing data collected
+using the developed IR sensing setup. Seven breathing patterns
+described in Section III were used as seven different data
+classes. After collecting breathing data, the system must decide
+whether the current data is useful or not, because faulty data
+might be collected due to the movement of the subject or
+other people at close proximity, too much noise from the
+surrounding or any system malfunction. To detect and discard
+or recollect such data, a separate data class called ‘faulty
+data’ was used. Therefore, the breathing anomaly detection
+problem was reduced to an eight-class classification task. The
+classes were eupnea, apnea, tachypnea, bradypnea, hyperpnea,
+hypopnea, Kussmaul’s breathing and faulty data. The general
+process followed for anomaly detection will be described here.
+Specific details will be provided in Section V.
+Breathing data belonging to the eight data classes mentioned
+above were collected in a controlled environment using the
+robot and the IR sensing setup. The collected data x[m] con-
+tained noises from the environment and the setup itself. The
+noise was filtered using k-points moving average technique
+using the following equation:
+yk[n] =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+k
+n+k−1
+�
+m=n
+x[m],
+if k′ ≥ k,
+1
+k′
+n+k′−1
+�
+m=n
+x[m],
+otherwise,
+(4)
+where k′ is the number of points in the data from the current
+data point x[n] to the end.
+Some data had a drift or trend in them which added
+DC component to the data and increased the peak-to-peak
+amplitude. Thus, those data were prone to underestimating
+breathing rate and overestimating breathing depth. To get
+around this problem, detrending was performed by subtracting
+a p-th order polynomial
+wp[n] = c1xp + c2xp−1 + ... + cnx + cn+1
+(5)
+from the data to produce
+z[n] = yk[n] − wp[n],
+(6)
+where c1, c2, .., cn+1 were the co-efficients of the polynomial
+wp[n] that ensured the best least-squares fit [75] to the data
+yk[n]. When p = 0, it becomes equivalent to subtracting the
+overall mean value from the data. The breathing data had
+higher order trends, so p > 0 was used.
+After detrending, the following four handcrafted features
+were extracted from the data:
+1) Peak-to-peak Amplitude: If the maximum and minimum
+values of the data z[n] are zmax and zmin, then, peak-to-peak
+amplitude A = zmax − zmin. It roughly represents a number
+proportional to the breathing depth in each class (except the
+faulty data class), when the test distance remains constant.
+When the test distance was changed, the value of this feature
+was shifted to a different range even though the data belonged
+to the same class, because of changed levels of received light
+intensity and noise.
+2) Breathing Rate: Breathing rate is the frequency with the
+highest spectral amplitude in the frequency domain represen-
+tation of the signal (again, it is not applicable in apnea and
+faulty data classes). Discrete Fourier Transform (DFT) of the
+time domain data z[n] is taken using
+Z[k] =
+L−1
+�
+n=0
+z[n]e
+−j2πkn
+L
+,
+(7)
+where L is the length of the dataset and its DFT. Z[k] contains
+different frequency components present in the signal z[n]. The
+frequency index with the maximum magnitude is found to be
+kmax = arg max
+k
+|Z[k]|.
+(8)
+
+三tk
+X
+Offset
+Amp
+BPM
+0
+60
+14
+Isin]
+Start 
+Update
+Basic Operation Insturctions:
+Press start to begin robot motion.
+Start also works as a pause button. The currently selected values will be read in when start is pressed
+Update will read in the currently selected values without pausing.
+The robot will reset to the selected offset when updating or starting/unpausing before beginning the selected motion.
+Operation Warning:
+Make sure the Amp and offset sliders do not add to higher than 1oo when reading into the calculations.
+The program will give an error message and pause until valid values are passed in.
+OtherUseful Info:
+BPM stands for breaths per minute.
+Amp and offset are input as a percentage of the maximum range of motion.
+The maximum range of motion is approximately 3cmARXIV VERSION UPLOADED: JANUARY 09, 2023
+5
+Fig. 3: The experimental setup diagram of the overall system used for data collection.
+Fig. 4: The hardware setup of the overall system used for data
+collection.
+Then, the corresponding frequency or the breathing rate,
+fmax, is calculated using
+fmax = kmaxfs
+L
+Hz = 60kmaxfs
+L
+BPM,
+(9)
+where fs is the sampling frequency used to collect data.
+Breathing rate stays in the same range for data collected at
+all distances. In contrast, error in breathing rate estimation
+increases with increasing distance because of increasing noise
+level.
+Fig. 5: Frequency domain plots of sample processed data from
+class 1 and 2.
+3) Effective Spectral Amplitude: It is the percentage count
+of frequencies in the spectrum of data after detrending whose
+spectral amplitude is greater than or equal to a predefined
+threshold value. The threshold is determined as a percentage
+of peak spectral amplitude. If the peak magnitude in the data
+spectrum is H, t is the percentage threshold, and A is the
+spectrum vector, then effective spectral amplitude
+S = l(B)
+l(A) × 100%, B = {x ∈ A|x ≥ tH},
+(10)
+where l() represents the length of its parameter vector. This
+number helps differentiating between a very clean spectrum
+with just a few large peaks (which are the breathing rate and
+its harmonics) and a spectrum with multiple peaks of similar
+magnitude at different frequencies.
+As an example, in the first subplot of Fig. 5, the spectrum
+of one data from Apnea class has the highest peak at 23 BPM
+which happened by chance. It has many other peaks of
+
+Keysight 33500B function generator
+SIOHT335eOB SId
+BNC to BNC cable
+BNC to BNC cable
+IRLED array
+0.00
+00001
+0000
+0
+Disply
+口
+0口
+SR830 Lock-in amplifier
+Distance
+BNC to grabber cable
+BNC to BNC cable
+Thorlabs
+photodetector
+Breathing robot
+(Controlled by a Raspberry Pi)
+Raspberry Pi
+DAQC2Piplate ADC
+(For data collection)IRlightsource&
+Photodetector
+Function
+generator
+Lock-in
+amplifier
+Rpi for data
+collection
+Powersupply
+units
+Breathing robotClass = 1 (Apnea), Distance = 1 m
+×10-4
+X 23
+Y0.000195041
+Spectral Amplitude (V)
+2
+Peak magnitude
+0.2*(Peak magnitude)
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Frequency(BPM)
+Class = 2 (Tachypnea), Frequency = 23 BPM, Depth = 32%, Distance = 1 m
+X23
+Peak magnitude
+0.01
+nplitude
+Y0.0113069
+0.008
+Am
+0.006
+ctr
+0.2*(Peak magnitude)
+0.002
+S
+A
+0
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Frequency (BPM)ARXIV VERSION UPLOADED: JANUARY 09, 2023
+6
+magnitude greater than the threshold (marked by red line).
+Hence, it is not representing breathing at 23 BPM which can
+be understood from its larger effective spectral amplitude
+(S = 51). An actual breathing data of Tachypnea class with
+breathing frequency 23 BPM has a cleaner peak at 23 BPM in
+its spectrum with S = 2, as shown in the second subplot of
+the same figure. The threshold is assumed to be 20% of the
+peak magnitude in both cases.
+Fig. 6: Flow diagram of the approach towards the breathing
+anomaly detection problem.
+4) Signal to Noise Ratio: Average signal to noise ratio
+(SNR) of a signal is an indicator of signal quality and is
+defined as the ratio of average signal power and average noise
+power. The photodector was a power detector which detected
+the signal power and noise power together. If z[n] is the mean-
+subtracted data received by the photodetector at a particular
+distance and xnoise[n] is the mean-subtracted noise data at that
+distance, then SNR in decibel can approximately be defined
+by
+SNRdB = 20 log10
+�
+�
+�
+�
+�
+�
+�
+N
+�
+n=1
+|z[n]|2
+�
+N
+�
+n=1
+|xnoise[n]|2
+�
+�
+�
+�
+�
+�
+,
+(11)
+considering root-mean-square averages of the signals. Here,
+both of z[n] and xnoise[n] had the same length N.
+SNR value will decrease with increasing distance in the
+same data class because of increasing level of noise. Also,
+SNR will decrease when the breathing depth is decreased, as
+in class 3 (Bradypnea), because of lower signal power.
+With these four handcrafted features, data were classified
+using decision tree and random forest models. These models
+are introduced in Section IV-D. In order to see the effect of
+distance on classification accuracy, classification was initially
+done separately on data collected at one particular distance
+at a time. But in realistic environments, users may not be at
+a particular distance every time. Hence, mixed sets of data
+collected at different distances were used later for feature
+extraction and classification. The overall plan for breathing
+anomaly detection is depicted in Fig. 6.
+D. Machine Learning Models
+Two renowned machine learning models - decision tree and
+random forest - were used for data classification in the process
+of anomaly detection. The data classes, defined in Table I,
+were differentiated by two categorical variables - breathing
+rate and breathing depth - which can be represented by a
+structure like decision tree. Random forest is a collection of
+multiple uncorrelated decision trees. Hence, these two models
+did the classification better than the other machine learning
+models with the breathing dataset.
+1) Decision Tree: Decision tree is one of the most effective
+non-parametric supervised machine learning models used for
+classification, prediction and data mining. This model starts
+the classification process by partitioning the dataset into two or
+more mutually exclusive subsets based on the values of a root
+node, then continues partitioning through internal nodes and
+ends at leaf nodes. Thus, it forms a tree-shaped graph overall.
+One of the popular algorithms to create binary decision trees
+(i.e. tree with exactly two branches at each node) is CART
+(classification and regression trees) algorithm [76], [77] which
+is used in Scikit-learn [78], [79], a machine learning library
+of Python. To describe this algorithm, say, the split at node m
+of the decision tree is done based on parameter θ = (fj, tm),
+where fj is the j-th feature and tm is the threshold considered
+for that feature. At node m, feature fj is tested against tm to
+partition the dataset Xm into subsets
+Xleft
+m
+(θ) = {(x, y) |fj ≤ tm}
+(12)
+and
+Xright
+m
+(θ) = {(x, y) |fj > tm} = Xm\Xleft
+m
+(θ),
+(13)
+where x is the training data instance and y is the corresponding
+label. If nm is the number of data at node m, then the quality
+of the split is calculated as follows using the information gain
+IG(Xm) = I(Xm) − G(Xm, θ),
+(14)
+where I() is the impurity function and G(Xm, θ) is a measure
+of overall impurity after the split defined as
+G(Xm, θ) = nleft
+m
+nm
+I(Xleft
+m
+(θ)) + nright
+m
+nm
+I(Xright
+m
+(θ)). (15)
+Common measures of impurity functions I() for construct-
+ing the decision tree are Gini impurity
+I(Xm) =
+K
+�
+i=1
+pi,m(1 − pi,m) = 1 −
+K
+�
+i=1
+(pi,m)2
+(16)
+
+Raw data
+collection
+Moving average
+filtering
+Mean
+Detrending
+subtraction
+Feature 1
+Feature 2
+Feature 3
+Feature 4
+Breathing
+Peak to peak
+Effective spectral
+Signal to
+amplitude
+rate
+amplitude
+noise ratio
+Classification
+Classification
+using Decision
+using Random
+Tree
+ForestARXIV VERSION UPLOADED: JANUARY 09, 2023
+7
+TABLE I: Characteristics and number of data for each class
+Class
+Class name
+Breathing rate
+(BPM)
+Breathing
+depth (%)
+Number of data collected
+Total
+at 0.5 m
+at 1 m
+at 1.5 m
+0
+Eupnea
+12-20
+30-58
+50
+50
+50
+150
+1
+Apnea
+0
+0
+50
+50
+50
+150
+2
+Tachypnea
+21-50
+30-58
+50
+50
+50
+150
+3
+Bradypnea
+1-11
+30-58
+50
+50
+50
+150
+4
+Hyperpnea
+12-20
+59-100
+50
+50
+50
+150
+5
+Hypopnea
+12-20
+1-29
+50
+50
+50
+150
+6
+Kussmaul’s
+21-50
+59-100
+50
+50
+50
+150
+7
+Faulty data
+Any
+Any
+50
+50
+50
+150
+Total
+400
+400
+400
+1200
+Fig. 7: Time domain representation of sample data from each class (the distance between the source-photodetector and the
+robot was 1 m).
+and log loss or Shannon entropy
+I(Xm) = −
+K
+�
+i=1
+pi,m log2(pi,m),
+(17)
+where K is the number of classes, i ∈ {1, 2, 3, ..., K} and
+pi,m = ni,m/nm is the proportion of the observations that
+belong to class i at node m. Gini impurity is used as the
+impurity function in CART algorithm.
+The feature-threshold pair denoted by θ∗ that maximizes
+the information gain, IG(Xm) or equivalently minimizes
+G(Xm, θ) is chosen for splitting the data at node m using
+θ∗ = arg min
+θ
+{G(Xm, θ)}.
+(18)
+Subsets Xleft
+m
+and Xright
+m
+are used to repeat the process
+until the maximum allowable depth (distance from the root
+node to the farthest leaf node) is reached, or nm becomes
+less than the pre-defined minimum number of samples at each
+node, or any other stopping criterion is met. If the tree is not
+stopped early, then the splitting continues until nm = 1.
+Decision trees are highly prone to overfitting. Overfitting is
+a situation when the model yields very high training accuracy
+by learning some particular training dataset extremely well,
+but fails to learn the general trend in the training data. Hence,
+the model becomes susceptible to perform poorly on unseen
+test data. To prevent a decision tree from overfitting, it needs
+to be regularised by stopping it early before it learns the
+noise prevalent in the training data. One of the methods of
+regularising the decision tree model is to limit the maximum
+depth of the tree. The optimum value of the maximum depth
+can be decided by testing trees with different depths on
+validation dataset [80], a process known as hyperparameter
+
+Class=0(Eupnea),Frequency=18BPM,Depth=43%
+Class=1(Apnea),Frequency=0BPM,Depth=0%
+M
+M
+0.1
+0.1
+Voltage (
+0.05
+Voltag
+0
+-0.05
+-0.05 
+0
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+40
+50
+60
+Time (s)
+Time (s)
+Class=2(Tachypnea),Frequency=35BPM,Depth=44%
+Class=3(Bradypnea),Frequency=8BPM,Depth=47%
+(V)
+M
+0.1
+0.1
+ge
+0.05
+AAAAAAA
+Voltag
+-0.05
+-0.05
+0
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+40
+50
+60
+Time (s)
+Time (s)
+Class = 4 (Hyperpnea), Frequency = 17 BPM, Depth = 91%
+Class=5(Hypopnea),Frequency=13BPM,Depth=2o%
+M
+M
+0.1
+0.1
+0.05
+Voltag
+0
+0
+-0.05
+-0.05
+0
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+40
+50
+60
+Time (s)
+Time (s)
+Class = 6 (Kussmaul's), Frequency = 26 BPM, Depth = 89%
+Class = 7 (Faulty Data)
+(V)
+M 0.1
+0.1
+0.05
+Itage
+0
+0
+-0.1
+-0.05
+-0.2
+0
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+40
+50
+60
+Time (s)
+Time (s)ARXIV VERSION UPLOADED: JANUARY 09, 2023
+8
+tuning.
+2) Random Forest: The learning technique related to build-
+ing multiple machine learning models and taking majority
+vote (for classification) or an average (for regression) of the
+predictions is referred to as ensemble learning [81]. It prevents
+the overall model from overfitting by reducing the variance
+in model prediction without affecting the bias much. Random
+forest is the ensemble version of decision tree model. It makes
+use of Bootstrap Aggregating or Bagging technique [82] in
+the following two ways to build multiple uncorrelated decision
+trees and finally takes majority vote to predict the class labels:
+(i) If the dataset is x1, x2, ..., xn, then random forest takes
+random samples of size n with replacement from this
+dataset m times and builds m different decision trees
+with them parallelly.
+(ii) If the features defining the dataset are f1, f2, ..., fp, then
+it considers a random subset of features of size k (k <
+p) instead of all features, while splitting each node of
+decision trees to create new branches.
+An important hyperparameter of the random forest model is
+the number of decision trees in the forest, which can be tuned
+over validation dataset to find its optimum value.
+V. EXPERIMENTAL EVALUATION
+A. Data Collection and Processing
+As the first step of breathing anomaly detection plan, data
+were collected using the IR light wave sensing setup and the
+robot. sin6 pattern was used for generating breathing data
+because of its visual similarity with real human breathing. The
+time constant at the lock-in amplifier was kept at 100 ms. The
+gain at the photodetector was selected to be 40 dB (halfway
+across the full span) so that it did not saturate at the time of
+data collection. The sensitivity of the lock-in amplifier was set
+in such a way so that the resting voltage level stays close to
+the middle of its voltage range in the lock-in amplifier to allow
+maximum voltage swing possible without being saturated.
+Breathing offset or the initial position of the chest of the robot
+was set to zero. Data were collected at three different distances
+(0.5 m, 1 m and 1.5 m) between the photodetector and the robot
+during the daytime. Windows of the room were unshaded
+and internal lighting was common for an office environment
+during data collection. The ranges of breathing rate and depth
+chosen for each class, as per the discussion in Section III,
+and the number of data collected are summarized in Table I.
+Faulty data for class 7 were generated by walking closely
+around the setup, manually interrupting the line of sight, or
+creating intentional system malfunction intermittently during
+data collection. Each data and corresponding timestamps were
+collected for 60 s duration and stored in the Raspberry Pi.
+The sampling frequency used was 100 Hz which was more
+than sufficient to capture even the fastest breathing frequency,
+50 BPM or 0.8 Hz as per Table I. Later, a MATLAB script
+was used to save the voltage amplitudes in a single CSV
+(Comma Separated Value) file along with their corresponding
+class labels. In that CSV file, each row denoted one instance of
+data that contained 60 s × 100 Hz = 6000 data points and one
+integer between 0 to 7 (inclusive) for denoting its class label.
+Timestamps corresponding to each data were saved in another
+file in the same order for plotting the data against them later.
+Fig. 8: Time domain and frequency domain plots of sample
+raw and processed data from class 4 (at 1.5 m distance).
+To reduce the noise from the collected data, moving average
+filtering was done as per equation (4) in MATLAB. It was
+identified by trial and error that 50 points moving average
+worked best for all data since it smoothed out the data to
+the right amount keeping the required sinusoidal variation
+of different frequencies due to breathing intact. Next, data
+was detrended by fitting a polynomial of order 5 to the data
+and subtracting that polynomial trend from the data as per
+equation (5) and (6). Detrending made the data vary around
+0 V and removed most of the drifts. Fig. 7 shows a few samples
+of raw data (one from each class, collected at a 1 m distance).
+For doing better comparison among data classes, first 7 graphs
+were plotted with same range of voltages in y axes. The last
+graph (faulty data) needed larger span in y axis to show the
+data signal completely.
+Fig. 8 shows one data from class 4 collected at a distance of
+1.5 m along with its processed versions and frequency domain
+representation. Since this data was collected from a farther
+distance, the raw data in the first subplot (green colored)
+was seen to be more affected by noise than those in Fig. 7.
+Also, this data clearly had some upward trend in it. In the
+second subplot, the effect of detrending became clear when
+the detrended data (red colored) was compared with the mean-
+subtracted version (blue colored) of the filtered data. The
+peak-to-peak amplitude without detrending was 0.022 V for
+this data which was higher than the actual peak-to-peak of
+the periodic variation present in the data. After detrending, a
+peak-to-peak amplitude 0.011 V was found which was a better
+estimation. In the third subplot, the highest spectral amplitude
+was found at 1 BPM without detrending which was erroneous.
+The actual breathing frequency was at the secondary peak in
+the spectrum occurred at 20 BPM. Detrending improved the
+breathing frequency estimation by suppressing the false peaks
+at low frequencies.
+
+Data Class = 4 (Hyperpnea), Breathing Rate = 20 BPM, Depth = 67%
+4.14
+4.135
+4.13
+4.125
+4.12
+Raw data
+4.115
+Moving-averageddata
+4.12
+0
+10
+20
+30
+40
+50
+60
+Time
+0.015
+0.01
+Voltage
+0.005
+0.005
+Mean-subtracted data
+0.01
+Detrended data
+-0.015
+0
+10
+20
+30
+40
+50
+60
+Time
+Single-sided Amplitude Spectrum (FFT)
+SpectralAmplitude (V)
+×10-3
+FFTof regulardata
+FFTofdetrendeddata
+S
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Frequency (BPM)ARXIV VERSION UPLOADED: JANUARY 09, 2023
+9
+Fig. 9: Training and validation accuracy plots for decision tree model for different distances.
+Fig. 10: Training and validation accuracy plots for random forest model for different distances.
+B. Feature Extraction
+Feature extraction is an important step in machine learning
+based data classification. All four handcrafted features de-
+scribed in Section IV-C were extracted from the data using
+MATLAB code for the following three cases:
+1) One distance at a time or at 0.5 m, 1 m and 1.5 m
+independently.
+2) Two consecutive distances at a time i.e. 0.5 m and 1 m
+together and also 1 m and 1.5 m together.
+3) All three distances together.
+Fast Fourier Transform (FFT) algorithm was used to find
+DFT of the processed data in order to extract feature 2 -
+breathing rate. For feature 3 - effective spectral amplitude,
+threshold t was chosen to be 20% and only first 100 frequency
+points in the spectrum were considered (l(A) = 100 in equa-
+tion (10)), because the rest of the frequencies had negligible
+amplitudes. The first three features did not need the distance
+information for being calculated, hence they were calculated
+once and used in all three cases above as needed. But the last
+feature, SNR, was dependent on noise power which varied
+with distance. Noise data for 60 s were collected by the IR
+sensing setup at the chosen three distances by keeping the
+robot at standstill. When data collected at only one distance
+was considered for classification, noise data of that distance
+was used to calculate SNR. For mixed set of data (either 2 or
+3 distances together), an average noise was taken and applied
+for all data in general.The features for each data were saved
+in separate rows in CSV files along with class label for each
+row. Thus, labeled features were prepared for the classification
+task next.
+C. Data Classification and Results
+Breathing data classification for anomaly detection was
+performed using decision tree and random forest models. The
+feature vectors extracted from the data were fed to these
+machine learning models using Scikit-learn library of Python.
+Both of these two models are highly prone to overfitting,
+hence hyperparameter tuning was performed to regularize the
+models.
+Classification was done initially by considering only one
+distance at a time, then two consecutive distances and finally
+all three distances together. Each time, the feature set for the
+data were divided into train-validation-test sets randomly with
+60% data in training set, and 20% data in each of validation
+and test sets. Decision tree model was fit with training dataset
+by varying the depth of tree from 1 to 25 and tested on val-
+idation dataset each time. Training and validation accuracies
+were plotted in Fig. 9 for all different cases considered based
+on test distance. The optimum depth of the tree will be close to
+the point from which the accuracies start saturating. According
+
+100
+100
+90
+80
+Validation accuracy (%)
+80
+(%)
+accuracy
+70
+60
+60
+Training 
+50
+0.5m
+40
+0.5m
+1m
+1m
+40
+1.5m
+1.5m
+0.5m & 1m
+0.5m & 1m
+30
+1m& 1.5m
+1m & 1.5m
+20
+0.5m, 1m & 1.5m
+0.5m, 1m & 1.5m
+0
+5
+10
+15
+20
+25
+5
+10
+15
+20
+25
+Maximum depth of decision tree
+Maximum depth of decision tree100
+100.0
+97.5
+90
+95.0
+80
+92.5
+70
+Training 
+90.0
+0.5m
+0.5m
+87.5
+60
+1m
+1m
+1.5m
+1.5m
+0.5m & 1m
+0.5m & 1m
+85.0
+1m & 1.5m
+1m & 1.5m
+50
+0.5m. 1m & 1.5m
+0.5m. 1m & 1.5m
+82.5
+0
+5
+10
+15
+20
+25
+0
+5
+10
+15
+20
+25
+Number of decision trees
+Number of decision treesARXIV VERSION UPLOADED: JANUARY 09, 2023
+10
+TABLE II: Training, validation and test accuracies using
+decision tree model
+Distances
+Training
+accuracy
+Validation
+accuracy
+Test
+accuracy
+CV accuracy
+Training
+Test
+0.5 m
+100%
+97.5%
+96.3%
+100%
+96.6%
+1 m
+100%
+96.3%
+96.3%
+100%
+94.1%
+1.5 m
+98.75%
+61.3%
+78.8%
+96.6%
+71.9%
+0.5 m, 1 m
+96%
+92.5%
+88.8%
+96.6%
+86.6%
+1 m, 1.5 m
+95.8%
+78.8%
+83.1%
+95.8%
+80.8%
+0.5 m,
+1 m, 1.5 m
+95%
+81.3%
+80%
+94.7%
+80.7%
+to the plots in Fig. 9, the optimum depth can be close to 5
+when only 0.5 m and 1 m distances were considered, but it
+was decided to be 10 considering 1.5 m distance too. Similar
+hyperparameter tuning was performed on random forest model
+where the hyperparameter was the number of decision trees.
+The results for different cases were plotted in Fig. 10 and the
+optimum number of decision trees was found to be 12 from
+these plots.
+Models with the optimum hyperparameters were built using
+training dataset and training, validation and test accuracies
+were calculated using those models. K-fold cross-validation is
+a preferred method for assessing model performance because
+it evaluates the model K times (typically K = 10) by allowing
+each data to be sometimes in the training dataset, while
+some other times in the test dataset and finally, takes the
+average result of all the models. 10-fold cross-validation was
+performed on the dataset comprising of training and validation
+data (80% of the total data) together and the models were
+tested on test data (the rest 20%). All of the accuracies were
+reported in Table II for decision tree and Table III for random
+forest. The test accuracies obtained through cross-validation,
+displayed again in a bar chart in Fig. 11, represented the model
+performance on unseen test data more reliably than the other
+accuracy numbers. In that bar chart, both models were seen
+to yield similar accuracies with the data collected at 0.5 m
+and 1 m distances which expressed that a single decision tree
+was sufficient to classify the data with > 94% accuracy at
+these distances. Though in other cases, intra-class variances
+increased because of higher level of noise and the presence of
+the data collected at multiple distances together which caused
+the accuracies to decrease (< 90%). Random forest performed
+better than decision tree in later cases because of taking the
+combined result from multiple uncorrelated decision trees and
+thus compensating for the wrong predictions on noisy data
+examples made by each of them. Hence, random forest were
+able to generate the most generalized model for respiratory
+anomaly detection as confirmed through cross-validation.
+VI. CONCLUSION AND FUTURE DIRECTIONS
+In conclusion, the infrared light wave sensing system was
+able to detect faulty data and breathing anomalies through
+machine learning models, with a highest accuracy of 96.6%
+obtained by decision tree model. Random forest model was
+TABLE III: Training, validation and test accuracies using
+random forest model
+Distances
+Training
+accuracy
+Validation
+accuracy
+Test
+accuracy
+CV accuracy
+Training
+Test
+0.5 m
+99.6%
+97.5%
+91.3%
+99.6%
+96.3%
+1 m
+100%
+97.5%
+92.5%
+99.7%
+94.1%
+1.5 m
+100%
+68.8%
+81.3%
+99%
+75.3%
+0.5 m, 1 m
+99.6%
+94.4%
+90.6%
+99.6%
+88.9%
+1 m, 1.5 m
+99.4%
+85%
+84.4%
+99.4%
+83.4%
+0.5 m,
+1 m, 1.5 m
+99.3%
+85.4%
+80.8%
+99.4%
+84%
+Fig. 11: Cross-validation based test accuracy plots for decision
+tree and random forest models for different distances.
+found to be more effective than decision tree model in
+classifying the data that were collected at multiple distances.
+Further refinement of models is necessary to make it more
+realistic and prevent false alarms by including training data
+that has more variances in breathing frequencies, depths,
+distances from the LWS setup and ambient lighting conditions.
+Human breathing will shift in nature when in deep sleep or
+exercising, hence these data could be included in the normal
+breathing class of the training data to make it even more real-
+istic. More handcrafted features can be extracted to improve
+the classification accuracy further, especially in the mixed
+datasets. Performance between machine learning models and
+deep learning models (convolutional neural network, recurrent
+neural network etc.) can be compared to see which one per-
+forms better in classifying respiration data. While the proposed
+work establishes the baseline of respiratory anomaly detection
+using light wave sensing, it’s applicability can be further
+tested in a realistic environment with human subjects. Also,
+more complicated anomalous breathing patterns like Cheyne-
+Stoke’s, Apneustic and Biot’s breathing can be included as
+separate classes in future similar efforts to evaluate the work
+in a more comprehensive way.
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+
+ARXIV VERSION UPLOADED: JANUARY 09, 2023
+13
+Md Zobaer Islam received his B.Sc. degree in
+Electrical and Electronic Engineering in 2012 from
+Bangladesh University of Engineering and Technol-
+ogy, Dhaka, Bangladesh. He joined Oklahoma State
+University, Stillwater, OK as a graduate teaching and
+research assistant to pursue his Ph.D. degree at the
+School of Electrical and Computer Engineering in
+Spring 2020. He has industry experience of 4 years
+at Bangladesh Telecommunications Company Ltd.
+in telecommunication and information technology
+(IT) sector and 3 years at Samsung R&D Institute
+Bangladesh in software sector. His current research interests include wireless
+light-wave sensing and machine learning.
+Brenden Martin was born in Muskogee, Oklahoma,
+in the summer of 1999. There he spent his childhood
+disassembling electronics, drawing, programming,
+and playing musical instruments. He received his BS
+in Electrical Engineering in 2021 from Oklahoma
+State University (OSU). He is now conducting his
+PhD studies in OSU’s Ultrafast Terahertz Optoelec-
+tronics Laboratory (UTOL) at OSU, where he has
+been involved since 2017. His research interests
+include ultrafast optoelectronics, materials science,
+and condensed matter physics.
+Carly Gotcher is currently working toward BS
+degree with the School of Electrical and Computer
+Engineering, Oklahoma State University, Stillwater,
+Oklahoma, USA. Her major is Electrical Engineer-
+ing. She is an undergraduate researcher working at
+Ultrafast THz Optoelectronic Laboratory (UTOL) at
+Oklahoma State University.
+Tyler Martinez is currently working toward BS
+degree with the School of Electrical and Computer
+Engineering, Oklahoma State University, Stillwater,
+Oklahoma, USA. His major is Electrical Engineer-
+ing. He worked as an undergraduate researcher at
+OSU Wireless Lab (OWL) and Ultrafast THz Op-
+toelectronic Laboratory (UTOL) at Oklahoma State
+University.
+John F. O’Hara (SM’19) received his BSEE degree
+from the University of Michigan in 1998 and his
+Ph.D. (electrical engineering) from Oklahoma State
+University in 2003. He was a Director of Central
+Intelligence Postdoctoral Fellow at Los Alamos Na-
+tional Laboratory (LANL) until 2006. From 2006-
+2011, he was with the Center for Integrated Nan-
+otechnologies (LANL) and worked on numerous
+metamaterial projects involving dynamic control
+over chirality, resonance frequency, polarization, and
+modulation of terahertz waves. In 2011, he founded
+a IoT, automation, and consulting/research company, Wavetech, LLC. In 2017,
+he joined Oklahoma State University as an Assistant Professor, where he now
+studies IoT, metamaterials, terahertz communications, and photonic sensing
+technologies. He has around 100 publications in journals and conference
+proceedings.
+Sabit Ekin (M’12, SM’21) is currently working
+as an Associate Professor of the Department of
+Engineering Technology and Industrial Distribution
+at Texas A&M University, College Station, Texas,
+USA. Previously, he was an associate professor of
+Electrical and Computer Engineering at Oklahoma
+State University (OSU). He was the founding direc-
+tor of OSU Wireless Lab (OWL) at OSU. He re-
+ceived the B.Sc. degree in electrical and electronics
+engineering from Eskis¸ehir Osmangazi University,
+Turkey, in 2006, the M.Sc. degree in electrical
+engineering from New Mexico Tech, Socorro, NM, USA, in 2008, and
+the Ph.D. degree in electrical and computer engineering from Texas A&M
+University, College Station, TX, USA, in 2012. He has four years of industrial
+experience as a Senior Modem Systems Engineer at Qualcomm Inc., where
+he has received numerous Qualstar awards for his achievements/contributions
+on cellular modem receiver design. His research interests include the design
+and analysis of wireless systems including mmWave and terahertz communi-
+cations in both theoretical and practical point of views, visible light sensing,
+communications and applications, non-contact health monitoring, and Internet
+of Things applications.
+
diff --git a/sdE2T4oBgHgl3EQfLgYA/content/tmp_files/load_file.txt b/sdE2T4oBgHgl3EQfLgYA/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1303 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf,len=1302
+page_content='ARXIV VERSION UPLOADED: JANUARY 09, 2023  1  Non-contact Respiratory Anomaly Detection using  Infrared Light Wave Sensing  Md Zobaer Islam, Brenden Martin, Carly Gotcher, Tyler Martinez, John F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' O’Hara, Senior Member, IEEE,  and Sabit Ekin, Senior Member, IEEE  Abstract—Human respiratory rate and its pattern convey  important information about the physical and psychological  states of the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Abnormal breathing can be a sign of fatal  health issues which may lead to further diagnosis and treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Wireless light wave sensing (LWS) using incoherent infrared light  turns out to be promising in human breathing monitoring in  a safe, discreet, efficient and non-invasive way without raising  any privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The regular breathing patterns of each  individual are unique, hence the respiration monitoring system  needs to learn the subject’s usual pattern in order to raise flags for  breathing anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Additionally, the system needs to be capable  of validating that the collected data is a breathing waveform,  since any faulty data generated due to external interruption or  system malfunction should be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In order to serve both of  these needs, breathing data of normal and abnormal breathing  were collected using infrared light wave sensing technology in  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Two machine learning algorithms, decision tree and  random forest, were applied to detect breathing anomalies and  faulty data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Finally, model performance was evaluated using  average classification accuracies found through cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The highest classification accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='6% was achieved with  the data collected at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m distance using decision tree model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Ensemble models like random forest were found to perform better  than a single model in classifying the data that were collected at  multiple distances from the light wave sensing setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Index Terms—Non-contact vitals monitoring, respiration mon-  itoring, light wave sensing, machine learning, anomaly detection,  data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' INTRODUCTION  P  HYSICAL and psychological states of human beings are  reflected in their respiratory patterns which can be moni-  tored to raise general health-consciousness and generate alarm  for anomalous breathing before more serious health issues  occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Anomalous breathing can occur due to various respi-  ratory illnesses like asthma, obstructive sleep apnea, chronic  bronchitis, chronic obstructive pulmonary disease (COPD),  emphysema and COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Also, it can be a symptom of  unstable mental conditions including stress, panic, anxiety,  fatigue, anger and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Doctors evaluate the breathing quality  This work was supported by the National Science Foundation under Grant  2008556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' (Corresponding author: Md Zobaer Islam, Sabit Ekin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=')  Md Zobaer Islam, Brenden Martin, Carly Gotcher, Tyler Martinez, and John  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' O’Hara are with the School of Electrical and Computer Engineering, Okla-  homa State University, Oklahoma, USA (e-mail: zobaer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='islam, brenden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='martin,  carly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='gotcher, tyler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='martinez, oharaj {@okstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='edu})  Sabit Ekin is with the Department of Engineering Technology and Industrial  Distribution, Texas A&M University, College Station, Texas, USA (e-mail:  sabitekin@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='edu)  This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Copyright may be transferred without notice, after which this version may  no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  of patients most often by manual observation which may  generate wrong perception due to its subjective nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' This  approach is not suitable for continuous monitoring either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Automated breathing monitoring solutions are rarely utilized  and they mostly use contact-based or wearable sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' When  a person is aware that their breathing is being monitored, they  may undergo a shift in breathing pattern from their typical  signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Moreover, the subject can be too young (infants at  neonatal intensive care unit or NICU) or too ill (burn unit  patients) to apply contact-based approaches [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' For patients  with contagious diseases like COVID-19, any contact-based  respiration monitoring can spread the disease further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Hence,  there is a need for non-contact, discreet respiration monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Research and development on non-contact vitals monitoring  using electromagnetic (EM) signals have been ongoing for  many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Received signal strength (RSS) and channel  state information (CSI) of WiFi networks were utilized for  human breathing detection, estimation and monitoring [2]–  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Remote monitoring of breathing has also been studied  using different types of radar including microwave bands [6],  [7], mmWave [8], ultra-wideband (UWB) [9], [10], frequency-  modulated continuous wave (FMCW) [11], [12], step fre-  quency continuous wave (SFCW) [13] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' These radio-  frequency (RF) based respiration monitoring approaches are  prone to electromagnetic interference with EM signals from  nearby devices [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Also, continuous exposure to RF  signals can be detrimental to the human body [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  WiFi-based approaches usually require a receiver device held  at chest, hence they are not fully contactless methods [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Besides RF signal based approaches, researchers also exploited  recorded videos or images from RGB cameras [18]–[25]  and thermal infrared cameras [26]–[32] to extract breathing  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' These approaches may raise privacy concerns  among users because the subjects’ image data is discreetly  captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Also, post-processing of video and image data is  computationally more expensive than one dimensional time  series data [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Non-contact vitals monitoring using light wave sensing may  prove superior to existing technologies because of the safe,  ubiquitous and harmless nature of light as well as the absence  of privacy issues as there are with camera-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Non-contact vitals monitoring using visible light has already  been performed as a proof of concept that showed >94%  accuracy in breathing rate measurement in comparison to FDA  approved contact-based counterpart [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' But visible light  can be troublesome to subjects in dark environments, espe-  cially during sleep, hence more subtle means are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='03713v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='SP]  9 Jan 2023  ARXIV VERSION UPLOADED: JANUARY 09, 2023  2  the current study, respiration has been monitored using widely  available incoherent infrared (IR) light in a discreet way (since  IR light is not visible to the naked human eye) and machine  learning algorithms were applied on the collected data to detect  breathing anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The main contributions of this project are  summarized below:  1) Development of a novel IR based light wave sensing sys-  tem model for human respiration monitoring and anomaly  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  2) Breathing data collection using the developed system in a  controlled environment, with precision and repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  3) Handcrafted feature extraction from the collected breath-  ing data for classifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  4) Smart detection of anomalous breathing and faulty data  from the collected breathing data using machine learning  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The remainder of this manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Section II includes an overview of related works on the  application of machine learning to breathing data collected  by different technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Section III describes various human  breathing patterns from the literature to be used as breathing  classes for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Section IV presents the system  model, relevant theory, hardware setup and the overall process  to detect breathing anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Section V presents the exper-  imental evaluation of the approach described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  That includes details of data collection and processing, feature  extraction, data classification and the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Finally, Sec-  tion VI presents the conclusions drawn from the whole effort  and forecasts future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' RELATED WORKS  Researchers have been applying machine learning and arti-  ficial intelligence on human breathing data collected through  various technologies, both contact-based and non-contact,  for multiple applications like posture detection [35], [36],  identity authentication [37]–[39], activity classification [40],  stress classification [41], exercise detection [42], breathing  anomaly detection [12], [43] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Most of these efforts made  use of handcrafted features which were mostly dependent on  their system model and hardware setup [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Some of the  common categories of features that were used for breathing  data classification in the literature were statistical features from  the data (mean, standard deviation, skewness, kurtosis, root  mean-square value, range etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' ), signal-processing based fea-  tures (Fourier co-efficients, autoregressive integrated moving  average co-efficients, wavelet decomposition, mel-frequency  cepstral coefficients, linear predictive coding etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' ), and respi-  ration related features (breathing rate, amplitude, inspiratory  time, expiratory time etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=') [12], [26], [35], [41]–[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In some  research efforts, convolutional neural networks were trained to  recognize subtle features from breathing data (2-dimensional  image or 1-dimensional time series data) thus making manual  feature extraction redundant [48]–[52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Some past classification efforts involved one-class classifi-  cation or outlier detection, as in [42] where the model was  trained using human breathing data in resting condition to  predict if the person was exercising in new examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Binary  classification between normal breathing and apnea were per-  formed in [43] to detect obstructive sleep apnea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Multiclass  breathing classification efforts considered different types of  breathing anomalies like tachypnea, bradypnea, hyperpnea,  hypopnea etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' and sometimes more complicated anomalies like  Cheyne-Stoke’s, Biot’s and Apneustic breathing as separate  classes [26], [44], [49], [50], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' These breathing patterns are  explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Data for these efforts were collected  from human subjects who are generally unable to breathe  using precise frequency, amplitude and pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Sometimes,  breathing data from patients with breathing disorder were  used which had limitations too because even the patients  might not breathe in consistent abnormal pattern all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  In the current study, more reliable data were generated by  using a programmable robot with precise human-like breath-  ing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The machine learning techniques used in the  literature for classifying breathing data were decision tree,  random forest, support vector machine, XGBoost, K-nearest  neighbours, feedforward neural network, logistic regression,  ensemble learning etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Model performances have been anal-  ysed in these works by using confusion matrices, K-fold cross-  validation, accuracy, precision, sensitivity or recall, specificity,  F1-score and so on [12], [26], [35], [37]–[41], [43], [44], [46],  [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' HUMAN BREATHING PATTERNS  Various types of human breathing patterns (both normal and  abnormal) have been identified from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The major  7 types of human breathing are described as follows:  1) Eupnea: This is regular human breathing with a uniform  depth, rate and pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The depths and rates that are  considered as normal vary according to the age and  activity level of human being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' For adult people, the  regular breathing rate is 12-20 BPM (breaths per minute)  at resting conditions [12], [44], [54], [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Breathing  depth is measured from rib cage movement and it is  expressed as a percentage of the maximum movement  of the rib cage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' For healthy adults of 20-39 years, the  breathing depth was experimentally found to be 44±14%  or 30-58% of the maximum rib cage movement in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  2) Apnea: Temporary cessation of breathing is known as  apnea [54], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' It often occurs during sleep which is  known as sleep apnea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  3) Tachypnea: This is an anomalous breathing condition  where the human breaths faster than usual [26], [45], [54],  [57]–[59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' When the breathing rate becomes higher than  20 BPM, then it can be considered as Tachypnea [44],  [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Sometimes, the threshold frequency for Tachypnea  is taken to be 25 BPM [60] or 30 BPM [57] too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  4) Bradypnea: Bradypnea is defined as slow breathing [12],  [26], [53], [54], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' So, any breathing rate less than  12 BPM can be considered as Bradypnea [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  5) Hyperpnea: Hyperpnea is a breathing pattern with in-  creased depth of breathing at normal rate [12], [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  6) Hypopnea: In hypopnea, the breathing becomes shallow  with at least 50% decrease in the regular air flow volume  for ≥10 seconds [43], [61], [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  ARXIV VERSION UPLOADED: JANUARY 09, 2023  3  7) Kussmaul’s breathing: When tachypnea and hyperpnea  occur together and thus the breathing becomes rapid, deep  and labored, then it is known as Kussmaul’s breathing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  So, in Kussmaul’s breathing, both of breathing depth  and rate will be higher than those of Eupnea or normal  breathing [45], [53]–[55], [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The above 7 types of breathing will be addressed in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' There are a few other more complicated breathing pat-  terns too like Cheyne-Stoke’s, Biot’s and apneustic breathing  which are composed of periods of these 7 base classes [12],  [44], [45], [53], [55], [63], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 1: Proposed system model for respiration monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' SYSTEM DESIGN AND IMPLEMENTATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Theory of Operation  The proposed system model for non-contact respiration  monitoring using IR sensing and anomaly detection is depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In this model, modulated IR light is sent towards the  chest of a human being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Light propagation from light emitting  diodes (LED) follows Lambertian propagation model [65],  [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' According to this model, if Pt is the optical power  transmitted from a point source, then power at distance d,  is given by  Pd = (n + 1)APt  2πdγ  cosn(φ) cos(θ), ∀ θ < φ1/2,  (1)  where A is the area intercepted, γ is the empirical path-loss  exponent, and φ and θ are irradiance and incident angles,  respectively [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' φ1/2 is the half-power angle in the field  of view of the light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' n is the order of the Lambertian  model and is given by n =  − ln(2)  ln{cos(φ 1  2 )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Propagated optical power intercepts the subject’s chest, part  of which is absorbed by the clothing material while the rest is  further modulated by the chest movement due to inhaling and  exhaling and is scattered back to the photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' If Pr is the  scattered optical power received by the photodetector, Rpd is  the responsivity and id is the dark current of the photodetector,  then the generated photocurrent ipd = id + RpdPr [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  transimpedance amplifier in the photodetector converts this  current into voltage  Vsig = gpd (id + RpdPr) ,  (2)  where gpd is the transimpedance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  This signal is contaminated by noise from the photodetector  and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' When the signal of interest is buried  in noise, lock-in detection technique can be used to detect  the signal accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Hence, the output voltage from the  photodetector (Vsig) is fed to the lock-in amplifier [68] as  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' If Vsig is a sinusoidal wave with only one frequency ωs  and phase φs i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Vsig = Vs sin(ωst + φs) and the reference  input to the lock-in amplifier is a periodic signal of the same  frequency ωs and amplitude Vr, then the output magnitude of  the lock-in amplifier can be shown to be R = 1  2VsVr, which  is proportional to Vs [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' R can be scaled to the desired level  by varying the sensitivity S = Vfs  G  of the lock-in amplifier,  where Vfs is the full-scale voltage (generally 10V) and G is  the overall gain of the lock-in amplifier, to produce  Rscaled = VfsVr  2S  Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  (3)  In practice, the signal Vsig is distorted by high frequency  noises which will be attenuated by the low pass filters inside  the lock-in amplifier1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The output signal of the lock-in ampli-  fier is collected, stored and processed for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The extracted features are fed to machine learning models for  breathing anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Experimental Setup  To collect respiration data in a controlled, precise and  repeatable way, a robot was developed to emulate human  respiration using 3-D printed structures and two servo motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The breathing rate, pattern, and depth could be set using a  Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' External and internal views of the robot are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 2a and 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The robot could be controlled by one  of two programs written in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The first program allowed  the robot to breathe following a list of predefined waveforms  including |sin|, sin2, sin4 and sin6 patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The breathing  rate could be varied from 0 to 50 BPM with maximum  depth ranging from 0 to 30 mm (denoted by 0% to 100%  in the program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Also, an offset could be provided to set the  initial position of the chest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 2c presents the graphical  user interface of this software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' A secondary control software  could actuate arbitrary waveforms from a text file generated  by MATLAB code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  A light wave sensing system was developed for collecting  respiration data using infrared light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' An IRLED matrix source  (FY-48 940 nm IR lamp board [70]) consisting of 48 IRLEDs  was used as the light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' This source gives the highest  intensity of light when it is connected to ≈12 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The light  source was modulated by a 1 kHz sinusoidal voltage wave  of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='8 V peak-to-peak amplitude and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1 V DC offset from  a Keysight 33500B function generator [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The voltage am-  plitude and DC offset were chosen as such to keep the light  intensity high enough (by applying 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='9 V peak voltage), while  1The noise frequencies that are very close to the reference frequency (ωs)  may exist in the filtered signal depending on the filter bandwidth and roll-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Filter bandwidth can be adjusted by varying the time constant, τ =  1  2πfc of  the lock-in amplifier, where fc is the 3-dB cut-off frequency of the low pass  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' A narrowband filter will suppress most of the noises while keeping the  signal to be detected intact as a true DC component  Distance  Warning if  anomaly  source  detected  Inhale  Classification  Exhale  Photodetector  using ML  Enlarged view of the  respiration monitoring device  Lock-in  Data  Data  Feature  detection  collection  processing  extractionARXIV VERSION UPLOADED: JANUARY 09, 2023  4  (a) External view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  (b) Internal view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  (c) User interface  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 2: External and internal views and the graphical user  interface of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  maintaining linear LED operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' For collecting the reflected  light from the chest of the breathing robot, Thorlabs PDA100A  photodetector [72] with a converging lens of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='4 mm focal  length was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The photodetector comprised of a p-i-n  photodiode and transimpedance amplifier with adjustable gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  For lock-in detection, SR830DSP frequency lock-in ampli-  fier [73] was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Finally, for data collection and storage,  a second Raspberry Pi preceded by analog to digital converter  (DAQC2Piplate ADC [74]) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' A diagram of the setup  used to generate and collect breathing data is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Also, a picture of the overall setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Anomaly Detection Process  Breathing anomaly detection was performed using machine  learning based classification of labeled breathing data collected  using the developed IR sensing setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Seven breathing patterns  described in Section III were used as seven different data  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' After collecting breathing data, the system must decide  whether the current data is useful or not, because faulty data  might be collected due to the movement of the subject or  other people at close proximity, too much noise from the  surrounding or any system malfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' To detect and discard  or recollect such data, a separate data class called ‘faulty  data’ was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Therefore, the breathing anomaly detection  problem was reduced to an eight-class classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  classes were eupnea, apnea, tachypnea, bradypnea, hyperpnea,  hypopnea, Kussmaul’s breathing and faulty data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The general  process followed for anomaly detection will be described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Specific details will be provided in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Breathing data belonging to the eight data classes mentioned  above were collected in a controlled environment using the  robot and the IR sensing setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The collected data x[m] con-  tained noises from the environment and the setup itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  noise was filtered using k-points moving average technique  using the following equation:  yk[n] =  �  �  �  �  �  �  �  �  �  �  �  �  �  1  k  n+k−1  �  m=n  x[m],  if k′ ≥ k,  1  k′  n+k′−1  �  m=n  x[m],  otherwise,  (4)  where k′ is the number of points in the data from the current  data point x[n] to the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Some data had a drift or trend in them which added  DC component to the data and increased the peak-to-peak  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Thus, those data were prone to underestimating  breathing rate and overestimating breathing depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' To get  around this problem, detrending was performed by subtracting  a p-th order polynomial  wp[n] = c1xp + c2xp−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' + cnx + cn+1  (5)  from the data to produce  z[n] = yk[n] − wp[n],  (6)  where c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='., cn+1 were the co-efficients of the polynomial  wp[n] that ensured the best least-squares fit [75] to the data  yk[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' When p = 0, it becomes equivalent to subtracting the  overall mean value from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The breathing data had  higher order trends, so p > 0 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  After detrending, the following four handcrafted features  were extracted from the data:  1) Peak-to-peak Amplitude: If the maximum and minimum  values of the data z[n] are zmax and zmin, then, peak-to-peak  amplitude A = zmax − zmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' It roughly represents a number  proportional to the breathing depth in each class (except the  faulty data class), when the test distance remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  When the test distance was changed, the value of this feature  was shifted to a different range even though the data belonged  to the same class, because of changed levels of received light  intensity and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  2) Breathing Rate: Breathing rate is the frequency with the  highest spectral amplitude in the frequency domain represen-  tation of the signal (again, it is not applicable in apnea and  faulty data classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Discrete Fourier Transform (DFT) of the  time domain data z[n] is taken using  Z[k] =  L−1  �  n=0  z[n]e  −j2πkn  L  ,  (7)  where L is the length of the dataset and its DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Z[k] contains  different frequency components present in the signal z[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  frequency index with the maximum magnitude is found to be  kmax = arg max  k  |Z[k]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  (8)  三tk  X  Offset  Amp  BPM  0  60  14  Isin]  Start  Update  Basic Operation Insturctions:  Press start to begin robot motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Start also works as a pause button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The currently selected values will be read in when start is pressed  Update will read in the currently selected values without pausing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The robot will reset to the selected offset when updating or starting/unpausing before beginning the selected motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Operation Warning:  Make sure the Amp and offset sliders do not add to higher than 1oo when reading into the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The program will give an error message and pause until valid values are passed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  OtherUseful Info:  BPM stands for breaths per minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Amp and offset are input as a percentage of the maximum range of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The maximum range of motion is approximately 3cmARXIV VERSION UPLOADED: JANUARY 09, 2023  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 3: The experimental setup diagram of the overall system used for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 4: The hardware setup of the overall system used for data  collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Then, the corresponding frequency or the breathing rate,  fmax, is calculated using  fmax = kmaxfs  L  Hz = 60kmaxfs  L  BPM,  (9)  where fs is the sampling frequency used to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Breathing rate stays in the same range for data collected at  all distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In contrast, error in breathing rate estimation  increases with increasing distance because of increasing noise  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 5: Frequency domain plots of sample processed data from  class 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  3) Effective Spectral Amplitude: It is the percentage count  of frequencies in the spectrum of data after detrending whose  spectral amplitude is greater than or equal to a predefined  threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The threshold is determined as a percentage  of peak spectral amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' If the peak magnitude in the data  spectrum is H, t is the percentage threshold, and A is the  spectrum vector, then effective spectral amplitude  S = l(B)  l(A) × 100%, B = {x ∈ A|x ≥ tH},  (10)  where l() represents the length of its parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' This  number helps differentiating between a very clean spectrum  with just a few large peaks (which are the breathing rate and  its harmonics) and a spectrum with multiple peaks of similar  magnitude at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  As an example, in the first subplot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 5, the spectrum  of one data from Apnea class has the highest peak at 23 BPM  which happened by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' It has many other peaks of  Keysight 33500B function generator  SIOHT335eOB SId  BNC to BNC cable  BNC to BNC cable  IRLED array  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='00  00001  0000  0  Disply  口  0口  SR830 Lock-in amplifier  Distance  BNC to grabber cable  BNC to BNC cable  Thorlabs  photodetector  Breathing robot  (Controlled by a Raspberry Pi)  Raspberry Pi  DAQC2Piplate ADC  (For data collection)IRlightsource&  Photodetector  Function  generator  Lock-in  amplifier  Rpi for data  collection  Powersupply  units  Breathing robotClass = 1 (Apnea), Distance = 1 m  ×10-4  X 23  Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='000195041  Spectral Amplitude (V)  2  Peak magnitude  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='2*(Peak magnitude)  0  10  20  30  40  50  60  70  80  90  100  Frequency(BPM)  Class = 2 (Tachypnea), Frequency = 23 BPM, Depth = 32%, Distance = 1 m  X23  Peak magnitude  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='01  nplitude  Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='0113069  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='008  Am  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='006  ctr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='2*(Peak magnitude)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='002  S  A  0  0  10  20  30  40  50  60  70  80  90  100  Frequency (BPM)ARXIV VERSION UPLOADED: JANUARY 09, 2023  6  magnitude greater than the threshold (marked by red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Hence, it is not representing breathing at 23 BPM which can  be understood from its larger effective spectral amplitude  (S = 51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' An actual breathing data of Tachypnea class with  breathing frequency 23 BPM has a cleaner peak at 23 BPM in  its spectrum with S = 2, as shown in the second subplot of  the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The threshold is assumed to be 20% of the  peak magnitude in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 6: Flow diagram of the approach towards the breathing  anomaly detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  4) Signal to Noise Ratio: Average signal to noise ratio  (SNR) of a signal is an indicator of signal quality and is  defined as the ratio of average signal power and average noise  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The photodector was a power detector which detected  the signal power and noise power together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' If z[n] is the mean-  subtracted data received by the photodetector at a particular  distance and xnoise[n] is the mean-subtracted noise data at that  distance, then SNR in decibel can approximately be defined  by  SNRdB = 20 log10  �  �  �  �  �  �  �  N  �  n=1  |z[n]|2  �  N  �  n=1  |xnoise[n]|2  �  �  �  �  �  �  ,  (11)  considering root-mean-square averages of the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Here,  both of z[n] and xnoise[n] had the same length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  SNR value will decrease with increasing distance in the  same data class because of increasing level of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Also,  SNR will decrease when the breathing depth is decreased, as  in class 3 (Bradypnea), because of lower signal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  With these four handcrafted features, data were classified  using decision tree and random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' These models  are introduced in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In order to see the effect of  distance on classification accuracy, classification was initially  done separately on data collected at one particular distance  at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' But in realistic environments, users may not be at  a particular distance every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Hence, mixed sets of data  collected at different distances were used later for feature  extraction and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The overall plan for breathing  anomaly detection is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Machine Learning Models  Two renowned machine learning models - decision tree and  random forest - were used for data classification in the process  of anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The data classes, defined in Table I,  were differentiated by two categorical variables - breathing  rate and breathing depth - which can be represented by a  structure like decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Random forest is a collection of  multiple uncorrelated decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Hence, these two models  did the classification better than the other machine learning  models with the breathing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  1) Decision Tree: Decision tree is one of the most effective  non-parametric supervised machine learning models used for  classification, prediction and data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' This model starts  the classification process by partitioning the dataset into two or  more mutually exclusive subsets based on the values of a root  node, then continues partitioning through internal nodes and  ends at leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Thus, it forms a tree-shaped graph overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  One of the popular algorithms to create binary decision trees  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' tree with exactly two branches at each node) is CART  (classification and regression trees) algorithm [76], [77] which  is used in Scikit-learn [78], [79], a machine learning library  of Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' To describe this algorithm, say, the split at node m  of the decision tree is done based on parameter θ = (fj, tm),  where fj is the j-th feature and tm is the threshold considered  for that feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' At node m, feature fj is tested against tm to  partition the dataset Xm into subsets  Xleft  m  (θ) = {(x, y) |fj ≤ tm}  (12)  and  Xright  m  (θ) = {(x, y) |fj > tm} = Xm\\Xleft  m  (θ),  (13)  where x is the training data instance and y is the corresponding  label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' If nm is the number of data at node m, then the quality  of the split is calculated as follows using the information gain  IG(Xm) = I(Xm) − G(Xm, θ),  (14)  where I() is the impurity function and G(Xm, θ) is a measure  of overall impurity after the split defined as  G(Xm, θ) = nleft  m  nm  I(Xleft  m  (θ)) + nright  m  nm  I(Xright  m  (θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' (15)  Common measures of impurity functions I() for construct-  ing the decision tree are Gini impurity  I(Xm) =  K  �  i=1  pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='m(1 − pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='m) = 1 −  K  �  i=1  (pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='m)2  (16)  Raw data  collection  Moving average  filtering  Mean  Detrending  subtraction  Feature 1  Feature 2  Feature 3  Feature 4  Breathing  Peak to peak  Effective spectral  Signal to  amplitude  rate  amplitude  noise ratio  Classification  Classification  using Decision  using Random  Tree  ForestARXIV VERSION UPLOADED: JANUARY 09,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 2023  7  TABLE I: Characteristics and number of data for each class  Class  Class name  Breathing rate  (BPM)  Breathing  depth (%)  Number of data collected  Total  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m  at 1 m  at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m  0  Eupnea  12-20  30-58  50  50  50  150  1  Apnea  0  0  50  50  50  150  2  Tachypnea  21-50  30-58  50  50  50  150  3  Bradypnea  1-11  30-58  50  50  50  150  4  Hyperpnea  12-20  59-100  50  50  50  150  5  Hypopnea  12-20  1-29  50  50  50  150  6  Kussmaul’s  21-50  59-100  50  50  50  150  7  Faulty data  Any  Any  50  50  50  150  Total  400  400  400  1200  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 7: Time domain representation of sample data from each class (the distance between the source-photodetector and the  robot was 1 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  and log loss or Shannon entropy  I(Xm) = −  K  �  i=1  pi,m log2(pi,m),  (17)  where K is the number of classes, i ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=', K} and  pi,m = ni,m/nm is the proportion of the observations that  belong to class i at node m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Gini impurity is used as the  impurity function in CART algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The feature-threshold pair denoted by θ∗ that maximizes  the information gain, IG(Xm) or equivalently minimizes  G(Xm, θ) is chosen for splitting the data at node m using  θ∗ = arg min  θ  {G(Xm, θ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  (18)  Subsets Xleft  m  and Xright  m  are used to repeat the process  until the maximum allowable depth (distance from the root  node to the farthest leaf node) is reached, or nm becomes  less than the pre-defined minimum number of samples at each  node, or any other stopping criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' If the tree is not  stopped early, then the splitting continues until nm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Decision trees are highly prone to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Overfitting is  a situation when the model yields very high training accuracy  by learning some particular training dataset extremely well,  but fails to learn the general trend in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Hence,  the model becomes susceptible to perform poorly on unseen  test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' To prevent a decision tree from overfitting, it needs  to be regularised by stopping it early before it learns the  noise prevalent in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' One of the methods of  regularising the decision tree model is to limit the maximum  depth of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The optimum value of the maximum depth  can be decided by testing trees with different depths on  validation dataset [80], a process known as hyperparameter  Class=0(Eupnea),Frequency=18BPM,Depth=43%  Class=1(Apnea),Frequency=0BPM,Depth=0%  M  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  Voltage (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  Voltag  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  0  10  20  30  40  50  60  0  10  20  30  40  50  60  Time (s)  Time (s)  Class=2(Tachypnea),Frequency=35BPM,Depth=44%  Class=3(Bradypnea),Frequency=8BPM,Depth=47%  (V)  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  ge  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  AAAAAAA  Voltag  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  0  10  20  30  40  50  60  0  10  20  30  40  50  60  Time (s)  Time (s)  Class = 4 (Hyperpnea), Frequency = 17 BPM, Depth = 91%  Class=5(Hypopnea),Frequency=13BPM,Depth=2o%  M  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  Voltag  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content="05  0  10  20  30  40  50  60  0  10  20  30  40  50  60  Time (s)  Time (s)  Class = 6 (Kussmaul's), Frequency = 26 BPM, Depth = 89%  Class = 7 (Faulty Data)  (V)  M 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  Itage  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='2  0  10  20  30  40  50  60  0  10  20  30  40  50  60  Time (s)  Time (s)ARXIV VERSION UPLOADED: JANUARY 09, 2023  8  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  2) Random Forest: The learning technique related to build-  ing multiple machine learning models and taking majority  vote (for classification) or an average (for regression) of the  predictions is referred to as ensemble learning [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' It prevents  the overall model from overfitting by reducing the variance  in model prediction without affecting the bias much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Random  forest is the ensemble version of decision tree model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' It makes  use of Bootstrap Aggregating or Bagging technique [82] in  the following two ways to build multiple uncorrelated decision  trees and finally takes majority vote to predict the class labels:  (i) If the dataset is x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=', xn, then random forest takes  random samples of size n with replacement from this  dataset m times and builds m different decision trees  with them parallelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  (ii) If the features defining the dataset are f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=', fp, then  it considers a random subset of features of size k (k <  p) instead of all features, while splitting each node of  decision trees to create new branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  An important hyperparameter of the random forest model is  the number of decision trees in the forest, which can be tuned  over validation dataset to find its optimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' EXPERIMENTAL EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Data Collection and Processing  As the first step of breathing anomaly detection plan, data  were collected using the IR light wave sensing setup and the  robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' sin6 pattern was used for generating breathing data  because of its visual similarity with real human breathing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  time constant at the lock-in amplifier was kept at 100 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  gain at the photodetector was selected to be 40 dB (halfway  across the full span) so that it did not saturate at the time of  data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The sensitivity of the lock-in amplifier was set  in such a way so that the resting voltage level stays close to  the middle of its voltage range in the lock-in amplifier to allow  maximum voltage swing possible without being saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Breathing offset or the initial position of the chest of the robot  was set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Data were collected at three different distances  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m, 1 m and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m) between the photodetector and the robot  during the daytime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Windows of the room were unshaded  and internal lighting was common for an office environment  during data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The ranges of breathing rate and depth  chosen for each class, as per the discussion in Section III,  and the number of data collected are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Faulty data for class 7 were generated by walking closely  around the setup, manually interrupting the line of sight, or  creating intentional system malfunction intermittently during  data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Each data and corresponding timestamps were  collected for 60 s duration and stored in the Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The sampling frequency used was 100 Hz which was more  than sufficient to capture even the fastest breathing frequency,  50 BPM or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='8 Hz as per Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Later, a MATLAB script  was used to save the voltage amplitudes in a single CSV  (Comma Separated Value) file along with their corresponding  class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In that CSV file, each row denoted one instance of  data that contained 60 s × 100 Hz = 6000 data points and one  integer between 0 to 7 (inclusive) for denoting its class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Timestamps corresponding to each data were saved in another  file in the same order for plotting the data against them later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 8: Time domain and frequency domain plots of sample  raw and processed data from class 4 (at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  To reduce the noise from the collected data, moving average  filtering was done as per equation (4) in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' It was  identified by trial and error that 50 points moving average  worked best for all data since it smoothed out the data to  the right amount keeping the required sinusoidal variation  of different frequencies due to breathing intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Next, data  was detrended by fitting a polynomial of order 5 to the data  and subtracting that polynomial trend from the data as per  equation (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Detrending made the data vary around  0 V and removed most of the drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 7 shows a few samples  of raw data (one from each class, collected at a 1 m distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  For doing better comparison among data classes, first 7 graphs  were plotted with same range of voltages in y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The last  graph (faulty data) needed larger span in y axis to show the  data signal completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 8 shows one data from class 4 collected at a distance of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m along with its processed versions and frequency domain  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Since this data was collected from a farther  distance, the raw data in the first subplot (green colored)  was seen to be more affected by noise than those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Also, this data clearly had some upward trend in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In the  second subplot, the effect of detrending became clear when  the detrended data (red colored) was compared with the mean-  subtracted version (blue colored) of the filtered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  peak-to-peak amplitude without detrending was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='022 V for  this data which was higher than the actual peak-to-peak of  the periodic variation present in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' After detrending, a  peak-to-peak amplitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='011 V was found which was a better  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In the third subplot, the highest spectral amplitude  was found at 1 BPM without detrending which was erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The actual breathing frequency was at the secondary peak in  the spectrum occurred at 20 BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Detrending improved the  breathing frequency estimation by suppressing the false peaks  at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Data Class = 4 (Hyperpnea), Breathing Rate = 20 BPM, Depth = 67%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='135  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='125  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='12  Raw data  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='115  Moving-averageddata  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='12  0  10  20  30  40  50  60  Time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='01  Voltage  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='005  Mean-subtracted data  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='01  Detrended data  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='015  0  10  20  30  40  50  60  Time  Single-sided Amplitude Spectrum (FFT)  SpectralAmplitude (V)  ×10-3  FFTof regulardata  FFTofdetrendeddata  S  10  20  30  40  50  60  70  80  90  100  Frequency (BPM)ARXIV VERSION UPLOADED: JANUARY 09, 2023  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 9: Training and validation accuracy plots for decision tree model for different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 10: Training and validation accuracy plots for random forest model for different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Feature Extraction  Feature extraction is an important step in machine learning  based data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' All four handcrafted features de-  scribed in Section IV-C were extracted from the data using  MATLAB code for the following three cases:  1) One distance at a time or at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m, 1 m and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m  independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  2) Two consecutive distances at a time i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m and 1 m  together and also 1 m and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  3) All three distances together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Fast Fourier Transform (FFT) algorithm was used to find  DFT of the processed data in order to extract feature 2 -  breathing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' For feature 3 - effective spectral amplitude,  threshold t was chosen to be 20% and only first 100 frequency  points in the spectrum were considered (l(A) = 100 in equa-  tion (10)), because the rest of the frequencies had negligible  amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The first three features did not need the distance  information for being calculated, hence they were calculated  once and used in all three cases above as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' But the last  feature, SNR, was dependent on noise power which varied  with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Noise data for 60 s were collected by the IR  sensing setup at the chosen three distances by keeping the  robot at standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' When data collected at only one distance  was considered for classification, noise data of that distance  was used to calculate SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' For mixed set of data (either 2 or  3 distances together), an average noise was taken and applied  for all data in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='The features for each data were saved  in separate rows in CSV files along with class label for each  row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Thus, labeled features were prepared for the classification  task next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Data Classification and Results  Breathing data classification for anomaly detection was  performed using decision tree and random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The  feature vectors extracted from the data were fed to these  machine learning models using Scikit-learn library of Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Both of these two models are highly prone to overfitting,  hence hyperparameter tuning was performed to regularize the  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Classification was done initially by considering only one  distance at a time, then two consecutive distances and finally  all three distances together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Each time, the feature set for the  data were divided into train-validation-test sets randomly with  60% data in training set, and 20% data in each of validation  and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Decision tree model was fit with training dataset  by varying the depth of tree from 1 to 25 and tested on val-  idation dataset each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Training and validation accuracies  were plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 9 for all different cases considered based  on test distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The optimum depth of the tree will be close to  the point from which the accuracies start saturating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' According  100  100  90  80  Validation accuracy (%)  80  (%)  accuracy  70  60  60  Training  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  1m  1m  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m & 1m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m & 1m  30  1m& 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  1m & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m, 1m & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m, 1m & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  0  5  10  15  20  25  5  10  15  20  25  Maximum depth of decision tree  Maximum depth of decision tree100  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='0  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5  90  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='0  80  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5  70  Training  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5m  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5  60  1m  1m  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
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+page_content='5 m  95%  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
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+page_content='7%  to the plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 9, the optimum depth can be close to 5  when only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m and 1 m distances were considered, but it  was decided to be 10 considering 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m distance too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Similar  hyperparameter tuning was performed on random forest model  where the hyperparameter was the number of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  The results for different cases were plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 10 and the  optimum number of decision trees was found to be 12 from  these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Models with the optimum hyperparameters were built using  training dataset and training, validation and test accuracies  were calculated using those models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' K-fold cross-validation is  a preferred method for assessing model performance because  it evaluates the model K times (typically K = 10) by allowing  each data to be sometimes in the training dataset, while  some other times in the test dataset and finally, takes the  average result of all the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 10-fold cross-validation was  performed on the dataset comprising of training and validation  data (80% of the total data) together and the models were  tested on test data (the rest 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' All of the accuracies were  reported in Table II for decision tree and Table III for random  forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' The test accuracies obtained through cross-validation,  displayed again in a bar chart in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 11, represented the model  performance on unseen test data more reliably than the other  accuracy numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In that bar chart, both models were seen  to yield similar accuracies with the data collected at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='5 m  and 1 m distances which expressed that a single decision tree  was sufficient to classify the data with > 94% accuracy at  these distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Though in other cases, intra-class variances  increased because of higher level of noise and the presence of  the data collected at multiple distances together which caused  the accuracies to decrease (< 90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Random forest performed  better than decision tree in later cases because of taking the  combined result from multiple uncorrelated decision trees and  thus compensating for the wrong predictions on noisy data  examples made by each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Hence, random forest were  able to generate the most generalized model for respiratory  anomaly detection as confirmed through cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' CONCLUSION AND FUTURE DIRECTIONS  In conclusion, the infrared light wave sensing system was  able to detect faulty data and breathing anomalies through  machine learning models, with a highest accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='6%  obtained by decision tree model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Random forest model was  TABLE III: Training, validation and test accuracies using  random forest model  Distances  Training  accuracy  Validation  accuracy  Test  accuracy  CV accuracy  Training  Test  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
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+page_content='4%  84%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 11: Cross-validation based test accuracy plots for decision  tree and random forest models for different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  found to be more effective than decision tree model in  classifying the data that were collected at multiple distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Further refinement of models is necessary to make it more  realistic and prevent false alarms by including training data  that has more variances in breathing frequencies, depths,  distances from the LWS setup and ambient lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Human breathing will shift in nature when in deep sleep or  exercising, hence these data could be included in the normal  breathing class of the training data to make it even more real-  istic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' More handcrafted features can be extracted to improve  the classification accuracy further, especially in the mixed  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Performance between machine learning models and  deep learning models (convolutional neural network, recurrent  neural network etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=') can be compared to see which one per-  forms better in classifying respiration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' While the proposed  work establishes the baseline of respiratory anomaly detection  using light wave sensing, it’s applicability can be further  tested in a realistic environment with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Also,  more complicated anomalous breathing patterns like Cheyne-  Stoke’s, Apneustic and Biot’s breathing can be included as  separate classes in future similar efforts to evaluate the work  in a more comprehensive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  REFERENCES  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
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+page_content=' Wang, Bootstrap Aggregating and Random  Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Cham: Springer International Publishing, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' 389–429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='1007/978-3-030-31150-6 13  ARXIV VERSION UPLOADED: JANUARY 09, 2023  13  Md Zobaer Islam received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' degree in  Electrical and Electronic Engineering in 2012 from  Bangladesh University of Engineering and Technol-  ogy, Dhaka, Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He joined Oklahoma State  University, Stillwater, OK as a graduate teaching and  research assistant to pursue his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' degree at the  School of Electrical and Computer Engineering in  Spring 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He has industry experience of 4 years  at Bangladesh Telecommunications Company Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  in telecommunication and information technology  (IT) sector and 3 years at Samsung R&D Institute  Bangladesh in software sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' His current research interests include wireless  light-wave sensing and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Brenden Martin was born in Muskogee, Oklahoma,  in the summer of 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' There he spent his childhood  disassembling electronics, drawing, programming,  and playing musical instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He received his BS  in Electrical Engineering in 2021 from Oklahoma  State University (OSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He is now conducting his  PhD studies in OSU’s Ultrafast Terahertz Optoelec-  tronics Laboratory (UTOL) at OSU, where he has  been involved since 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' His research interests  include ultrafast optoelectronics, materials science,  and condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Carly Gotcher is currently working toward BS  degree with the School of Electrical and Computer  Engineering, Oklahoma State University, Stillwater,  Oklahoma, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Her major is Electrical Engineer-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' She is an undergraduate researcher working at  Ultrafast THz Optoelectronic Laboratory (UTOL) at  Oklahoma State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Tyler Martinez is currently working toward BS  degree with the School of Electrical and Computer  Engineering, Oklahoma State University, Stillwater,  Oklahoma, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' His major is Electrical Engineer-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He worked as an undergraduate researcher at  OSU Wireless Lab (OWL) and Ultrafast THz Op-  toelectronic Laboratory (UTOL) at Oklahoma State  University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  John F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' O’Hara (SM’19) received his BSEE degree  from the University of Michigan in 1998 and his  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' (electrical engineering) from Oklahoma State  University in 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He was a Director of Central  Intelligence Postdoctoral Fellow at Los Alamos Na-  tional Laboratory (LANL) until 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' From 2006-  2011, he was with the Center for Integrated Nan-  otechnologies (LANL) and worked on numerous  metamaterial projects involving dynamic control  over chirality, resonance frequency, polarization, and  modulation of terahertz waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In 2011, he founded  a IoT, automation, and consulting/research company, Wavetech, LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' In 2017,  he joined Oklahoma State University as an Assistant Professor, where he now  studies IoT, metamaterials, terahertz communications, and photonic sensing  technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He has around 100 publications in journals and conference  proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='  Sabit Ekin (M’12, SM’21) is currently working  as an Associate Professor of the Department of  Engineering Technology and Industrial Distribution  at Texas A&M University, College Station, Texas,  USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' Previously, he was an associate professor of  Electrical and Computer Engineering at Oklahoma  State University (OSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He was the founding direc-  tor of OSU Wireless Lab (OWL) at OSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He re-  ceived the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' degree in electrical and electronics  engineering from Eskis¸ehir Osmangazi University,  Turkey, in 2006, the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' degree in electrical  engineering from New Mexico Tech, Socorro, NM, USA, in 2008, and  the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' degree in electrical and computer engineering from Texas A&M  University, College Station, TX, USA, in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' He has four years of industrial  experience as a Senior Modem Systems Engineer at Qualcomm Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=', where  he has received numerous Qualstar awards for his achievements/contributions  on cellular modem receiver design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
+page_content=' His research interests include the design  and analysis of wireless systems including mmWave and terahertz communi-  cations in both theoretical and practical point of views, visible light sensing,  communications and applications, non-contact health monitoring, and Internet  of Things applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQfLgYA/content/2301.03713v1.pdf'}
diff --git a/sdE2T4oBgHgl3EQffQc2/content/tmp_files/2301.03924v1.pdf.txt b/sdE2T4oBgHgl3EQffQc2/content/tmp_files/2301.03924v1.pdf.txt
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@@ -0,0 +1,1528 @@
+Data-driven adjoint-based calibration of port-Hamiltonian
+systems in time domain
+Michael Günther∗, Birgit Jacob†, Claudia Totzeck‡
+IMACM, School of Mathematics and Natural Sciences,
+University of Wuppertal, Germany
+January 11, 2023
+Abstract
+We present a gradient-based calibration algorithm to identify the system matrices of a linear
+port-Hamiltonian system from given input-output time data.
+Aiming for a direct structure-
+preserving approach, we employ techniques from optimal control with ordinary differential equa-
+tions and define a constrained optimization problem. The input-to-state stability is discussed
+which is the key step towards the existence of optimal controls. Further, we derive the first-order
+optimality system taking into account the port-Hamiltonian structure. Indeed, the proposed
+method preserves the skew-symmetry and positive (semi)-definiteness of the system matrices
+throughout the optimization iterations. Numerical results with perturbed and unperturbed syn-
+thetic data, as well as an example from the PHS benchmark collection [Sch22] demonstrate the
+feasibility of the approach.
+AMS classification: 37J06, 37M99, 49J15, 49K15, 49M29, 49Q12, 65P10, 93A30, 93B30, 93C05
+Keywords: Port-Hamiltonian systems, data-driven approach, optimal control, adjoint-based calibration,
+time domain, coupled dynamical systems, structure preservation
+1
+Introduction
+In structure-preserving modelling of coupled dynamical systems the port-Hamiltonian framework
+allows for constructing overall port-Hamiltonian systems (PHS) provided that (a) all subsystems are
+PHS and (b) a power-conserving interconnection between the input and outputs of the subsystems is
+provided [MM19, Sch06, EMv07, DMSB09]. In realistic applications this approach reaches its limits
+if for a specific subsystem, either no knowledge that would allow the definition of a physics-based
+PHS is available, or one is forced to use user-specified simulation packages with no information
+of the intrinsic dynamics, and thus only the input-output characteristics are available.
+In both
+cases a remedy is to generate input-output data either by physical measurements or evaluation of
+the simulation package, and derive a PHS surrogate that fits these input-output data best. This
+PHS surrogate can then be used to model the subsystem, and overall leads to a coupled PHS with
+structure-preserving properties.
+∗Research Group Applied and Computational Mathematics, guenther@uni-wuppertal.de
+†Research Group Functional Analysis, bjacob@uni-wuppertal.de
+‡Research Group Optimization, totzeck@uni-wuppertal.de
+1
+arXiv:2301.03924v1  [math.OC]  10 Jan 2023
+
+Data-driven port-Hamiltonian realizations of dynamic systems from input-output data have been
+researched using various approaches. The Loewner framework has been used to construct a port-
+Hamiltonian realization from frequency domain data in [ALI17, BGVD20]. Another frequency do-
+main approach is proposed in [Sch21], where a parametrization of the class of PHS is used to permit
+the usage of unconstrained optimization solvers during identification. In [CGaB22] the frequency
+response data are inferred using the time-domain input-output data and then frequency domain
+methods are used to construct a port-Hamiltonian realization. Further, in [CMH19] a best-fit linear
+state-space model is derived from time-domain data and then, in a post-processing step, a nearest
+port-Hamiltonian realization is inferred. Based on input, state and output time-domain data and a
+given quadratic Hamiltonian the authors of [MNU22] construct a port-Hamiltonian realization using
+dynamic mode decompositions.
+We propose a direct time-domain approach for constructing a best-fit PHS model in one step
+using input-output data. Techniques from optimal control with ordinary differential equations and a
+constrained optimization problem are employed. We derive the first-order optimality system taking
+into account the port-Hamiltonian structure. The proposed method preserves the skew-symmetry
+and positive (semi)-definiteness of the system matrices throughout the optimization iterations. Our
+approach does not generate frequency data as an intermediate step and data of the state variable are
+not needed. We remark, that our method is overdetermined because we identify all system param-
+eters including the quadratic Hamiltonian. This is interesting for example when the Hamiltonian
+contains unknown physical parameters as the spring constant or masses. However, if the Hamiltonian
+is known it is possible to use our approach to derive a best-fit PHS with a given Hamiltonian.
+The paper is organized as follows: in the next section, we define the calibration problem of
+computing the best-fit of a PHS to given input-output data, prove the continuous dependence of
+the state on the input which is the key step to obtain the existence of solutions to the calibration
+problem. The adjoints for all system matrices, i.e., the positive definite scaling matrix Q, the fixed-
+rank semi-definite dissipation matrix R, the skew-symmetric system matrix J, the input matrix B
+and the initial value, are derived in section three. The gradient-descent algorithm and numerical
+schemes for the calibration process are stated in chapter four. Before we show numerical results
+for synthetic data with and without perturbation in section five to validate our algorithm. The
+numerical examples are rounded off with a single mass-spring-damper chain example from the PHS
+benchmark collection [Sch22]. We conclude with an outlook.
+2
+Calibration problem
+For a given input u: [0, T] → Rm, a data set ydata : [0, T] → Rm which can consist of continuous
+data on the interval [0, T] for T > 0 or continuous interpolation of measurements at discrete time
+points and reference values for the calibration wref, we consider the calibration problem with cost
+functional
+J (y, w) = 1
+2
+� T
+0
+|y(t) − ydata(t)|2dt + λ
+2 |w − wref|2,
+λ ≥ 0,
+(1)
+subject to the state constraint
+d
+dtx = (J − R)Qx + Bu,
+x(0) = x0 ∈ Rn,
+(2a)
+y = B⊤Qx.
+(2b)
+2
+
+where w = (J, R, Q, B, x0) contains the matrices and initial data to be identified having the proper-
+ties
+J, R, Q ∈ Rn×n,
+J⊤ = −J,
+R ≽ 0,
+Q ≻ 0,
+B ∈ Rn×m.
+(3)
+Note that only output data and for λ > 0 reference values are considered in the cost functional.
+Hence, the dimension n of the internal state is unknown. We assume in the following that n is
+chosen a priori and remark that a deliberate small choice can also be interpreted as a model order
+reduction for the internal state. For notational convenience, we introduce the notation
+J1(y) := 1
+2
+� T
+0
+|y(t) − ydata(t)|2dt,
+J2(w) := λ
+2 |w − wref|2.
+In applications often m ≤ n hence we obtain only partial information about the internal state x. In
+particular, we have no information about the initial condition x0. This is the reason why we include
+the initial data in the identification process for w. As we will use techniques from optimal control
+theory, we may refer to w as control in the following and introduce the space
+W = Rn×n × Rn×n × Rn×n × Rn×m × Rn
+with set of admissible controls as
+Uad = {w = (J, R, Q, B, x0) ∈ W : J⊤ = −J, R ≽ 0, Q ≻ 0}.
+For later reference we define the identification task at hand:
+Problem 1. We seek to find system matrices and initial data, w = (J, R, Q, B, x0), which solve the
+problem
+min
+(y,w)∈Y ×Uad
+J (y, w)
+subject to
+(2).
+(IdP)
+In [GJT23], we proposed a sensitivity-based approach to identify the system matrices of a port-
+Hamiltonian system and argued that this is a valid ansatz for small-scale systems, as it requires to
+solve auxilliary problems for each basis element of the matrix spaces. Here, we derive an adjoint-
+based approach. Before we begin the derivation of the algorithm, we analyze the well-posedness of
+the calibration problem. First, we prove that the state solution depends continuously on the data.
+Lemma 2. For every w ∈ Uad and u ∈ C([0, T], Rm) the state equation (2) admits a unique solution
+x ∈ C1([0, T], Rn). Moreover, the solution depends continuously on the data, in more detail, for
+w, w′ ∈ Uad and corresponding solutions x, x′ there exists a constant C > 0 such that
+∥x − x′∥H1((0,T),Rn) ≤ C∥w − w′∥Uad.
+Proof. For given w ∈ Uad and u ∈ C([0, T], Rm) the existence of a unique solution to (2) follows
+by standard ODE theory [Tes12]. For the second statement we estimate ∥x − x′∥L2((0,T),Rn) and
+∥ d
+dtx − d
+dtx′∥L2((0,T),Rn) separately.
+We obtain
+|x(t) − x′(t)|2 ≤ 2|x0 − x′
+0|2 + 2T
+� t
+0
+|(J − R)Qx(s) − (J′ − R′)Q′x′(s)|2ds
+≤ C1|w − w′|2 +
+� t
+0
+C2|x(s) − x′(s)|2ds
+3
+
+An application of Gronwall inequality yields
+|x(t) − x′(t)|2 ≤ C3|w − w′|2.
+Integration over [0, T] we obtain
+∥x − x′∥2
+L2((0,T),Rn) ≤ C4|w − w′|2.
+(4)
+Moreover, we obtain for
+| d
+dt
+�
+x(t) − x′(t)
+�
+|2 = C5|w − w′|2 + C6
+� t
+0
+| d
+ds
+�
+x(s) − x′(s)
+�
+|2ds.
+Again, Gronwall and integration over [0, T] yields
+∥ d
+dtx(t) − x′(t)∥2
+L2((0,T),Rn) ≤ C7|w − w′|2
+(5)
+Adding (4) and (5) and taking square root yield the desired result.
+The well-posedness of the state equation allows us to introduce the solution operator
+S : Uad → H1((0, T), Rn),
+S(w) = x.
+Moreover, we define the reduced cost functional
+ˆ
+J (w) = 1
+2
+� T
+0
+|B⊤QS(w)(t) − ydata(t)|2dt + λ
+2 |w − wref|2.
+Both will be helpful in the proof of the well-posedness result of the identification problem and also
+play a major role in the derivation of the gradient-descent algorithm later on.
+The next ingredient for the well-posedness result is the weak convergence of the state operator
+e: H1((0, T), Rn) × Uad → L2((0, T), Rn),
+(x, w) �→
+� d
+dtx − (J − R)Qx − Bu
+x(0) − x0
+�
+.
+(6)
+Lemma 3. The state operator e defined in (6) is weakly continuous.
+Proof. We consider {xn}n ⊂ H1((0, T), Rn) with xn ⇀ x and {wn}n ⊂ Uad with wn ⇀ w. Note that
+Uad is finite dimensional, thus weak and strong convergence coincide. Then for ϕ ∈ L2((0, T), Rn)
+we obtain
+� T
+0
+� d
+dtxn − (Jn − Rn)Qnxn(t) − Bnu(t)
+�
+ϕ(t)dt −
+� T
+0
+� d
+dtx − (J − R)Qx(t) − Bnu(t)
+�
+ϕ(t)dt
+=
+� T
+0
+� d
+dtxn − d
+dtx
+�
+ϕ(t) dt −
+� T
+0
+((Jn − J) − (Rn − R))Qnxn(t)ϕ(t)dt
+−
+� T
+0
+(J − R)(Qn − Q)xn(t)ϕ(t) + (J − R)Q(xn(t) − x(t))ϕ(t) + (Bn − B)u(t)ϕ(t)dt.
+By the weak convergence of xn and wn we conclude that all terms tends to zero as n → ∞. Note that
+weak convergence of {xn}n implies boundedness of the sequence {xn}n. Moreover, by the embedding
+H1((0, T), Rn) �→ C([0, T], Rn) we conclude that xn(0) → x(0) and further we obtain (x0)n → x0
+by the convergence of wn. Altogether, this yields the desired result.
+4
+
+Theorem 4. Let λ > 0. Then (IdP) admits a global minimum.
+Proof. We follow the lines of [Trö10, HPUU08] and consider a minimizing sequence {wn}n, that
+means lim
+n→∞
+ˆ
+J (wn) =
+inf
+w∈Uad
+ˆ
+J (w).
+The assumption λ > 0 yields that ˆ
+J (w) is coercive w.r.t. w and hence the boundedness of the
+minimization sequence. As Uad is a subset of a reflexive, finite-dimensional space, we can extract a
+convergent subsequence {wnk}k with wnk → w∗. Note that w = 0 implies z ≡ 0, hence the continuous
+dependence on the data shown in Lemma 2 implies the boundedness of {S(wnk)}k ⊂ H1((0, T), Rn)
+and we can extract a weakly convergent subsequence S(wnkℓ) ⇀ x in H1((0, T), Rn). The weak
+continuity of the state operator shown in Lemma 3 yields
+e(S(wnkℓ), wnkℓ) ⇀ e(x, w)
+and the weak lower semicontinuity of the norm allows us to estimate
+∥e(x, w)∥L2((0,T),Rn) ≤ lim inf
+ℓ→∞ ∥e(S(wnkℓ), wnkℓ)∥L2((0,T),Rn) = 0,
+which proves that S(w) = x. We note that the weak lower semicontinuity of J implies the weak
+lower semicontinuity of ˆ
+J , hence
+ˆ
+J (w) ≤ lim
+ℓ→∞
+ˆ
+J (wnkℓ) =
+inf
+w∈Uad
+ˆ
+J (w).
+This proves the existence of a minimizer.
+Remark. We want to emphasize that even though the state equation is linear in x, it is non-linear
+in w. We therefore cannot expect to obtain a uniqueness result for (IdP).
+Here, we aim for a general approach, also feasible for high-dimensional systems, and therefore
+discuss an adjoint-based approach in the following. A challenge in this context is the structure of the
+system matrices (3) which we aim to preserve in each step of the calibration process. To this end,
+we employ techniques from optimization on manifolds to compute the retraction for Q and preserve
+its positive definiteness in each iteration. To the authors’ knowledge, there is no explicit formula for
+the retraction in the space of semi-definite matrices. Hence, we use the flat metric for R and as the
+space of skew-symmetric matrices as well as Rn×m and Rn are vector spaces, we are naturally in the
+flat case for J, B and x0, respectively.
+One of our objectives is the derivation of a gradient-based algorithm for the calibration. We
+derive the adjoint-based gradient descent scheme step-wise, i.e., we discuss the computation for
+each of the different matrix structures separately and discuss a proof of concept with (randomly
+perturbed) synthetic data.
+3
+First-order optimality system
+As the system matrices have different structure, we organize the derivation of the adjoints in sub-
+sections. At the end of the section we combine the intermediate results to derive the optimality
+system for Problem 1. We emphasize that we can resort to the subproblems for the derivation of
+the adjoint, since the controls appear bilinearly in the state equation.
+5
+
+3.1
+Positive definite matrices Q
+In this section we consider a toy problem with positive definite system matrix Q and employ find-
+ings of [SH15] where conic geometric optimization on the manifold of positive definite matrices is
+discussed. Let Pn denote the set of Hermitian positive definite (HPD) matrices and Hn the set of
+n × n Hermitian matrices. For simplicity we consider
+d
+dtz = Qz,
+z(0) = z0,
+(7)
+y = B⊤Qz
+(8)
+with control Q ∈ Pn together with the cost functional
+J (z, Q) = 1
+2
+� T
+0
+∥y(t) − ydata(t)∥2dt + J2(Q)
+and denote the first part by J1(z) =
+1
+2
+� T
+0 ∥y(t) − ydata(t)∥dt and the second part by J2(Q) =
+λ∥ log(Q)∥2
+2. Note that the simple structure allows to solve (7) explicity. Indeed, z(t) = exp(Qt)z0,
+where exp denotes the matrix exponential
+exp(Qt) =
+∞
+�
+k=0
+(Qt)k
+k!
+= I + Qt + (Qt)2
+2
++ (Qt)3
+6
++ · · ·
+From [SH15] we know that the geodesic along the manifold of HPD matrices and passing through
+γ(0) = Q with ˙γ = ξ ∈ Hn is given by
+γ(t) = Q1/2 exp(tQ−1/2ξQ−1/2)Q1/2.
+(9)
+In the following we use this information to derive the first-order optimality system of our optimization
+problem. In particular, we will use the expansion
+γ(t) = Q + ξt + ξQ−1ξt2 + o(t2).
+Let us begin with the well-posedness result of the state problem, which is a direct application of
+Picard-Lindelöf.
+Theorem 5. Let Q ∈ Pn and z0 ∈ Rn be given.
+The state problem (7) has a unique solution
+z ∈ C1([0, T], Rn).
+The uniqueness result in Theorem 5 allows us to define the solution operator for the toy problem
+S : {Q ∈ Rn×n : Q ≻ 0} → H1((0, T), Rn),
+S(Q; z0) = z.
+Moreover, we obtain the reduced cost functional
+ˆ
+J (Q) = 1
+2
+� T
+0
+∥B⊤QS(Q; z0)(t) − ydata(t)∥2dt + λ∥ log Q∥2
+2.
+The second term of the cost functional measures the distance of Q to the identity matrix w.r.t. the
+Thompson metric [SH15].
+6
+
+We aim to derive a gradient descent scheme for ˆ
+J (Q). The directional derivative of ˆ
+J is given
+by
+lim
+ϵ→0
+ˆ
+J (γ(ϵ)) − ˆ
+J (γ(0))
+ϵ
+= lim
+ϵ→0
+J1(zϵ) − J1(z) + J2(γ(ϵ)) − J2(γ(0))
+ϵ
+.
+Since it is not a priori clear how variations in Q influence the solution S(Q; z0) we use an adjoint
+approach. The first task is therefore to identify the adjoint equation.
+Let zϵ denote the solution of
+d
+dtzϵ = γ(ϵ)zϵ,
+zϵ(0) = z0,
+where γ is defined in (9).
+Note that it holds
+d
+dt(zϵ − z) = Q(zϵ − z) + ϵξzϵ + o(e)
+Moreover, let zh be the solution to
+d
+dtzh = (Q + ϵξ)zh,
+zh(0) = z0.
+(10)
+Then
+d
+dt(zϵ − zh) = (Q + ϵξ)(zϵ − zh) + o(ϵ)
+and by Gronwall inequality
+∥zϵ(t) − zh(t)∥ ≤ o(ϵ) exp(t∥Q + ϵξ∥) ϵ→0
+−→ 0.
+(11)
+Note that the difference ψh(t) = zh(t) − z(t) satisfies the equation
+d
+dtψh = Qψh + ϵξzh,
+ψh(0) = 0.
+Standard ODE theory [Tes12] yields the expression
+ψh(t) = ϵ
+� t
+0
+exp(Q(t − s))ξzh(s)ds
+(12)
+and for ϵ → 0 we obtain ψh ≡ 0. We are now able to state the first-order necessary condition for
+(z, Q) to be a stationary point.
+Theorem 6. Let (¯z, ¯Q) be an optimal pair. Then for any ξ ∈ Hn it holds that
+dJ2( ¯Q)[ξ] +
+� T
+0
+⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0,
+(13)
+where ψ is the solution to
+d
+dtψ = Qψ + ξz, ψ(0) = 0.
+Proof. Let γ(t) be the geodesic through ¯Q with
+d
+dtγ(t) = ξ, zϵ the solution to
+d
+dtzϵ = γ(ϵ)zϵ with
+zϵ(0) = z0, d
+dtzh = ( ¯Q + ϵξ)zh, zh(0) = z0 and ψh as in (12). We obtain
+J1(zϵ) − J1(¯z) = J1(zh) − J1(¯z) + o(ϵ)
+=
+� T
+0
+(¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)
+= ϵ
+� T
+0
+�
+¯y(t) − ydata(t), B⊤Qψh(t)
+�
+dt + o(ϵ)
+7
+
+Owing to the minimality of ¯Q we find
+0 ≤
+ˆ
+J (γ(ϵ)) − ˆ
+J ( ¯Q)
+ϵ
+= dJ2( ¯Q)[ξ] +
+� T
+0
+�
+y(t) − ydata(t), B⊤Qψh(t)
+�
+dt + O(ϵ).
+Passing to the limit ϵ → 0+ yields
+0 ≤ dJ2( ¯Q)[ξ] +
+� T
+0
+�
+y(t) − ydata(t), B⊤Qψ(t)
+�
+dt
+for any ξ ∈ Hn. Note that a sign change in ξ leads to a change of the sign of ψ, which yields the
+desired equality.
+We aim for an adjoint-based representation of the first-order optimality system in order to provide
+a gradient descent algorithm for the numerical approximation. The dual problem corresponding to
+d
+dtψ = Qψ + ξz with ψ(0) = 0 is obtained by testing with p
+0 =
+� d
+dtψ − Qψ − ξz, p
+�
+=
+�
+p · ψ
+�T
+0 −
+� T
+0
+d
+dtp · ψ + ψ · Q⊤p + z · ξ⊤p dt.
+Choosing p to satisfy the adjoint equation
+− d
+dtp = Q⊤p + QB(y(t) − ydata(t)),
+p(T) = 0,
+yields together with the optimality condition (13) that
+dJ2( ¯Q)[ξ] −
+� T
+0
+p · ξz dt = 0
+for all
+ξ ∈ Hn.
+For the choice J2(Q) = 1
+2∥ log(Q)∥2
+2 we obtain dJ2( ¯Q)[ξ] = (Q−1 log(Q), ξ) (see [SH15]) and hence
+we can identify the optimality condition as
+∇J2( ¯Q) −
+� T
+0
+¯z ⊗ ¯p dt = 0,
+where a ⊗ b denotes the dyadic product ab⊤ for a, b ∈ Rn. For J2(Q) = 1
+2∥ log(Q)∥2
+2 we obtain
+∇J2(Q) = Q−1 log ¯Q.
+3.2
+Fixed-rank semi-definite matrices R
+Our preferred approach in case of semi-definite matrices would follow the same lines as above for
+positive definite matrices. First of all, in full generality there is no hope, hence we have to restrict
+to the case of fixed-rank semi-definite matrices. This would lead us to the polar factorization of the
+semi-definite matrix R given by R = GU with G ∈ {Rn×r : det(GG⊤) ̸= 0} and U ∈ St(n, r) with
+St(n, r) = {Rn×r : UU⊤ = I} is the Stiefel manifold of r-dimensional orthonormal bases in Rn and
+P ∈ S+(r). Unfortunately, there is no explicit expression for the retraction of U [MBS11], hence
+we consider the flat metric and the decomposition R = GG⊤ with G as above and consider the toy
+problem with cost functional
+ˆ
+J (G) = 1
+2
+� T
+0
+∥y(t) − ydata(t)∥2dt + J2(GG⊤)
+(14)
+8
+
+and state equation
+d
+dtz = −Rz,
+z(0) = z0,
+(15)
+y = B⊤Qz
+(16)
+with control R ∈ S+(r, n) a fixed-rank positive semi-definite matrix. Note that this implies the
+assumption that we know the number of conserved and dissipative variables a priori. Note further
+that we already stated the reduced cost functional only depending on G as we obtain the existence
+and uniqueness result and therefore the control-to-state map analogous to the previous section.
+A horizontal tangent vector ξG at G is given by
+ξG = Sym(∆)G,
+∆ ∈ Rd×d,
+where Sym(∆) denotes the symmetric part of ∆. Since we consider the flat metric, the exponential
+mapping reads
+ExpG(ξG) = G + ξG.
+Let us denote the geodesic through G by
+γ(t) = G + tξG.
+It satisfies ˙γ = ξG. We proceed as above and define zϵ as the solution to
+d
+dtzϵ = −γ(ϵ)γ(ϵ)⊤zϵ,
+zϵ(0) = z0.
+Then it holds
+d
+dt(zϵ − z) = −
+�
+(GG⊤)(zϵ − z) + ϵ(Gξ⊤
+G + ξGG⊤)zϵ�
++ o(ϵ)
+We note that zh and zϵ of the previous section coincide here, as we are in the flat metric. Following
+the lines above, we find
+d
+dtψϵ = −GG⊤ψϵ − (Gξ⊤
+G + ξGG⊤)zϵ,
+ψϵ(0) = 0
+(17)
+and ψϵ ≡ 0 for ϵ → 0. The optimality condition reads as follows.
+Theorem 7. Let (¯z, ¯R) be an optimal pair. Then for any ξG = Sym(∆)G with ∆ ∈ Rd×d it holds
+that
+dJ2( ¯R)[ξG] +
+� T
+0
+⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0
+(18)
+where ψ is the solution to
+d
+dtψ = GG⊤ψ + (Gξ⊤
+G + ξ⊤
+GG)z, ψ(0) = 0.
+Proof. Let zϵ the solution to d
+dtzϵ = −γ(ϵ)γ(ϵ)⊤zϵ with zϵ(0) = z0 and ψϵ as in (17). We obtain
+J1(zϵ) − J1(¯z) = J1(zh) − J1(¯z) + o(ϵ) =
+� T
+0
+(¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)
+= ϵ
+� T
+0
+�
+¯y(t) − ydata(t), B⊤Qψϵ(t)
+�
+dt + o(ϵ)
+9
+
+Owing to the minimality of ¯R we find
+0 ≤
+ˆ
+J (γ(ϵ)) − ˆ
+J ( ¯R)
+ϵ
+= dJ2( ¯R)[ξG] +
+� T
+0
+�
+y(t) − ydata(t), B⊤Qψϵ(t)
+�
+dt + O(ϵ).
+Passing to the limit ϵ → 0+ yields
+0 ≤ dJ2( ¯R)[ξG] +
+� T
+0
+�
+y(t) − ydata(t), B⊤Qψ(t)
+�
+dt
+for any ξ ∈ Hn. Note that a sign change in ξ leads to a change of the sign of ψ, which yields the
+desired equality.
+To derive the adjoint equation we test by p and obtain
+0 = ⟨ d
+dtψ + GG⊤ψ + (Gξ⊤
+G + ξGG⊤)z, p⟩
+=
+�
+p · ψ
+�T
+0 −
+� T
+0
+d
+dtp · ψ − GG⊤p · ψ − p · (Gξ⊤
+G + ξGG⊤)z dt
+Choosing p to satisfy the dual problem
+− d
+dtp = −GG⊤p + QB(y − ydata),
+p(T) = 0.
+allows us to rewrite the optimality condition (18) as
+dJ2( ¯R)[ξ] +
+� T
+0
+p · (Gξ⊤
+G + ξGG⊤)z dt = 0
+for all ξG,
+or
+dJ2( ¯R)[Sym(∆)G] +
+� T
+0
+p · (GG⊤ Sym(∆) + Sym(∆)GG⊤)z dt = 0
+for all Sym(∆).
+Using R = GG⊤ we find
+∇J2( ¯J) −
+� T
+0
+z ⊗ GG⊤p + p ⊗ GG⊤z dt = ∇J2( ¯J) −
+� T
+0
+z ⊗ Rp + p ⊗ Rz dt = 0.
+3.3
+Skew-symmetric matrices J
+The skew-symmetric matrices of size n × n, n ∈ N, are a vector space, hence we are naturally in the
+flat metric. By analogy with above, we consider the toy problem with cost functional
+ˆ
+J (J) = 1
+2
+� T
+0
+∥y(t) − ydata(t)∥2dt + J2(J)
+(19)
+and state equation
+d
+dtz = Jz,
+z(0) = z0
+(20)
+y = B⊤Qz
+10
+
+with control J ∈ Rn×n skew-symmetric, i.e. J⊤ = −J. As we are in the vector space setting we
+let ∆ ∈ Rn×n and consider
+d
+dtzh = (J + ϵ Skew(∆))zh,
+zh(0) = z0.
+Then ψh = zh − z satisfies the equation
+d
+dtψh = Jψh + ϵ Skew(∆)zh,
+ψ(0) = 0.
+(21)
+and ψh ≡ 0 as ϵ → 0. We obtain the optimality condition
+Theorem 8. Let (¯z, ¯J) be an optimal pair. Then for any ξJ = Skew(∆)G with ∆ ∈ Rn×n it holds
+that
+dJ2( ¯J)[Skew(∆)] +
+� T
+0
+⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0
+(22)
+where ψ is the solution to
+d
+dtψ = Jψ + Skew(∆)z, ψ(0) = 0.
+Proof. Let zh the solution to
+d
+dtzh = (J + ϵ Skew(∆))zh with zh(0) = z0 and ψh as in (21). We
+obtain
+J1(zϵ) − J1(¯z) = J1(zh) − J1(¯z) + o(ϵ) =
+� T
+0
+(¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)
+= ϵ
+� T
+0
+�
+¯y(t) − ydata(t), B⊤Qψh(t)
+�
+dt + o(ϵ)
+Owing to the minimality of ¯J we find
+0 ≤
+ˆ
+J ( ¯J + ϵ Skew(∆)) − ˆ
+J ( ¯J)
+ϵ
+= dJ2( ¯J)[Skew(∆)] +
+� T
+0
+�
+y(t) − ydata(t), B⊤Qψh(t)
+�
+dt + O(ϵ).
+Passing to the limit ϵ → 0+ yields
+0 ≤ dJ2( ¯J)[Skew(∆)] +
+� T
+0
+�
+y(t) − ydata(t), B⊤Qψ(t)
+�
+dt
+for any ξ ∈ Hn. Note that a sign change in ξ leads to a change of the sign of ψ, which yields the
+desired equality.
+To derive the adjoint equation we test by p and obtain
+0 = ⟨ d
+dtψ − Jψ − Skew(∆)z, p⟩
+=
+�
+p · ψ
+�T
+0 −
+� T
+0
+d
+dtp · ψ + J⊤p · ψ + Skew(∆)⊤p · z dt
+=
+�
+p · ψ
+�T
+0 −
+� T
+0
+d
+dtp · ψ − Jp · ψ − Skew(∆)p · z dt
+Choosing p to satisfy the dual problem
+− d
+dtp = J⊤p + QB(y − ydata),
+p(T) = 0,
+11
+
+we obtain the optimality condition
+0 = dJ2( ¯J)[Skew(∆)] −
+� T
+0
+Skew(∆)p · zdt
+for all ∆ ∈ Rn×n
+leading to the expression
+∇J2( ¯J) −
+� T
+0
+p ⊗ z dt = 0.
+3.4
+Input matrix B
+The input matrix B plays a special role in the dynamics as is acts only on the input u and not
+directly on the state. Let H ∈ Rn×m be arbitrary, we compute the directional derivative of the
+output w.r.t. the control B in direction H as
+d
+dB y(t)[H] = H⊤Qx(t) + B⊤Q
+� t
+0
+Hu(s)ds.
+Hence, we obtain for the cost functional
+J (y, w) = 1
+2
+� T
+0
+∥y(t) − ydata(t)∥2dt + J2(B)
+the variation in direction H by
+d
+dB J (y, w)[H] =
+�
+y − ydata, H⊤BQx + B⊤Q
+� t
+0
+u(s)ds
+�
+Thus, the variational lemma, allows to identify the optimality condition w.r.t. B as
+∇J2(B) +
+� T
+0
+Qx ⊗ (y − ydata) +
+� t
+0
+u(s)ds ⊗ Q⊤B(y − ydata)dt = 0.
+3.5
+Initial data z0 and optimality system
+Our calibration problem is assumed to be data-driven, i.e., we only have knowledge about the output
+y. In the above derivations, the initial condition z0 were assumed to be given. We consider two cases:
+first, we assume that the equilibrium of the system is zero, i.e., as long as there is no input it is
+reasonable to set z0 = 0 and the input can steer the system out of equilibrium. Second, we estimate
+z0 from data.
+For the second case, we derive the optimaliy conditions as follows. Consider the dynamics
+d
+dtz = (J − R)z + Bu,
+z(0) = z0
+y = B⊤Qz
+and z0 is to be estimated from data. By analogy with above, we consider the cost functional
+ˆ
+J (z0) = 1
+2
+� T
+0
+∥y(t) − ydata(t)∥2dt + J2(z0).
+(23)
+12
+
+The control variable is now z0 hence we obtain the linearization
+d
+dtzh = (J − R)Qzh,
+zh(0) = z0 + ϵξ0
+leading us to ψh with dynamics
+d
+dtψh = (J − R)Qψh,
+ψh(0) = ϵξ0
+(24)
+and ψh ≡ 0 as ϵ → 0.
+Theorem 9. Let (¯z, ¯z0) be an optimal pair. Then for any ξ0 ∈ Rn it holds that
+dJ2(¯x0)[ξ0] +
+� T
+0
+⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0
+(25)
+where ψ is the solution to
+d
+dtψ = Q(J⊤ − R⊤)z, ψ(0) = ξ0.
+Proof. Let zh the solution to d
+dtzh = (J −R)Qzh with zz(0) = z0 +ϵξ0 and ψh as in (24). We obtain
+ˆ
+J (z0 + ϵξ0) − ˆ
+J (¯z0) =
+� T
+0
+(¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)
+= ϵ
+� T
+0
+�
+¯y(t) − ydata(t), B⊤Qψh(t)
+�
+dt + o(ϵ)
+Owing to the minimality of ¯z0 we find
+0 ≤
+ˆ
+J (¯z0 + ϵξ0) − ˆ
+J (¯z0)
+ϵ
+=
+� T
+0
+�
+¯y(t) − ydata(t), B⊤Qψh(t)
+�
+dt + O(ϵ).
+Passing to the limit ϵ → 0+ yields
+0 ≤
+� T
+0
+�
+¯y(t) − ydata(t), B⊤Qψ(t)
+�
+dt
+for any ξ0 ∈ Rn. Note that a sign change in ξ0 leads to a change of the sign of ψ, which yields the
+desired equality.
+Testing the (24) with p yields
+0 = ⟨ d
+dtψ − (J − R)Qψ, p⟩
+= p(T) · ψ(T) − p(0) · ξ0 −
+� T
+0
+d
+dtp · ψ + Q(J⊤ − R⊤)p · ψ dt
+Choosing p to satisfy the adjoint equation
+− d
+dtp = Q(J⊤ − R⊤)p + QB(y − ydata),
+p(T) = 0.
+which can be rewritten using the matrix properties as
+− d
+dtp = −Q(J + R)p + QB(y − ydata),
+p(T) = 0
+yields the optimality condition
+∇J2(x0) − p(0) = 0.
+13
+
+3.6
+Derivation of the gradient
+We are now well-prepared to identify the gradient of our calibration problem. We discuss the case
+including the estimation of the initial state, if this is needless the gradient update with respect to z0
+can be neglected. We have already seen that the solution operator allows us to define the reduced
+cost function which which leads us to an reformulation of (IdP) as unconstrained optimization
+problem, i.e., the state constraint is treated implicitly. The gradient we derive in the following is
+the one of the reduced cost functional. In fact, the whole gradient-descent algorithm will only vary
+w and thereby vary the state implicitly.
+Let w = (Q, R, J, B, z0) the control variables or, to be more precise, the matrices and initial
+data to be identified. We denote by ⟨·, ·⟩W ∗,W : W ∗, W → R the dual pairing of W and its dual W ∗.
+Moreover, A∗ denotes the adjoint of the operator A. To identify the gradient of the reduced cost
+functional we first note that the state operator yields
+0 = dye(S(w), w)S′(w)[h] + dwe(S(w), w)[h]
+⇒
+S′(w) = −dye(S(w), w)−1dwe(S(w), w)[h].
+This allows us to find
+⟨ ˆ
+J ′(w), h⟩W ∗,W = ⟨dyJ (y, w), B⊤QS′(w)[h]⟩H−1,H1 + ⟨dwJ (y, w), h⟩W ∗,W
+= ⟨dwJ (y, w) − dwe(S(w), w)∗dye(S(w), w)−∗[QB dyJ (y, w)], h⟩W ∗,W
+Since W is a Hilbert space and h was chosen arbitrarily, Riesz representation theorem allows us to
+identify
+(∇J (w), h)W = ⟨dwJ (y, w) − dwe(S(w), w)∗dye(S(w), w)−∗[QB dyJ (y, w)], h⟩W ∗,W .
+(26)
+In the previous subsections we computed the adjoint equation already and the gradient compo-
+nents corresponding (Q, R, J, B, x0), respectively. We summarize the result:
+Theorem 10. A stationary point ¯w = ( ¯Q, ¯R, ¯J, ¯B, ¯x0) of Problem 1 satisfies the optimality condition
+∇QJ2( ¯w) +
+� T
+0
+¯x ⊗ ¯p dt = 0,
+(27a)
+∇RJ2( ¯w) −
+� T
+0
+¯x ⊗ ¯R¯p + ¯p ⊗ ¯R¯x dt = 0,
+(27b)
+∇JJ2( ¯w) +
+� T
+0
+¯p ⊗ ¯x dt = 0,
+(27c)
+∇x0J2( ¯w) + ¯p(0) = 0,
+(27d)
+∇BJ2( ¯w) +
+� T
+0
+¯Q¯x ⊗ (¯y − ydata) +
+� t
+0
+u(s)ds ⊗ ¯Q⊤ ¯B(¯y − ydata)dt = 0,
+(27e)
+where ¯p satisfies the adjoint equation
+− d
+dt ¯p = ¯Q( ¯J⊤ − ¯R⊤)¯p + ¯Q ¯B(¯y − ydata),
+¯p(T) = 0.
+(28)
+and ¯x the state equation with output ¯y given by
+d
+dt ¯x = ( ¯J − ¯R) ¯Q¯x + ¯Bu,
+¯x(0) = ¯x0,
+¯y = ¯B⊤ ¯Q¯x.
+14
+
+We emphasize that the computation (26) shows that the left-hand side of system (27) is in fact
+the gradient of the reduced cost functional at ¯w. We will use this information in the gradient-descent
+algorithm proposed in the following section.
+4
+Algorithm and numerical schemes
+An optimal solution has to satisfy the conditions in Theorem 10 all at once. However, due to the
+forward/backward structure of coupled state and adjoint equation it is in general difficult to solve the
+system directly. This is the reason why we propose an iterative approach based on the gradient (27).
+In fact, for an initial guess of systems matrices and initial condition w we solve the state equation
+(2) and obtain S(w), using this information we solve the corresponding adjoint equation (28). The
+information of the state and adjoint solutions allows us to evaluate the gradient at w with ∇ ˆ
+J (w)
+we update our initial guess for the second iteration using a step size obtained by Armijo rule with
+initial stepsize σ0 [HPUU08]. We proceed with this iteration until an appropriate stopping criterion
+is fulfilled.
+As stopping criterion we check the relative cost: First, we can stop is the gradient
+vanishes numerically, i.e., ∥∇ ˆ
+J (w)∥ < ϵ for 0 < ϵ ≪ 1. Another option is to stop the iteration,
+if the update from one to the next gradient step ∥∇ ˆ
+J (wk) − ∇J (wk+1)∥ < ϵ is sufficiently small.
+Additionally, we can impose a maximal number of iterations to stop the iteration. We summarize
+the steps in Algorihm 1.
+Algorithm 1: Gradient-descent algorithm for the identification process
+Data: initial guess w, algorithmic parameters, stopping criterion;
+Result: optimized matrices and initial data ¯w
+1) solve state equation → S(w);
+2) solve adjoint equation → p;
+3) evaluate gradient (27)→ ∇J (w);
+4) find admissible step size with Armijo rule → σ;
+5) update control w �→ w + σ∇ ˆ
+J (w);
+6) if stopping criterion is not fulfilled → go to 1);
+else return optimal control;
+For the numerical results presented in the next section, we use the following algorithmic parame-
+ters unless explicitly stated otherwise: maximal number of gradient iterations 100, initial step size for
+Armijo rule 1. The time step size of the explicit Euler discretizations of the state and adjoint ODE,
+respectively, is 0.001 and the upper bound of the time interval considered for the ODEs is T = 1.
+The integrals in the gradient expressions are computed with an simple left-endpoint approximation.
+In order to avoid numerical errors to interfere with the structure of the matrices, we check
+for skew-symmetry or symmetry, respectively, and if needed overwrite the new iterate for J by
+J �→ 1
+2(J − J⊤) and analogous Q by Q �→ 1
+2(Q + Q⊤) and R by R �→ 1
+2(R + R⊤). As we assume to
+have no knowledge about the systems, we use no reference values for the tests and set J2(w) ≡ 0.
+Moreover, we set m = 2 for all the numerical tests and use the inputs (see Figure 1)
+u: [0, T] → R2,
+u(t) =
+�10 sin(2πt)
+5 cos(2πt)
+�
+.
+We initialize B with an identity block in the first m × m entries and zero everywhere else. The
+matrix Q representing the Hamiltonian is initialized as the n × n identity matrix. For R we set
+the first n/10 × n/10 block to identity and zero everywhere else. Of course it is nontrivial to find
+15
+
+Figure 1: Input signals u (left) and corresponding (uncontrolled) output (right).
+a good initial guess for R in the dynamics as in particular the number of dissipative elements is
+unknown. Furthermore, we assume that the system is in equilibrium if no input applies, therefore
+we refrain to estimate the initial state x0 and set x0 ≡ 0. To generate a feasible initial J we draw an
+n × n matrix with independent uniformly distributed entries in [−1, 1], take the upper part of this
+matrix (starting with the first off-diagonal) and fill the lower part of the matrix by the negative of
+the transpose of the upper part.
+5
+Proof of concept by numerical tests
+To underline the feasibility of the proposed approach, we show three test cases. First, we generate
+synthetic data and show that Algorithm 1 is able to find system matrices that approximate the
+data set. Then we repeat the test with a randomly perturbed data set. Our third test case is the
+single mass-spring-damper chain from the PHS benchmark collection [Sch22]. The code is publicly
+available on github1.
+5.1
+Synthetic data (deterministic)
+In the following we show results for a synthetic data set with n = 20 and m = 2. The reference
+output is generated using random matrices with appropriate structure. The output from the data
+set is plotted in Figure 2 (left) and the output of the optimized system is shown in Figure 2 (right).
+A visual comparison shows that the dynamics is well-approximated, only small variations can be
+seen at the beginning and at the end of the simulation. The algorithm terminated after the maximal
+number of iterations as can be seen in Figure 3, where we show the evolution of the cost functional
+on the left-hand side and the evolution of the Frobenius-norm differences on the right. The evolution
+of the cost shows the typical structure of optimal control problems, the cost decays fast in the first
+optimization iterations and later only small improvements are made. The optimized cost value is
+only 0.276% of the cost for the initial guess.
+The graphs of the norm differences indicate that the optimal matrices found in the optimization
+procedure differ from the matrices we used to obtain the reference data. This underpins the non-
+uniqueness of optimal controls we already discussed in the theory section.
+1https://github.com/ctotzeck/PHScalibration
+16
+
+10.0
+7.5 
+5.0 
+2.5
+(*)n
+0.0
+2.5
+5.0
+7.5
+10.0
+0.0
+0.2 
+0.4 
+0.6
+0.8
+1.0
+t3.0 -
+2.5
+2.0 -
+Vinitial(t)
+1.5
+1.0 -
+0.5
+0'0
+0.5
+0'0
+0.2 
+0.4
+0.6
+8'0
+1.0
+tFigure 2: Left: ydata generated with reference system matrices. Center: y resulting from
+the approximated system matrices. Right: Difference of the outputs.
+Figure 3: Left: evolution of the cost over the optimization iterations. Right: evolution
+of the difference of the system matrices and the reference matrices over the optimization
+iterations.
+5.2
+Synthetic data (randomly perturbed)
+Let us consider the same setting as in the previous subsection but with randomly perturb the output
+data. In every time-step we add independent normally distributed vectors ni ∈ Rm leading us to
+the perturbed data given by
+ydata,σ(ti) = ydata(ti) + σni.
+We show results for σ ∈ {0.01, 0.05, 0.25}. Figure 4 shows a study of the outputs for different noise
+levels. The increasing noise level is clearly visible in the plots of the reference date (left column) and
+the difference (right column). The approximations of the output signals in the middle are very similar
+for the different noise levels, which proves the robustness of the approach with respect to random
+perturbations. For σ = 0.25 the noise superimposes the difference of the outputs.
+In Figure 5
+(left) the evolution of the cost functional for the different noise levels is shown. The influence of the
+noise is clearly visible, the higher the noise level the higher the cost values. However, the optimal
+cost is only 0.27% of the initial cost for σ = 0.01, it is 0.6% of the initial cost for σ = 0.05 and 7%
+for σ = 0.25. As the optimal outputs are very robust, we conclude that the difference in the cost
+is mainly driven by the noise. The Frobenius norm differences of the optimized system matrices
+and the matrices used for the data generation are plotted in Figure 5 (right). The noise has only
+marginal influence on these graphs. Moreover, the graphs look very similar to the deterministic case
+in Figure 3 (right). This underlines the robustness of the algorithms w.r.t. noisy data.
+17
+
+2.5
+2.0 -
+1.5 -
+(2)pf
+1.0
+0.5
+0.0 -
+0.5
+0.0
+0.2
+0.4
+0.6
+8'0
+1.0
+t2.5
+2.0 -
+1.5 -
+1.0
+0.5 .
+0.0 
+0.5 
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+t0.05 
+0.00 
+(0)PR
+0.05
+0.10 -
+0.15
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t0.7
+0.6 -
+0.5 
+0.4 -
+0.3
+0.2 
+0.1 -
+0.0 -
+0
+1
+20
+40
+09
+08
+100
+iterationB  Brer  r
+  l
+WJ  Jrerl F
+3.5
+IIR  Rre ll 
+3.0 
+2.5
+2.0
+10
+20
+40
+60
+08
+100
+tFigure 4: Left: randomly perturbed output data. Center: output of optimized model.
+Right: difference of the optimized model output and the perturbed output.
+Figure 5: Left: evolution of the cost functional for difference noise levels over the op-
+timization iterations. Right: evolution of the difference of the system matrices and the
+reference matrices over the optimization iterations.
+18
+
+0.10
+0.05
+0.00
+()fi
+0.05
+0.10
+0.15
+0.0
+0.2
+0.4
+0.6
+0.8
+1.02.5 -
+2.0 
+1.5 -
+1.0
+0.5 -
+0'0
+0.5 -
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t2.5
+2.0
+1.5 -
+1.0
+0.5
+0.0 
+0.5 
+0'0
+0.2 
+0.4
+0.6
+0.8
+1.00.2
+0.1
+0.0
+(>)fi
+0.1
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.03.0
+2.5
+2.0
+1.5
+4
+1.0
+P
+0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.02.5
+2.0 -
+1.5 -
+1.0
+0.5 
+0.0 -
+0.5
+0.0 
+0.2 
+0.4
+0.6
+0.8
+1.00.75
+0.50
+0.25
+ Ydatal
+0.00 
+0.25
+0.50
+0.75
+-1.00
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0100
+ = 0.01
+o = 0.05
+o = 0.25
+10-1
+102
+0
+20
+40
+09
+08
+100
+iteration = 0.01
+o = 0.05
+3.5 
+o = 0.25
+Frobenius norm
+3.0
+2.5
+2.0
+0
+20
+40
+09
+08
+100
+iteration2.5
+2.0 -
+1.5
+Pf
+1.0
+0.5 -
+0.0 
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t2.5
+2.0 -
+1.5 -
+1.0
+0.5
+0.0 -
+0.5 
+0'0
+0.2 
+0.4
+0.6
+0.8
+1.05.3
+Single mass-spring-damper chain
+The last test case we consider is taken from the PHS benchmark collection [Sch22]. We generated the
+standard example with 50 mass-spring-damper cells that are each connected with their neighboring
+masses by springs. The last mass is connected to a wall via a spring while at the two first masses
+external forces are applied. These are two dimensional inputs leading to m = 2. Moreover, each
+mass is connected with the ground with a damper. All masses are set to 4, the damping coefficient
+to 1 and the stiffness to 4. The output y are the velocities of the masses which are controlled. For an
+illustration and more details we refer to the documentation of the MSD Chain example referenced
+at [Sch22].
+Figure 6: Input signals u (left) and corresponding (uncontrolled) output (right).
+We extracted the system matrices J, Q, R ∈ R100×100 and the input matrix B ∈ R100×2 from the
+julia implementation and stored them in python npy-format to use them with our implementation
+of the algorithm.
+To obtain interesting dynamics we increase the time interval to [0, 5] which yields T = 5 and
+5000 time steps to retain the time step size 0.001. The input on the larger time interval and the
+corresponding uncontrolled output are shown in Figure 6.
+The evolution of the cost functional and the difference of the Frobenius norms for the different
+system matrices are illustrated in Figure 3. The cost in the last iteration is only 0.033% of the cost
+of the initial guess. The graphs on the left are interesting, as they show that the norm difference is
+approximately constant over the optimization iterations. However, as the cost decrease rapidly the
+system matrices must change. Figure 7 (right) therefore indicates that the matrices change but the
+norm difference to the reference matrices are conserved.
+In Figure 8 we compare the output of the reference matrices, with the output of the optimized
+model and their difference. From a visual comparision of y (center) and ydata (left) it is difficult to
+tell differences. This is confirmed by the plot on the (right) which shows that the difference of the
+two outputs is rather small.
+6
+Conclusion and outlook
+In this paper we have proposed a way of calibrating linear PHS models from given input-output time
+data in a direct approach: structure preservation defines a constrained optimization problem, which
+is solved by an adjoint-based gradient descent algorithm — there is no need of first deriving a best-fit
+19
+
+10.0
+7.5
+5.0 
+2.5
+()n
+0'0
+2.5
+5.0
+7.5
+10.0
+0
+1
+2
+3
+4
+51.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1
+2
+3
+5Figure 7: Left: evolution of the cost function over the optimization iterations. Right:
+Difference of the reference and model matrices over the optimization iterations measures
+in Frobenius norm.
+Figure 8: Left: output for given reference system. Center: output for optimized model.
+Right: difference of the output.
+linear state-space model and then, in a post-processing step, finding the nearest PHS realization. In
+addition, there is no need for parameterization to transfer the constrained optimization problem into
+an unconstrained one. Numerical results underpin the feasibility of our approach for both, synthetic
+data and a mass-spring-damper benchmark problem. In future work the authors plan to generalize
+the approach from linear PHS-ODE models to nonlinear PHS-ODE models and linear PHS-DAE
+models.
+References
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+tion, pages 335–376. Society for Industrial and Applied Mathematics, 2017.
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+
+3.5
+3.0 -
+2.5
+2.0 -
+1.5
+1.0
+0.5 -
+0.0 -
+1
+0
+5
+10
+15
+20
+25
+iteration09
+50
+40 
+II B  Brerll 
+MJ  Jrerll r
+WR  Rrer F
+20
+10
+F 0
+5
+0
+10 
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+iteration0.6
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+()f
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+A. Schaft. Port-Hamiltonian systems: an introductory survey. In Proceedings on the
+International Congress of Mathematicians, Vol. 3, pages 1339–1366, 2006.
+[Sch21]
+Paul Schwerdtner. Port-Hamiltonian system identification from noisy frequency response
+data. ArXiv, abs/2106.11355, 2021.
+[Sch22]
+Paul
+Schwerdtner.
+Port-Hamiltonian
+benchmark
+systems,
+8.12.2022.
+https://github.com/Algopaul/PortHamiltonianBenchmarkSystems.jl.
+[SH15]
+S. Sra and R. Hosseini. Conic geometric optimization on the manifold of positive definite
+matrices. SIAM Optim., 25:713–739, 2015.
+[Tes12]
+Gerald Teschl. Ordinary differential equations and dynamical systems, volume 140. Amer-
+ican Mathematical Soc., 2012.
+[Trö10]
+Fredi Tröltzsch. Optimal control of partial differential equations: theory, methods, and
+applications, volume 112. American Mathematical Soc., 2010.
+21
+
diff --git a/sdE2T4oBgHgl3EQffQc2/content/tmp_files/load_file.txt b/sdE2T4oBgHgl3EQffQc2/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf,len=729
+page_content='Data-driven adjoint-based calibration of port-Hamiltonian  systems in time domain  Michael Günther∗, Birgit Jacob†, Claudia Totzeck‡  IMACM, School of Mathematics and Natural Sciences,  University of Wuppertal, Germany  January 11, 2023  Abstract  We present a gradient-based calibration algorithm to identify the system matrices of a linear  port-Hamiltonian system from given input-output time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Aiming for a direct structure-  preserving approach, we employ techniques from optimal control with ordinary differential equa-  tions and define a constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The input-to-state stability is discussed  which is the key step towards the existence of optimal controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Further, we derive the first-order  optimality system taking into account the port-Hamiltonian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Indeed, the proposed  method preserves the skew-symmetry and positive (semi)-definiteness of the system matrices  throughout the optimization iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Numerical results with perturbed and unperturbed syn-  thetic data, as well as an example from the PHS benchmark collection [Sch22] demonstrate the  feasibility of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  AMS classification: 37J06,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 37M99,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 49J15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 49K15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 49M29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 49Q12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 65P10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 93A30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 93B30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' 93C05  Keywords: Port-Hamiltonian systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' data-driven approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' optimal control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' adjoint-based calibration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  time domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' coupled dynamical systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' structure preservation  1  Introduction  In structure-preserving modelling of coupled dynamical systems the port-Hamiltonian framework  allows for constructing overall port-Hamiltonian systems (PHS) provided that (a) all subsystems are  PHS and (b) a power-conserving interconnection between the input and outputs of the subsystems is  provided [MM19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Sch06,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' EMv07,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' DMSB09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In realistic applications this approach reaches its limits  if for a specific subsystem, either no knowledge that would allow the definition of a physics-based  PHS is available, or one is forced to use user-specified simulation packages with no information  of the intrinsic dynamics, and thus only the input-output characteristics are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  In both  cases a remedy is to generate input-output data either by physical measurements or evaluation of  the simulation package, and derive a PHS surrogate that fits these input-output data best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This  PHS surrogate can then be used to model the subsystem, and overall leads to a coupled PHS with  structure-preserving properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  ∗Research Group Applied and Computational Mathematics, guenther@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='de  †Research Group Functional Analysis, bjacob@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='de  ‡Research Group Optimization, totzeck@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='de  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='03924v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='OC]  10 Jan 2023  Data-driven port-Hamiltonian realizations of dynamic systems from input-output data have been  researched using various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The Loewner framework has been used to construct a port-  Hamiltonian realization from frequency domain data in [ALI17, BGVD20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Another frequency do-  main approach is proposed in [Sch21], where a parametrization of the class of PHS is used to permit  the usage of unconstrained optimization solvers during identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In [CGaB22] the frequency  response data are inferred using the time-domain input-output data and then frequency domain  methods are used to construct a port-Hamiltonian realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Further, in [CMH19] a best-fit linear  state-space model is derived from time-domain data and then, in a post-processing step, a nearest  port-Hamiltonian realization is inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Based on input, state and output time-domain data and a  given quadratic Hamiltonian the authors of [MNU22] construct a port-Hamiltonian realization using  dynamic mode decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  We propose a direct time-domain approach for constructing a best-fit PHS model in one step  using input-output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Techniques from optimal control with ordinary differential equations and a  constrained optimization problem are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We derive the first-order optimality system taking  into account the port-Hamiltonian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The proposed method preserves the skew-symmetry  and positive (semi)-definiteness of the system matrices throughout the optimization iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Our  approach does not generate frequency data as an intermediate step and data of the state variable are  not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We remark, that our method is overdetermined because we identify all system param-  eters including the quadratic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This is interesting for example when the Hamiltonian  contains unknown physical parameters as the spring constant or masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' However, if the Hamiltonian  is known it is possible to use our approach to derive a best-fit PHS with a given Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The paper is organized as follows: in the next section, we define the calibration problem of  computing the best-fit of a PHS to given input-output data, prove the continuous dependence of  the state on the input which is the key step to obtain the existence of solutions to the calibration  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The adjoints for all system matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=', the positive definite scaling matrix Q, the fixed-  rank semi-definite dissipation matrix R, the skew-symmetric system matrix J, the input matrix B  and the initial value, are derived in section three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The gradient-descent algorithm and numerical  schemes for the calibration process are stated in chapter four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Before we show numerical results  for synthetic data with and without perturbation in section five to validate our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The  numerical examples are rounded off with a single mass-spring-damper chain example from the PHS  benchmark collection [Sch22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We conclude with an outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  2  Calibration problem  For a given input u: [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' T] → Rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' a data set ydata : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' T] → Rm which can consist of continuous  data on the interval [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' T] for T > 0 or continuous interpolation of measurements at discrete time  points and reference values for the calibration wref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' we consider the calibration problem with cost  functional  J (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' w) = 1  2  � T  0  |y(t) − ydata(t)|2dt + λ  2 |w − wref|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  λ ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (1)  subject to the state constraint  d  dtx = (J − R)Qx + Bu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  x(0) = x0 ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (2a)  y = B⊤Qx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (2b)  2  where w = (J, R, Q, B, x0) contains the matrices and initial data to be identified having the proper-  ties  J, R, Q ∈ Rn×n,  J⊤ = −J,  R ≽ 0,  Q ≻ 0,  B ∈ Rn×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (3)  Note that only output data and for λ > 0 reference values are considered in the cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Hence, the dimension n of the internal state is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We assume in the following that n is  chosen a priori and remark that a deliberate small choice can also be interpreted as a model order  reduction for the internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For notational convenience, we introduce the notation  J1(y) := 1  2  � T  0  |y(t) − ydata(t)|2dt,  J2(w) := λ  2 |w − wref|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  In applications often m ≤ n hence we obtain only partial information about the internal state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In  particular, we have no information about the initial condition x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This is the reason why we include  the initial data in the identification process for w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' As we will use techniques from optimal control  theory, we may refer to w as control in the following and introduce the space  W = Rn×n × Rn×n × Rn×n × Rn×m × Rn  with set of admissible controls as  Uad = {w = (J, R, Q, B, x0) ∈ W : J⊤ = −J, R ≽ 0, Q ≻ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  For later reference we define the identification task at hand:  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We seek to find system matrices and initial data, w = (J, R, Q, B, x0), which solve the  problem  min  (y,w)∈Y ×Uad  J (y, w)  subject to  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (IdP)  In [GJT23], we proposed a sensitivity-based approach to identify the system matrices of a port-  Hamiltonian system and argued that this is a valid ansatz for small-scale systems, as it requires to  solve auxilliary problems for each basis element of the matrix spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Here, we derive an adjoint-  based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Before we begin the derivation of the algorithm, we analyze the well-posedness of  the calibration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' First, we prove that the state solution depends continuously on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For every w ∈ Uad and u ∈ C([0, T], Rm) the state equation (2) admits a unique solution  x ∈ C1([0, T], Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Moreover, the solution depends continuously on the data, in more detail, for  w, w′ ∈ Uad and corresponding solutions x, x′ there exists a constant C > 0 such that  ∥x − x′∥H1((0,T),Rn) ≤ C∥w − w′∥Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For given w ∈ Uad and u ∈ C([0, T], Rm) the existence of a unique solution to (2) follows  by standard ODE theory [Tes12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For the second statement we estimate ∥x − x′∥L2((0,T),Rn) and  ∥ d  dtx − d  dtx′∥L2((0,T),Rn) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  We obtain  |x(t) − x′(t)|2 ≤ 2|x0 − x′  0|2 + 2T  � t  0  |(J − R)Qx(s) − (J′ − R′)Q′x′(s)|2ds  ≤ C1|w − w′|2 +  � t  0  C2|x(s) − x′(s)|2ds  3  An application of Gronwall inequality yields  |x(t) − x′(t)|2 ≤ C3|w − w′|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Integration over [0, T] we obtain  ∥x − x′∥2  L2((0,T),Rn) ≤ C4|w − w′|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (4)  Moreover, we obtain for  | d  dt  �  x(t) − x′(t)  �  |2 = C5|w − w′|2 + C6  � t  0  | d  ds  �  x(s) − x′(s)  �  |2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Again, Gronwall and integration over [0, T] yields  ∥ d  dtx(t) − x′(t)∥2  L2((0,T),Rn) ≤ C7|w − w′|2  (5)  Adding (4) and (5) and taking square root yield the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The well-posedness of the state equation allows us to introduce the solution operator  S : Uad → H1((0, T), Rn),  S(w) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Moreover, we define the reduced cost functional  ˆ  J (w) = 1  2  � T  0  |B⊤QS(w)(t) − ydata(t)|2dt + λ  2 |w − wref|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Both will be helpful in the proof of the well-posedness result of the identification problem and also  play a major role in the derivation of the gradient-descent algorithm later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The next ingredient for the well-posedness result is the weak convergence of the state operator  e: H1((0, T), Rn) × Uad → L2((0, T), Rn),  (x, w) �→  � d  dtx − (J − R)Qx − Bu  x(0) − x0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (6)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The state operator e defined in (6) is weakly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We consider {xn}n ⊂ H1((0, T), Rn) with xn ⇀ x and {wn}n ⊂ Uad with wn ⇀ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that  Uad is finite dimensional, thus weak and strong convergence coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Then for ϕ ∈ L2((0, T), Rn)  we obtain  � T  0  � d  dtxn − (Jn − Rn)Qnxn(t) − Bnu(t)  �  ϕ(t)dt −  � T  0  � d  dtx − (J − R)Qx(t) − Bnu(t)  �  ϕ(t)dt  =  � T  0  � d  dtxn − d  dtx  �  ϕ(t) dt −  � T  0  ((Jn − J) − (Rn − R))Qnxn(t)ϕ(t)dt  −  � T  0  (J − R)(Qn − Q)xn(t)ϕ(t) + (J − R)Q(xn(t) − x(t))ϕ(t) + (Bn − B)u(t)ϕ(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  By the weak convergence of xn and wn we conclude that all terms tends to zero as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that  weak convergence of {xn}n implies boundedness of the sequence {xn}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Moreover, by the embedding  H1((0, T), Rn) �→ C([0, T], Rn) we conclude that xn(0) → x(0) and further we obtain (x0)n → x0  by the convergence of wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Altogether, this yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  4  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Then (IdP) admits a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We follow the lines of [Trö10, HPUU08] and consider a minimizing sequence {wn}n, that  means lim  n→∞  ˆ  J (wn) =  inf  w∈Uad  ˆ  J (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The assumption λ > 0 yields that ˆ  J (w) is coercive w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' w and hence the boundedness of the  minimization sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' As Uad is a subset of a reflexive, finite-dimensional space, we can extract a  convergent subsequence {wnk}k with wnk → w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that w = 0 implies z ≡ 0, hence the continuous  dependence on the data shown in Lemma 2 implies the boundedness of {S(wnk)}k ⊂ H1((0, T), Rn)  and we can extract a weakly convergent subsequence S(wnkℓ) ⇀ x in H1((0, T), Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The weak  continuity of the state operator shown in Lemma 3 yields  e(S(wnkℓ), wnkℓ) ⇀ e(x, w)  and the weak lower semicontinuity of the norm allows us to estimate  ∥e(x, w)∥L2((0,T),Rn) ≤ lim inf  ℓ→∞ ∥e(S(wnkℓ), wnkℓ)∥L2((0,T),Rn) = 0,  which proves that S(w) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We note that the weak lower semicontinuity of J implies the weak  lower semicontinuity of ˆ  J , hence  ˆ  J (w) ≤ lim  ℓ→∞  ˆ  J (wnkℓ) =  inf  w∈Uad  ˆ  J (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  This proves the existence of a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We want to emphasize that even though the state equation is linear in x, it is non-linear  in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We therefore cannot expect to obtain a uniqueness result for (IdP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Here, we aim for a general approach, also feasible for high-dimensional systems, and therefore  discuss an adjoint-based approach in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' A challenge in this context is the structure of the  system matrices (3) which we aim to preserve in each step of the calibration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' To this end,  we employ techniques from optimization on manifolds to compute the retraction for Q and preserve  its positive definiteness in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' To the authors’ knowledge, there is no explicit formula for  the retraction in the space of semi-definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Hence, we use the flat metric for R and as the  space of skew-symmetric matrices as well as Rn×m and Rn are vector spaces, we are naturally in the  flat case for J, B and x0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  One of our objectives is the derivation of a gradient-based algorithm for the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We  derive the adjoint-based gradient descent scheme step-wise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=', we discuss the computation for  each of the different matrix structures separately and discuss a proof of concept with (randomly  perturbed) synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  3  First-order optimality system  As the system matrices have different structure, we organize the derivation of the adjoints in sub-  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' At the end of the section we combine the intermediate results to derive the optimality  system for Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We emphasize that we can resort to the subproblems for the derivation of  the adjoint, since the controls appear bilinearly in the state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='1  Positive definite matrices Q  In this section we consider a toy problem with positive definite system matrix Q and employ find-  ings of [SH15] where conic geometric optimization on the manifold of positive definite matrices is  discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let Pn denote the set of Hermitian positive definite (HPD) matrices and Hn the set of  n × n Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For simplicity we consider  d  dtz = Qz,  z(0) = z0,  (7)  y = B⊤Qz  (8)  with control Q ∈ Pn together with the cost functional  J (z, Q) = 1  2  � T  0  ∥y(t) − ydata(t)∥2dt + J2(Q)  and denote the first part by J1(z) =  1  2  � T  0 ∥y(t) − ydata(t)∥dt and the second part by J2(Q) =  λ∥ log(Q)∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that the simple structure allows to solve (7) explicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Indeed, z(t) = exp(Qt)z0,  where exp denotes the matrix exponential  exp(Qt) =  ∞  �  k=0  (Qt)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  = I + Qt + (Qt)2  2  + (Qt)3  6  + · · ·  From [SH15] we know that the geodesic along the manifold of HPD matrices and passing through  γ(0) = Q with ˙γ = ξ ∈ Hn is given by  γ(t) = Q1/2 exp(tQ−1/2ξQ−1/2)Q1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (9)  In the following we use this information to derive the first-order optimality system of our optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In particular, we will use the expansion  γ(t) = Q + ξt + ξQ−1ξt2 + o(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Let us begin with the well-posedness result of the state problem, which is a direct application of  Picard-Lindelöf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let Q ∈ Pn and z0 ∈ Rn be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The state problem (7) has a unique solution  z ∈ C1([0, T], Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The uniqueness result in Theorem 5 allows us to define the solution operator for the toy problem  S : {Q ∈ Rn×n : Q ≻ 0} → H1((0, T), Rn),  S(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' z0) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Moreover, we obtain the reduced cost functional  ˆ  J (Q) = 1  2  � T  0  ∥B⊤QS(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' z0)(t) − ydata(t)∥2dt + λ∥ log Q∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The second term of the cost functional measures the distance of Q to the identity matrix w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' the  Thompson metric [SH15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  6  We aim to derive a gradient descent scheme for ˆ  J (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The directional derivative of ˆ  J is given  by  lim  ϵ→0  ˆ  J (γ(ϵ)) − ˆ  J (γ(0))  ϵ  = lim  ϵ→0  J1(zϵ) − J1(z) + J2(γ(ϵ)) − J2(γ(0))  ϵ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Since it is not a priori clear how variations in Q influence the solution S(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' z0) we use an adjoint  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The first task is therefore to identify the adjoint equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Let zϵ denote the solution of  d  dtzϵ = γ(ϵ)zϵ,  zϵ(0) = z0,  where γ is defined in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Note that it holds  d  dt(zϵ − z) = Q(zϵ − z) + ϵξzϵ + o(e)  Moreover, let zh be the solution to  d  dtzh = (Q + ϵξ)zh,  zh(0) = z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (10)  Then  d  dt(zϵ − zh) = (Q + ϵξ)(zϵ − zh) + o(ϵ)  and by Gronwall inequality  ∥zϵ(t) − zh(t)∥ ≤ o(ϵ) exp(t∥Q + ϵξ∥) ϵ→0  −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (11)  Note that the difference ψh(t) = zh(t) − z(t) satisfies the equation  d  dtψh = Qψh + ϵξzh,  ψh(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Standard ODE theory [Tes12] yields the expression  ψh(t) = ϵ  � t  0  exp(Q(t − s))ξzh(s)ds  (12)  and for ϵ → 0 we obtain ψh ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We are now able to state the first-order necessary condition for  (z, Q) to be a stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let (¯z, ¯Q) be an optimal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Then for any ξ ∈ Hn it holds that  dJ2( ¯Q)[ξ] +  � T  0  ⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0,  (13)  where ψ is the solution to  d  dtψ = Qψ + ξz, ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let γ(t) be the geodesic through ¯Q with  d  dtγ(t) = ξ, zϵ the solution to  d  dtzϵ = γ(ϵ)zϵ with  zϵ(0) = z0, d  dtzh = ( ¯Q + ϵξ)zh, zh(0) = z0 and ψh as in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We obtain  J1(zϵ) − J1(¯z) = J1(zh) − J1(¯z) + o(ϵ)  =  � T  0  (¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)  = ϵ  � T  0  �  ¯y(t) − ydata(t), B⊤Qψh(t)  �  dt + o(ϵ)  7  Owing to the minimality of ¯Q we find  0 ≤  ˆ  J (γ(ϵ)) − ˆ  J ( ¯Q)  ϵ  = dJ2( ¯Q)[ξ] +  � T  0  �  y(t) − ydata(t), B⊤Qψh(t)  �  dt + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Passing to the limit ϵ → 0+ yields  0 ≤ dJ2( ¯Q)[ξ] +  � T  0  �  y(t) − ydata(t), B⊤Qψ(t)  �  dt  for any ξ ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that a sign change in ξ leads to a change of the sign of ψ, which yields the  desired equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  We aim for an adjoint-based representation of the first-order optimality system in order to provide  a gradient descent algorithm for the numerical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The dual problem corresponding to  d  dtψ = Qψ + ξz with ψ(0) = 0 is obtained by testing with p  0 =  � d  dtψ − Qψ − ξz, p  �  =  �  p · ψ  �T  0 −  � T  0  d  dtp · ψ + ψ · Q⊤p + z · ξ⊤p dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Choosing p to satisfy the adjoint equation  − d  dtp = Q⊤p + QB(y(t) − ydata(t)),  p(T) = 0,  yields together with the optimality condition (13) that  dJ2( ¯Q)[ξ] −  � T  0  p · ξz dt = 0  for all  ξ ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  For the choice J2(Q) = 1  2∥ log(Q)∥2  2 we obtain dJ2( ¯Q)[ξ] = (Q−1 log(Q), ξ) (see [SH15]) and hence  we can identify the optimality condition as  ∇J2( ¯Q) −  � T  0  ¯z ⊗ ¯p dt = 0,  where a ⊗ b denotes the dyadic product ab⊤ for a, b ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For J2(Q) = 1  2∥ log(Q)∥2  2 we obtain  ∇J2(Q) = Q−1 log ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='2  Fixed-rank semi-definite matrices R  Our preferred approach in case of semi-definite matrices would follow the same lines as above for  positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' First of all, in full generality there is no hope, hence we have to restrict  to the case of fixed-rank semi-definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This would lead us to the polar factorization of the  semi-definite matrix R given by R = GU with G ∈ {Rn×r : det(GG⊤) ̸= 0} and U ∈ St(n, r) with  St(n, r) = {Rn×r : UU⊤ = I} is the Stiefel manifold of r-dimensional orthonormal bases in Rn and  P ∈ S+(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Unfortunately, there is no explicit expression for the retraction of U [MBS11], hence  we consider the flat metric and the decomposition R = GG⊤ with G as above and consider the toy  problem with cost functional  ˆ  J (G) = 1  2  � T  0  ∥y(t) − ydata(t)∥2dt + J2(GG⊤)  (14)  8  and state equation  d  dtz = −Rz,  z(0) = z0,  (15)  y = B⊤Qz  (16)  with control R ∈ S+(r, n) a fixed-rank positive semi-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that this implies the  assumption that we know the number of conserved and dissipative variables a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note further  that we already stated the reduced cost functional only depending on G as we obtain the existence  and uniqueness result and therefore the control-to-state map analogous to the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  A horizontal tangent vector ξG at G is given by  ξG = Sym(∆)G,  ∆ ∈ Rd×d,  where Sym(∆) denotes the symmetric part of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Since we consider the flat metric, the exponential  mapping reads  ExpG(ξG) = G + ξG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Let us denote the geodesic through G by  γ(t) = G + tξG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  It satisfies ˙γ = ξG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We proceed as above and define zϵ as the solution to  d  dtzϵ = −γ(ϵ)γ(ϵ)⊤zϵ,  zϵ(0) = z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Then it holds  d  dt(zϵ − z) = −  �  (GG⊤)(zϵ − z) + ϵ(Gξ⊤  G + ξGG⊤)zϵ�  + o(ϵ)  We note that zh and zϵ of the previous section coincide here, as we are in the flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Following  the lines above, we find  d  dtψϵ = −GG⊤ψϵ − (Gξ⊤  G + ξGG⊤)zϵ,  ψϵ(0) = 0  (17)  and ψϵ ≡ 0 for ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The optimality condition reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let (¯z, ¯R) be an optimal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Then for any ξG = Sym(∆)G with ∆ ∈ Rd×d it holds  that  dJ2( ¯R)[ξG] +  � T  0  ⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0  (18)  where ψ is the solution to  d  dtψ = GG⊤ψ + (Gξ⊤  G + ξ⊤  GG)z, ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let zϵ the solution to d  dtzϵ = −γ(ϵ)γ(ϵ)⊤zϵ with zϵ(0) = z0 and ψϵ as in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We obtain  J1(zϵ) − J1(¯z) = J1(zh) − J1(¯z) + o(ϵ) =  � T  0  (¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)  = ϵ  � T  0  �  ¯y(t) − ydata(t), B⊤Qψϵ(t)  �  dt + o(ϵ)  9  Owing to the minimality of ¯R we find  0 ≤  ˆ  J (γ(ϵ)) − ˆ  J ( ¯R)  ϵ  = dJ2( ¯R)[ξG] +  � T  0  �  y(t) − ydata(t), B⊤Qψϵ(t)  �  dt + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Passing to the limit ϵ → 0+ yields  0 ≤ dJ2( ¯R)[ξG] +  � T  0  �  y(t) − ydata(t), B⊤Qψ(t)  �  dt  for any ξ ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that a sign change in ξ leads to a change of the sign of ψ, which yields the  desired equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  To derive the adjoint equation we test by p and obtain  0 = ⟨ d  dtψ + GG⊤ψ + (Gξ⊤  G + ξGG⊤)z, p⟩  =  �  p · ψ  �T  0 −  � T  0  d  dtp · ψ − GG⊤p · ψ − p · (Gξ⊤  G + ξGG⊤)z dt  Choosing p to satisfy the dual problem  − d  dtp = −GG⊤p + QB(y − ydata),  p(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  allows us to rewrite the optimality condition (18) as  dJ2( ¯R)[ξ] +  � T  0  p · (Gξ⊤  G + ξGG⊤)z dt = 0  for all ξG,  or  dJ2( ¯R)[Sym(∆)G] +  � T  0  p · (GG⊤ Sym(∆) + Sym(∆)GG⊤)z dt = 0  for all Sym(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Using R = GG⊤ we find  ∇J2( ¯J) −  � T  0  z ⊗ GG⊤p + p ⊗ GG⊤z dt = ∇J2( ¯J) −  � T  0  z ⊗ Rp + p ⊗ Rz dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='3  Skew-symmetric matrices J  The skew-symmetric matrices of size n × n, n ∈ N, are a vector space, hence we are naturally in the  flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' By analogy with above, we consider the toy problem with cost functional  ˆ  J (J) = 1  2  � T  0  ∥y(t) − ydata(t)∥2dt + J2(J)  (19)  and state equation  d  dtz = Jz,  z(0) = z0  (20)  y = B⊤Qz  10  with control J ∈ Rn×n skew-symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' J⊤ = −J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' As we are in the vector space setting we  let ∆ ∈ Rn×n and consider  d  dtzh = (J + ϵ Skew(∆))zh,  zh(0) = z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Then ψh = zh − z satisfies the equation  d  dtψh = Jψh + ϵ Skew(∆)zh,  ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (21)  and ψh ≡ 0 as ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We obtain the optimality condition  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let (¯z, ¯J) be an optimal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Then for any ξJ = Skew(∆)G with ∆ ∈ Rn×n it holds  that  dJ2( ¯J)[Skew(∆)] +  � T  0  ⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0  (22)  where ψ is the solution to  d  dtψ = Jψ + Skew(∆)z, ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let zh the solution to  d  dtzh = (J + ϵ Skew(∆))zh with zh(0) = z0 and ψh as in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We  obtain  J1(zϵ) − J1(¯z) = J1(zh) − J1(¯z) + o(ϵ) =  � T  0  (¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)  = ϵ  � T  0  �  ¯y(t) − ydata(t), B⊤Qψh(t)  �  dt + o(ϵ)  Owing to the minimality of ¯J we find  0 ≤  ˆ  J ( ¯J + ϵ Skew(∆)) − ˆ  J ( ¯J)  ϵ  = dJ2( ¯J)[Skew(∆)] +  � T  0  �  y(t) − ydata(t), B⊤Qψh(t)  �  dt + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Passing to the limit ϵ → 0+ yields  0 ≤ dJ2( ¯J)[Skew(∆)] +  � T  0  �  y(t) − ydata(t), B⊤Qψ(t)  �  dt  for any ξ ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that a sign change in ξ leads to a change of the sign of ψ, which yields the  desired equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  To derive the adjoint equation we test by p and obtain  0 = ⟨ d  dtψ − Jψ − Skew(∆)z, p⟩  =  �  p · ψ  �T  0 −  � T  0  d  dtp · ψ + J⊤p · ψ + Skew(∆)⊤p · z dt  =  �  p · ψ  �T  0 −  � T  0  d  dtp · ψ − Jp · ψ − Skew(∆)p · z dt  Choosing p to satisfy the dual problem  − d  dtp = J⊤p + QB(y − ydata),  p(T) = 0,  11  we obtain the optimality condition  0 = dJ2( ¯J)[Skew(∆)] −  � T  0  Skew(∆)p · zdt  for all ∆ ∈ Rn×n  leading to the expression  ∇J2( ¯J) −  � T  0  p ⊗ z dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='4  Input matrix B  The input matrix B plays a special role in the dynamics as is acts only on the input u and not  directly on the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let H ∈ Rn×m be arbitrary, we compute the directional derivative of the  output w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' the control B in direction H as  d  dB y(t)[H] = H⊤Qx(t) + B⊤Q  � t  0  Hu(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Hence, we obtain for the cost functional  J (y, w) = 1  2  � T  0  ∥y(t) − ydata(t)∥2dt + J2(B)  the variation in direction H by  d  dB J (y, w)[H] =  �  y − ydata, H⊤BQx + B⊤Q  � t  0  u(s)ds  �  Thus, the variational lemma, allows to identify the optimality condition w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' B as  ∇J2(B) +  � T  0  Qx ⊗ (y − ydata) +  � t  0  u(s)ds ⊗ Q⊤B(y − ydata)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='5  Initial data z0 and optimality system  Our calibration problem is assumed to be data-driven, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=', we only have knowledge about the output  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In the above derivations, the initial condition z0 were assumed to be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We consider two cases:  first, we assume that the equilibrium of the system is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=', as long as there is no input it is  reasonable to set z0 = 0 and the input can steer the system out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Second, we estimate  z0 from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  For the second case, we derive the optimaliy conditions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Consider the dynamics  d  dtz = (J − R)z + Bu,  z(0) = z0  y = B⊤Qz  and z0 is to be estimated from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' By analogy with above, we consider the cost functional  ˆ  J (z0) = 1  2  � T  0  ∥y(t) − ydata(t)∥2dt + J2(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (23)  12  The control variable is now z0 hence we obtain the linearization  d  dtzh = (J − R)Qzh,  zh(0) = z0 + ϵξ0  leading us to ψh with dynamics  d  dtψh = (J − R)Qψh,  ψh(0) = ϵξ0  (24)  and ψh ≡ 0 as ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let (¯z, ¯z0) be an optimal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Then for any ξ0 ∈ Rn it holds that  dJ2(¯x0)[ξ0] +  � T  0  ⟨y(t) − ydata(t), B⊤Qψ(t)⟩dt = 0  (25)  where ψ is the solution to  d  dtψ = Q(J⊤ − R⊤)z, ψ(0) = ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Let zh the solution to d  dtzh = (J −R)Qzh with zz(0) = z0 +ϵξ0 and ψh as in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We obtain  ˆ  J (z0 + ϵξ0) − ˆ  J (¯z0) =  � T  0  (¯y(t) − ydata(t), B⊤Q(zh(t) − ¯z(t)))dt + o(ϵ)  = ϵ  � T  0  �  ¯y(t) − ydata(t), B⊤Qψh(t)  �  dt + o(ϵ)  Owing to the minimality of ¯z0 we find  0 ≤  ˆ  J (¯z0 + ϵξ0) − ˆ  J (¯z0)  ϵ  =  � T  0  �  ¯y(t) − ydata(t), B⊤Qψh(t)  �  dt + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Passing to the limit ϵ → 0+ yields  0 ≤  � T  0  �  ¯y(t) − ydata(t), B⊤Qψ(t)  �  dt  for any ξ0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Note that a sign change in ξ0 leads to a change of the sign of ψ, which yields the  desired equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Testing the (24) with p yields  0 = ⟨ d  dtψ − (J − R)Qψ, p⟩  = p(T) · ψ(T) − p(0) · ξ0 −  � T  0  d  dtp · ψ + Q(J⊤ − R⊤)p · ψ dt  Choosing p to satisfy the adjoint equation  − d  dtp = Q(J⊤ − R⊤)p + QB(y − ydata),  p(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  which can be rewritten using the matrix properties as  − d  dtp = −Q(J + R)p + QB(y − ydata),  p(T) = 0  yields the optimality condition  ∇J2(x0) − p(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='6  Derivation of the gradient  We are now well-prepared to identify the gradient of our calibration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We discuss the case  including the estimation of the initial state, if this is needless the gradient update with respect to z0  can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We have already seen that the solution operator allows us to define the reduced  cost function which which leads us to an reformulation of (IdP) as unconstrained optimization  problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=', the state constraint is treated implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The gradient we derive in the following is  the one of the reduced cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In fact, the whole gradient-descent algorithm will only vary  w and thereby vary the state implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Let w = (Q, R, J, B, z0) the control variables or, to be more precise, the matrices and initial  data to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We denote by ⟨·, ·⟩W ∗,W : W ∗, W → R the dual pairing of W and its dual W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Moreover, A∗ denotes the adjoint of the operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' To identify the gradient of the reduced cost  functional we first note that the state operator yields  0 = dye(S(w), w)S′(w)[h] + dwe(S(w), w)[h]  ⇒  S′(w) = −dye(S(w), w)−1dwe(S(w), w)[h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  This allows us to find  ⟨ ˆ  J ′(w), h⟩W ∗,W = ⟨dyJ (y, w), B⊤QS′(w)[h]⟩H−1,H1 + ⟨dwJ (y, w), h⟩W ∗,W  = ⟨dwJ (y, w) − dwe(S(w), w)∗dye(S(w), w)−∗[QB dyJ (y, w)], h⟩W ∗,W  Since W is a Hilbert space and h was chosen arbitrarily, Riesz representation theorem allows us to  identify  (∇J (w), h)W = ⟨dwJ (y, w) − dwe(S(w), w)∗dye(S(w), w)−∗[QB dyJ (y, w)], h⟩W ∗,W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (26)  In the previous subsections we computed the adjoint equation already and the gradient compo-  nents corresponding (Q, R, J, B, x0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We summarize the result:  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' A stationary point ¯w = ( ¯Q, ¯R, ¯J, ¯B, ¯x0) of Problem 1 satisfies the optimality condition  ∇QJ2( ¯w) +  � T  0  ¯x ⊗ ¯p dt = 0,  (27a)  ∇RJ2( ¯w) −  � T  0  ¯x ⊗ ¯R¯p + ¯p ⊗ ¯R¯x dt = 0,  (27b)  ∇JJ2( ¯w) +  � T  0  ¯p ⊗ ¯x dt = 0,  (27c)  ∇x0J2( ¯w) + ¯p(0) = 0,  (27d)  ∇BJ2( ¯w) +  � T  0  ¯Q¯x ⊗ (¯y − ydata) +  � t  0  u(s)ds ⊗ ¯Q⊤ ¯B(¯y − ydata)dt = 0,  (27e)  where ¯p satisfies the adjoint equation  − d  dt ¯p = ¯Q( ¯J⊤ − ¯R⊤)¯p + ¯Q ¯B(¯y − ydata),  ¯p(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  (28)  and ¯x the state equation with output ¯y given by  d  dt ¯x = ( ¯J − ¯R) ¯Q¯x + ¯Bu,  ¯x(0) = ¯x0,  ¯y = ¯B⊤ ¯Q¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  14  We emphasize that the computation (26) shows that the left-hand side of system (27) is in fact  the gradient of the reduced cost functional at ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We will use this information in the gradient-descent  algorithm proposed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  4  Algorithm and numerical schemes  An optimal solution has to satisfy the conditions in Theorem 10 all at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' However, due to the  forward/backward structure of coupled state and adjoint equation it is in general difficult to solve the  system directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This is the reason why we propose an iterative approach based on the gradient (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  In fact, for an initial guess of systems matrices and initial condition w we solve the state equation  (2) and obtain S(w), using this information we solve the corresponding adjoint equation (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The  information of the state and adjoint solutions allows us to evaluate the gradient at w with ∇ ˆ  J (w)  we update our initial guess for the second iteration using a step size obtained by Armijo rule with  initial stepsize σ0 [HPUU08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We proceed with this iteration until an appropriate stopping criterion  is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  As stopping criterion we check the relative cost: First, we can stop is the gradient  vanishes numerically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=', ∥∇ ˆ  J (w)∥ < ϵ for 0 < ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Another option is to stop the iteration,  if the update from one to the next gradient step ∥∇ ˆ  J (wk) − ∇J (wk+1)∥ < ϵ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Additionally, we can impose a maximal number of iterations to stop the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We summarize  the steps in Algorihm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Algorithm 1: Gradient-descent algorithm for the identification process  Data: initial guess w, algorithmic parameters, stopping criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Result: optimized matrices and initial data ¯w  1) solve state equation → S(w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  2) solve adjoint equation → p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  3) evaluate gradient (27)→ ∇J (w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  4) find admissible step size with Armijo rule → σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  5) update control w �→ w + σ∇ ˆ  J (w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  6) if stopping criterion is not fulfilled → go to 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  else return optimal control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  For the numerical results presented in the next section, we use the following algorithmic parame-  ters unless explicitly stated otherwise: maximal number of gradient iterations 100, initial step size for  Armijo rule 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The time step size of the explicit Euler discretizations of the state and adjoint ODE,  respectively, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='001 and the upper bound of the time interval considered for the ODEs is T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The integrals in the gradient expressions are computed with an simple left-endpoint approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  In order to avoid numerical errors to interfere with the structure of the matrices, we check  for skew-symmetry or symmetry, respectively, and if needed overwrite the new iterate for J by  J �→ 1  2(J − J⊤) and analogous Q by Q �→ 1  2(Q + Q⊤) and R by R �→ 1  2(R + R⊤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' As we assume to  have no knowledge about the systems, we use no reference values for the tests and set J2(w) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Moreover, we set m = 2 for all the numerical tests and use the inputs (see Figure 1)  u: [0, T] → R2,  u(t) =  �10 sin(2πt)  5 cos(2πt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  We initialize B with an identity block in the first m × m entries and zero everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The  matrix Q representing the Hamiltonian is initialized as the n × n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For R we set  the first n/10 × n/10 block to identity and zero everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Of course it is nontrivial to find  15  Figure 1: Input signals u (left) and corresponding (uncontrolled) output (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  a good initial guess for R in the dynamics as in particular the number of dissipative elements is  unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Furthermore, we assume that the system is in equilibrium if no input applies, therefore  we refrain to estimate the initial state x0 and set x0 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' To generate a feasible initial J we draw an  n × n matrix with independent uniformly distributed entries in [−1, 1], take the upper part of this  matrix (starting with the first off-diagonal) and fill the lower part of the matrix by the negative of  the transpose of the upper part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  5  Proof of concept by numerical tests  To underline the feasibility of the proposed approach, we show three test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' First, we generate  synthetic data and show that Algorithm 1 is able to find system matrices that approximate the  data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Then we repeat the test with a randomly perturbed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Our third test case is the  single mass-spring-damper chain from the PHS benchmark collection [Sch22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The code is publicly  available on github1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='1  Synthetic data (deterministic)  In the following we show results for a synthetic data set with n = 20 and m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The reference  output is generated using random matrices with appropriate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The output from the data  set is plotted in Figure 2 (left) and the output of the optimized system is shown in Figure 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  A visual comparison shows that the dynamics is well-approximated, only small variations can be  seen at the beginning and at the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The algorithm terminated after the maximal  number of iterations as can be seen in Figure 3, where we show the evolution of the cost functional  on the left-hand side and the evolution of the Frobenius-norm differences on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The evolution  of the cost shows the typical structure of optimal control problems, the cost decays fast in the first  optimization iterations and later only small improvements are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The optimized cost value is  only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='276% of the cost for the initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The graphs of the norm differences indicate that the optimal matrices found in the optimization  procedure differ from the matrices we used to obtain the reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This underpins the non-  uniqueness of optimal controls we already discussed in the theory section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='com/ctotzeck/PHScalibration  16  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content='0  tFigure 2: Left: ydata generated with reference system matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Center: y resulting from  the approximated system matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Right: Difference of the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Figure 3: Left: evolution of the cost over the optimization iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Right: evolution  of the difference of the system matrices and the reference matrices over the optimization  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='2  Synthetic data (randomly perturbed)  Let us consider the same setting as in the previous subsection but with randomly perturb the output  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In every time-step we add independent normally distributed vectors ni ∈ Rm leading us to  the perturbed data given by  ydata,σ(ti) = ydata(ti) + σni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  We show results for σ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='25}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Figure 4 shows a study of the outputs for different noise  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The increasing noise level is clearly visible in the plots of the reference date (left column) and  the difference (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The approximations of the output signals in the middle are very similar  for the different noise levels, which proves the robustness of the approach with respect to random  perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='25 the noise superimposes the difference of the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  In Figure 5  (left) the evolution of the cost functional for the different noise levels is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The influence of the  noise is clearly visible, the higher the noise level the higher the cost values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' However, the optimal  cost is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='27% of the initial cost for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='01, it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='6% of the initial cost for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='05 and 7%  for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' As the optimal outputs are very robust, we conclude that the difference in the cost  is mainly driven by the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The Frobenius norm differences of the optimized system matrices  and the matrices used for the data generation are plotted in Figure 5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The noise has only  marginal influence on these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Moreover, the graphs look very similar to the deterministic case  in Figure 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This underlines the robustness of the algorithms w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content='0 -  0  1  20  40  09  08  100  iterationB  Brer  r  l  WJ  Jrerl F  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='3  Single mass-spring-damper chain  The last test case we consider is taken from the PHS benchmark collection [Sch22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' We generated the  standard example with 50 mass-spring-damper cells that are each connected with their neighboring  masses by springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The last mass is connected to a wall via a spring while at the two first masses  external forces are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' These are two dimensional inputs leading to m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Moreover, each  mass is connected with the ground with a damper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' All masses are set to 4, the damping coefficient  to 1 and the stiffness to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The output y are the velocities of the masses which are controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' For an  illustration and more details we refer to the documentation of the MSD Chain example referenced  at [Sch22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Figure 6: Input signals u (left) and corresponding (uncontrolled) output (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  We extracted the system matrices J, Q, R ∈ R100×100 and the input matrix B ∈ R100×2 from the  julia implementation and stored them in python npy-format to use them with our implementation  of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  To obtain interesting dynamics we increase the time interval to [0, 5] which yields T = 5 and  5000 time steps to retain the time step size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The input on the larger time interval and the  corresponding uncontrolled output are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  The evolution of the cost functional and the difference of the Frobenius norms for the different  system matrices are illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The cost in the last iteration is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='033% of the cost  of the initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' The graphs on the left are interesting, as they show that the norm difference is  approximately constant over the optimization iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' However, as the cost decrease rapidly the  system matrices must change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Figure 7 (right) therefore indicates that the matrices change but the  norm difference to the reference matrices are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  In Figure 8 we compare the output of the reference matrices, with the output of the optimized  model and their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' From a visual comparision of y (center) and ydata (left) it is difficult to  tell differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' This is confirmed by the plot on the (right) which shows that the difference of the  two outputs is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  6  Conclusion and outlook  In this paper we have proposed a way of calibrating linear PHS models from given input-output time  data in a direct approach: structure preservation defines a constrained optimization problem, which  is solved by an adjoint-based gradient descent algorithm — there is no need of first deriving a best-fit  19  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content="5  ()n  0'0  2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content='0  0  1  2  3  4  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='0  0  1  2  3  5Figure 7: Left: evolution of the cost function over the optimization iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Right:  Difference of the reference and model matrices over the optimization iterations measures  in Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Figure 8: Left: output for given reference system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Center: output for optimized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Right: difference of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  linear state-space model and then, in a post-processing step, finding the nearest PHS realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In  addition, there is no need for parameterization to transfer the constrained optimization problem into  an unconstrained one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Numerical results underpin the feasibility of our approach for both, synthetic  data and a mass-spring-damper benchmark problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In future work the authors plan to generalize  the approach from linear PHS-ODE models to nonlinear PHS-ODE models and linear PHS-DAE  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  References  [ALI17]  Athanasios C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Antoulas, Sanda Lefteriu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Cosmin Ionita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Chapter 8: A tutorial  introduction to the Loewner framework for model reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' In Peter Benner, Mario  Ohlberger, Albert Cohen, and Karen Willcox, editors, Model Reduction and Approxima-  tion, pages 335–376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Society for Industrial and Applied Mathematics, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  [BGVD20] Peter Benner, Pawan Goyal, and Paul Van Dooren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Identification of port-Hamiltonian  systems from frequency response data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Systems & Control Letters, 143:104741, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  [CGaB22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Cherifi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Goyal, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' and Benner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' A non-intrusive method to inferring lin-  ear port-Hamiltonian realizations using time-domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Electronic Transactions on  Numerical Analysis: Special Issue SciML, 56:102–116, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content='0 -  1  0  5  10  15  20  25  iteration09  50  40  II B  Brerll  MJ  Jrerll r  WR  Rrer F  20  10  F 0  5  0  10  15  20  25  iteration0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+page_content='00  (3)fi  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='04  0  3  5[CMH19]  Karim Cherifi, Volker Mehrmann, and Kamel Hariche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  Numerical methods to com-  pute a minimal realization of a port-Hamiltonian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  arXiv, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='07042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  [DMSB09] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Duindam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Macchelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Stramigioli, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Bruyninckx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Modeling and Control of  Complex Physical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='  [EMv07]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content=' Eberard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE2T4oBgHgl3EQffQc2/content/2301.03924v1.pdf'}
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+arXiv:2301.01427v1  [math.NA]  4 Jan 2023
+Entropy dissipative higher order accurate positivity
+preserving time-implicit discretizations for nonlinear
+degenerate parabolic equations
+Fengna Yan1,2, J. J. W. Van der Vegt2, Yinhua Xia3, Yan Xu3
+Abstract
+We develop entropy dissipative higher order accurate local discontinuous Galerkin
+(LDG) discretizations coupled with Diagonally Implicit Runge-Kutta (DIRK) meth-
+ods for nonlinear degenerate parabolic equations with a gradient flow structure.
+Using the simple alternating numerical flux, we construct DIRK-LDG discretiza-
+tions that combine the advantages of higher order accuracy, entropy dissipation and
+proper long-time behavior. The implicit time-discrete methods greatly alleviate
+the time-step restrictions needed for the stability of the numerical discretizations.
+Also, the larger time step significantly improves computational efficiency. We the-
+oretically prove the unconditional entropy dissipation of the implicit Euler-LDG
+discretization. Next, in order to ensure the positivity of the numerical solution,
+we use the Karush-Kuhn-Tucker (KKT) limiter, which couples the positivity in-
+equality constraint with higher order accurate DIRK-LDG discretizations using
+Lagrange multipliers. In addition, mass conservation of the positivity-limited so-
+lution is ensured by imposing a mass conservation equality constraint to the KKT
+equations. The unique solvability and unconditional entropy dissipation for an im-
+plicit first order accurate in time, but higher order accurate in space, KKT-LDG
+Email address: fnyan@hfut.edu.cn (F. Yan), j.j.w.vandervegt@utwente.nl (J. J. W. Van der Vegt),
+yhxia@ustc.edu.cn (Y. Xia), yxu@ustc.edu.cn (Y. Xu).
+1 School of Mathematics, Hefei University of Technology, Hefei, Anhui, 230000, PR China.
+2 Department of Applied Mathematics, Mathematics of Computational Science Group, University
+of Twente, Enschede, 7500 AE, The Netherlands.
+3 School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR
+China.
+1
+
+discretizations are proved, which provides a first theoretical analysis of the KKT
+limiter. Finally, numerical results demonstrate the higher order accuracy and en-
+tropy dissipation of the KKT-DIRK-LDG discretizations for problems requiring a
+positivity limiter.
+Keywords: Local discontinuous Galerkin discretizations, DIRK methods, Nonlinear
+degenerate parabolic equations, Unconditional entropy dissipation, KKT limiter.
+1
+Introduction
+Consider the following degenerate parabolic equation [5]
+
+
+
+ut = ∇ · (f(u)∇(φ(xxx) + H′(u))),
+in Ω × (0, T],
+u(xxx, 0) = u0(xxx),
+in Ω,
+(1.1)
+with zero-flux boundary condition
+∇(φ(xxx) + H′(u)) · ννν = 0,
+on ∂Ω × (0, T],
+(1.2)
+where Ω is an open bounded domain in Rd, d = 1, 2, with unit outward normal vector ννν
+at the boundary ∂Ω, u(xxx, t) ⩾ 0 is a nonnegative density with time derivative denoted
+as ut, φ(xxx) is a given potential function for xxx ∈ Rd, f, H are given functions such that
+f : R+ −→ R+,
+H : R+ −→ R,
+f(u)H′′(u) ⩾ 0,
+(1.3)
+where R+ is the nonnegative real space. Here f(u)H′′(u) can vanish for certain values
+of u, resulting in degenerate cases. The entropy corresponding to (1.1) is defined by
+E(u) =
+�
+Ω
+(uφ(xxx) + H(u))dΩ.
+(1.4)
+Multiplying (1.1) with φ(xxx)+H′(u) and integrating over Ω, with the zero-flux boundary
+condition (1.2), together with (1.4), we obtain that the time derivative of the entropy
+satisfies
+d
+dtE(u) = −
+�
+Ω
+f(u)|∇(φ(xxx) + H′(u))|2dΩ ⩽ 0.
+(1.5)
+System (1.1) can represent different physical problems, such as the porous media equation
+[31, 33], the nonlinear nonlocal equation with a double-well potential [7], the nonlinear
+Fokker-Plank model for fermion and boson gases [1, 9, 29].
+2
+
+Recently, many numerical discretizations have been proposed for (1.1); e.g. mixed
+finite element methods [6], finite volume methods [5, 7], DG methods [19, 20, 21] and
+LDG methods [33]. Regarding positivity preserving discretizations, Liu and Yu devel-
+oped in [20, 21], respectively, for the linear Fokker-Plank equation a maximum preserving
+DG scheme and an entropy satisfying DG scheme, but these discretizations can not be
+directly applied to the general case given by (1.1).
+Liu and Wang subsequently de-
+veloped in [19] an explicit Runge-Kutta (RK) time-discrete method for (1.1) in one
+dimension together with a positivity preserving high order accurate DG scheme under
+some Courant-Friedrichs-Lewy (CFL) constraints. For the porous media equation, an
+LDG discretization coupled with an explicit RK method was considered in [33], which
+is similar to the DG method in [19]. Still, it uses a special numerical flux to ensure the
+non-negativity of the numerical solution. Cheng and Shen in [10] propose a Lagrange
+multiplier approach to construct positivity preserving schemes for a class of parabolic
+equations, which is different from (1.1), but contains the porous media equation.
+For the time-step τ and mesh size h, the condition τ = O(h2) is needed for stability
+in [19] and [33]. Therefore, these explicit time discretizations suffer from severe time step
+restrictions, but there are currently no feasible positivity preserving time-implicit LDG
+discretizations for (1.1).
+In this paper, we present higher order accurate Diagonally
+Implicit Runge-Kutta (DIRK) LDG discretizations, which ensure positivity and mass
+conservation of the numerical solution without the severe time step restrictions of explicit
+methods.
+The LDG method proposed by Cockburn and Shu in [12] has many advantages,
+including high parallelizability, high order accuracy, a simple choice of trial and test
+spaces and easy handling of complicated geometries. We refer to [11, 15, 28, 36] for
+examples of applications of the LDG method.
+For many physical problems, it is crucial that the numerical discretization preserves
+the positivity properties of the partial differential equations (PDEs). Not only is this nec-
+essary to obtain physically meaningful solutions, but also negative values may result in ill-
+posedness of the problem and divergence of the numerical discretization. Positivity pre-
+serving DG methods have been extensively studied by many mathematicians. However,
+most positivity preserving DG methods are combined with explicit time-discretizations
+[19, 32, 34, 35], for which numerical stability frequently imposes severe time step restric-
+tions. These severe time-step constraints make explicit methods impractical for parabolic
+3
+
+PDEs, such as (1.1).
+Recently, Qin and Shu extended in [25] the general framework for establishing positivity-
+preserving schemes, proposed in [34, 35], from explicit to implicit time discretizations.
+They developed for one-dimensional conservation laws a positivity preserving DG method
+with high-order spatial accuracy combined with the first-order backward Euler implicit
+temporal discretization. This approach requires, however, a detailed analysis of the nu-
+merical discretization to ensure positivity and it is not straightforward to extend this
+approach to higher order accurate time-implicit methods. Huang and Shen in [17] con-
+structed higher order linear bound preserving implicit discretizations for the Keller-Segel
+and Poisson-Nernst-Planck equations. Van der Vegt, Xia and Xu proposed in [30] the
+KKT limiter concept to construct positivity preserving time-implicit discretizations. The
+KKT limiter in [30] is obtained by coupling the inequality and equality constraints im-
+posed by the physical problem with higher order accurate DIRK-DG discretizations using
+Lagrange multipliers. The resulting semi-smooth nonlinear equations are solved by an
+efficient active set semi-smooth Newton method.
+In this paper, we consider a general class of nonlinear degenerate parabolic equations
+given by (1.1) and aim at developing higher order accurate entropy dissipative and pos-
+itivity preserving time-implicit LDG discretizations. For the spatial discretization, we
+use an LDG method with simple alternating numerical fluxes, which results in entropy
+dissipation of the semi-discrete LDG discretization. For the temporal discretization, we
+consider DIRK methods, which significantly enlarge the time step for stability.
+The
+unconditional entropy dissipation of the LDG discretization combined with an implicit
+Euler time integration method is proved theoretically. We construct positivity preserving
+discretizations using the KKT limiter by imposing the positivity constraint on the nu-
+merical discretization using Lagrange multipliers. The unique solvability of the resulting
+positivity preserving KKT system is proved. We will also prove the unconditional entropy
+dissipation of the positivity preserving LDG discretization when it is combined with the
+backward Euler time integration method. Numerical results demonstrate the accuracy
+and entropy dissipation of the higher order accurate positivity preserving DIRK-LDG
+discretizations.
+This paper is organized as follows. In Section 2, we present the semi-discrete LDG
+discretization with simple alternating numerical fluxes for the nonlinear degenerate
+parabolic equation stated in (1.1) and prove that the numerical approximation is en-
+4
+
+tropy dissipative. Higher order accurate DIRK-LDG discretizations, which enlarge the
+stable time step to a great extent, are discussed in Section 3. The unconditional en-
+tropy dissipation of the implicit Euler LDG discretizations is proved in Section 3.1. In
+order to ensure positivity of the numerical solution and mass conservation of the pos-
+itivity limited numerical discretizations, we introduce in Section 4.1 the KKT system.
+The higher order DIRK-LDG discretizations with positivity and mass conservation con-
+straints are formulated in Section 4.2 as a KKT mixed complementarity problem. The
+unique solvability and unconditional entropy dissipation of the resulting algebraic system
+are proved in Section 4.3. In Section 5, numerical results are provided to demonstrate
+the higher order accuracy, positivity and entropy dissipation of the positivity preserving
+KKT-DIRK-LDG discretizations. Concluding remarks are given in Section 6.
+2
+Semi-discrete LDG schemes
+2.1
+Definitions, Notations
+Let Th be a shape-regular tessellation of Ω ⊂ Rd, d = 1, 2, with line or convex
+quadrilateral elements K. Given the reference element �K = [−1, 1]d. Let Qk( �K) denote
+the space composed of the tensor product of Legendre polynomials Pk( �K) on [−1, 1]
+of degree at most k ⩾ 0.
+The space Qk(K) is obtained by using an isoparametric
+transformation from element K to the reference element �K. The finite element spaces
+V k
+h and W
+W
+W k
+h are defined by
+V k
+h = {v ∈ L2(Ω) : v|K ∈ Qk(K), ∀K ∈ Th},
+W
+W
+W k
+h = {www ∈ [L2(Ω)]d : www|K ∈ [Qk(K)]d, ∀K ∈ Th},
+and are allowed to have discontinuities across element interfaces. Let e be an interior
+edge connected to the “left” and “right” elements denoted, respectively, by KL and KR.
+If u is a function on KL and KR, we set uL := (u|KL) |e and uR := (u|KR)|e for the left
+and right trace of u at e.
+Note that L1(Ω), L2(Ω) and L∞(Ω) are standard Sobolev spaces, ∥u∥L2(Ω) is the
+L2(Ω)-norm and (·, ·)Ω is the L2(Ω) inner product. For simplicity, we denote the inner
+product as (u, v) := (u, v)Ω.
+5
+
+2.2
+LDG discretization in space
+For the LDG discretization of (1.1), we first rewrite this equation as a first order
+system
+ut =∇ · qqq,
+qqq =f(u)sss,
+sss =∇p,
+p =φ(xxx) + H′(u).
+Then the LDG discretization can be readily obtained by multiplying the above equations
+with arbitrary test functions, integrating by parts over each element K ∈ Th, and finally
+a summation of element and face contributions. The LDG discretization can be stated
+as: find uh, ph ∈ V k
+h , qqqh,sssh ∈ W
+W
+W k
+h, such that for all ρ, ϕ ∈ V k
+h and θθθ,ηηη ∈ W
+W
+W k
+h, we have
+(uht, ρ) + L1
+h(qqqh; ρ) = 0,
+(2.1a)
+(qqqh,θθθ) + L2
+h(uh,sssh;θθθ) = 0,
+(2.1b)
+(sssh,ηηη) + L3
+h(ph;ηηη) = 0,
+(2.1c)
+(ph, ϕ) + L4
+h(uh; ϕ) = 0,
+(2.1d)
+where
+L1
+h(qqqh; ρ) :=(qqqh, ∇ρ) −
+�
+K∈Th
+(�qqqh · ννν, ρ)∂K,
+(2.2a)
+L2
+h(uh,sssh;θθθ) := − (f(uh)sssh,θθθ),
+(2.2b)
+L3
+h(ph;ηηη) :=(ph, ∇ · ηηη) −
+�
+K∈Th
+(�ph,ννν · ηηη)∂K,
+(2.2c)
+L4
+h(uh; ϕ) := − (φ(xxx) + H′(uh), ϕ) .
+(2.2d)
+Note that ννν is the unit outward normal vector of an element K at its boundary ∂K.
+The “hat” terms in L1
+h and L3
+h are the so-called “numerical fluxes”, whose choices play
+an important role in ensuring stability. We remark that the choices for the numerical
+fluxes are not unique. Here we use the alternating numerical fluxes
+�qqqh =qqqR
+h ,
+�ph = pL
+h,
+(2.3)
+6
+
+or
+�qqqh =qqqL
+h,
+�ph = pR
+h .
+(2.4)
+Considering the zero-flux boundary condition ∇(φ(xxx) + H′(u)) · ννν = 0, we take
+�qqqh · ννν = 0,
+ph = (ph)in
+(2.5)
+at ∂Ω, where “in” refers to the value obtained by taking the boundary trace from the
+inside of the domain Ω.
+2.3
+Entropy dissipation
+Theorem 2.1. For uh ∈ V k
+h , sssh ∈ W
+W
+W k
+h, the LDG scheme (2.1)-(2.5) with f satisfying
+(1.3) is entropy dissipative and satisfies
+d
+dtE(uh) = −(f(uh)sssh,sssh) ⩽ 0,
+which is consistent with the entropy dissipation property (1.5) of the PDE (1.1).
+Proof. By taking
+ρ = ph,
+θθθ = −sssh,
+ηηη = qqqh,
+ϕ = −uht,
+in (2.1a)-(2.1d), respectively, and after integration by parts, we have
+(φ(xxx) + H′(uh), uht)
+= − (f(uh)sssh,sssh) − (qqqh, ∇ph) +
+�
+K∈Th
+(�qqqh · ννν, ph)∂K − (ph, ∇ · qqqh) +
+�
+K∈Th
+(�ph,ννν · qqqh)∂K
+= − (f(uh)sssh,sssh) −
+�
+K∈Th
+(qqqh · ννν, ph)∂K +
+�
+K∈Th
+(�qqqh · ννν, ph)∂K +
+�
+K∈Th
+(�ph,ννν · qqqh)∂K.
+(2.6)
+Assume that e is an interior edge shared by elements KL and KR, then νννR = −νννL, and
+together with the numerical fluxes (2.3), we obtain
+−
+�
+KL
+� KR
+(qqqh · ννν, ph)e +
+�
+KL
+� KR
+(�qqqh · ννν, ph)e +
+�
+KL
+� KR
+(�ph,ννν · qqqh)e
+= − (qqqL
+h · νννL, pL
+h)e + (qqqR
+h · νννL, pR
+h )e + (qqqR
+h · νννL, pL
+h)e − (qqqR
+h · νννL, pR
+h )e
++ (qqqL
+h · νννL, pL
+h)e − (qqqR
+h · νννL, pL
+h)e = 0.
+(2.7)
+7
+
+Combining (2.6)-(2.7), using (1.4), boundary conditions (2.5) and the condition on f
+(1.3), we get
+d
+dtE(uh) = (φ(xxx) + H′(uh), uht) = −(f(uh)sssh,sssh) ⩽ 0.
+Remark 2.1. For brevity, we will only consider in the remaining article the numerical
+fluxes (2.3) and omit the discussion of the numerical fluxes (2.4), but all results also
+apply to the numerical fluxes (2.4).
+Remark 2.2. Compared to the spatial discretizations in [19, 33], we choose the simpler
+alternating numerical fluxes (2.3) and (2.4), which significantly simplifies the theoretical
+analysis of the entropy dissipation property of the LDG discretization.
+3
+Time-implicit LDG schemes
+The numerical discretization of the nonlinear parabolic equations (1.1) using explicit
+time discretization methods suffers from the rather severe time-step constraint τ =
+O(h2). In this section, we will discuss implicit time discretizations coupled with positivity
+constraints in Section 4.
+We divide the time interval [0, T] into N parts 0 = t0 < t1 < ... < tN = T, with τ n =
+tn − tn−1 (n = 1, 2, . . . , N). For n = 0, 1, . . . , N, let un = u(·, tn) and un
+h, respectively,
+denote the exact and approximate values of u at time tn.
+3.1
+Backward Euler LDG discretization
+Discretizing (2.1) in time with the implicit Euler method gives the following discrete
+system
+�un+1
+h
+− un
+h
+τ n+1
+, ρ
+�
++ L1
+h(qqqn+1
+h
+; ρ) = 0,
+(3.1a)
+(qqqn+1
+h
+,θθθ) + L2
+h(un+1
+h
+,sssn+1
+h
+;θθθ) = 0,
+(3.1b)
+(sssn+1
+h
+,ηηη) + L3
+h(pn+1
+h
+;ηηη) = 0,
+(3.1c)
+(pn+1
+h
+, ϕ) + L4
+h(un+1
+h
+; ϕ) = 0.
+(3.1d)
+8
+
+Define the discrete entropy as
+Eh(un
+h) =
+�
+Ω
+(un
+hφ(xxx) + H(un
+h))dx.
+(3.2)
+We have the following relation for the discrete entropy dissipation.
+Theorem 3.1. For all time levels n, the numerical solutions un
+h, un+1
+h
+∈ V k
+h of the
+LDG discretization (3.1), with boundary condition (2.5) and conditions on f, H stated
+in (1.3), satisfy the following entropy dissipation relation
+Eh(un+1
+h
+) ⩽ Eh(un
+h),
+(3.3)
+which implies that the LDG discretization is unconditionally entropy dissipative.
+Proof. By choosing, respectively, in (3.1a)-(3.1d) the following test functions
+ρ = pn+1
+h
+,
+θθθ = −sssn+1
+h
+,
+ηηη = qqqn+1
+h
+,
+ϕ = −un+1
+h
+− un
+h
+τ n+1
+,
+we get
+�
+φ(xxx), un+1
+h
+− un
+h
+τ n+1
+�
++
+�
+H′(un+1
+h
+), un+1
+h
+− un
+h
+τ n+1
+�
+= −
+�
+f(un+1
+h
+)sssn+1
+h
+,sssn+1
+h
+�
+−
+�
+qqqn+1
+h
+, ∇pn+1
+h
+�
++
+�
+K∈Th
+(�qqqn+1
+h
+· ννν, pn+1
+h
+)∂K
+− (pn+1
+h
+, ∇ · qqqn+1
+h
+) +
+�
+K∈Th
+(�pn+1
+h
+,ννν · qqqn+1
+h
+)∂K
+= − (f(un+1
+h
+)sssn+1
+h
+,sssn+1
+h
+) −
+�
+K∈Th
+(qqqn+1
+h
+· ννν, pn+1
+h
+)∂K +
+�
+K∈Th
+(�qqqn+1
+h
+· ννν, pn+1
+h
+)∂K
++
+�
+K∈Th
+(�pn+1
+h
+,ννν · qqqn+1
+h
+)∂K.
+Together with (2.7), the numerical fluxes (2.3) and the boundary condition (2.5), we
+obtain then
+�
+φ(xxx), un+1
+h
+− un
+h
+τ n+1
+�
++
+�
+H′(un+1
+h
+), un+1
+h
+− un
+h
+τ n+1
+�
+= −
+�
+f(un+1
+h
+)sssn+1
+h
+,sssn+1
+h
+�
+.
+Because of the following Taylor expansion
+H(un
+h) =H(un+1
+h
+) + H′(un+1
+h
+)(un
+h − un+1
+h
+) + 1
+2H′′(ξn+1)(un+1
+h
+− un
+h)2,
+ξn+1 ∈ (un
+h, un+1
+h
+),
+9
+
+we have, using the conditions on f, H stated in (1.3) and the definition of Eh in (3.2),
+Eh(un+1
+h
+) − Eh(un
+h) =
+�
+φ(xxx), un+1
+h
+− un
+h
+�
++
+�
+H(un+1
+h
+) − H(un
+h), 1
+�
+= − τ n+1 �
+f(un+1
+h
+)sssn+1
+h
+,sssn+1
+h
+�
+− 1
+2
+�
+H′′(ξn+1),
+�
+un+1
+h
+− un
+h
+�2�
+⩽ 0.
+3.2
+Higher order DIRK-LDG discretizations
+For higher order accurate implicit in time discretizations of the system (2.1), we
+use a Diagonally Implicit Runge-Kutta (DIRK) method [16]. Assuming we know the
+numerical solution at time level n, we obtain the solution at time level n + 1 with a
+DIRK method by solving for each DIRK stage i, i = 1, 2, · · · , s the following equations.
+�
+un+1,i
+h
+− un
+h
+τ n+1
+, ρ
+�
++
+i
+�
+j=1
+aijL1
+h(qqqn+1,j
+h
+; ρ) = 0,
+(3.4a)
+(qqqn+1,i
+h
+,θθθ) + L2
+h(un+1,i
+h
+,sssn+1,i
+h
+;θθθ) = 0,
+(3.4b)
+(sssn+1,i
+h
+,ηηη) + L3
+h(pn+1,i
+h
+;ηηη) = 0,
+(3.4c)
+(pn+1,i
+h
+, ϕ) + L4
+h(un+1,i
+h
+; ϕ) = 0.
+(3.4d)
+Then the solution at time tn+1 is
+(un+1
+h
+, ρ) =(un
+h, ρ) − τ
+s
+�
+i=1
+biL1
+h(qqqn+1,i
+h
+; ρ).
+(3.5)
+The coefficient matrices (aij) in (3.4a) and (bi) in (3.5) are defined in the Butcher tableau.
+We choose for polynomials of order k = 1 and k = 2, 3 the DIRK methods introduced in
+[3] and [26], respectively, that satisfy asi = bi, i = 1, 2, ···, s, which implies un+1
+h
+= un+1,s
+h
+.
+The order of these DIRK methods is k + 1. The above time discretization methods are
+easy to implement since the matrix (aij) in the DIRK methods has a lower triangular
+structure, which means that we can compute the DIRK stages one after another, starting
+from i = 1 up to i = s. For detailed information about the DIRK time integration
+method, we refer to [16].
+10
+
+4
+Higher order accurate positivity preserving DIRK-
+LDG discretizations
+The positivity constraints on the LDG solution will be enforced by transforming the
+DIRK-LDG equations with positivity constraints into a mixed complementarity problem
+using the Karush-Kuhn-Tucker (KKT) equations [14].
+In the following sections, we
+will first define the positivity preserving KKT-DIRK-LDG discretization. Next, we will
+consider the unique solvability and unconditional entropy dissipation of the discrete KKT
+system.
+4.1
+KKT-system
+For the KKT equations [14], we define the set
+K := {�U ∈ Rdof| h(�U) = 0, g(�U) ⩽ 0},
+(4.1)
+with equality constraints h : Rdof → Rl and inequality constraints g : Rdof → Rm
+being vector-valued continuously differentiable functions. The inequality constraints are
+used to ensure positivity. The equality constraint ensures that the limited DIRK-LDG
+discretization is mass conservative. Mass conservation is a property of the unlimited
+DIRK-LDG discretization, but one has to ensure that this property also holds after
+applying the positivity preserving limiter.
+Let L be the LDG discretization (3.4) for each DIRK stage i = 1, 2, · · · , s, without a
+positivity preserving limiter. We assume that L is a continuously differentiable function
+from K to Rdof. The corresponding KKT-system [14] then is
+L(�U) + ∇ �Uh(�U)Tµ + ∇ �Ug(�U)Tλ = 0,
+(4.2a)
+−h(�U) = 0,
+(4.2b)
+0 ⩾ g(�U)⊥λ ⩾ 0,
+(4.2c)
+where µ ∈ Rl and λ ∈ Rm are the Lagrange multipliers used to ensure h(�U) = 0
+and g(�U) ⩽ 0, respectively, �U ∈ Rdof are the LDG coefficients in the KKT-DIRK-
+LDG discretization, and ∇ �U denotes the gradient with respect to �U. The compatibility
+condition (4.2c) is equivalent to
+gj(�U) ⩽ 0,
+λj ⩾ 0,
+and
+gj(�U)λj = 0,
+j = 1, 2, · · ·, m,
+11
+
+which can be expressed as
+min(−gj(�U), λj) = 0,
+j = 1, 2, · · ·, m.
+The KKT-system then can be formulated as
+0 = F(z) =
+
+
+
+
+L(�U) + ∇ �Uh(�U)Tµ + ∇ �Ug(�U)Tλ
+−h(�U)
+min(−g(�U), λ)
+
+
+
+ .
+(4.3)
+Here z = (�U, µ, λ) ∈ Rdof+l+m, and F : Rdof+l+m → Rdof+l+m represents the DIRK-LDG
+discretization combined with the positivity and mass conservation constraints. Note, the
+KKT system (4.3) is nonlinear and F(z) is not continuously differentiable, as is necessary
+for standard Newton methods, but semi-smooth. We will therefore solve (4.3) with the
+active set semi-smooth Newton method presented in [30].
+4.2
+Positivity preserving LDG discretizations
+In this section, we will provide the details of the higher order accurate positivity
+preserving DIRK-LDG discretizations (3.4) coupled with the positivity and mass con-
+servation constraints using Lagrange multipliers as stated in (4.2).
+Let Nk be the number of basis functions in one element. Let Ne be the number of
+elements K in the tessellation Th of the domain Ω. We introduce the following notation
+for the element-wise positivity preserving LDG solution
+Uh|K :=
+Nk
+�
+j=1
+�UK
+j φK
+j ,
+QQQh|K :=
+Nk
+�
+j=1
+�QQQ
+K
+j φK
+j
+with K ∈ Th, φK
+j
+the tensor product Legendre basis functions in Qk(K), and LDG
+coefficients �UK
+j
+∈ R, �QQQ
+K
+j
+∈ Rd.
+Taking in each element K ∈ Th the test function
+ρ = φK
+j , j = 1, 2, · · · , Nk in the operator L1
+h(QQQh; ρ), stated in (2.2a), we can define
+L1
+h(�QQQ) := L1
+h(QQQh; ρ) ∈ RNkNe,
+(4.4)
+with similar definitions of Lk
+h for Lk
+h, k = 2, 3, 4 stated in (2.2b)-(2.2d).
+Representing the block-diagonal mass matrices for the scalar and vector variables as
+M ∈ RNkNe×NkNe and M
+M
+M ∈ RdNkNe×dNkNe, respectively, the operator L for DIRK stage
+12
+
+i (i = 1, 2, · · · , s), as stated in (3.4a), can be expressed as
+L(�Un+1,i) :=M(�Un+1,i − �Un) + τ n+1
+i
+�
+j=1
+aijL1
+h(�QQQ
+n+1,j),
+(4.5)
+with LDG coefficients �Un+1,i ∈ RNkNe. Similarly, using (3.4b), (3.4c) and (3.4d), we have
+�QQQ
+n+1,i = − M
+M
+M −1L2
+h(�Un+1,i, �SSS
+n+1,i),
+(4.6a)
+�SSS
+n+1,i = − M
+M
+M −1L3
+h( �P n+1,i),
+(4.6b)
+�P n+1,i = − M−1L4
+h(�Un+1,i),
+(4.6c)
+with LDG coefficients �QQQ
+n+1,i ∈ RdNkNe, �SSS
+n+1,i ∈ RdNkNe, �P n+1,i ∈ RNkNe.
+The constraints on the DIRK-LDG discretization can be directly imposed on the
+DG coefficients for each DIRK stage using the equality and inequality constraints in the
+KKT-system (4.3). We obtain for each DIRK stage i, with i = 1, 2, · · · , s, the LDG
+coefficients �Un+1,i by solving the following KKT system for �Un+1,i,
+
+
+
+
+L(�Un+1,i) + ∇ �Uh(�Un+1,i)Tµ + ∇ �Ug(�Un+1,i)Tλ
+−h(�Un+1,i)
+min(−g(�Un+1,i), λ)
+
+
+
+ = 0,
+(4.7)
+where the positivity preserving inequality constraint g(�Un+1,i) and the mass conservation
+equality constraint h(�Un+1,i) are defined as follows.
+1. Positivity preserving inequality constraint
+In each element K ∈ Th, we define the function g stated in (4.7) as
+gK
+p (�Un+1,i) = umin −
+Nk
+�
+j=1
+�UK,(n+1,i)
+j
+φK
+j (xxxp),
+p = 1, · · ·, Np,
+(4.8)
+with Np the number of Gauss-Lobatto quadrature points, and xxxp the Gauss-Lobatto
+quadrature points where the inequality constraints Uh(xxxp) ⩾ umin are imposed. The use
+of Gauss-Lobatto quadrature rules ensures that the positivity constraint is also imposed
+in the computation of the numerical fluxes at the element edges where Gauss-Lobatto
+rules have, next to the element itself, also quadrature points. Note, the Gauss-Lobatto
+quadrature points xxxp are the only points used in the LDG discretization and the positivity
+constraint umin therefore only needs to be enforced at these points.
+13
+
+2. Mass conservation equality constraint
+In order to ensure mass conservation of the LDG discretization when the positivity
+constraint is enforced, we impose the following equality constraint, which is obtained by
+setting ρ = 1 in (3.4a) and using the numerical flux (2.3) or (2.4).
+h(�Un+1,i) =
+�
+K∈Th
+�
+K
+Un
+h dK + τ n+1
+i
+�
+j=1
+aij
+�
+K∈Th
+∂K∩∂Ω̸=∅
+(�QQQ
+n+1,j
+h
+· ννν, 1)∂K
+−
+�
+K∈Th
+Nk
+�
+j=1
+�UK,(n+1,i)
+j
+�
+K
+φK
+j (xxx)dK,
+(4.9)
+with Un
+h the KKT-DIRK-LDG solution at time tn.
+For each DIRK stage i, the KKT-system (4.7) for the higher order accurate positivity
+preserving LDG discretization is now defined. After solving the KKT equations (4.7) for
+i = 1, · · · , s, the numerical solution at time tn+1 is directly obtained from the last DIRK
+stage, Un+1
+h
+= Un+1,s
+h
+since we use DIRK methods with asi = bi.
+Remark 4.1. In order to ensure the positivity of the discrete initial solution U0
+h, we
+use the L2-projection coupled with the positivity constraint (4.8), which is obtained
+by replacing �Un+1,i with �U0. The equality constraint ensures mass conservation of the
+positivity limited initial solution
+h(�U0) =
+�
+K∈Th
+�
+K
+u0(xxx)dK −
+�
+K∈Th
+Nk
+�
+j=1
+�UK,0
+j
+�
+K
+φK
+j (xxx)dK.
+The constraints on the L2-projection are imposed using KKT equations similar to (4.3).
+To prevent pathological cases, we assume that the limited initial solution satisfies
+1
+|Ω|
+�
+K∈Th
+�
+K
+u0(xxx)dK ⩾ umin.
+Remark 4.2. We emphasize that umin must be chosen strictly positive to ensure that
+errors do not violate the positivity of the numerical solution due to the finite precision
+of the computer arithmetic.
+4.3
+Unique solvability and stability of the positivity preserving
+LDG discretization
+In Section 4.2, we have presented the positivity preserving LDG discretization for
+(1.1). In this section, we will consider the unique solvability of the algebraic equations
+14
+
+resulting from the backward Euler KKT-LDG discretization. In the theoretical analysis
+we will also consider the entropy dissipation of the positivity preserving backward Euler
+LDG discretization and use periodic boundary conditions.
+With (4.5)-(4.9), the positivity preserving backward Euler LDG discretization results
+now in the following KKT system,
+L(�Un+1) + ∇ �Uh(�Un+1)Tµn+1 + ∇ �Ug(�Un+1)Tλn+1 = 0,
+(4.10a)
+−h(�Un+1) = 0,
+(4.10b)
+min(−g(�Un+1), λn+1) = 0.
+(4.10c)
+Here L : RNkNe → RNkNe and
+L(�Un+1) :=M(�Un+1 − �Un) + τ n+1B �QQQ
+n+1,
+(4.11)
+M
+M
+M �QQQ
+n+1 =Cd(�Un+1)�SSS
+n+1,
+(4.12)
+M
+M
+M �SSS
+n+1 =A �P n+1,
+(4.13)
+M �P n+1 =D(�Un+1).
+(4.14)
+From (4.4)-(4.6), we obtain that
+B �QQQ
+n+1 =L1
+h(�QQQ
+n+1) ∈ RNkNe,
+(4.15)
+Cd(�Un+1)�SSS
+n+1 = − L2
+h(�Un+1, �SSS
+n+1) ∈ RdNkNe,
+(4.16)
+A �P n+1 = − L3
+h( �P n+1) ∈ RdNkNe,
+(4.17)
+D(�Un+1) = − L4
+h(�Un+1) ∈ RNkNe,
+(4.18)
+where
+Cd(�Un+1) =
+
+
+
+
+C(�Un+1)
+...
+C(�Un+1)
+
+
+
+ ∈ RdNkNe×dNkNe, C(�Un+1) ∈ RNkNe. (4.19)
+The constraints h : RNkNe → R, g : RNkNe → RNpNe are defined by
+h(�Un+1) :=
+�
+K∈Th
+�
+K
+U0
+hdK −
+�
+K∈Th
+Nk
+�
+j=1
+�UK,(n+1)
+j
+�
+K
+φK
+j (xxx)dK,
+(4.20)
+g(�Un+1) :=(gK1
+1 (�Un+1), · · · , gK1
+Np(�Un+1), · · · , g
+KNe
+1
+(�Un+1), · · · , g
+KNe
+Np (�Un+1)),
+(4.21)
+with the definition of the constraints g
+Kj
+p , 1 ⩽ p ⩽ Np, 1 ⩽ j ⩽ Ne given in (4.8).
+15
+
+4.3.1
+Auxiliary results used to prove the solvability of the KKT-system
+In this section, we will introduce some auxiliary results, which will be used in Section
+4.3.2 to prove the unique solvability of the KKT-system (4.10).
+Definition 4.3. [14, Sections 1.1, 3.2] Let K be given by (4.1), given a map L : K →
+Rdof. The Variational Inequality (VI(K, L)) is to find �U ∈ K such that
+(y − �U)TL(�U) ⩾ 0,
+y ∈ K.
+(4.22)
+The solution of VI(K, L) (4.22) is denoted by SOL(K, L).
+Using the nodal basis function and the definition of g in (4.21) and (4.8), the inequal-
+ity constraint set in (4.1) can be written as
+Kb := {�U ∈ Rdof| �Umin
+i
+⩽ �Ui ⩽ �Umax
+i
+, i ∈ {1, · · · , dof}},
+(4.23)
+and we write Kb as
+Kb =
+N
+�
+ϑ=1
+Knϑ,
+(4.24)
+where Knϑ is a subset of Rnϑ with
+N
+�
+ϑ=1
+nϑ = dof. Thus for a vector �U ∈ Kb, we write
+�U = (�Uϑ), where each �Uϑ belongs to Knϑ.
+Definition 4.4. [14, Section 3.5.2] Let Kb be given by (4.23), a map L : Kb → Rdof is
+said to be
+a) a P-function on Kb if for all pairs of distinct vectors �U and �U′ in Kb,
+max
+1⩽ϑ⩽N(�Uϑ − �U′
+ϑ)T(Lϑ(�U) − Lϑ(�U′)) > 0,
+b) a uniformly P-function on Kb if there exists a constant ̟ > 0 such that for all
+pairs of distinct vectors �U and �U′ in Kb,
+max
+1⩽ϑ⩽N(�Uϑ − �U′
+ϑ)T(Lϑ(�U) − Lϑ(�U′)) ⩾ ̟∥�U − �U′∥2.
+Lemma 4.1. [14, Proposition 3.5.10] Let Kb be given by (4.23).
+a) If L is a P-function on Kb, then VI(Kb, L) has at most one solution.
+b) If each Knϑ is closed convex and L is a continuous uniformly P-function on Kb,
+then the VI(Kb, L) has a unique solution.
+16
+
+Lemma 4.2. [14, Proposition 1.3.4] Let �U ∈ SOL(K, L) solve (4.22) with K given by
+(4.1). If Abadie’s Constraint Qualification holds at �U, then there exist vectors µ ∈ Rl
+and λ ∈ Rm satisfying the KKT system (4.10).
+Conversely, if each function hj (1 ⩽ j ⩽ l) is affine and each function gi (1 ⩽ i ⩽ m)
+is convex, and if (�U, µ λ) satisfies (4.10), then �U solves VI(K, L) given by (4.22) with K
+given by (4.1).
+4.3.2
+Existence and uniqueness of LDG discretization with positivity and
+mass conservation constraints
+In this section, we will prove the existence and uniqueness of the KKT system (4.10)-
+(4.21) using the unique solvability conditions discussed in Section 4.3.1.
+Lemma 4.3. For periodic boundary conditions, the matrices B in (4.15) and A in (4.17)
+satisfy BT = A.
+Proof. In order to prove the symmetry of B in (4.15) and A in (4.17), we define the
+bilinear function a : (V k
+h × W
+W
+W k
+h) × (V k
+h × W
+W
+W k
+h) → R by
+a(P n+1
+h
+,QQQn+1
+h
+; ρ,θθθ) =(QQQn+1
+h
+, ∇ρ) −
+�
+K∈Th
+(�QQQ
+n+1
+h
+· ννν, ρ)∂K
+− (P n+1
+h
+, ∇ · θθθ) +
+�
+K∈Th
+( �P n+1
+h
+,ννν · θθθ)∂K.
+Based on the definition of B in (4.15) using (2.2a), A in (4.17) using (2.2c), we rewrite
+the above bilinear function a as follows:
+a(P n+1
+h
+,QQQn+1
+h
+; ρ,θθθ) =(̺, Θ)
+�
+0
+B
+A
+0
+�
+( �P n+1, �QQQ
+n+1)T,
+with ̺, Θ the LDG coefficients of ρ,θθθ and �P n+1, �QQQ
+n+1 the LDG coefficients of P n+1
+h
+,QQQn+1
+h
+,
+respectively.
+Interchanging the arguments of a, we get
+a(ρ,θθθ; P n+1
+h
+,QQQn+1
+h
+) =(θθθ, ∇P n+1
+h
+) −
+�
+K∈Th
+(�θθθ · ννν, P n+1
+h
+)∂K
+− (ρ, ∇ · QQQn+1
+h
+) +
+�
+K∈Th
+(�ρ,ννν · QQQn+1
+h
+)∂K
+17
+
+= − (P n+1
+h
+, ∇ · θθθ) +
+�
+K∈Th
+(θθθ · ννν, P n+1
+h
+)∂K −
+�
+K∈Th
+(�θθθ · ννν, P n+1
+h
+)∂K
++ (QQQn+1
+h
+, ∇ρ) −
+�
+K∈Th
+(ρ,ννν · QQQn+1
+h
+)∂K +
+�
+K∈Th
+(�ρ,ννν · QQQn+1
+h
+)∂K,
+Using equality (2.7), the alternating numerical fluxes for �θθθ and �ρ in (2.3) or (2.4), and
+the periodic boundary conditions, we obtain
+�
+K∈Th
+(θθθ · ννν, P n+1
+h
+)∂K −
+�
+K∈Th
+(�θθθ · ννν, P n+1
+h
+)∂K =
+�
+K∈Th
+( �P n+1
+h
+,ννν · θθθ)∂K,
+−
+�
+K∈Th
+(ρ,ννν · QQQn+1
+h
+)∂K +
+�
+K∈Th
+(�ρ,ννν · QQQn+1
+h
+)∂K = −
+�
+K∈Th
+(�QQQ
+n+1
+h
+· ννν, ρ)∂K.
+Hence,
+a(P n+1
+h
+,QQQn+1
+h
+; ρ,θθθ) = a(ρ,θθθ; P n+1
+h
+,QQQn+1
+h
+),
+which implies
+(̺, Θ)
+�
+0
+B
+A
+0
+�
+( �P n+1, �QQQ
+n+1)T =( �P n+1, �QQQ
+n+1)
+�
+0
+B
+A
+0
+�
+(̺, Θ)T
+=(̺, Θ)
+�
+0
+AT
+BT
+0
+�
+( �P n+1, �QQQ
+n+1)T.
+(4.25)
+Since (P n+1
+h
+,QQQn+1
+h
+) ∈ V k
+h × W
+W
+W k
+h and (ρ,θθθ) ∈ V k
+h × W
+W
+W k
+h are arbitrary functions, relation
+(4.25) implies that A = BT.
+Using (4.12)-(4.14) and Lemma 4.3, the operator L(�Un+1) in (4.11) can be written
+as
+L(�Un+1) = M(�Un+1 − �Un) + τ n+1BM
+M
+M −1Cd(�Un+1)M
+M
+M −1BTM−1D(�Un+1).
+(4.26)
+Lemma 4.4. Given �Un, the operator L in (4.26) is a uniformly P-function on Kb.
+Proof. Using relation (4.26) for L, for arbitrary �Un+1
+I
+, �Un+1
+II
+∈ Kb, there holds
+L(�Un+1
+I
+) − L(�Un+1
+II
+) =M(�Un+1
+I
+− �Un+1
+II
+) + τ n+1BM
+M
+M −1Cd(�Un+1
+I
+)M
+M
+M −1BT M−1D(�Un+1
+I
+)
+− τ n+1BM
+M
+M −1Cd(�Un+1
+II
+)M
+M
+M −1BTM−1D(�Un+1
+II
+).
+(4.27)
+18
+
+After subtracting and adding τ n+1BM
+M
+M −1Cd(�Un+1
+I
+)M
+M
+M −1BTM−1D(�Un+1
+II
+) in (4.27), we
+obtain
+L(�Un+1
+I
+) − L(�Un+1
+II
+)
+=M(�Un+1
+I
+− �Un+1
+II
+) + τ n+1BM
+M
+M −1Cd(�Un+1
+I
+)M
+M
+M −1BTM−1(D(�Un+1
+I
+) − D(�Un+1
+II
+))
++ τ n+1BM
+M
+M −1(Cd(�Un+1
+I
+) − Cd(�Un+1
+II
+))M
+M
+M−1BTM−1D(�Un+1
+II
+).
+(4.28)
+With the definition of D in (4.18) using (2.2d), we obtain that
+(D(�Un+1
+I
+) − D(�Un+1
+II
+))i =
+�
+Ω
+�
+H′
+�NkNe
+�
+j=1
+�Un+1
+I,j φj
+�
+− H′
+�NkNe
+�
+j=1
+�Un+1
+II,j φj
+��
+φidΩ
+=
+NkNe
+�
+j=1
+(�Un+1
+I,j
+− �Un+1
+II,j )
+�
+Ω
+H′′(ξn+1
+1
+)φjφidΩ, i ∈ {1, · · · , NkNe}, ξn+1
+1
+∈ (Un+1
+h,I , Un+1
+h,II ),
+and write
+D(�Un+1
+I
+) − D(�Un+1
+II
+) :=D �U(ξn+1
+1
+)(�Un+1
+I
+) − �Un+1
+II
+).
+(4.29)
+Similarly, from the definition of Cd in (4.16), (4.19) using (2.2b), we obtain that
+Cd(�Un+1
+I
+) − Cd(�Un+1
+II
+) =
+
+
+
+
+C(�Un+1
+I
+) − C(�Un+1
+II
+)
+...
+C(�Un+1
+I
+) − C(�Un+1
+II
+)
+
+
+
+ ,
+(C(�Un+1
+I
+) − C(�Un+1
+II
+))ij =
+�
+Ω
+�
+f
+�NkNe
+�
+k=1
+�Un+1
+I,k φk
+�
+− f
+�NkNe
+�
+k=1
+�Un+1
+II,k φk
+��
+φjφidΩ
+=
+NkNe
+�
+k=1
+(�Un+1
+I,k − �Un+1
+II,k )
+�
+Ω
+f ′(ξn+1
+2
+)φkφjφidΩ, i, j, k ∈ {1, · · · , NkNe}, ξn+1
+2
+∈ (Un+1
+h,I , Un+1
+h,II ),
+and write
+C(�Un+1
+I
+) − C(�Un+1
+II
+) :=
+NkNe
+�
+k=1
+[Cd�U(ξn+1
+2
+)]k(�Un+1
+I,k ) − �Un+1
+II,k ).
+(4.30)
+Assume for arbitrary �U ∈ Kb in (4.23), that
+|C(�U)ij| ⩽c,
+|D(�U)i| ⩽ c,
+19
+
+|[C �U(�U)ij]k| ⩽c,
+|D �U(�U)ij| ⩽ c, i, j, k ∈ {1, · · · , NkNe},
+(4.31)
+with c a positive constant, independent of �U. In the remainder of this section, c is a
+positive constant, but not necessarily the same.
+Using (4.29)-(4.30) and assumption (4.31), we obtain the following two estimates
+(�Un+1
+I
+− �Un+1
+II
+)TBM
+M
+M −1Cd(�Un+1
+I
+)M
+M
+M −1BTM−1(D(�Un+1
+I
+) − D(�Un+1
+II
+))
+⩽∥B∥∥M
+M
+M −1∥∥Cd(�Un+1
+I
+)∥∥M
+M
+M−1∥∥BT∥∥M−1∥∥D �U(ξn+1
+1
+)∥∥�Un+1
+I
+− �Un+1
+II
+∥2
+⩽c∥�Un+1
+I
+− �Un+1
+II
+∥2,
+and
+(�Un+1
+I
+− �Un+1
+II
+)TBM
+M
+M−1(Cd(�Un+1
+I
+) − Cd(�Un+1
+II
+))M
+M
+M −1BT M−1D(�Un+1
+II
+)
+⩽∥B∥∥M
+M
+M −1∥
+NkNe
+�
+k=1
+∥[Cd�U(ξn+1
+2
+)]k∥∥M
+M
+M−1∥∥BT∥∥M−1∥∥D(�Un+1
+II
+)∥∥�Un+1
+I
+− �Un+1
+II
+∥2
+⩽c∥�Un+1
+I
+− �Un+1
+II
+∥2.
+Then multiplying (4.28) with (�Un+1
+I
+− �Un+1
+II
+)T gives
+(�Un+1
+I
+− �Un+1
+II
+)T(L(�Un+1
+I
+) − L(�Un+1
+II
+)) = (�Un+1
+I
+− �Un+1
+II
+)TM(�Un+1
+I
+− �Un+1
+II
+)
++ τ n+1(�Un+1
+I
+− �Un+1
+II
+)TBM
+M
+M−1Cd(�Un+1
+I
+)M
+M
+M−1BTM−1(D(�Un+1
+I
+) − D(�Un+1
+II
+))
++ τ n+1(�Un+1
+I
+− �Un+1
+II
+)TBM
+M
+M−1(Cd(�Un+1
+I
+) − Cd(�Un+1
+II
+))M
+M
+M −1BT M−1D(�Un+1
+II
+)
+⩾σ∥�Un+1
+I
+− �Un+1
+II
+∥2 − 2cτ n+1∥�Un+1
+I
+− �Un+1
+II
+∥2,
+(4.32)
+where σ > 0 is the smallest eigenvalue of the symmetric positive mass matrix M.
+Choosing 0 < τ n+1 ⩽ σ
+4c, we obtain that
+(�Un+1
+I
+− �Un+1
+I
+)T(L(�Un+1
+I
+) − L(�Un+1
+II
+)) ⩾ σ
+2 ∥�Un+1
+I
+− �Un+1
+II
+∥2, ∀�Un+1
+I
+, �Un+1
+II
+∈ Kb, (4.33)
+which implies that for τ n+1 sufficiently small L(�Un+1) is a uniformly function of Kb,
+From Lemmas 4.1, Lemma 4.2 and Lemma 4.4, we obtain the main result of this
+section.
+Theorem 4.5. Given the DG coefficients �Un and the positivity preserving backward Eu-
+ler KKT-LDG discretization (4.10)-(4.21) with equality constraint h ≡ 0. If assumption
+(4.31) is satisfied, then the KKT system (4.10)-(4.21) has only one solution.
+20
+
+Corollary 4.6. Given the DG coefficients �Un. If assumption (4.31) is satisfied, then
+for the degenerate parabolic equation (1.1) with periodic boundary conditions there exists
+only one solution satisfying the higher order accurate in time, positivity preserving KKT-
+DIRK-LDG discretizations (4.7) with equality constraint h ≡ 0.
+Proof. Since the DIRK coefficient matrix (aij) introduced in Section 3.2 is a lower tri-
+angular matrix, the structure of the DIRK-LDG discretizations is similar to the form
+obtained for the backward Euler LDG discretization. The analysis therefore is completely
+analogous to Theorem 4.5.
+4.3.3
+Stability of the KKT-LDG discretization
+Theorem 4.7. Given the numerical solution Un
+h ∈ V k
+h of the positivity preserving back-
+ward Euler KKT-LDG discretization (4.10)-(4.21). If assumption (4.31) is satisfied,
+then the discrete entropy Eh stated in (3.2) satisfies for n = 0, 1, · · ·,
+Eh(Un+1
+h
+) ⩽ Eh(Un
+h ),
+(4.34)
+which implies that the positivity preserving backward Euler KKT-LDG discretization is
+unconditionally entropy dissipative.
+Proof. From Lemma 4.2, we obtain that the LDG coefficients �Un+1 of the positivity
+preserving solution Un+1
+h
+solve
+(y − �Un+1)TL(�Un+1) ⩾ 0,
+∀y ∈ K,
+(4.35)
+with L given by (4.26) and K given by (4.1).
+From assumption (4.31), we have that there exists a positive constant c ⩾ c0 > 0
+such that
+�Un+1 − cM−1D(�Un+1) ∈ K.
+(4.36)
+Next, we choose y = �Un+1 − cM−1D(�Un+1) in (4.35), which implies
+−c(M−1D(�Un+1))TL(�Un+1) ⩾ 0.
+(4.37)
+Using (4.26) and the fact that c > 0, we obtain that (4.37) implies the inequality
+D(�Un+1)T(�Un+1 − �Un)
+21
+
++τ n+1D(�Un+1)TM−1BM
+M
+M −1Cd(�Un+1)M
+M
+M −1BTM−1D(�Un+1) ⩽ 0.
+(4.38)
+From the definition of Cd in (4.16), (4.19) using (2.2b) and the conditions on f stated
+in (1.3), we obtain that Cd(�Un+1) is symmetric positive definite. Hence using τ n+1 > 0,
+we have
+τ n+1D(�Un+1)TM−1BM
+M
+M −1Cd(�Un+1)M
+M
+M −1BTM−1D(�Un+1) ⩾ 0,
+which with (4.38) yields
+D(�Un+1)T(�Un+1 − �Un) ⩽ 0.
+(4.39)
+From the definition of D in (4.18) using (2.2d) and (4.39), we obtain the bound
+�
+φ(xxx), Un+1
+h
+− Un
+h
+�
++
+�
+H′(Un+1
+h
+), Un+1
+h
+− Un
+h
+�
+⩽ 0.
+(4.40)
+Using the following Taylor expansion
+H(Un
+h ) =H(Un+1
+h
+) + H′(Un+1
+h
+)(Un
+h − Un+1
+h
+)
++ 1
+2H′′(ξn+1
+3
+)(Un+1
+h
+− Un
+h )2, ξn+1
+3
+∈ (Un
+h , Un+1
+h
+),
+we obtain that (4.40) gives
+�
+φ(xxx), Un+1
+h
+− Un
+h
+�
++
+�
+H(Un+1
+h
+) − H(Un
+h ), 1
+�
++ 1
+2
+�
+H′′(ξn+1
+3
+),
+�
+Un+1
+h
+− Un
+h
+�2�
+⩽ 0,
+which implies, using the definition of Eh in (3.2), that
+Eh(Un+1
+h
+) − Eh(Un
+h ) =
+�
+φ(xxx), Un+1
+h
+− Un
+h
+�
++
+�
+H(Un+1
+h
+) − H(Un
+h ), 1
+�
+⩽ 0,
+since (1.3) gives H′′(ξn+1
+3
+) ⩾ 0. This proves (4.34).
+5
+Numerical tests
+In this section, we will discuss several numerical experiments to demonstrate the
+performance of the KKT-DIRK-LDG positivity preserving algorithm for the degenerate
+parabolic equation (1.1).
+In the computations, we will consider the porous medium
+equation, the nonlinear diffusion equation with a double-well potential and the nonlinear
+Fokker-Plank equation for fermion and boson gases. Firstly, we will present in Section
+22
+
+5.1 the order of accuracy of the DIRK-LDG discretizations with and without positivity
+preserving limiter to investigate if the limiter negatively affects the accuracy of the
+discretizations. Next, we will present in Sections 5.3-5.5 test cases for which the positivity
+preserving limiter is essential. Without the positivity constraint, obtaining a numerical
+solution or only for extremely small time steps is impossible.
+In the computations, we take τ = α · h.
+If the Newton method during strongly
+nonlinear stages requires a large number of iterations, it is generally more efficient to
+reduce the time step to 1
+2τ and restart the Newton iterations. When the Newton method
+converges well, then τ is increased each time step to 1.2τ, till the maximum predefined
+time step is obtained.
+In order to avoid round-off effects, a positivity bound umin = 10−10 is used in the
+numerical simulations, except for Section 5.1 where umin = 10−14. If it is not stated
+otherwise, the numerical results for 1D problems are obtained on a mesh containing 100
+elements and Legendre polynomials of order 2. For 2D problems, a mesh consisting of
+30×30 square elements and tensor product Legendre polynomial basis functions of order
+2 are used.
+5.1
+Accuracy tests
+For the accuracy test, we use a uniform mesh with M elements and positivity bound
+umin = 10−14.
+Example 5.1. We consider (1.1) on the domain Ω = [−1, 1] with Dirichlet boundary
+conditions based on the exact solution and select the following parameters
+f(u) = u,
+H′(u) = u2,
+φ(x) = 0,
+x ∈ Ω.
+Then (1.1) with a properly chosen source term has the nonnegative solution
+u(x, t) = exp(−t)(1 − x4)5,
+x ∈ Ω.
+We take α in the definition of the time step as α = 1. Tables 5.1-5.2 show that the
+DIRK-LDG discretizations with and without positivity preserving limiter are convergent
+at the rate O(hk+1) for basis functions with polynomial order ranging from 1 to 3. The
+errors and orders of accuracy presented in Tables 5.1-5.2 indicate that the positivity
+preserving limiter is necessary and does not negatively affect accuracy.
+23
+
+Table 5.1:
+Error in L∞− and L1− norms for Example 5.1 at time T = 1 without positivity
+preserving limiter.
+Pk
+M
+∥un − un
+h∥L∞(Ω)
+Order
+∥un − un
+h∥L1(Ω)
+Order
+min un
+h
+40
+7.33E-003
+–
+1.03E-003
+–
+-8.87e-005
+1
+80
+1.24e-003
+2.56
+2.27e-004
+2.18
+-1.08e-005
+160
+2.63e-004
+2.24
+5.44e-005
+2.06
+-4.41e-007
+320
+6.05e-005
+2.12
+1.35e-005
+2.01
+-1.57e-008
+40
+1.70E-003
+–
+8.73E-005
+–
+-1.60e-005
+2
+80
+1.43e-004
+3.57
+8.07e-006
+3.44
+-1.79e-007
+160
+1.36e-005
+3.39
+9.40e-007
+3.10
+-6.24e-009
+320
+1.34e-006
+3.34
+1.16e-007
+3.02
+-2.07e-010
+40
+1.45e-004
+–
+6.00e-006
+–
+-2.14e-006
+3
+80
+9.87e-006
+3.88
+3.11e-007
+4.27
+-9.56e-008
+160
+5.51e-007
+4.16
+1.76e-008
+4.14
+-3.51e-009
+320
+3.50e-008
+3.98
+1.11e-009
+3.99
+-1.19e-010
+Table 5.2:
+Error in L∞− and L1− norms for Example 5.1 at time T = 1 with positivity
+preserving limiter.
+Pk
+M
+∥un − Un
+h ∥L∞(Ω)
+Order
+∥un − Un
+h ∥L1(Ω)
+Order
+min Un
+h
+40
+7.33E-003
+–
+1.05E-003
+–
+2.05e-005
+1
+80
+1.24e-003
+2.56
+2.27e-004
+2.21
+8.15e-007
+160
+2.63e-004
+2.24
+5.44e-005
+2.06
+2.77e-008
+320
+6.05e-005
+2.12
+1.35e-005
+2.01
+8.55e-010
+40
+1.70E-003
+–
+8.73E-005
+–
+6.15e-008
+2
+80
+1.43e-004
+3.57
+8.08e-006
+3.43
+3.03e-007
+160
+1.36e-005
+3.39
+9.40e-007
+3.10
+1.08e-008
+320
+1.34e-006
+3.34
+1.16e-007
+3.02
+4.55e-010
+40
+1.45e-004
+–
+6.02e-006
+–
+1.00e-014
+3
+80
+9.87e-006
+3.88
+3.13e-007
+4.27
+4.45e-008
+160
+5.51e-007
+4.16
+1.77e-008
+4.14
+1.21e-009
+320
+3.50e-008
+3.98
+1.11e-009
+4.00
+2.55e-011
+24
+
+5.2
+Porous media equation
+For the porous media equation, f(u)H′′(u) can locally vanish, resulting in degenerate
+cases [5].
+We test the asymptotic behavior of the numerical solution and will show
+that the KKT limiter is necessary. The entropy defined in (1.4), which should be non-
+increasing, is also computed.
+Example 5.2. In order to test degenerate cases, we choose the following parameters in
+(1.1) on the domain Ω = [0, 1] with zero-flux boundary conditions
+f(u) = u,
+H′(u) = 4
+3
+�
+u − 1
+2
+�3
+max
+�
+u, 1
+2
+�
+,
+φ(x) = 0,
+x ∈ Ω,
+and initial data
+u(x, 0) = 1
+2 − 1
+2 cos(2πx),
+x ∈ Ω.
+During the computations, the value of α for optimal convergence of the semi-smooth
+Newton algorithm is usually close to 0.1. We present the numerical solution in Fig.
+5.1 for basis functions with polynomial order ranging from 1 to 3 and with the KKT
+limiter enforced. Values of the Lagrange multiplier λ larger than 10−10 are shown in Fig.
+5.1, which indicate that the positivity constraint works well since it is only active at
+locations where the solution is close to the minimum value. The entropy decay using the
+KKT limiter and polynomial basis functions of order 3 is presented in Fig. 5.2, in which
+the result is consistent with the stability analysis. In Fig. 5.3, the numerical solution
+without KKT limiter and for polynomial basis functions with order 3 is plotted. This
+computation breaks down due to unphysical oscillations.
+Example 5.3. We consider a 2D test case on the domain Ω = [−6, 6]2 with zero-flux
+boundary conditions by choosing in (1.1) the following parameters
+f(u) = u,
+H′(u) = 2u,
+φ(xxx) = 0,
+xxx ∈ Ω,
+and initial data
+u(xxx, 0) = exp
+�
+−1
+2|xxx|2
+�
+,
+xxx ∈ Ω.
+The value of α in the definition of the time step ranges in this case between 0.1 and
+1. Fig. 5.4 presents the numerical solution with the KKT limiter active and also the
+25
+
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+x
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Uh
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+10-3
+Time 3
+Uh
+(a) P1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+x
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Uh
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+10-3
+Time 3
+Uh
+(b) P2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+x
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Uh
+0
+0.5
+1
+1.5
+10-3
+Time 3
+Uh
+(c) P3
+Figure 5.1: (Example 5.2) Numerical solution Uh for different orders of polynomial basis
+functions P1-P3 with the KKT limiter enforced and Lagrange multiplier λ (red dots).
+Figure 5.2: (Example 5.2) Entropy Eh for P3 basis functions with the KKT limiter enforced.
+26
+
+Entropy
+X10~3
+3.5
+3
+2.5 米
+E
+2
+迷
+米
+米
+米
+1.5
+******米
+0.5
+0
+0
+0.5
+1
+1.5
+2
+2.5
+3
+t0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+x
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+Uh
+Time 0.29317
+Figure 5.3: (Example 5.2) Numerical solution Uh for P3 basis functions without KKT limiter
+just before blow up.
+Lagrange multiplier λ. Considering the position of the non-zero Lagrange multipliers,
+we can see that the limiter also works well in the two-dimensional case since it is only
+active in areas where positivity must be enforced. The entropy decay is plotted in Fig.
+5.5, which is consistent with the stability result of the numerical solution. Without the
+KKT limiter, there will be unphysical oscillations, and the computation will break down
+at some point in the computations.
+5.3
+Nonlinear diffusion with a double-well potential
+Consider the nonlinear diffusion equation with double-well potential [18] on the do-
+main Ω = [−1.4, 1.4], which is obtained by choosing in (1.1) zero-flux boundary condi-
+tions and the following parameters
+f(u) = u,
+H′(u) = u,
+φ(x) = 1
+4x4 − 1
+2x2,
+x ∈ Ω.
+(5.1)
+This model is taken from [7]. We will test the evolution of the numerical solution with
+and without KKT limiter, and also the decay of the entropy (1.4). The value of α to
+compute the time step ranges between 0.01 to 0.1.
+Example 5.4. We consider (1.1) with (5.1) and the initial data
+u(x, 0) =
+0.2
+√
+0.4π exp
+�
+− x2
+0.4
+�
+,
+x ∈ Ω.
+27
+
+Figure 5.4: (Example 5.3) Numerical solution Uh for P2 basis functions with KKT limiter
+enforced (Left) and Lagrange multiplier λ (Right).
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+t
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+E
+Entropy
+Figure 5.5: (Example 5.3) Entropy Eh for P2 basis functions with KKT limiter enforced.
+28
+
+Time5
+X10~5
+8
+6
+lambda
+4
+0
+-6
+-4
+2
+6
+0
+4
+2
+2
+0
+4
+-2
+4
+X
+6
+-6
+yTime 5
+0.2
+0.18
+0.25
+0.16
+0.2
+0.14
+0.15
+0.12
+0.1
+0.1
+0.05
+0.08
+0.06
+-6
+-4
+0.04
+-2
+6
+4
+0
+2
+0.02
+2
+0
+4
+-2
+X
+-4
+6
+9-
+yThe numerical solution with the KKT limiter enforced and the values of the Lagrange
+multiplier λ larger than 10−10 are shown in Fig. 5.6. These results indicate that the
+numerical solution tends to a steady state and that the KKT limiter is only active at
+places where the positivity constraint needs to be imposed. The entropy dissipation is
+presented in Fig. 5.7, in which uniform decay coincides with our theoretical analysis. For
+the numerical solution without the KKT limiter, we observe that violating the positivity
+constraint will result in discontinuities in the solution and a computation breakdown,
+even for a very small CFL number.
+-1
+-0.5
+0
+0.5
+1
+x
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+Uh
+0
+0.5
+1
+1.5
+10-4
+Time 10.2418
+Uh
+-1
+-0.5
+0
+0.5
+1
+x
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+Uh
+0
+0.5
+1
+1.5
+10-4
+Time 20
+Uh
+Figure 5.6: (Example 5.4) Numerical solution Uh for P2 basis functions with KKT limiter
+enforced and Lagrange multiplier λ (red dots).
+Figure 5.7: (Example 5.4) Entropy Eh for P2 basis functions with KKT limiter enforced.
+29
+
+Entropy
+0
+-0.005
+-0.01
+E-0.015
+-0.02
+-0.025
+-0.03
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+t5.4
+Nonlinear Fokker-Plank equation for fermion gases
+We consider the nonlinear Fokker-Plank equation for fermion gases [5] on the domain
+Ω = [−10, 10]2, for which we select the following parameters in (1.1)
+f(u) = u(1 − u),
+H′(u) = log
+u
+1 − u,
+φ(xxx) = 1
+2|xxx|2,
+xxx ∈ Ω,
+(5.2)
+together with zero-flux boundary conditions.
+Example 5.5. We consider (1.1) with (5.2) and initial data
+u(xxx, 0) =
+1
+2
+√
+2π
+�
+exp
+�
+−1
+2|xxx − (2, 2)|2
+�
++ exp
+�
+−1
+2|xxx − (2, −2)|2
+�
++ exp
+�
+−1
+2|xxx − (−2, 2)|2
+�
++ exp
+�
+−1
+2|xxx − (−2, −2)|2
+��
+,
+xxx ∈ Ω.
+During the computations, the value of α in the definition of the time step ranges
+between 0.1 and 1, but for most time steps α = 1. The numerical solutions at several
+time levels with the KKT limiter enforced and the entropy dissipation are presented in
+Figs 5.8 and 5.9, respectively, showing the time-asymptotic convergence of the numerical
+solution towards a steady state.
+Without the KKT limiter, the computations break
+down, even for very small CFL numbers.
+5.5
+Nonlinear Fokker-Plank equation for boson gases
+Example 5.6. We consider a nonlinear Fokker-Plank equation for boson gases with
+zero-flux boundary condition on a domain Ω = [−10, 10], which requires the following
+parameters in (1.1)
+f(u) = u(1 + u3),
+H′(u) = log
+u
+(1 + u3)
+1
+3 ,
+φ(x) = x2
+2 ,
+x ∈ Ω.
+The initial data is [5, 19]
+u(x, 0) = M
+2
+√
+2π
+�
+exp
+�
+−(x − 2)2
+2
+�
++ exp
+�
+−(x + 2)2
+2
+��
+,
+x ∈ Ω,
+where M ⩾ 0 is the mass of u(x, 0).
+For most time steps, the value of α in the definition of the time step is 1. For the case
+M = 1, Fig. 5.10 displays the numerical solution at various times. Also, the locations
+30
+
+Figure 5.8: (Example 5.5) Numerical solution Uh for P2 basis functions with KKT limiter
+enforced.
+31
+
+Time20
+0.5
+0.6
+0.45
+0.5
+0.4
+0.4
+0.35
+0.3
+0.3
+0.2.
+0.25
+0.1
+0.2
+-10
+0.15
+-5
+10
+0.1
+0
+5
+0.05
+0
+5
+-5
+X
+10
+-10
+yTime10.9604
+0.5
+0.6
+0.45
+0.5
+0.4
+0.4
+0.35
+0.3
+0.3
+0.2
+0.25
+0.1元
+0.2
+0
+-10
+0.15
+-5
+10
+0.1
+0
+5
+0.05
+5
+-5
+x
+10
+-10
+yTime 0.49611
+0.2
+0.18
+0.25
+0.16
+0.2
+0.14
+0.15
+0.12
+0.1
+0.1
+0.05
+0.08
+0.06
+-10
+-5
+0.04
+10
+0
+5
+0.02
+0
+5
+-5
+X
+10
+-10
+yTime 0
+0.18
+0.2
+0.16
+0.14
+0.15
+0.12
+0.1
+0.1
+0.05
+0.08
+0.06
+-10
+0.04
+-5
+10
+0
+5
+0.02
+0
+5
+-5
+X
+10
+-10
+y0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+t
+-6
+-4
+-2
+0
+2
+4
+6
+8
+E
+Entropy
+Figure 5.9: (Example 5.5) Entropy Eh for P2 basis functions with KKT limiter enforced.
+and values of the Lagrange multiplier λ and the entropy with the KKT limiter enforced
+are shown. The results in Figs 5.10 and 5.11 indicate that the numerical solution tends to
+a steady state, and that the Lagrange multiplier λ is needed to ensure that the positivity
+constraint is satisfied. Without the KKT limiter, the computations break down, even
+for very small CFL numbers.
+For this model equation, there is a critical mass phenomenon [1], which states that
+solutions with a large initial mass blows-up in a finite time, while solutions with a small
+mass at an initial time will not. The numerical solutions with sub-critical mass M = 1
+at times t = 5 and t = 10 and with super-critical mass M = 10 at times t = 0.2 and
+t = 1 are shown in Fig. 5.12 and Fig. 5.13, respectively, and agree with the results
+shown in [1] and the numerical observation in [5, 19].
+6
+Conclusions
+The main topic of this paper is the formulation of higher order accurate positivity
+preserving DIRK-LDG discretizations for the nonlinear degenerate parabolic equation
+(1.1).
+The presented numerical discretizations allow the combination of a positivity
+preserving limiter and time-implicit numerical discretizations for PDEs and alleviate
+the time step restrictions of currently available positivity preserving DG discretizations,
+which generally require the use of explicit time integration methods. For the spatial
+32
+
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Uh
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Time 0
+Uh
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+Uh
+0.99
+0.995
+1
+1.005
+1.01
+1.015
+1.02
+1.025
+10-9
+Time 0.32998
+Uh
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+Uh
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+10-9
+Time 5.0713
+Uh
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+Uh
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+10-9
+Time 10
+Uh
+Figure 5.10: (Example 5.6): Numerical solution Uh for P2 basis functions with KKT limiter
+enforced.
+33
+
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+t
+4
+4.5
+5
+5.5
+E
+Entropy
+Figure 5.11: (Example 5.6): Entropy Eh for P2 basis functions with KKT limiter enforced.
+discretization an LDG method combined with a simple alternating numerical flux is used,
+which simplifies the theoretical analysis for the entropy dissipation. For the temporal
+discretization, the implicit DIRK methods significantly enlarge the time-step required
+for stability of the numerical discretization. We prove the existence, uniqueness and
+unconditional entropy dissipation of the positivity preserving high order accurate KKT-
+LDG discretization combined with an implicit Euler time discretization.
+Numerical
+results are presented to demonstrate the accuracy of the higher order accurate positivity
+preserving KKT-DIRK-LDG discretizations, which are of optimal order and not affected
+by the positivity preserving KKT limiter. The numerical solutions satisfy the entropy
+decay condition.
+Acknowledgement
+The research of Fengna Yan was funded by a fellowship from the China Scholarship
+Council (No. 201806340058). The research of J.J.W. van der Vegt was partially sup-
+ported by the University of Science and Technology of China (USTC), Hefei, Anhui,
+China, while the author was in residence at USTC. The research of Yinhua Xia was par-
+tially supported by National Natural Science Foundation of China grant No. 12271498.
+The research of Yan Xu was partially supported by National Natural Science Foundation
+of China grant No. 12071455.
+34
+
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+Uh
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+10-9
+Time 5.0713
+Uh
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+Uh
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+10-9
+Time 10
+Uh
+Figure 5.12: (Example 5.6: M = 1): Numerical solution Uh for P2 basis functions with KKT
+limiter enforced.
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+5
+10
+15
+20
+25
+Uh
+9.999796
+9.999798
+9.9998
+9.999802
+9.999804
+9.999806
+9.999808
+9.99981
+9.999812
+9.999814
+10-10
+Time 0.25832
+Uh
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+x
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Uh
+9.9992
+9.99925
+9.9993
+9.99935
+9.9994
+9.99945
+10-10
+Time 1
+Uh
+Figure 5.13: (Example 5.6: M = 10) Numerical solution Uh for P2 basis functions with KKT
+limiter enforced.
+35
+
+References
+[1] N. B. Abdallah, I. M. Gamba, and G. Toscani, On the minimization problem
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+nonlinear Fokker-Planck equations, Journal of Scientific Computing, 68 (2016),
+1217–1240.
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+puting, 36 (2014), A2296–A2325.
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+Fokker-Planck equations, Journal of Scientific Computing, 62 (2015), 803–830.
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+local discontinuous Galerkin method for the Navier-Stokes-Korteweg equations, Jour-
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+time-implicit higher order accurate discontinuous Galerkin discretizations, SIAM
+Journal on Scientific Computing, 41 (2019), A2037–A2063.
+[31] J. L. V´azquez, The porous medium equation: mathematical theory, Oxford Uni-
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+consensus models and pressureless Euler equations, Journal of Computational
+Physics, 252 (2013), 109–127.
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+[33] Q. Zhang and Z.-L. Wu, Numerical simulation for porous medium equation by
+local discontinuous Galerkin finite element method, Journal of Scientific Computing,
+38 (2009), 127–148.
+[34] X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes
+for scalar conservation laws, Journal of Computational Physics, 229 (2010), 3091–
+3120.
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+Galerkin schemes for compressible Euler equations on rectangular meshes, Journal
+of Computational Physics, 229 (2010), 8918–8934.
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+tral deferred correction coupled with local discontinuous Galerkin method for linear
+convection–diffusion equations, Journal of Scientific Computing, 77 (2018), 1001–
+1029.
+39
+
diff --git a/xdAzT4oBgHgl3EQfdvz8/content/tmp_files/load_file.txt b/xdAzT4oBgHgl3EQfdvz8/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cb0ba74bde023f1a4869ec56d3acc606f05e98ab
--- /dev/null
+++ b/xdAzT4oBgHgl3EQfdvz8/content/tmp_files/load_file.txt
@@ -0,0 +1,1343 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf,len=1342
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='01427v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='NA]  4 Jan 2023  Entropy dissipative higher order accurate positivity  preserving time-implicit discretizations for nonlinear  degenerate parabolic equations  Fengna Yan1,2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Van der Vegt2, Yinhua Xia3, Yan Xu3  Abstract  We develop entropy dissipative higher order accurate local discontinuous Galerkin  (LDG) discretizations coupled with Diagonally Implicit Runge-Kutta (DIRK) meth-  ods for nonlinear degenerate parabolic equations with a gradient flow structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Using the simple alternating numerical flux, we construct DIRK-LDG discretiza-  tions that combine the advantages of higher order accuracy, entropy dissipation and  proper long-time behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The implicit time-discrete methods greatly alleviate  the time-step restrictions needed for the stability of the numerical discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Also, the larger time step significantly improves computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We the-  oretically prove the unconditional entropy dissipation of the implicit Euler-LDG  discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Next, in order to ensure the positivity of the numerical solution,  we use the Karush-Kuhn-Tucker (KKT) limiter, which couples the positivity in-  equality constraint with higher order accurate DIRK-LDG discretizations using  Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In addition, mass conservation of the positivity-limited so-  lution is ensured by imposing a mass conservation equality constraint to the KKT  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The unique solvability and unconditional entropy dissipation for an im-  plicit first order accurate in time, but higher order accurate in space, KKT-LDG  Email address: fnyan@hfut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='cn (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Yan), j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='vandervegt@utwente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='nl (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Van der Vegt),  yhxia@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='cn (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Xia), yxu@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='cn (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Xu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  1 School of Mathematics, Hefei University of Technology, Hefei, Anhui, 230000, PR China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  2 Department of Applied Mathematics, Mathematics of Computational Science Group, University  of Twente, Enschede, 7500 AE, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  3 School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  1  discretizations are proved, which provides a first theoretical analysis of the KKT  limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Finally, numerical results demonstrate the higher order accuracy and en-  tropy dissipation of the KKT-DIRK-LDG discretizations for problems requiring a  positivity limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Keywords: Local discontinuous Galerkin discretizations, DIRK methods, Nonlinear  degenerate parabolic equations, Unconditional entropy dissipation, KKT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  1  Introduction  Consider the following degenerate parabolic equation [5]  \uf8f1  \uf8f2  \uf8f3  ut = ∇ · (f(u)∇(φ(xxx) + H′(u))),  in Ω × (0, T],  u(xxx, 0) = u0(xxx),  in Ω,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1)  with zero-flux boundary condition  ∇(φ(xxx) + H′(u)) · ννν = 0,  on ∂Ω × (0, T],  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2)  where Ω is an open bounded domain in Rd, d = 1, 2, with unit outward normal vector ννν  at the boundary ∂Ω, u(xxx, t) ⩾ 0 is a nonnegative density with time derivative denoted  as ut, φ(xxx) is a given potential function for xxx ∈ Rd, f, H are given functions such that  f : R+ −→ R+,  H : R+ −→ R,  f(u)H′′(u) ⩾ 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3)  where R+ is the nonnegative real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Here f(u)H′′(u) can vanish for certain values  of u, resulting in degenerate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The entropy corresponding to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) is defined by  E(u) =  �  Ω  (uφ(xxx) + H(u))dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4)  Multiplying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) with φ(xxx)+H′(u) and integrating over Ω, with the zero-flux boundary  condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2), together with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4), we obtain that the time derivative of the entropy  satisfies  d  dtE(u) = −  �  Ω  f(u)|∇(φ(xxx) + H′(u))|2dΩ ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5)  System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) can represent different physical problems, such as the porous media equation  [31, 33], the nonlinear nonlocal equation with a double-well potential [7], the nonlinear  Fokker-Plank model for fermion and boson gases [1, 9, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  2  Recently, many numerical discretizations have been proposed for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' mixed  finite element methods [6], finite volume methods [5, 7], DG methods [19, 20, 21] and  LDG methods [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Regarding positivity preserving discretizations, Liu and Yu devel-  oped in [20, 21], respectively, for the linear Fokker-Plank equation a maximum preserving  DG scheme and an entropy satisfying DG scheme, but these discretizations can not be  directly applied to the general case given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Liu and Wang subsequently de-  veloped in [19] an explicit Runge-Kutta (RK) time-discrete method for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) in one  dimension together with a positivity preserving high order accurate DG scheme under  some Courant-Friedrichs-Lewy (CFL) constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For the porous media equation, an  LDG discretization coupled with an explicit RK method was considered in [33], which  is similar to the DG method in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Still, it uses a special numerical flux to ensure the  non-negativity of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Cheng and Shen in [10] propose a Lagrange  multiplier approach to construct positivity preserving schemes for a class of parabolic  equations, which is different from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1), but contains the porous media equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  For the time-step τ and mesh size h, the condition τ = O(h2) is needed for stability  in [19] and [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Therefore, these explicit time discretizations suffer from severe time step  restrictions, but there are currently no feasible positivity preserving time-implicit LDG  discretizations for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  In this paper, we present higher order accurate Diagonally  Implicit Runge-Kutta (DIRK) LDG discretizations, which ensure positivity and mass  conservation of the numerical solution without the severe time step restrictions of explicit  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The LDG method proposed by Cockburn and Shu in [12] has many advantages,  including high parallelizability, high order accuracy, a simple choice of trial and test  spaces and easy handling of complicated geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We refer to [11, 15, 28, 36] for  examples of applications of the LDG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  For many physical problems, it is crucial that the numerical discretization preserves  the positivity properties of the partial differential equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Not only is this nec-  essary to obtain physically meaningful solutions, but also negative values may result in ill-  posedness of the problem and divergence of the numerical discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Positivity pre-  serving DG methods have been extensively studied by many mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' However,  most positivity preserving DG methods are combined with explicit time-discretizations  [19, 32, 34, 35], for which numerical stability frequently imposes severe time step restric-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' These severe time-step constraints make explicit methods impractical for parabolic  3  PDEs, such as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Recently, Qin and Shu extended in [25] the general framework for establishing positivity-  preserving schemes, proposed in [34, 35], from explicit to implicit time discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  They developed for one-dimensional conservation laws a positivity preserving DG method  with high-order spatial accuracy combined with the first-order backward Euler implicit  temporal discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' This approach requires, however, a detailed analysis of the nu-  merical discretization to ensure positivity and it is not straightforward to extend this  approach to higher order accurate time-implicit methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Huang and Shen in [17] con-  structed higher order linear bound preserving implicit discretizations for the Keller-Segel  and Poisson-Nernst-Planck equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Van der Vegt, Xia and Xu proposed in [30] the  KKT limiter concept to construct positivity preserving time-implicit discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The  KKT limiter in [30] is obtained by coupling the inequality and equality constraints im-  posed by the physical problem with higher order accurate DIRK-DG discretizations using  Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The resulting semi-smooth nonlinear equations are solved by an  efficient active set semi-smooth Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  In this paper, we consider a general class of nonlinear degenerate parabolic equations  given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) and aim at developing higher order accurate entropy dissipative and pos-  itivity preserving time-implicit LDG discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For the spatial discretization, we  use an LDG method with simple alternating numerical fluxes, which results in entropy  dissipation of the semi-discrete LDG discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For the temporal discretization, we  consider DIRK methods, which significantly enlarge the time step for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The  unconditional entropy dissipation of the LDG discretization combined with an implicit  Euler time integration method is proved theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We construct positivity preserving  discretizations using the KKT limiter by imposing the positivity constraint on the nu-  merical discretization using Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The unique solvability of the resulting  positivity preserving KKT system is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We will also prove the unconditional entropy  dissipation of the positivity preserving LDG discretization when it is combined with the  backward Euler time integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Numerical results demonstrate the accuracy  and entropy dissipation of the higher order accurate positivity preserving DIRK-LDG  discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In Section 2, we present the semi-discrete LDG  discretization with simple alternating numerical fluxes for the nonlinear degenerate  parabolic equation stated in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) and prove that the numerical approximation is en-  4  tropy dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Higher order accurate DIRK-LDG discretizations, which enlarge the  stable time step to a great extent, are discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The unconditional en-  tropy dissipation of the implicit Euler LDG discretizations is proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In  order to ensure positivity of the numerical solution and mass conservation of the pos-  itivity limited numerical discretizations, we introduce in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 the KKT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The higher order DIRK-LDG discretizations with positivity and mass conservation con-  straints are formulated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2 as a KKT mixed complementarity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The  unique solvability and unconditional entropy dissipation of the resulting algebraic system  are proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In Section 5, numerical results are provided to demonstrate  the higher order accuracy, positivity and entropy dissipation of the positivity preserving  KKT-DIRK-LDG discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Concluding remarks are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  2  Semi-discrete LDG schemes  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  Definitions, Notations  Let Th be a shape-regular tessellation of Ω ⊂ Rd, d = 1, 2, with line or convex  quadrilateral elements K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Given the reference element �K = [−1, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Let Qk( �K) denote  the space composed of the tensor product of Legendre polynomials Pk( �K) on [−1, 1]  of degree at most k ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The space Qk(K) is obtained by using an isoparametric  transformation from element K to the reference element �K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The finite element spaces  V k  h and W  W  W k  h are defined by  V k  h = {v ∈ L2(Ω) : v|K ∈ Qk(K), ∀K ∈ Th},  W  W  W k  h = {www ∈ [L2(Ω)]d : www|K ∈ [Qk(K)]d, ∀K ∈ Th},  and are allowed to have discontinuities across element interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Let e be an interior  edge connected to the “left” and “right” elements denoted, respectively, by KL and KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  If u is a function on KL and KR, we set uL := (u|KL) |e and uR := (u|KR)|e for the left  and right trace of u at e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Note that L1(Ω), L2(Ω) and L∞(Ω) are standard Sobolev spaces, ∥u∥L2(Ω) is the  L2(Ω)-norm and (·, ·)Ω is the L2(Ω) inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For simplicity, we denote the inner  product as (u, v) := (u, v)Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  LDG discretization in space  For the LDG discretization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1), we first rewrite this equation as a first order  system  ut =∇ · qqq,  qqq =f(u)sss,  sss =∇p,  p =φ(xxx) + H′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Then the LDG discretization can be readily obtained by multiplying the above equations  with arbitrary test functions, integrating by parts over each element K ∈ Th, and finally  a summation of element and face contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The LDG discretization can be stated  as: find uh, ph ∈ V k  h , qqqh,sssh ∈ W  W  W k  h, such that for all ρ, ϕ ∈ V k  h and θθθ,ηηη ∈ W  W  W k  h, we have  (uht, ρ) + L1  h(qqqh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ) = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1a)  (qqqh,θθθ) + L2  h(uh,sssh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='θθθ) = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1b)  (sssh,ηηη) + L3  h(ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='ηηη) = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1c)  (ph, ϕ) + L4  h(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ϕ) = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1d)  where  L1  h(qqqh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ) :=(qqqh, ∇ρ) −  �  K∈Th  (�qqqh · ννν, ρ)∂K,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2a)  L2  h(uh,sssh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='θθθ) := − (f(uh)sssh,θθθ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2b)  L3  h(ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='ηηη) :=(ph, ∇ · ηηη) −  �  K∈Th  (�ph,ννν · ηηη)∂K,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2c)  L4  h(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ϕ) := − (φ(xxx) + H′(uh), ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2d)  Note that ννν is the unit outward normal vector of an element K at its boundary ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The “hat” terms in L1  h and L3  h are the so-called “numerical fluxes”, whose choices play  an important role in ensuring stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We remark that the choices for the numerical  fluxes are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Here we use the alternating numerical fluxes  �qqqh =qqqR  h ,  �ph = pL  h,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3)  6  or  �qqqh =qqqL  h,  �ph = pR  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4)  Considering the zero-flux boundary condition ∇(φ(xxx) + H′(u)) · ννν = 0, we take  �qqqh · ννν = 0,  ph = (ph)in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5)  at ∂Ω, where “in” refers to the value obtained by taking the boundary trace from the  inside of the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3  Entropy dissipation  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For uh ∈ V k  h , sssh ∈ W  W  W k  h, the LDG scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5) with f satisfying  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) is entropy dissipative and satisfies  d  dtE(uh) = −(f(uh)sssh,sssh) ⩽ 0,  which is consistent with the entropy dissipation property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5) of the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' By taking  ρ = ph,  θθθ = −sssh,  ηηη = qqqh,  ϕ = −uht,  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1a)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1d), respectively, and after integration by parts, we have  (φ(xxx) + H′(uh), uht)  = − (f(uh)sssh,sssh) − (qqqh, ∇ph) +  �  K∈Th  (�qqqh · ννν, ph)∂K − (ph, ∇ · qqqh) +  �  K∈Th  (�ph,ννν · qqqh)∂K  = − (f(uh)sssh,sssh) −  �  K∈Th  (qqqh · ννν, ph)∂K +  �  K∈Th  (�qqqh · ννν, ph)∂K +  �  K∈Th  (�ph,ννν · qqqh)∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6)  Assume that e is an interior edge shared by elements KL and KR, then νννR = −νννL, and  together with the numerical fluxes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3), we obtain  −  �  KL  � KR  (qqqh · ννν, ph)e +  �  KL  � KR  (�qqqh · ννν, ph)e +  �  KL  � KR  (�ph,ννν · qqqh)e  = − (qqqL  h · νννL, pL  h)e + (qqqR  h · νννL, pR  h )e + (qqqR  h · νννL, pL  h)e − (qqqR  h · νννL, pR  h )e  + (qqqL  h · νννL, pL  h)e − (qqqR  h · νννL, pL  h)e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7)  7  Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7), using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4), boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5) and the condition on f  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3), we get  d  dtE(uh) = (φ(xxx) + H′(uh), uht) = −(f(uh)sssh,sssh) ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For brevity, we will only consider in the remaining article the numerical  fluxes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) and omit the discussion of the numerical fluxes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4), but all results also  apply to the numerical fluxes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Compared to the spatial discretizations in [19, 33], we choose the simpler  alternating numerical fluxes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4), which significantly simplifies the theoretical  analysis of the entropy dissipation property of the LDG discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  3  Time-implicit LDG schemes  The numerical discretization of the nonlinear parabolic equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) using explicit  time discretization methods suffers from the rather severe time-step constraint τ =  O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In this section, we will discuss implicit time discretizations coupled with positivity  constraints in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  We divide the time interval [0, T] into N parts 0 = t0 < t1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' < tN = T, with τ n =  tn − tn−1 (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' , N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' , N, let un = u(·, tn) and un  h, respectively,  denote the exact and approximate values of u at time tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  Backward Euler LDG discretization  Discretizing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) in time with the implicit Euler method gives the following discrete  system  �un+1  h  − un  h  τ n+1  , ρ  �  + L1  h(qqqn+1  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1a)  (qqqn+1  h  ,θθθ) + L2  h(un+1  h  ,sssn+1  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='θθθ) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1b)  (sssn+1  h  ,ηηη) + L3  h(pn+1  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='ηηη) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1c)  (pn+1  h  , ϕ) + L4  h(un+1  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1d)  8  Define the discrete entropy as  Eh(un  h) =  �  Ω  (un  hφ(xxx) + H(un  h))dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2)  We have the following relation for the discrete entropy dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For all time levels n, the numerical solutions un  h, un+1  h  ∈ V k  h of the  LDG discretization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1), with boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5) and conditions on f, H stated  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3), satisfy the following entropy dissipation relation  Eh(un+1  h  ) ⩽ Eh(un  h),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3)  which implies that the LDG discretization is unconditionally entropy dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' By choosing, respectively, in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1a)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1d) the following test functions  ρ = pn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  θθθ = −sssn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  ηηη = qqqn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  ϕ = −un+1  h  − un  h  τ n+1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  we get  �  φ(xxx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' un+1  h  − un  h  τ n+1  �  +  �  H′(un+1  h  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' un+1  h  − un  h  τ n+1  �  = −  �  f(un+1  h  )sssn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='sssn+1  h  �  −  �  qqqn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ∇pn+1  h  �  +  �  K∈Th  (�qqqn+1  h  ννν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' pn+1  h  )∂K  − (pn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ∇ · qqqn+1  h  ) +  �  K∈Th  (�pn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='ννν · qqqn+1  h  )∂K  = − (f(un+1  h  )sssn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='sssn+1  h  ) −  �  K∈Th  (qqqn+1  h  ννν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' pn+1  h  )∂K +  �  K∈Th  (�qqqn+1  h  ννν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' pn+1  h  )∂K  +  �  K∈Th  (�pn+1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='ννν · qqqn+1  h  )∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7), the numerical fluxes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) and the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5), we  obtain then  �  φ(xxx), un+1  h  − un  h  τ n+1  �  +  �  H′(un+1  h  ), un+1  h  − un  h  τ n+1  �  = −  �  f(un+1  h  )sssn+1  h  ,sssn+1  h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Because of the following Taylor expansion  H(un  h) =H(un+1  h  ) + H′(un+1  h  )(un  h − un+1  h  ) + 1  2H′′(ξn+1)(un+1  h  − un  h)2,  ξn+1 ∈ (un  h, un+1  h  ),  9  we have, using the conditions on f, H stated in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) and the definition of Eh in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2),  Eh(un+1  h  ) − Eh(un  h) =  �  φ(xxx), un+1  h  − un  h  �  +  �  H(un+1  h  ) − H(un  h), 1  �  = − τ n+1 �  f(un+1  h  )sssn+1  h  ,sssn+1  h  �  − 1  2  �  H′′(ξn+1),  �  un+1  h  − un  h  �2�  ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  Higher order DIRK-LDG discretizations  For higher order accurate implicit in time discretizations of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1), we  use a Diagonally Implicit Runge-Kutta (DIRK) method [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Assuming we know the  numerical solution at time level n, we obtain the solution at time level n + 1 with a  DIRK method by solving for each DIRK stage i, i = 1, 2, · · · , s the following equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  �  un+1,i  h  − un  h  τ n+1  , ρ  �  +  i  �  j=1  aijL1  h(qqqn+1,j  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4a)  (qqqn+1,i  h  ,θθθ) + L2  h(un+1,i  h  ,sssn+1,i  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='θθθ) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4b)  (sssn+1,i  h  ,ηηη) + L3  h(pn+1,i  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='ηηη) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4c)  (pn+1,i  h  , ϕ) + L4  h(un+1,i  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4d)  Then the solution at time tn+1 is  (un+1  h  , ρ) =(un  h, ρ) − τ  s  �  i=1  biL1  h(qqqn+1,i  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5)  The coefficient matrices (aij) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4a) and (bi) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5) are defined in the Butcher tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  We choose for polynomials of order k = 1 and k = 2, 3 the DIRK methods introduced in  [3] and [26], respectively, that satisfy asi = bi, i = 1, 2, ···, s, which implies un+1  h  = un+1,s  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The order of these DIRK methods is k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The above time discretization methods are  easy to implement since the matrix (aij) in the DIRK methods has a lower triangular  structure, which means that we can compute the DIRK stages one after another, starting  from i = 1 up to i = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For detailed information about the DIRK time integration  method, we refer to [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  10  4  Higher order accurate positivity preserving DIRK-  LDG discretizations  The positivity constraints on the LDG solution will be enforced by transforming the  DIRK-LDG equations with positivity constraints into a mixed complementarity problem  using the Karush-Kuhn-Tucker (KKT) equations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  In the following sections, we  will first define the positivity preserving KKT-DIRK-LDG discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Next, we will  consider the unique solvability and unconditional entropy dissipation of the discrete KKT  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  KKT-system  For the KKT equations [14], we define the set  K := {�U ∈ Rdof| h(�U) = 0, g(�U) ⩽ 0},  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1)  with equality constraints h : Rdof → Rl and inequality constraints g : Rdof → Rm  being vector-valued continuously differentiable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The inequality constraints are  used to ensure positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The equality constraint ensures that the limited DIRK-LDG  discretization is mass conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Mass conservation is a property of the unlimited  DIRK-LDG discretization, but one has to ensure that this property also holds after  applying the positivity preserving limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Let L be the LDG discretization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4) for each DIRK stage i = 1, 2, · · · , s, without a  positivity preserving limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We assume that L is a continuously differentiable function  from K to Rdof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The corresponding KKT-system [14] then is  L(�U) + ∇ �Uh(�U)Tµ + ∇ �Ug(�U)Tλ = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2a)  −h(�U) = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2b)  0 ⩾ g(�U)⊥λ ⩾ 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2c)  where µ ∈ Rl and λ ∈ Rm are the Lagrange multipliers used to ensure h(�U) = 0  and g(�U) ⩽ 0, respectively, �U ∈ Rdof are the LDG coefficients in the KKT-DIRK-  LDG discretization, and ∇ �U denotes the gradient with respect to �U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The compatibility  condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2c) is equivalent to  gj(�U) ⩽ 0,  λj ⩾ 0,  and  gj(�U)λj = 0,  j = 1, 2, · · ·, m,  11  which can be expressed as  min(−gj(�U), λj) = 0,  j = 1, 2, · · ·, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The KKT-system then can be formulated as  0 = F(z) =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  L(�U) + ∇ �Uh(�U)Tµ + ∇ �Ug(�U)Tλ  −h(�U)  min(−g(�U), λ)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3)  Here z = (�U, µ, λ) ∈ Rdof+l+m, and F : Rdof+l+m → Rdof+l+m represents the DIRK-LDG  discretization combined with the positivity and mass conservation constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Note, the  KKT system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) is nonlinear and F(z) is not continuously differentiable, as is necessary  for standard Newton methods, but semi-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We will therefore solve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) with the  active set semi-smooth Newton method presented in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  Positivity preserving LDG discretizations  In this section, we will provide the details of the higher order accurate positivity  preserving DIRK-LDG discretizations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4) coupled with the positivity and mass con-  servation constraints using Lagrange multipliers as stated in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Let Nk be the number of basis functions in one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Let Ne be the number of  elements K in the tessellation Th of the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We introduce the following notation  for the element-wise positivity preserving LDG solution  Uh|K :=  Nk  �  j=1  �UK  j φK  j ,  QQQh|K :=  Nk  �  j=1  �QQQ  K  j φK  j  with K ∈ Th, φK  j  the tensor product Legendre basis functions in Qk(K), and LDG  coefficients �UK  j  ∈ R, �QQQ  K  j  ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Taking in each element K ∈ Th the test function  ρ = φK  j , j = 1, 2, · · · , Nk in the operator L1  h(QQQh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ), stated in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2a), we can define  L1  h(�QQQ) := L1  h(QQQh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ) ∈ RNkNe,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4)  with similar definitions of Lk  h for Lk  h, k = 2, 3, 4 stated in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2b)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Representing the block-diagonal mass matrices for the scalar and vector variables as  M ∈ RNkNe×NkNe and M  M  M ∈ RdNkNe×dNkNe, respectively, the operator L for DIRK stage  12  i (i = 1, 2, · · · , s), as stated in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4a), can be expressed as  L(�Un+1,i) :=M(�Un+1,i − �Un) + τ n+1  i  �  j=1  aijL1  h(�QQQ  n+1,j),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5)  with LDG coefficients �Un+1,i ∈ RNkNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Similarly, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4b), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4c) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4d), we have  �QQQ  n+1,i = − M  M  M −1L2  h(�Un+1,i, �SSS  n+1,i),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6a)  �SSS  n+1,i = − M  M  M −1L3  h( �P n+1,i),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6b)  �P n+1,i = − M−1L4  h(�Un+1,i),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6c)  with LDG coefficients �QQQ  n+1,i ∈ RdNkNe, �SSS  n+1,i ∈ RdNkNe, �P n+1,i ∈ RNkNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The constraints on the DIRK-LDG discretization can be directly imposed on the  DG coefficients for each DIRK stage using the equality and inequality constraints in the  KKT-system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We obtain for each DIRK stage i, with i = 1, 2, · · · , s, the LDG  coefficients �Un+1,i by solving the following KKT system for �Un+1,i,  \uf8eb  \uf8ec  \uf8ec  \uf8ed  L(�Un+1,i) + ∇ �Uh(�Un+1,i)Tµ + ∇ �Ug(�Un+1,i)Tλ  −h(�Un+1,i)  min(−g(�Un+1,i), λ)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7)  where the positivity preserving inequality constraint g(�Un+1,i) and the mass conservation  equality constraint h(�Un+1,i) are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Positivity preserving inequality constraint  In each element K ∈ Th, we define the function g stated in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7) as  gK  p (�Un+1,i) = umin −  Nk  �  j=1  �UK,(n+1,i)  j  φK  j (xxxp),  p = 1, · · ·, Np,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8)  with Np the number of Gauss-Lobatto quadrature points, and xxxp the Gauss-Lobatto  quadrature points where the inequality constraints Uh(xxxp) ⩾ umin are imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The use  of Gauss-Lobatto quadrature rules ensures that the positivity constraint is also imposed  in the computation of the numerical fluxes at the element edges where Gauss-Lobatto  rules have, next to the element itself, also quadrature points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Note, the Gauss-Lobatto  quadrature points xxxp are the only points used in the LDG discretization and the positivity  constraint umin therefore only needs to be enforced at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Mass conservation equality constraint  In order to ensure mass conservation of the LDG discretization when the positivity  constraint is enforced, we impose the following equality constraint, which is obtained by  setting ρ = 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4a) and using the numerical flux (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  h(�Un+1,i) =  �  K∈Th  �  K  Un  h dK + τ n+1  i  �  j=1  aij  �  K∈Th  ∂K∩∂Ω̸=∅  (�QQQ  n+1,j  h  ννν, 1)∂K  −  �  K∈Th  Nk  �  j=1  �UK,(n+1,i)  j  �  K  φK  j (xxx)dK,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='9)  with Un  h the KKT-DIRK-LDG solution at time tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  For each DIRK stage i, the KKT-system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7) for the higher order accurate positivity  preserving LDG discretization is now defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' After solving the KKT equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7) for  i = 1, · · · , s, the numerical solution at time tn+1 is directly obtained from the last DIRK  stage, Un+1  h  = Un+1,s  h  since we use DIRK methods with asi = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In order to ensure the positivity of the discrete initial solution U0  h, we  use the L2-projection coupled with the positivity constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8), which is obtained  by replacing �Un+1,i with �U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The equality constraint ensures mass conservation of the  positivity limited initial solution  h(�U0) =  �  K∈Th  �  K  u0(xxx)dK −  �  K∈Th  Nk  �  j=1  �UK,0  j  �  K  φK  j (xxx)dK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The constraints on the L2-projection are imposed using KKT equations similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  To prevent pathological cases, we assume that the limited initial solution satisfies  1  |Ω|  �  K∈Th  �  K  u0(xxx)dK ⩾ umin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We emphasize that umin must be chosen strictly positive to ensure that  errors do not violate the positivity of the numerical solution due to the finite precision  of the computer arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3  Unique solvability and stability of the positivity preserving  LDG discretization  In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2, we have presented the positivity preserving LDG discretization for  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In this section, we will consider the unique solvability of the algebraic equations  14  resulting from the backward Euler KKT-LDG discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In the theoretical analysis  we will also consider the entropy dissipation of the positivity preserving backward Euler  LDG discretization and use periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  With (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='9), the positivity preserving backward Euler LDG discretization results  now in the following KKT system,  L(�Un+1) + ∇ �Uh(�Un+1)Tµn+1 + ∇ �Ug(�Un+1)Tλn+1 = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10a)  −h(�Un+1) = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10b)  min(−g(�Un+1), λn+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10c)  Here L : RNkNe → RNkNe and  L(�Un+1) :=M(�Un+1 − �Un) + τ n+1B �QQQ  n+1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='11)  M  M  M �QQQ  n+1 =Cd(�Un+1)�SSS  n+1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='12)  M  M  M �SSS  n+1 =A �P n+1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='13)  M �P n+1 =D(�Un+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='14)  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6), we obtain that  B �QQQ  n+1 =L1  h(�QQQ  n+1) ∈ RNkNe,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='15)  Cd(�Un+1)�SSS  n+1 = − L2  h(�Un+1, �SSS  n+1) ∈ RdNkNe,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='16)  A �P n+1 = − L3  h( �P n+1) ∈ RdNkNe,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='17)  D(�Un+1) = − L4  h(�Un+1) ∈ RNkNe,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='18)  where  Cd(�Un+1) =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  C(�Un+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  C(�Un+1)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ∈ RdNkNe×dNkNe, C(�Un+1) ∈ RNkNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='19)  The constraints h : RNkNe → R, g : RNkNe → RNpNe are defined by  h(�Un+1) :=  �  K∈Th  �  K  U0  hdK −  �  K∈Th  Nk  �  j=1  �UK,(n+1)  j  �  K  φK  j (xxx)dK,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='20)  g(�Un+1) :=(gK1  1 (�Un+1), · · · , gK1  Np(�Un+1), · · · , g  KNe  1  (�Un+1), · · · , g  KNe  Np (�Un+1)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21)  with the definition of the constraints g  Kj  p , 1 ⩽ p ⩽ Np, 1 ⩽ j ⩽ Ne given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  Auxiliary results used to prove the solvability of the KKT-system  In this section, we will introduce some auxiliary results, which will be used in Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2 to prove the unique solvability of the KKT-system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' [14, Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2] Let K be given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1), given a map L : K →  Rdof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The Variational Inequality (VI(K, L)) is to find �U ∈ K such that  (y − �U)TL(�U) ⩾ 0,  y ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='22)  The solution of VI(K, L) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='22) is denoted by SOL(K, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Using the nodal basis function and the definition of g in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8), the inequal-  ity constraint set in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) can be written as  Kb := {�U ∈ Rdof| �Umin  i  ⩽ �Ui ⩽ �Umax  i  , i ∈ {1, · · · , dof}},  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='23)  and we write Kb as  Kb =  N  �  ϑ=1  Knϑ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='24)  where Knϑ is a subset of Rnϑ with  N  �  ϑ=1  nϑ = dof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Thus for a vector �U ∈ Kb, we write  �U = (�Uϑ), where each �Uϑ belongs to Knϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' [14, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2] Let Kb be given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='23), a map L : Kb → Rdof is  said to be  a) a P-function on Kb if for all pairs of distinct vectors �U and �U′ in Kb,  max  1⩽ϑ⩽N(�Uϑ − �U′  ϑ)T(Lϑ(�U) − Lϑ(�U′)) > 0,  b) a uniformly P-function on Kb if there exists a constant ̟ > 0 such that for all  pairs of distinct vectors �U and �U′ in Kb,  max  1⩽ϑ⩽N(�Uϑ − �U′  ϑ)T(Lϑ(�U) − Lϑ(�U′)) ⩾ ̟∥�U − �U′∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' [14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10] Let Kb be given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  a) If L is a P-function on Kb, then VI(Kb, L) has at most one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  b) If each Knϑ is closed convex and L is a continuous uniformly P-function on Kb,  then the VI(Kb, L) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  16  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' [14, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4] Let �U ∈ SOL(K, L) solve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='22) with K given by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' If Abadie’s Constraint Qualification holds at �U, then there exist vectors µ ∈ Rl  and λ ∈ Rm satisfying the KKT system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Conversely, if each function hj (1 ⩽ j ⩽ l) is affine and each function gi (1 ⩽ i ⩽ m)  is convex, and if (�U, µ λ) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10), then �U solves VI(K, L) given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='22) with K  given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  Existence and uniqueness of LDG discretization with positivity and  mass conservation constraints  In this section, we will prove the existence and uniqueness of the KKT system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10)-  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21) using the unique solvability conditions discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For periodic boundary conditions, the matrices B in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='15) and A in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='17)  satisfy BT = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In order to prove the symmetry of B in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='15) and A in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='17), we define the  bilinear function a : (V k  h × W  W  W k  h) × (V k  h × W  W  W k  h) → R by  a(P n+1  h  ,QQQn+1  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ,θθθ) =(QQQn+1  h  , ∇ρ) −  �  K∈Th  (�QQQ  n+1  h  ννν, ρ)∂K  − (P n+1  h  , ∇ · θθθ) +  �  K∈Th  ( �P n+1  h  ,ννν · θθθ)∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Based on the definition of B in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='15) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2a), A in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='17) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2c), we rewrite  the above bilinear function a as follows:  a(P n+1  h  ,QQQn+1  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ,θθθ) =(̺, Θ)  �  0  B  A  0  �  ( �P n+1, �QQQ  n+1)T,  with ̺, Θ the LDG coefficients of ρ,θθθ and �P n+1, �QQQ  n+1 the LDG coefficients of P n+1  h  ,QQQn+1  h  ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Interchanging the arguments of a, we get  a(ρ,θθθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' P n+1  h  ,QQQn+1  h  ) =(θθθ, ∇P n+1  h  ) −  �  K∈Th  (�θθθ · ννν, P n+1  h  )∂K  − (ρ, ∇ · QQQn+1  h  ) +  �  K∈Th  (�ρ,ννν · QQQn+1  h  )∂K  17  = − (P n+1  h  , ∇ · θθθ) +  �  K∈Th  (θθθ · ννν, P n+1  h  )∂K −  �  K∈Th  (�θθθ · ννν, P n+1  h  )∂K  + (QQQn+1  h  , ∇ρ) −  �  K∈Th  (ρ,ννν · QQQn+1  h  )∂K +  �  K∈Th  (�ρ,ννν · QQQn+1  h  )∂K,  Using equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7), the alternating numerical fluxes for �θθθ and �ρ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4), and  the periodic boundary conditions, we obtain  �  K∈Th  (θθθ · ννν, P n+1  h  )∂K −  �  K∈Th  (�θθθ · ννν, P n+1  h  )∂K =  �  K∈Th  ( �P n+1  h  ,ννν · θθθ)∂K,  −  �  K∈Th  (ρ,ννν · QQQn+1  h  )∂K +  �  K∈Th  (�ρ,ννν · QQQn+1  h  )∂K = −  �  K∈Th  (�QQQ  n+1  h  ννν, ρ)∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Hence,  a(P n+1  h  ,QQQn+1  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' ρ,θθθ) = a(ρ,θθθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' P n+1  h  ,QQQn+1  h  ),  which implies  (̺, Θ)  �  0  B  A  0  �  ( �P n+1, �QQQ  n+1)T =( �P n+1, �QQQ  n+1)  �  0  B  A  0  �  (̺, Θ)T  =(̺, Θ)  �  0  AT  BT  0  �  ( �P n+1, �QQQ  n+1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='25)  Since (P n+1  h  ,QQQn+1  h  ) ∈ V k  h × W  W  W k  h and (ρ,θθθ) ∈ V k  h × W  W  W k  h are arbitrary functions, relation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='25) implies that A = BT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='12)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='14) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3, the operator L(�Un+1) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='11) can be written  as  L(�Un+1) = M(�Un+1 − �Un) + τ n+1BM  M  M −1Cd(�Un+1)M  M  M −1BTM−1D(�Un+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='26)  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Given �Un, the operator L in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='26) is a uniformly P-function on Kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Using relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='26) for L, for arbitrary �Un+1  I  , �Un+1  II  ∈ Kb, there holds  L(�Un+1  I  ) − L(�Un+1  II  ) =M(�Un+1  I  − �Un+1  II  ) + τ n+1BM  M  M −1Cd(�Un+1  I  )M  M  M −1BT M−1D(�Un+1  I  )  − τ n+1BM  M  M −1Cd(�Un+1  II  )M  M  M −1BTM−1D(�Un+1  II  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='27)  18  After subtracting and adding τ n+1BM  M  M −1Cd(�Un+1  I  )M  M  M −1BTM−1D(�Un+1  II  ) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='27), we  obtain  L(�Un+1  I  ) − L(�Un+1  II  )  =M(�Un+1  I  − �Un+1  II  ) + τ n+1BM  M  M −1Cd(�Un+1  I  )M  M  M −1BTM−1(D(�Un+1  I  ) − D(�Un+1  II  ))  + τ n+1BM  M  M −1(Cd(�Un+1  I  ) − Cd(�Un+1  II  ))M  M  M−1BTM−1D(�Un+1  II  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='28)  With the definition of D in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='18) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2d), we obtain that  (D(�Un+1  I  ) − D(�Un+1  II  ))i =  �  Ω  �  H′  �NkNe  �  j=1  �Un+1  I,j φj  �  − H′  �NkNe  �  j=1  �Un+1  II,j φj  ��  φidΩ  =  NkNe  �  j=1  (�Un+1  I,j  − �Un+1  II,j )  �  Ω  H′′(ξn+1  1  )φjφidΩ, i ∈ {1, · · · , NkNe}, ξn+1  1  ∈ (Un+1  h,I , Un+1  h,II ),  and write  D(�Un+1  I  ) − D(�Un+1  II  ) :=D �U(ξn+1  1  )(�Un+1  I  ) − �Un+1  II  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='29)  Similarly, from the definition of Cd in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='19) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2b), we obtain that  Cd(�Un+1  I  ) − Cd(�Un+1  II  ) =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  C(�Un+1  I  ) − C(�Un+1  II  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  C(�Un+1  I  ) − C(�Un+1  II  )  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  (C(�Un+1  I  ) − C(�Un+1  II  ))ij =  �  Ω  �  f  �NkNe  �  k=1  �Un+1  I,k φk  �  − f  �NkNe  �  k=1  �Un+1  II,k φk  ��  φjφidΩ  =  NkNe  �  k=1  (�Un+1  I,k − �Un+1  II,k )  �  Ω  f ′(ξn+1  2  )φkφjφidΩ, i, j, k ∈ {1, · · · , NkNe}, ξn+1  2  ∈ (Un+1  h,I , Un+1  h,II ),  and write  C(�Un+1  I  ) − C(�Un+1  II  ) :=  NkNe  �  k=1  [Cd�U(ξn+1  2  )]k(�Un+1  I,k ) − �Un+1  II,k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='30)  Assume for arbitrary �U ∈ Kb in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='23), that  |C(�U)ij| ⩽c,  |D(�U)i| ⩽ c,  19  |[C �U(�U)ij]k| ⩽c,  |D �U(�U)ij| ⩽ c, i, j, k ∈ {1, · · · , NkNe},  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='31)  with c a positive constant, independent of �U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In the remainder of this section, c is a  positive constant, but not necessarily the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='29)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='30) and assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' we obtain the following two estimates  (�Un+1  I  − �Un+1  II  )TBM  M  M −1Cd(�Un+1  I  )M  M  M −1BTM−1(D(�Un+1  I  ) − D(�Un+1  II  ))  ⩽∥B∥∥M  M  M −1∥∥Cd(�Un+1  I  )∥∥M  M  M−1∥∥BT∥∥M−1∥∥D �U(ξn+1  1  )∥∥�Un+1  I  − �Un+1  II  ∥2  ⩽c∥�Un+1  I  − �Un+1  II  ∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  and  (�Un+1  I  − �Un+1  II  )TBM  M  M−1(Cd(�Un+1  I  ) − Cd(�Un+1  II  ))M  M  M −1BT M−1D(�Un+1  II  )  ⩽∥B∥∥M  M  M −1∥  NkNe  �  k=1  ∥[Cd�U(ξn+1  2  )]k∥∥M  M  M−1∥∥BT∥∥M−1∥∥D(�Un+1  II  )∥∥�Un+1  I  − �Un+1  II  ∥2  ⩽c∥�Un+1  I  − �Un+1  II  ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Then multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='28) with (�Un+1  I  − �Un+1  II  )T gives  (�Un+1  I  − �Un+1  II  )T(L(�Un+1  I  ) − L(�Un+1  II  )) = (�Un+1  I  − �Un+1  II  )TM(�Un+1  I  − �Un+1  II  )  + τ n+1(�Un+1  I  − �Un+1  II  )TBM  M  M−1Cd(�Un+1  I  )M  M  M−1BTM−1(D(�Un+1  I  ) − D(�Un+1  II  ))  + τ n+1(�Un+1  I  − �Un+1  II  )TBM  M  M−1(Cd(�Un+1  I  ) − Cd(�Un+1  II  ))M  M  M −1BT M−1D(�Un+1  II  )  ⩾σ∥�Un+1  I  − �Un+1  II  ∥2 − 2cτ n+1∥�Un+1  I  − �Un+1  II  ∥2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='32)  where σ > 0 is the smallest eigenvalue of the symmetric positive mass matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Choosing 0 < τ n+1 ⩽ σ  4c, we obtain that  (�Un+1  I  − �Un+1  I  )T(L(�Un+1  I  ) − L(�Un+1  II  )) ⩾ σ  2 ∥�Un+1  I  − �Un+1  II  ∥2, ∀�Un+1  I  , �Un+1  II  ∈ Kb, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='33)  which implies that for τ n+1 sufficiently small L(�Un+1) is a uniformly function of Kb,  From Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4, we obtain the main result of this  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Given the DG coefficients �Un and the positivity preserving backward Eu-  ler KKT-LDG discretization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21) with equality constraint h ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' If assumption  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='31) is satisfied, then the KKT system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21) has only one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  20  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Given the DG coefficients �Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' If assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='31) is satisfied, then  for the degenerate parabolic equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) with periodic boundary conditions there exists  only one solution satisfying the higher order accurate in time, positivity preserving KKT-  DIRK-LDG discretizations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7) with equality constraint h ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Since the DIRK coefficient matrix (aij) introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2 is a lower tri-  angular matrix, the structure of the DIRK-LDG discretizations is similar to the form  obtained for the backward Euler LDG discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The analysis therefore is completely  analogous to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3  Stability of the KKT-LDG discretization  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Given the numerical solution Un  h ∈ V k  h of the positivity preserving back-  ward Euler KKT-LDG discretization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' If assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='31) is satisfied,  then the discrete entropy Eh stated in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2) satisfies for n = 0, 1, · · ·,  Eh(Un+1  h  ) ⩽ Eh(Un  h ),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='34)  which implies that the positivity preserving backward Euler KKT-LDG discretization is  unconditionally entropy dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2, we obtain that the LDG coefficients �Un+1 of the positivity  preserving solution Un+1  h  solve  (y − �Un+1)TL(�Un+1) ⩾ 0,  ∀y ∈ K,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='35)  with L given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='26) and K given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  From assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='31), we have that there exists a positive constant c ⩾ c0 > 0  such that  �Un+1 − cM−1D(�Un+1) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='36)  Next, we choose y = �Un+1 − cM−1D(�Un+1) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='35), which implies  −c(M−1D(�Un+1))TL(�Un+1) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='37)  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='26) and the fact that c > 0, we obtain that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='37) implies the inequality  D(�Un+1)T(�Un+1 − �Un)  21  +τ n+1D(�Un+1)TM−1BM  M  M −1Cd(�Un+1)M  M  M −1BTM−1D(�Un+1) ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='38)  From the definition of Cd in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='19) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2b) and the conditions on f stated  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3), we obtain that Cd(�Un+1) is symmetric positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Hence using τ n+1 > 0,  we have  τ n+1D(�Un+1)TM−1BM  M  M −1Cd(�Un+1)M  M  M −1BTM−1D(�Un+1) ⩾ 0,  which with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='38) yields  D(�Un+1)T(�Un+1 − �Un) ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='39)  From the definition of D in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='18) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2d) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='39), we obtain the bound  �  φ(xxx), Un+1  h  − Un  h  �  +  �  H′(Un+1  h  ), Un+1  h  − Un  h  �  ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='40)  Using the following Taylor expansion  H(Un  h ) =H(Un+1  h  ) + H′(Un+1  h  )(Un  h − Un+1  h  )  + 1  2H′′(ξn+1  3  )(Un+1  h  − Un  h )2, ξn+1  3  ∈ (Un  h , Un+1  h  ),  we obtain that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='40) gives  �  φ(xxx), Un+1  h  − Un  h  �  +  �  H(Un+1  h  ) − H(Un  h ), 1  �  + 1  2  �  H′′(ξn+1  3  ),  �  Un+1  h  − Un  h  �2�  ⩽ 0,  which implies, using the definition of Eh in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2), that  Eh(Un+1  h  ) − Eh(Un  h ) =  �  φ(xxx), Un+1  h  − Un  h  �  +  �  H(Un+1  h  ) − H(Un  h ), 1  �  ⩽ 0,  since (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) gives H′′(ξn+1  3  ) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5  Numerical tests  In this section, we will discuss several numerical experiments to demonstrate the  performance of the KKT-DIRK-LDG positivity preserving algorithm for the degenerate  parabolic equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  In the computations, we will consider the porous medium  equation, the nonlinear diffusion equation with a double-well potential and the nonlinear  Fokker-Plank equation for fermion and boson gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Firstly, we will present in Section  22  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 the order of accuracy of the DIRK-LDG discretizations with and without positivity  preserving limiter to investigate if the limiter negatively affects the accuracy of the  discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Next, we will present in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5 test cases for which the positivity  preserving limiter is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Without the positivity constraint, obtaining a numerical  solution or only for extremely small time steps is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  In the computations, we take τ = α · h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  If the Newton method during strongly  nonlinear stages requires a large number of iterations, it is generally more efficient to  reduce the time step to 1  2τ and restart the Newton iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' When the Newton method  converges well, then τ is increased each time step to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2τ, till the maximum predefined  time step is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  In order to avoid round-off effects, a positivity bound umin = 10−10 is used in the  numerical simulations, except for Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 where umin = 10−14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' If it is not stated  otherwise, the numerical results for 1D problems are obtained on a mesh containing 100  elements and Legendre polynomials of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For 2D problems, a mesh consisting of  30×30 square elements and tensor product Legendre polynomial basis functions of order  2 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  Accuracy tests  For the accuracy test, we use a uniform mesh with M elements and positivity bound  umin = 10−14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) on the domain Ω = [−1, 1] with Dirichlet boundary  conditions based on the exact solution and select the following parameters  f(u) = u,  H′(u) = u2,  φ(x) = 0,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) with a properly chosen source term has the nonnegative solution  u(x, t) = exp(−t)(1 − x4)5,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  We take α in the definition of the time step as α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Tables 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2 show that the  DIRK-LDG discretizations with and without positivity preserving limiter are convergent  at the rate O(hk+1) for basis functions with polynomial order ranging from 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The  errors and orders of accuracy presented in Tables 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2 indicate that the positivity  preserving limiter is necessary and does not negatively affect accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  23  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1:  Error in L∞− and L1− norms for Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 at time T = 1 without positivity  preserving limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Pk  M  ∥un − un  h∥L∞(Ω)  Order  ∥un − un  h∥L1(Ω)  Order  min un  h  40  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='33E-003  –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='03E-003  –  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='87e-005  1  80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='24e-003  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='56  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='27e-004  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='08e-005  160  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='63e-004  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='44e-005  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='06  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='41e-007  320  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='05e-005  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='35e-005  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='57e-008  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='70E-003  –  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='73E-005  –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='60e-005  2  80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='43e-004  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='57  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='07e-006  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='79e-007  160  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='36e-005  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='39  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='40e-007  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='24e-009  320  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='07e-010  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='14e-006  3  80  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='11e-007  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='56e-008  160  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='51e-007  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='51e-009  320  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='50e-008  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='11e-009  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='99  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='19e-010  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2:  Error in L∞− and L1− norms for Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 at time T = 1 with positivity  preserving limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Pk  M  ∥un − Un  h ∥L∞(Ω)  Order  ∥un − Un  h ∥L1(Ω)  Order  min Un  h  40  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='33E-003  –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='56  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='27e-004  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='15e-007  160  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='06  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='77e-008  320  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='35e-005  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='01  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='55e-010  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='70E-003  –  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='73E-005  –  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='15e-008  2  80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='43e-004  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='57  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='08e-006  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='03e-007  160  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='36e-005  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='39  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='40e-007  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='08e-008  320  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='34e-006  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='16e-007  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='02  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='55e-010  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='45e-004  –  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='02e-006  –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='00e-014  3  80  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='88  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='13e-007  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='27  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='45e-008  160  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='51e-007  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='77e-008  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='21e-009  320  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='50e-008  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='11e-009  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='55e-011  24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  Porous media equation  For the porous media equation, f(u)H′′(u) can locally vanish, resulting in degenerate  cases [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  We test the asymptotic behavior of the numerical solution and will show  that the KKT limiter is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The entropy defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4), which should be non-  increasing, is also computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In order to test degenerate cases, we choose the following parameters in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) on the domain Ω = [0, 1] with zero-flux boundary conditions  f(u) = u,  H′(u) = 4  3  �  u − 1  2  �3  max  �  u, 1  2  �  ,  φ(x) = 0,  x ∈ Ω,  and initial data  u(x, 0) = 1  2 − 1  2 cos(2πx),  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  During the computations, the value of α for optimal convergence of the semi-smooth  Newton algorithm is usually close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We present the numerical solution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 for basis functions with polynomial order ranging from 1 to 3 and with the KKT  limiter enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Values of the Lagrange multiplier λ larger than 10−10 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1, which indicate that the positivity constraint works well since it is only active at  locations where the solution is close to the minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The entropy decay using the  KKT limiter and polynomial basis functions of order 3 is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2, in which  the result is consistent with the stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3, the numerical solution  without KKT limiter and for polynomial basis functions with order 3 is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' This  computation breaks down due to unphysical oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We consider a 2D test case on the domain Ω = [−6, 6]2 with zero-flux  boundary conditions by choosing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) the following parameters  f(u) = u,  H′(u) = 2u,  φ(xxx) = 0,  xxx ∈ Ω,  and initial data  u(xxx, 0) = exp  �  −1  2|xxx|2  �  ,  xxx ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The value of α in the definition of the time step ranges in this case between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4 presents the numerical solution with the KKT limiter active and also the  25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='7  Uh  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4  10-3  Time 3  Uh  (a) P1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='9  1  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='7  Uh  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6  10-3  Time 3  Uh  (b) P2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='9  1  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7  Uh  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  10-3  Time 3  Uh  (c) P3  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2) Numerical solution Uh for different orders of polynomial basis  functions P1-P3 with the KKT limiter enforced and Lagrange multiplier λ (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2) Entropy Eh for P3 basis functions with the KKT limiter enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  26  Entropy  X10~3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5 米  E  2  迷  米  米  米  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  ******米  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  3  t0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='9  1  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8  1  Uh  Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='29317  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2) Numerical solution Uh for P3 basis functions without KKT limiter  just before blow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Lagrange multiplier λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Considering the position of the non-zero Lagrange multipliers,  we can see that the limiter also works well in the two-dimensional case since it is only  active in areas where positivity must be enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The entropy decay is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5, which is consistent with the stability result of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Without the  KKT limiter, there will be unphysical oscillations, and the computation will break down  at some point in the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3  Nonlinear diffusion with a double-well potential  Consider the nonlinear diffusion equation with double-well potential [18] on the do-  main Ω = [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4], which is obtained by choosing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) zero-flux boundary condi-  tions and the following parameters  f(u) = u,  H′(u) = u,  φ(x) = 1  4x4 − 1  2x2,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1)  This model is taken from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We will test the evolution of the numerical solution with  and without KKT limiter, and also the decay of the entropy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The value of α to  compute the time step ranges between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) and the initial data  u(x, 0) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4π exp  �  − x2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4  �  ,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  27  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) Numerical solution Uh for P2 basis functions with KKT limiter  enforced (Left) and Lagrange multiplier λ (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  5  t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  E  Entropy  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='3) Entropy Eh for P2 basis functions with KKT limiter enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  28  Time5  X10~5  8  6  lambda  4  0  6  4  2  6  0  4  2  2  0  4  2  4  X  6  6  yTime 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='06  6  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='04  2  6  4  0  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='02  2  0  4  2  X  4  6  9-  yThe numerical solution with the KKT limiter enforced and the values of the Lagrange  multiplier λ larger than 10−10 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' These results indicate that the  numerical solution tends to a steady state and that the KKT limiter is only active at  places where the positivity constraint needs to be imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The entropy dissipation is  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7, in which uniform decay coincides with our theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For  the numerical solution without the KKT limiter, we observe that violating the positivity  constraint will result in discontinuities in the solution and a computation breakdown,  even for a very small CFL number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='18  Uh  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  10-4  Time 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2418  Uh  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='18  Uh  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  10-4  Time 20  Uh  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4) Numerical solution Uh for P2 basis functions with KKT limiter  enforced and Lagrange multiplier λ (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='7: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4) Entropy Eh for P2 basis functions with KKT limiter enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  29  Entropy  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='01  E-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='03  0  2  4  6  8  10  12  14  16  18  20  t5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='4  Nonlinear Fokker-Plank equation for fermion gases  We consider the nonlinear Fokker-Plank equation for fermion gases [5] on the domain  Ω = [−10, 10]2, for which we select the following parameters in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1)  f(u) = u(1 − u),  H′(u) = log  u  1 − u,  φ(xxx) = 1  2|xxx|2,  xxx ∈ Ω,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2)  together with zero-flux boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='2) and initial data  u(xxx, 0) =  1  2  √  2π  �  exp  �  −1  2|xxx − (2, 2)|2  �  + exp  �  −1  2|xxx − (2, −2)|2  �  + exp  �  −1  2|xxx − (−2, 2)|2  �  + exp  �  −1  2|xxx − (−2, −2)|2  ��  ,  xxx ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  During the computations, the value of α in the definition of the time step ranges  between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1 and 1, but for most time steps α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The numerical solutions at several  time levels with the KKT limiter enforced and the entropy dissipation are presented in  Figs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='9, respectively, showing the time-asymptotic convergence of the numerical  solution towards a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Without the KKT limiter, the computations break  down, even for very small CFL numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  Nonlinear Fokker-Plank equation for boson gases  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We consider a nonlinear Fokker-Plank equation for boson gases with  zero-flux boundary condition on a domain Ω = [−10, 10], which requires the following  parameters in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='1)  f(u) = u(1 + u3),  H′(u) = log  u  (1 + u3)  1  3 ,  φ(x) = x2  2 ,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  The initial data is [5, 19]  u(x, 0) = M  2  √  2π  �  exp  �  −(x − 2)2  2  �  + exp  �  −(x + 2)2  2  ��  ,  x ∈ Ω,  where M ⩾ 0 is the mass of u(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  For most time steps, the value of α in the definition of the time step is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For the case  M = 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10 displays the numerical solution at various times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Also, the locations  30  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='8: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5) Numerical solution Uh for P2 basis functions with KKT limiter  enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  31  Time20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='  and values of the Lagrange multiplier λ and the entropy with the KKT limiter enforced  are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The results in Figs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='3  10-9  Time 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='0713  Uh  10  8  6  4  2  0  2  4  6  8  10  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='3  10-9  Time 10  Uh  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='10: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6): Numerical solution Uh for P2 basis functions with KKT limiter  enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  33  0  1  2  3  4  5  6  7  8  9  10  t  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='5  E  Entropy  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='11: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6): Entropy Eh for P2 basis functions with KKT limiter enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  discretization an LDG method combined with a simple alternating numerical flux is used,  which simplifies the theoretical analysis for the entropy dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' For the temporal  discretization, the implicit DIRK methods significantly enlarge the time-step required  for stability of the numerical discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' We prove the existence, uniqueness and  unconditional entropy dissipation of the positivity preserving high order accurate KKT-  LDG discretization combined with an implicit Euler time discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Numerical  results are presented to demonstrate the accuracy of the higher order accurate positivity  preserving KKT-DIRK-LDG discretizations, which are of optimal order and not affected  by the positivity preserving KKT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The numerical solutions satisfy the entropy  decay condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  Acknowledgement  The research of Fengna Yan was funded by a fellowship from the China Scholarship  Council (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 201806340058).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The research of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' van der Vegt was partially sup-  ported by the University of Science and Technology of China (USTC), Hefei, Anhui,  China, while the author was in residence at USTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' The research of Yinhua Xia was par-  tially supported by National Natural Science Foundation of China grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='  The research of Yan Xu was partially supported by National Natural Science Foundation  of China grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' 12071455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='3  10-9  Time 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='0713  Uh  10  8  6  4  2  0  2  4  6  8  10  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='45  Uh  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='3  10-9  Time 10  Uh  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='12: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6: M = 1): Numerical solution Uh for P2 basis functions with KKT  limiter enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  10  8  6  4  2  0  2  4  6  8  10  x  0  5  10  15  20  25  Uh  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='999796  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='999798  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='999814  10-10  Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='25832  Uh  10  8  6  4  2  0  2  4  6  8  10  x  0  5  10  15  20  25  30  35  40  45  50  Uh  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='9992  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='99925  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+page_content='9994  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='99945  10-10  Time 1  Uh  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='13: (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='6: M = 10) Numerical solution Uh for P2 basis functions with KKT  limiter enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  35  References  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Abdallah, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Gamba, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Toscani, On the minimization problem  of sub-linear convex functionals, Kinetic & Related Models, 4 (2011), 857–871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Agarwal and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' O’Regan, Ordinary and partial differential equations,  Springer Science & Business Media, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' Alexander, Diagonally implicit Runge-Kutta methods for stiff ODE’s, SIAM  Journal on Numerical Analysis, 14 (1977), 1006–1021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfdvz8/content/2301.01427v1.pdf'}
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+submitted to Journal of Space Weather and Space Climate
+© The author(s) under the Creative Commons Attribution 4.0 International License (CC BY 4.0)
+Is the F10.7cm – Sunspot Number relation linear and
+stable?
+F. Clette
+World Data Center SILSO, Royal Observatory of Belgium, 1180 Brussels, Belgium
+e-mail: frederic.clette@oma.be
+ABSTRACT
+The F10.7cm radio flux and the Sunspot Number are the most widely used long-term indices of
+solar activity. They are strongly correlated, which led to the publication of many proxy relations
+allowing to convert one index onto the other. However, those existing proxies show significant
+disagreements, in particular at low solar activity. Moreover, a temporal drift was recently found in
+the relative scale of those two solar indices.
+Our aim is to bring a global clarification of those many issues. We compute new polynomial
+regressions up to degree 4, in order to obtain a more accurate proxy over the whole range of solar
+activity. We also study the role of temporal averaging on the regression, and we investigate the
+issue of the all-quiet F10.7 background flux. Finally, we check for any change in the F10.7 – sunspot
+number relation over the entire period 1947–2015.
+We find that, with a 4th-degree polynomial, we obtain a more accurate proxy relation than all
+previous published ones, and we derive a formula giving standard errors. The relation is different
+for daily, monthly and yearly mean values, and it proves to be fully linear for raw non-averaged
+daily data. By a simple two-component model for daily values, we show how temporal averaging
+leads to non-linear proxy relations. We also show that the quiet–Sun F10.7 background is not abso-
+lute and actually depends on the duration of the spotless periods. Finally, we find that the F10.7cm
+time series is inhomogeneous, with an abrupt 10.5% upward jump occurring between 1980 and
+1981, and splitting the series in two stable intervals.
+Our new proxy relations bring a strong improvement and show the importance of temporal
+scale for choosing the appropriate proxy and the F10.7 quiet-Sun background level. From historical
+evidence, we conclude that the 1981 jump is most likely due to a unique change in the F10.7
+scientific team and the data processing, and that the newly re-calibrated sunspot number (version
+2) will probably provide the only possible reference to correct this inhomogeneity.
+Key words. Sun – Solar activity – Solar indices – Solar irradiance (radio) – Solar cycle
+1. Introduction
+The sunspot number (hereafter SN; symbol: S N ) (Clette et al., 2014; Clette and Lef`evre, 2016)
+and the F10.7cm radio flux (symbol: F10.7) (Tapping and Morton, 2013) are arguably the most widely
+1
+arXiv:2301.02588v1  [astro-ph.SR]  6 Jan 2023
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+used solar indices to characterize the long term evolution of the solar activity cycle and of the
+underlying dynamo mechanism. In order to be usable over duration of decades to centuries, those
+indices must guarantee a long term stability so that levels of activity at two widely spaced epochs
+can be compared on exactly the same scale. F10.7cm and the SN are both absolute indices. Indeed,
+they do not lean on external references, which anyway are absent over a large part of their temporal
+range (410 years for the SN, and 73 years for F10.7 ). Instead, the index values are only based on
+the knowledge of the different steps in their determination, from the raw measurements to the data
+processing method. For the SN, a full revision process was undertaken in 2011 and led to a first
+re-calibration of this multi-century series, with correction reaching up to 20% (Clette et al., 2014;
+Clette and Lef`evre, 2016). This re-calibration included a full re-construction of the SN from raw
+original data for the period 1981 to the present, while correction factors were applied to the original
+series built by the Z¨urich Observatory before 1981.
+In this article, we will take a closer look at the F10.7cm radio flux, which comes second in duration
+after the SN, among the global long-duration solar activity indices based on a single uninterrupted
+observing and processing technique, and on a single long-duration standard reference. Since the
+measurements of the F10.7cm radio flux were started in Ottawa in 1947 (Covington, 1948, 1952), this
+new solar index proved to be highly correlated with the SN. This can be explained by the fact that the
+background flux, outside flaring events, is associated with the thermal free-free and gyroresonance
+emission of electrons trapped in closed loops anchored in the active regions, and thus primarily
+in their sunspots (Tapping, 1987; Tapping and Detracey, 1990; Tapping and Zwaan, 2001). Those
+two indices ran in parallel over the last 73 years and they are produced by completely different and
+independent processes. Therefore, as they are supposed to retrace exactly the same evolution of
+the last 7 solar activity cycles, studying their mutual relation can give a prime diagnostic of their
+long-term stability. We will thus focus on the proxy relation between those two indices.
+It turns out that, given the excellent long-term correlation between the SN and the F10.7cm radio
+flux, various proxy relations were derived over past years by different authors, or they were estab-
+lished for operational purposes by solar data services like NOAA-SWPC (Space Weather Prediction
+Center) in the USA or the IPS (Ionospheric Prediction Service, part of the Bureau of Meteorology)
+in Australia. Those relations are motivated by two kinds of applications. One of them is the re-
+construction of the F10.7 time series before the actual measurements started in 1947. Indeed, as the
+sunspot number extends back over four centuries, it allows to extrapolate this radio index over a
+much longer period (Svalgaard, 2016). Another application is producing mid-term predictions of
+the future evolution of the F10.7cm flux. As such predictions often need to be calibrated and validated
+over many past solar cycles, they are typically based on the SN, and consequently, they produce
+their predictions in terms of this SN. Moreover, the number of sunspots and sunspot groups give a
+direct measure of magnetic flux emergence at the solar surface, and is thus directly related to the
+dynamo mechanism at work inside the Sun (Charbonneau, 2010; Hathaway, 2010; Stenflo, 2012).
+On the other hand, F10.7 is a chromospheric/coronal index that combines two kinds of emission:
+gyroresonance and free-free, the latter being associated with the magnetic decay of active regions
+under the action of the random convection, leading to the chromospheric plage component (Tapping,
+1987; Tapping and Zwaan, 2001). This is why, on timescales shorter than the average lifetime of
+individual active regions and their associated plages, daily values of F10.7 are less correlated with
+the sunspot number, as first found by Vitinsky and Petrova (1980), Vitinsky (1982), Kopecky
+(1982) and Kuklin (1986), and using more modern methods by Dudok de Wit et al. (2009) and
+2
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Dudok de Wit et al. (2014). On the other hand, the daily flux offers a better proxy for ultraviolet
+(UV) and X-ray fluxes produced in the chromosphere, the transition region and the solar corona.
+For this reason, F10.7 is used by preference to the sunspot number for short-term forecasts of solar
+irradiance in the UV to X-ray domain and of its influence on the Earth environment (ionosphere,
+stratospheric temperatures, chemistry of the upper atmosphere), and for the resulting applications
+(radio propagation, atmospheric drag on low-Earth orbiting satellites). This close relation with the
+solar UV irradiance also allows to produce backward reconstructions of past UV fluxes for epochs
+well before the advent of direct space-based measurements of those fluxes (Svalgaard, 2016).
+However, beyond their strong Sun-related similarities on long timescales, the two indices differ
+by two base characteristics that play a role primarily at the lowest levels of activity. Firstly, F10.7
+does not fall to 0 when the Sun is spotless. A base background flux exists even when the Sun is fully
+quiet. This background emission, which corresponds to the spatially diffuse component of F10.7, is
+probably associated with the small magnetic loops rooted in the quiet-Sun chromospheric network
+(Tapping and Zwaan, 2001). This lower limit is still a matter of debate, but it is generally estimated
+in the range between 64 and 67 solar flux units (sfu) (Tapping and Detracey, 1990; Tapping and
+Charrois, 1994).
+Secondly, by its definition (Wolf, 1856; Clette et al., 2014), the SN is quantized at the lowest
+values, as each new group (with at least one single spot), adds 11 to the index. So, for the first
+spot, the SN jumps from 0 directly to 11. This effect quickly decreases for values larger than 22,
+as contributions from several groups with multiple sunspots are then combined in the total number.
+However, this low jump stretches the SN scale near 0. As this SN feature is absent in F10.7, we can
+expect that it will break the proportionality between the two indices in the lowest range.
+In this article, we first review all F10.7 – S N proxy relations published in the literature or used
+by operational space weather services. Given the mismatches between those existing proxies, we
+build more carefully a new least-square polynomial regression, while exploring the effect of tem-
+poral averaging of the source data. We also investigate the issue of the quiet-Sun F10.7 background
+level. We then check the temporal stability of the relation between F10.7 and the SN over the entire
+duration of the series. We finally conclude on the new picture emerging from our analysis and on
+important aspects to be taken into account for future updates of this relation between those two
+most fundamental measures of the long-term solar activity.
+In this analysis, we use the sunspot number data provided by the World Data Center SILSO
+(Sunspot Index and Long-term Solar Observations) at http://www.sidc.be/silso/datafiles
+and the F10.7cm data series from the Dominion Radio Astrophysical Observatory, available via
+the Space Weather Canada service at https://www.spaceweather.gc.ca/solarflux/
+sx-5-en.php,
+and
+also
+accessible
+through
+NOAA
+(https://www.ngdc.noaa.gov/
+stp/space-weather/solar-data/solar-features/solar-radio/noontime-flux/
+penticton/). We use the adjusted F10.7cm flux, which reduces the flux to a fixed distance of 1
+Astronomical Unit (AU), and thus eliminates any annual modulation due to the orbital eccentricity
+of the Earth.
+3
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Table 1. List of the proxies included in our comparison. For each proxy, we indicate the temporal granularity
+of the series used to fit the proxy (day, month, year; simple means or running means), and the SN version on
+which the proxy was based. The label identifies the curves in the associated figures, and the corresponding
+formulae are given in Table 2.
+Source
+Temporal base
+SN version
+Plot label
+Kuklin (1984)
+Unknown
+1
+KU1984 V1
+Holland and Vaughn (1984)
+13-month smoothed
+1
+HV1984 V1
+Xanthakis and Poulakos (1984)
+1 day
+1
+XP1985 V1
+Hathaway et al. (2002)
+24-month Gaussian smoothed
+1
+HA2002 V1
+Zhao and Han (2008) formula 1
+1-year mean
+1
+ZH2008 F1
+Zhao and Han (2008) formula 3
+1-year mean
+1
+ZH2008 F3
+Svalgaard (2009)
+1-month mean
+1
+SV2009 V1
+IPS Australia (Thompson, 2010)
+1-month mean
+1
+IPS2011 V1
+Tapping and Vald´es (2011)
+1-year mean
+1
+T2011 V1
+Johnson (2011) formula 1 monthly
+1-month mean
+1
+J2011 F1ma
+Johnson (2011) formula 1 yearly
+1-year mean
+1
+J2011 F1ya
+Johnson (2011) formula 2 monthly
+1-month mean
+1
+J2011 F2ma
+Johnson (2011) formula 2 yearly
+1-year mean
+1
+J2011 F2y
+NOAA-SWPC (2016)
+Unknown
+2
+NOAA V2
+Tapping and Morgan (2017) S N version 1
+10-month smoothed
+1
+T2017 V1
+Tapping and Morgan (2017) S N version 2
+10-month smoothed
+2
+T2017 V2
+Tiwari and Kumar (2018) formula 1
+1-month mean
+2
+TK2018 F1
+Tiwari and Kumar (2018) formula 2
+1-month mean
+2
+TK2018 F2
+2. Past F10.7cm – SN proxy relations
+Quite a number of proxy relations were proposed in the past. Here, we first compile all relations
+accessible in the literature or documented with associated data products at data centers 1. They are
+listed in Table 1, and the corresponding formulae are given in Table 2. In Figure 1, we plot all those
+proxies together, superimposed on the monthly mean values of F10.7 versus S N version 2, the most
+recent re-calibration of this series (Clette and Lef`evre, 2016).
+1 The Svalgaard (2009) proxy was found in the Web source: https://wattsupwiththat.com/
+2009/05/18/why-the-swpc-10-7-radio-flux-graph-is-wrong/.
+The
+NOAA-SWPC
+(2016)
+proxy is used for solar cycle predictions provided at https://www.swpc.noaa.gov/products/
+predicted-sunspot-number-and-radio-flux
+and
+https://www.swpc.noaa.gov/products/
+solar-cycle-progression. This proxy formula was formerly mentioned in an on-line document
+(ftp://ftp.swpc.noaa.gov/pub/weekly/Predict.txt), which is not accessible anymore. Some
+related information can presently be found in https://www.swpc.noaa.gov/sites/default/files/
+images/u2/Usr_guide.pdf.
+4
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Table 2. List of the proxies (labels from Table 1) and the corresponding formulae.
+Plot label
+Formula
+KU1984 V1
+71.74 + 0.2970 S N + 0.005146 S 2
+N for S N < 100.5,
+104.63 + 0.3037 S N + 0.001817 S 2
+N for S N > 100.5
+HV1984 V1
+67.0 + 0.97 S N + 17.6 (e−0.035 S N − 1)
+XP1985 V1
+68.15 + 0.65 S N
+HA2002 V1
+58.52 + 0.926 S N
+ZH2008 F1
+60.1 + 0.932 S N
+ZH2008 F3
+65.2 + 0.633 S N + 3.76 10−3S 2
+N + 1.28 10−5S 3
+N
+SV2009 V1
+67.29 + 0.316 S N + 1.084 10−2S 2
+N + 6.813 10−5S 3
+N + 1.314 10−7S 4
+N
+IPS2011 V1
+67.0 + 0.572 S N + 3.31 10−3S 2
+N–9.13 10−6S 3
+N
+T2011 V1
+66 + 0.446 S N(2. − e−0.027 S N)
+J2011 F1ma
+60.72 + 0.900 S N + 0.0002 S 2
+N
+J2011 F1ya
+62.87 + 0.835 S N + 0.0005 S 2
+N
+J2011 F2ma
+62.72 + (0.686 S N)1.0642
+J2011 F2y
+64.98 + (0.582 S N)1.0970
+NOAA V2
+67.00 + 0.4903 S N for S N ≤ 50; 56.06 + 0.7092 S N for S N > 50
+T2017 V1
+67 + 0.44 S N [2. − e−0.031 S N]
+T2017 V2
+67 + 0.31 S N [2. − e−0.019 S N]
+TK2018 F1
+62.51 + 0.6422 S N
+TH2018 F2
+65.6605 + 0.500687 S N + 1.21647 10−3S 2
+N + 2.71853 10−6S 3
+N
+So far, only a few recent proxy relations were calculated for this new version of the SN series,
+S Nv2 released in July 2015. Therefore, for converting older proxies based on S Nv1 to the S Nv2 scale,
+we used the following re-scaling relation:
+S Nv2 = S Nv1/0.6/1.177
+(1)
+The 0.6 factor is associated with a change of reference observer, making the former 0.6 conven-
+tional Z¨urich factor obsolete. The second factor corresponds to a correction for an artificial inflation
+of the original SN values due to the use of a weighting according to the sunspot size artificially
+introduced in Z¨urich (Clette et al., 2016). This was affecting all the S Nv1 values after 1946. As the
+F10.7 series starts in 1947, this relation is thus valid for the entire time interval considered here. The
+1.177 factor is in fact an asymptotic values reached for medium to high levels of solar activity. It is
+thus variable at low solar activity, dropping to almost 1 near cycle minima. Therefore, Equation 1
+is not fully accurate for low-activity values. Still, in our analysis presented below, we did not find
+any significant deviation between version 1 and version 2 proxies due to this effect, at the level of
+precision associated with the data themselves. This can be seen in the closeup view (Fig. 2).
+Overall, we observe that some proxies are very crude. They are simple linear fits, ignoring the
+visible deviation from linearity at the lower end of the range. Strong deviations also appear at
+the high values, in particular for non-linear fits (polynomial or exponential models). This can be
+explained by the limited number of such high values in the past solar activity record, which thus
+leads to large uncertainties. We also note that in the low range below S N = 20, virtually all proxies
+fall below the observed values, and thus lead to systematic underestimates of the average F10.7 flux
+at low activity.
+5
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 1. Combined plot of past published proxy relations giving F10.7 as a function of the sunspot number.
+The curves are labeled according to the identification in column 4 of Table 1. The curves are superimposed
+on the observed monthly mean values (gray dots).
+Moreover, some proxies were derived using monthly means or yearly means. In that case, the
+upper range of values is more limited, and the fits should not be trusted beyond their calibration
+range. Unfortunately, while a few estimates of the error of individual daily F10.7 flux values were
+published (Nicolet and Bossy , 1985; Tapping and Charrois, 1994), we must note that most of the
+available proxy relations are given without any estimate of their uncertainties, and often without
+clear indication of the calibration range. Here, we conservatively derived the mean and standard
+deviation of all proxy models shown in the plot (black line and shaded band in Fig. 3), to get a
+rough first idea of their actual uncertainty.
+6
+
+350
+Data
+J2011_F1ya
+HV1984_V1
+J2011_F2ma
+KU1984 V1
+J2011_F2ya
+300-
+HA2002_V1
+T2011_V1
+ZH2008_F1
+T2017_V1
+ZH2008_F3
+NOAA_V2
+SV2009_V1
+TK2018_F1
+IPS2011_V1
+250
+TK2018F2
+J2011F1ma
+T2017_V2
+200
+150
+100
+50
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 2. Combined plot of past published proxy relations giving F10.7 as a function of the sunspot number:
+close-up view of the low-activity range of Figure 1. The curves are labeled according to the identification in
+column 4 of Table 1. The curves are superimposed on the observed monthly mean values (gray dots).
+The proxies giving the best fit to the non-linear section at low SN are those published by Tapping
+and Vald´es (2011) (based on S Nv1) and Tapping and Morgan (2017) (based on S Nv1 and S Nv2). The
+authors mention that those proxies were defined purely empirically, and they do not explain how
+they were adjusted on the data. Both proxies are shown in Figure 3. One can see that the S Nv1
+and S Nv2 proxies are almost identical, indicating that the conversion in Equation[1] is accurate.
+However, we note that all those proxies reach a value of 67 sfu for S N = 0, while almost all
+observed F10.7 values are above this lower limit. In fact, below in Section 6, we find that the most
+probable F10.7 value for a spotless Sun is 70.5 sfu. This mismatch indicates that this part of the
+curve was not derived by least-squares but was adjusted empirically to reach exactly a tie-point at
+7
+
+100
+Data
+J2011_F1ya
+HV1984_V1
+J2011_F2ma
+KU1984 V1
+J2011_F2ya
+95
+HA2002_V1
+T2011_V1
+ZH2008F1
+T2017_V1
+ZH2008 F3
+NOAA_V2
+06
+SV2009_V1
+TK2018_F1
+IPS2011_V1
+TK2018_F2
+J2011_F1ma
+T2017_V2
+85
+F10.7
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 3. Mean (purple dash-dotted curve) and standard deviation of all proxies in Figure 1 (gray shading).
+Three recent proxies by Tapping et al. (cf. Table 1) are also included, as well as the observed monthly mean
+values (gray dots). The lower plot is a zoomed-in view of the upper plot for low activity levels.
+8
+
+T2011V1
+Proxymean
+T2017V1
+Data
+300
+T2017 V2
+3 Stdv
+250
+7. 
+200
+F10.
+150
+100
+0
+50
+100
+150
+200
+250
+300
+350
+Sunspot number100
+.....
+T2011 V1
+Proxy mean
+T2017 V1
+Data
+T2017 V2
+3 Stdv
+95
+90
+85
+F10.7
+80
+75
+70
+65
+60
+0
+10
+20
+30
+40
+50
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+67 sfu, chosen as base quiet-Sun background when S N=0. Given the mismatch with the actual data,
+this choice seems questionable. Indeed, for real applications, users need the most probable F10.7
+flux, and not the lowest possible value, which is rarely reached. In Section 6, we will consider more
+closely the properties of this quiet-Sun F10.7 background.
+Still, the other published proxies are underestimating even more the F10.7 flux at low activity.
+Therefore, overall, none of the proxies proposed so far are providing a satisfactory representation
+of the relation at low solar activity. Moreover, we note that all past proxies used classical least-
+square fits, which assume that errors are present only in the fitted measurement (here F10.7), while
+the other quantity (S N) is considered as a parameter (without error). As S N is also affected by errors,
+this fitting model may thus lead to systematic biases. We also checked this aspect as explained in
+sub-Section 5.2 below.
+3. Mean profiles
+In order to extract the F10.7/S N relation without any parametric model, we first derive the mean of
+F10.7 values and σm, the standard error of the mean (SEM), for a given value of S N. As the temporal
+averaging of raw daily values will influence the relation between the two quantities, we repeated this
+calculation for raw daily value pairs, for monthly means, for 13-month smoothed monthly values
+and for yearly means. In order to include a sufficiently large sample of values and to reduce random
+noise effects, we derived the statistics over a limited S N range centered on each given S N value.
+The bin width was 3, 20, 20, 60 respectively for daily, monthly, smoothed and yearly values. As our
+analysis immediately showed that results for yearly means and 13-month smoothed data are almost
+identical, we will not further discuss the 13-month smoothed results here.
+Figures 4, 5 and 6 show the resulting mean curves and 3-σm band for the daily, monthly, and
+yearly calculations. While the standard deviation of the base daily data is quite large, in particular
+for raw daily values (16.7 sfu overall), the SEM value is rather small in the low and medium range
+(< 1 sfu for daily values), thanks to the fact that each mean value is based on a large number of
+data points within each S N slice (about 400 points). It strongly increases in the upper range, above
+S N = 200, as data points become sparse, indicating that the fits will become much less precise in
+that upper range.
+Now considering the raw daily values, we observe that the distribution of individual F10.7 values
+around the mean is asymmetrical with a more extended wing towards high values (Figure 7).This
+upper wing may result from different contributions. This excess flux may come from the incomplete
+elimination of eruptive events and from the temporal under-sampling of the 20h00UT ”spot” mea-
+surements, as indicated by Tapping and Charrois (1994). If any flaring emission was present over
+time intervals when the S-component background emission was extracted, it inevitably contributed
+to an overestimate of the background flux and thus to a net positive excess, contrary to simple ran-
+dom measurement noise. This thus leads to an upwards asymmetry of the random deviations, like
+we find here.
+Another consequence of this deviation from the assumed symmetrical Gaussian noise distribution
+should be a small upward bias in the estimate of the mean. Moreover, when considering temporal
+variations (see Section 8), we find that the distribution is also slightly higher around the maxima
+of solar cycles (time of maximum -2 years to +3 years; blue curve in Figure 7) than around solar
+9
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 4. Mean non-parametric profile (red line) with 3 σm range (gray shading), obtained by averaging all
+daily F10.7 values for S N within a narrow band centered on the S N in the bottom axis (See details in the main
+text). Gray dots are the daily observed values. The number of data points strongly decreases above S N = 300
+and F10.7 = 250 sfu, leading to a much larger SEM σm.
+minima (red curve), by about 12%. This thus suggests a significant change in the F10.7 – S N relation
+over each solar cycle. We will further examine this interesting property in Section 8.
+However, this min – max shift, as well as the asymmetry of the distribution strongly decreases
+when considering longer time scales, i.e. for monthly and yearly means. The distribution becomes
+Gaussian and constant over time over those longer timescales. We can observe this by comparing
+daily data with the plots for the monthly and yearly means (Figures 5 and 6). We find that all tempo-
+rally averaged values lead to very consistent means, and that those means are all slightly lower than
+the mean for daily values (though still marginally consistent within the SEM σm). This indicates
+10
+
+400
+Mean
+3 Stdv
+Data
+350
+300
+250
+F10.7
+200
+150
+100
+0
+100
+200
+300
+400
+500
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 5. Mean non-parametric profile (red line) with 3 σm range (gray shading), like in Figure 4 but obtained
+by averaging monthly mean F10.7 values over narrow S N bands. Gray dots are the monthly mean observed
+values.
+that the upward bias affecting raw daily values largely vanishes for longer time scales. Moreover,
+σm values are lower, and similar for the 13-month smoothed and yearly means, as expected: 10.0
+sfu (monthly), 6.2 sfu (smoothed), 6.3 sfu (yearly). This lower dispersion and higher correlation
+between F10.7 and the SN indicates that for long duration, beyond a single solar rotation, both in-
+dices record the level of solar activity (total magnetic flux emergence) essentially in the same way.
+The global emergence rate dominates the statistics, and the skewed randomness of the F10.7 flux on
+short times scales only plays a minor role. We will deepen this interpretation in Section 8.
+In all cases, we can see that over most of the observed range, the means trace a largely linear
+proportionality between the two indices. Only above S N = 250, F10.7 tends to fall slightly below the
+11
+
+300
+Mean
+3 Stdv
+Data
+250
+200
+F10.7
+150
+100
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 6. Mean non-parametric profile (red line) with 3 σm range (grey shading), like in Figure 4 but obtained
+by averaging yearly mean F10.7 values over narrow S N bands. Gray dots are the yearly mean observed values.
+linear relation. This slight curvature would suggest that for short periods of extreme activity during
+cycle maxima, F10.7 does not grow as fast relative to the sunspot number. However, taking into
+account the large uncertainty, this is only marginally significant, and a fully linear relation remains
+valid up to the highest observed F10.7 fluxes.
+When zooming in on the low values (Figures 8, 9, 10), we find that the means for monthly,
+smoothed and yearly data start to deviate from the main linear part for S N values below an inflection
+point at about S N = 35 (monthly values) or 50 (smoothed and yearly values). As there are almost
+no monthly or yearly periods with a 0 mean S N, the ordinate at S N = 0 can only be extrapolated,
+12
+
+250
+Mean
+3 Stdv
+Data
+225
+200
+175
+150
+125
+100
+75
+0
+50
+100
+150
+200
+250
+300
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 7. Histogram of the ratio between daily F10.7 data and the mean F10.7 value (after subtraction of a 67 sfu
+base quiet-Sun flux), for all data with S N > 15. The black curve is for the entire data series, while the blue
+and red histograms are respectively for the maxima of the solar cycles (time of maximum -2 to +3 years)
+and the minima (the rest of the data). The distributions are slightly asymmetrical with a longer upper wing.
+The means of the distributions are indicated by thick vertical lines, with the matching colors. The standard
+deviation of the distributions equals 35%, while the SEM equals 0.5% (based on more than 10000 daily ratios
+in each distribution). The distributions for cycle maxima and minima are significantly shifted by 12% above
+and below the global mean.
+and cannot be trusted. Only for monthly means, we find that the means tend towards 68 sfu for SN
+below 5.
+On the other hand, for raw daily values, we find that the means continue to follow linear propor-
+tionality down to very low values, around S N = 5. There are only a very few points between 11
+and 0, but at S N = 0 , the mean value is well defined at 70.5 sfu. We find that the means reach this
+values within the uncertainties for S N = 8, which is very low.
+13
+
+700-
+Min
+Max
+600-
+All data
+500
+Count
+400
+300
+200
+100
+0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+F10.7Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 8. Mean non-parametric profile (red line) with 3 σm range (grey shading) for daily values: enlarged view
+of the low activity part of Figure 4.
+We can thus draw three important conclusions:
+– F10.7 and the SN are fully proportional over the full range of observed values, and this propor-
+tionality continues down to an almost spotless Sun. This suggests that the variability of F10.7 is
+entirely determined by the level of magnetic activity also controlling the number of sunspots,
+without any other contribution to the radio flux. The excess associated with the presence of
+spots becomes negligible relative to the F10.7 flux distribution for a fully spotless Sun only below
+S N = 8. In other words, the distribution of the F10.7 is largely the same for a fully spotless Sun
+and when a single isolated and short-lived spot is present. On the other hand, once the number of
+spots grows beyond one (one group with a single spot), F10.7 increases fully proportionally with
+the sunspot number. So, in that sense, the quantization of the SN at low activity (the 0 to 11 jump)
+14
+
+120
+Mean
+3 Stdv
+Data
+110
+100
+7
+F10.
+90
+80
+70
+60
+0
+10
+20
+30
+40
+50
+60
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 9. Mean non-parametric profile (red line) with 3 σm range (grey shading) for monthly means: enlarged
+view of the low activity part of Figure 5.
+does not lead to a significant non-linearity between the two quantities, except for the values at
+S N = 0.
+– As in temporal means, solar activity varies during the chosen time interval, the linear relation
+will be changed near the origin, essentially because of this single deviating point at S N = 0. As
+the latter is above the overall linear trend, the means will be pushed upwards, and this effect will
+increase as the mean SN decreases towards 0. Indeed, the time interval used for each mean will
+contain a growing proportion of spotless days, dominated by the F10.7 = 70.5 sfu background.
+This is exactly what we find in monthly, smoothed and yearly means, with the non-linearity ex-
+tending progressively to higher minimum SN as the duration of the temporal averaging increases.
+In Section 7, we will build a simulation to validate this interpretation.
+15
+
+95
+Mean
+3 Stdv
+Data
+90
+85
+7.
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 10. Mean non-parametric profile (red line) with 3 σm range (grey shading) for yearly means: enlarged
+view of the low activity part of Figure 6.
+– The proxy relation is consistent with Gaussian statistics and is stable only for timescales equal
+to or longer than one month. Daily data and short timescales include an excess of high fluxes,
+which varies with the solar cycle. Those data are thus inappropriate for building a reliable proxy
+relation.
+4. New high-degree polynomial fits
+4.1. Ordinary least-square polynomial fit: monthly means
+In order to obtain a better fit to the data than the earlier, sometimes very crude, fits shown in Section
+2, we fitted polynomials with degrees up to 4 by least-square regression of F10.7 versus S N. Indeed,
+16
+
+95
+Mean
+3 Stdv
+Data
+90
+85
+F10.7
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Table 3. Coefficients of polynomials of order 1 to 4 fitted by the ordinary least-square regression on the
+monthly mean values. The coefficients Cn correspond to Equation 2, with their standard error σn.
+Coefficients
+Order 1
+Order 2
+Order 3
+Order 4
+(FCpol1 m)
+(FCpol2 m)
+(FCpol3 m)
+(FCpol4 m)
+C0
+62.31
+62.87
+65.64
+67.73
+σ0
+0.5743
+0.7692
+0.9457
+1.134
+C1
+0.6432
+0.6279
+0.4918
+0.3368
+σ1
+4.528 10−3
+1.478 10−2
+3.132 10−2
+5.649 10−2
+C2
+6.141 10−5
+1.304 10−3
+3.690 10−3
+σ2
+5.637 10−5
+2.592 10−4
+7.699 10−4
+C3
+−2.919 10−6
+−1.517 10−5
+σ3
+5.946 10−7
+3.773 10−6
+C4
+1.974 10−8
+σ4
+6.003 10−9
+the non-parametric mean curves shown in the previous section indicate that the actual relation is
+largely linear over a wide range, with a rather sharp bifurcation towards a constant background in
+the low range. The polynomials are of the form (here for a 4th degree polynomial):
+F10.7 = C0 + C1 S N + C2 S 2
+N + C3 S 3
+N + C4 S 4
+N
+(2)
+In Figure 11, we show the fits to the monthly mean values for degrees 1 (linear) to 4. As we know
+that the relation becomes strongly non-linear below S N = 25 , the linear fit (degree 1) was applied
+to a restricted range without the interval S N = 0 to 25. So, this fit gives a good model for the main
+linear section.
+From S N = 30 to 250, all fits are almost identical and remain within the uncertainty range of
+the mean values (gray shaded band). Only above 250, there is a slight deviation, with the higher
+degrees falling below the linear fit. But this is hardly significant, given the low number of data points
+in this upper range. This is confirmed by the fact that coefficients of the high-degree terms are only
+marginally significant. In particular for the degree-2 polynomial, only the linear term (degree 1) is
+significant. This curve indeed gives the worst fit to the data.
+This can be seen in the close-up of the low part (second plot), which shows that the fitted curve
+match progressively better the curved lowest part of as the polynomial degree increases. Only the
+4th-degree polynomial closely reproduces the low part and remains within the uncertainty of the
+mean values. The coefficients for fitted polynomials up to degree 4 are listed in Table 3.
+As the relation between the two indices is fully linear over a wide range, in this case above
+S N = 25, we also derived the linear fits to this linear section. The coefficients are given in Table
+4 (also for the orthogonal regression method described below in sub-Section 5.2). The linearity is
+confirmed by the fact that polynomials fits above degree 1 do not give stable solutions, once the
+lowest range is excluded (terms of degrees above 1 are not significant).
+17
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 11. Polynomial of order 1 (linear fit) to 4 fitted to the monthly mean data by ordinary least-square
+regression (OLS). The curves are superimposed on the corresponding non-parametric mean (cf. Fig. 5) to
+show the agreement within 3 σm, and on the base data (gray dots). The lower plot is a close-up view of the
+low range of the upper plot.
+18
+
+300
+Mean
+FCRpold1 m
+FCpold2 m
+Data
+FCpold3 m
+3 Stdv
+FCpold4 m
+250
+200
+F10.7
+150
+100
+50
+100
+150
+200
+250
+300
+350
+Sunspot number105
+Mean
+FCRpold1 m
+FCpold2 m
+Data
+FCpold3 m
+3 Stdv
+100
+FCpold4 m
+95
+90
+F10.7
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Table 4. Coefficients of the linear fits to the monthly mean data in the restricted linear range S N = 25 − 290
+by ordinary least-squares and by orthogonal distance regression. The two fits match closely.
+Coefficients
+Order 1
+Order 1 (ODR)
+(FCRpol1 m)
+C0
+59.66
+58.21
+σ0
+0.8801
+0.8831
+C1
+0.6601
+0.6720
+σ1
+6.313 10−3
+6.338 10−3
+Table 5. Coefficients of polynomials of order 1 to 4 fitted by the ordinary least-square regression on the
+yearly mean values. For the order 2 and 4 polynomials, coefficients for degree 2 and above are not significant
+(marked in italics), indicating that the proxy relation is essentially linear.
+Coefficients
+Order 1
+Order 2
+Order 3
+Order 4
+(FCpol1 y)
+(FCpol2 y)
+(FCpol3 y)
+(FCpol4 y)
+C0
+61.07
+62.64
+66.56
+66.85
+σ0
+1.323
+1.873
+2.404
+3.220
+C1
+0.6555
+0.6114
+0.4163
+0.3942
+σ1
+1.062 10−2
+3.893 10−2
+8.761 10−2
+1.816 10−1
+C2
+1.862 10−4
+2.129 10−3
+2.510 10−3
+σ2
+1.583 10−4
+8.031 10−4
+2.849 10−3
+C3
+−5.081 10−6
+−7.327 10−6
+σ3
+2.062 10−6
+1.623 10−5
+C4
+4.216 10−9
+σ4
+3.021 10−8
+4.2. Polynomial fits to yearly means
+We repeated the analysis on yearly values and found largely the same conclusions. The curves are
+shown in Figure 12, and the polynomial coefficients for the fits to yearly means are given in Tables
+5 and 6.
+In this case, the fit is also not significant at polynomial order 2, and the order-4 polynomial gives
+roughly the same quality of fit as order 3. Although the fits on yearly means are slightly different
+from the fits derived from monthly mean values, both are compatible within the uncertainties in
+yearly means. This difference is due to a lower non-linearity and the slightly wider range over
+which the relation is non-linear for yearly means, but is hardly significant.
+4.3. Polynomial fits to daily values
+Based on Section 3, we may expect slightly different results for daily values. The polynomial and
+linear regressions are shown in Figure 13 and the coefficients are listed in Tables 7 and 8. We find
+that for the main linear range up to S N = 250, the different fits match closely, like for yearly and
+monthly means. They then diverge from each other at higher values, which is again due to the
+steeply decreasing number of data points in this upper range. The non-linear fits at order 2 to 4 tend
+to fall below the linear fit, aligning better with the mean values. However, the linear fit still falls
+19
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 12. Polynomial of order 1 (linear fit) to 4 fitted to the yearly mean data by ordinary least-square regres-
+sion (OLS). The curves are superimposed on the corresponding non-parametric mean (cf. Fig. 6) to show the
+agreement within 3 σm, and on the base data (gray dots). The lower plot is a close-up view of the low range
+of the upper plot.
+20
+
+250
+Mean
+...
+FCRpold1_y
+FCpold2_y
+Data
+FCpold3_y
+3 Stdv
+225
+FCpold4_y
+200
+175
+10.7
+ 150
+125
+100
+75
+0
+50
+100
+150
+200
+250
+300
+Sunspot number105
+Mean
+FCRpold1_y
+FCpold2_y
+Data
+FCpold3_y
+3 Stdv
+100
+FCpold4_y
+95
+90
+F10.7
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Table 6. Coefficients of the linear fits to the yearly mean data in the restricted linear range S N = 30 - 220 by
+ordinary least-squares.
+Coefficients
+Order 1
+(FCRpol1 y)
+C0
+56.24
+σ0
+2.259
+C1
+0.6936
+σ1
+1.739 10−2
+Table 7. Coefficients of polynomials of order 1 to 4 fitted by the ordinary least-square regression on the
+daily values. The coefficients Cn correspond to Equation 2, with their standard error σn.
+Coefficients
+Order 1
+Order 2
+Order 3
+Order 4
+(FCpol1 d)
+(FCpol2 d)
+(FCpol3 d)
+(FCpol4 d)
+C0
+66.72
+65.97
+67.52
+69.41
+σ0
+0.1654
+0.2083
+0.2430
+0.2711
+C1
+0.6002
+0.6198
+0.5432
+0.3938
+σ1
+1.253 10−3
+3.572 10−3
+7.185 10−3
+1.204 10−2
+C2
+−7.068 10−5
+5.561 10−4
+2.613 10−3
+σ2
+1.207 10−5
+5.246 10−5
+1.432 10−4
+C3
+−1.260 10−6
+−1.033 10−5
+σ3
+1.027 10−7
+5.965 10−7
+C4
+1.225 10−8
+σ4
+7.937 10−10
+within the uncertainty range. So, the differences brought by higher degrees are not fully significant.
+This is again confirmed by the low level of significance of polynomial coefficients with degrees
+higher than 1.
+The low range is where the situation differs markedly from the monthly and yearly mean analyses.
+Here (Fig. 13), the linear fit (over the range above S N = 25) remains close to the mean values down
+to the lowest values. In the low range, the order-4 curve again gives the best fit, although it fails
+for S N values below 6, as the mean then deviates abruptly over a very small range. Ignoring this
+section, the order-3 polynomial gives the best fit overall. It reaches the mean background value for
+S N = 0 (70.5 sfu) at S N = 5. This suggests that for S N below 6, this background value can be used
+instead of the polynomial fit.
+Therefore, the fit on daily values helps to confirm the higher linearity of the F10.7 – S N relation
+down to very low levels. However, as expected, the fitted curves are significantly higher than the fits
+on monthly and yearly means, due to the upward bias characterizing raw daily values (see Section
+3). Beyond the linearity check, they should thus not be considered for a proxy relation.
+4.4. Comparison with the Tapping and Morgan (2017) proxy
+In order to check if indeed the new polynomial fits bring an improvement on past relations, in Figure
+14, we compare the order-4 polynomial with the best curve identified among the past published
+proxies, namely the curve by Tapping and Morgan (2017). Here, we consider the fit on monthly
+21
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 13. Polynomial of order 1 (linear fit) to 4 fitted to the daily data by ordinary least-square regression
+(OLS). The curves are superimposed on the corresponding non-parametric mean (cf. Fig. 4) to show the
+agreement within 3 σm, and on the base data (gray dots). The lower plot is a close-up view of the low range
+of the upper plot.
+22
+
+400
+Mean
+FCRpold1 d
+FCpold2 d
+Data
+FCpold3 d
+3 Stdv
+350
+FCpold4 d
+300
+250
+F10.7
+200
+150
+100
+0
+100
+200
+300
+400
+500
+Sunspot number105
+Mean
+FCRpold1 d
+FCpold2 d
+Data
+FCpold3 d
+3 Stdv
+100
+FCpold4 d
+95
+90
+F10.7
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Table 8. Coefficients of the linear fits to the daily data in the restricted linear range S N = 5 - 290 by ordinary
+least-squares and by orthogonal distance regression. The two fits match closely.
+Coefficients
+Order 1
+Order 1 (ODR)
+(FCRpol1 d)
+C0
+64.79
+62.23
+σ0
+0.2010
+0.2030
+C1
+0.6171
+0.6419
+σ1
+1.592 10−3
+1.610 10−3
+means, as most of the fits are based on temporally averaged numbers, in order to smooth out the
+random variations due to short-timescale solar variations.
+One can see that both fits match very closely, within 4 sfu over the whole linear range. The slope
+is slightly lower than the slope of a purely linear fit on the range S N = 25 to 290 (dotted line). This
+may be due to the influence of the upward deviation for S N below 30.
+Now, looking at the lowest range, we find that the 4th-degree polynomial tracks the data slightly
+better, and at least remain within the uncertainty range of the means, contrary to the relation by
+Tapping and Morgan (2017), which is too low. It reaches 68 sfu at S N = 0, instead of the 67 sfu
+background value used by Tapping and Morgan (2017). This is in agreement with our analysis of the
+background flux for a spotless Sun in Section 6 below: our polynomial fits the most probable flux
+instead of the assumed F10.7 background value at S N = 0, which is a lower boundary. Finally, we
+point out that our 4th-order polynomial is entirely defined by a least-square fit to the data, and is not
+attached to a predefined tie-point, like the Tapping and Morgan (2017) curve. It thus allows classical
+statistical tests on fitted polynomials, including the estimate of errors on polynomial coefficients and
+on the resulting proxy values.
+5. Polynomial error determination
+5.1. Uncertainties on polynomial values
+Regression methods allow to determine the standard errors σn on each polynomial coefficient.
+However, an exact derivation of the standard error σp on polynomial values themselves, based on
+those standard errors σn, does not exist in the literature, due to the mathematical complexity of this
+problem. Indeed, the errors on the coefficients for the different terms are actually inter-correlated,
+as they are determined together. Therefore, the total variance of the polynomial values is not the
+simple naive sum of the individual variances of all terms. However, a proper estimate of σp can be
+derived, based on the fact that the actual error for each term (each degree) is the conditional error
+on that coefficient, i.e. the uncertainty of that polynomial term given the values of the coefficients
+for all other terms (in the solution of the least-square regression). In order to estimate this condi-
+tional error, we can make a regression for only one term (one polynomial degree) at a time, after
+subtracting all other terms from the original observed F10.7 values, with the other coefficients set at
+the values given by the regression.
+In order to simplify this calculation, we considered that, as we go to higher degree terms, their
+contribution becomes smaller. Therefore, we derived the conditional error for each degree n by re-
+gressing for each degree separately (one-term model), after subtracting successively all polynomial
+23
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 14. Comparison of our 4th order polynomial with the proxy relation from Tapping and Morgan (2017).
+The curves are superimposed on the corresponding non-parametric mean (cf. Fig. 5) to show the agreement
+within 3 σm, and on the base data (gray dots). The lower plot is a close-up view of the low range of the upper
+plot.
+24
+
+105
+Mean
+FCRpold1 m
+T2017V2
+Data
+FCpold4 m
+3 Stdv
+100
+95
+90
+F10.7
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+Sunspot number300
+Mean
+.......
+FCRpold1 m
+T2017 V2
+Data
+FCpold4 m
+3 Stdv
+250
+200
+F10.7
+150
+100
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+contributions of lower degrees (< n) from the original F10.7 data, starting from the lowest degree.
+Thus, the single-term model of degree n is:
+Fn = Cn S n
+N
+n = 0, . . . , d
+(3)
+where Cn is the coefficient to be determined (with its error) and d is the degree of the polynomial.
+This model is fitted to the F10.7 data series, minus all fitted terms of lower degree:
+Fcorr = F10.7 −
+n−1
+�
+k=0
+Ck S k
+n
+(4)
+The fact that we did not subtract the terms of higher degrees leads to a slight overestimate of the
+residual error for each degree, as it also includes the residual uncertainties of all degrees above n.
+Therefore, the conditional errors calculated in this way for each separate degree give an upper limit.
+Deriving the above errors from the data requires a statistical processing and multiple regressions
+on the source data. So, for practical applications, this approach would be too heavy. Therefore, based
+on the data-based errors obtained by this procedure, we found a simple mathematical representation
+that gives a good approximation of the σp errors from the full determination described above, and
+that can be calculated directly for a polynomial value calculated at any given S N:
+σTot =
+�
+�
+� d
+�
+n=0
+�������
+S n
+N − S N
+n
+2 n−1 (d + 1 − n)2 σn
+�������
+2
+(5)
+where S N is the mean of all S N values in the data set (≈ 120 with the actual data).
+It consists in the sum of squared errors for each term, using the (non-conditional) standard error
+on each coefficient given by the least-square regression procedure, σn, but with a weight factor that
+decreases for increasing degree n (powers of 2), and also decreases for each term of a given degree
+n, as the degree d of the polynomial increases (and thus the number of degrees of freedom in the
+regression).
+We observe that those rather simple expressions already give a very good agreement with the real
+data-based errors. This rather simple empirical weighting thus probably reflects the dominant cor-
+rections associated with the inter-dependency of the least-square polynomial coefficients. A math-
+ematical demonstration goes well beyond the scope of this study, but the good match with the
+data-based errors indicates that we obtain here a reliable estimate of this polynomial error (Figure
+15). This marks a big improvement on all previous proxy relations, where the error was missing and
+thus entirely undetermined. The formula in Equation 5 conveniently allows a direct calculation of
+the error, without requiring to re-do the above extraction of conditional errors from the data them-
+selves. Finally, we point out that σTot gives the uncertainty on the proxy values, which combines
+both the errors in F10.7 and S N. This must be distinguished from the standard error of a single daily
+F10.7 measurement, which is globally estimated at about 2% by (Tapping and Charrois, 1994). As
+expected for such a regression, the smallest errors (±2 sfu) are found in the vicinity of the mean of
+all values, i.e. S N = 120 and F10.7 = 135 sfu, and the error grows in both directions away from this
+point.
+25
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 15. Order-4 polynomial with uncertainty range (1 standard error σp) on polynomial values, superim-
+posed on the corresponding non-parametric mean profile and SEM σm obtained in Section 3. The lower plot
+is a close-up view of the low-activity range in the upper plot.
+26
+
+Data
+300
+Mean
+Polynomial deg 4
++/- 1 sigma
+3 Stdv
+250
+F10.7
+200
+150
+100
+50
+100
+150
+200
+250
+300
+350
+400
+Sunspot number105
+Data
+Mean
+100
+Polynomial deg 4
++/- 1 sigma
+3 Stdv
+95
+90
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+70
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+5.2. Orthogonal-distance regression versus ordinary least-square regression
+In the ordinary least-square (OLS) regression, the model assumes that all errors are in the dependent
+variable (“response”, here F10.7) and not in the independent variable (“explanatory”, here S N). As
+we know that in our case, both quantities are actually affected by errors, we repeated the regression,
+but using instead the orthogonal distance regression (ODR) technique, which takes into account the
+uncertainties in both regressed variables.
+We find that the differences between the coefficients derived from the ordinary and ODR re-
+gressions are within the computed uncertainties, and are thus not significant. Likewise, Figure 16
+illustrates this close agreement for the 4th-degree polynomials derived by both methods. Therefore,
+we conclude that the ordinary regression gives valid fits in this case. This can be explained by the
+very high level of correlation between the two indices over long timescales.
+6. Background flux for a spotless Sun
+In the above curves, we noted that past relations found by Tapping and Vald´es (2011) and Tapping
+and Morgan (2017) assumed a base radio flux of 67 sfu at S N = 0. By contrast, our mean curves
+based on daily values indicate a higher mean F10.7 for all spotless days in the series, at 70.5 sfu.
+However, the monthly means tend to converge towards lower values near S N = 0, though still above
+67 sfu. We can thus wonder how to reconcile those apparently contradictory determinations of the
+same parameter. As the underlying temporal resolution is different in each case, we suspected that
+the temporal scale plays a central role.
+6.1. Dependency on spotless duration
+In order to investigate such a temporal effect, we extracted all spotless days in the SN series, and
+the corresponding daily F10.7 flux. Then, we also grouped uninterrupted sequences of contiguous
+spotless days. Finally, we computed the distribution of F10.7 values for all spotless sequences of
+the same length in the observed series. Figure 17 shows the mean values, standard deviations and
+extreme values of the F10.7cm flux for all sequence lengths found in the series. In the lower panel, we
+also plotted the number of days included in each category, and how many sequences were found for
+each duration. The longest sequence lasted 42 days, but most spotless days sequences last less than
+10 days, with many isolated spotless days.
+Our analysis shows that the mean F10.7 flux systematically increases as the duration of a spotless-
+day sequence decreases. For duration above 15 days, the most likely F10.7 value is near 68 or 69 sfu.
+On the other hand, it increases to 74 sfu for single spotless days immediately surrounded by active
+days with one or more sunspots. The lowest mean daily value is 67 sfu and is reached only for 6
+sequences with duration of 22, 27 and 28 days. The red dashed line at 70.5 sfu corresponds to the
+mean flux for all spotless days. Quite logically, it corresponds to the mean levels for duration 5 to
+10 days, which is near the mean duration of spotless intervals.
+Now, considering the extreme values, we also find a steady increase of the upper values (blue
+dots) with decreasing duration, up to as high as 95 sfu for a single spotless day. Based on the above
+regressions, such fluxes usually correspond to S N values of 50, i.e. to moderate levels of activity. On
+the other hand, the lowest values (green dots) do not show any dependency on duration. They stay
+around the 67 sfu level, with rare extremely low values down to 61.6 sfu (November 3, 1954). This
+27
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 16. Comparison of 4th-degree polynomials obtained by the ordinary least-square regression (OLS) and
+by the orthogonal-distance regression (ODR). The differences are small in comparison with the 1-σp standard
+error of the fit, and are thus not significant over the whole range of values.
+28
+
+Data
+300
+Polynomial deg 4
+- +/- 1 sigma
+Poly.ODR deg 4
++/- 1 sigma ODR
+250
+F10.7
+200
+150
+100
+50
+100
+150
+200
+250
+300
+350
+400
+Sunspot number105
+Data
+Polynomial deg 4
+100
+- +/-1 sigma
+Poly. ODR deg 4
++/- 1 sigma ODR
+95
+90
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+70
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 17. Plot of the mean F10.7 background flux for spotless days (red dots) as a function of the duration of the
+sequence of contiguous spotless days. The errors bars correspond to one standard deviation. The upper curve
+(blue dots) and lower curve (green dots) are the lowest and largest F10.7 values for each spotless duration.
+The red dashed line marks the overall mean spotless flux (70.5 sfu), while the lower black dashed line is the
+mean minimum flux (67 sfu). The lower panel gives the number of intervals for each spotless duration (red
+bar) and the number of days included in each duration (green).
+thus validates the choice of 67 sfu as the all-quiet base flux, shown in Figure 17 as the horizontal
+black dashed line.
+Actually, there are only 33 values below 66 sfu in the whole series. Moreover, all 8 values below
+65 sfu and 14 values out of 25 between 65 and 66 sfu appear exclusively in 1953 and 1954, some
+29
+
+95
+Meanflux
+Lowestvalues
+Highest values
+Mean Flux & std error
+90
+85
+80
+10.7
+山
+75
+70
+65
+60
+200
+Nb days
+Nb intervals
+Count
+100
+0
+0
+5
+10
+15
+20
+25
+30
+35
+40
+NbzerosdaysClette: F10.7cm proxy relation and temporal homogeneity
+of them even on days when the Sun was not spotless. By contrast, the lowest values recorded after
+1954 are always above 65.5 sfu, and almost all are occurring quite logically during the longest
+minimum recorded in the F10.7 series, between cycle 23 – 24 in 2008. By comparison, the cycle 18
+– 19 minimum in 1954 was not particularly low and protracted. This suggests that the record-low
+values in 1953 – 1954 are either spurious or suffer from a calibration problem. As indicated by
+Tapping (2013) and Tapping and Morgan (2017), those early data may indeed suffer from larger
+errors, as the calibration method was not fully standardized, and the location in Ottawa until 1962
+caused larger radio interferences. Therefore, a base background flux as low as 64 sfu, as suggested
+by Tapping and Detracey (1990) and Tapping and Charrois (1994), seems doubtful and too low for
+the real fully-quiet Sun.
+6.2. Interpretation
+We can explain this dependency of the background levels on the duration of spotless intervals by
+the presence of other sources of F10.7 emission, even when there are no associated sunspots. This
+includes various features in the chromosphere and lower corona associated with closed magnetic
+fields weaker than those concentrated in sunspots, like bright chromospheric plages, filaments, coro-
+nal condensations (Shimojo et al., 2006; Schonfeld et al. , 2015; Pevtsov et al. , 2014; Ermolli et
+al., 2014). In particular, when the Sun is very quiet (few isolated spots) or entirely spotless, the
+associated plages have a small extent but still contribute a significant excess in F10.7. Indeed, Figure
+17 shows that most of the values are below 80 sfu, although there are a few as high as 80 sfu or
+more. On the other hand, there are almost no values below 67 sfu.
+Such chromospheric plages are typically present just before the emergence of a first spot, or they
+remain after the decay of a last sunspot. Therefore, short spotless intervals surrounded by more ac-
+tive periods are most likely to show a significant excess in radio emission above the lowest all-quiet
+level. Conversely, only very long spotless periods can include days without any activity features on
+the Earth-facing solar disk. The only small active regions present just before the long spotless inter-
+val have enough time to decay entirely, well before new bright chromospheric structures develop,
+heralding the appearance of the first spots marking the end of the protracted spotless period.
+Consequently, the mean background level does not have an absolute value, as the lowest level
+of 67 sfu is almost never reached, even when there are no sunspots. We can thus expect to see a
+dependency of the asymptotic F10.7 value near S N = 0 when applying a temporal averaging to the
+raw daily data series. Without any averaging or when the averaging duration is short, days in long
+spotless intervals are mixed with those in short intervals, leading to a higher mean, rising to 70.5
+sfu for 1-day timescale. At the other extreme, averaging over very long duration, like one year,
+inevitably mixes spotless periods with active periods, as virtually no spotless periods have duration
+above about 30 days. Therefore, the lowest F10.7 yearly mean values are expected to be also higher
+than the 67 sfu minimum. It turns out that a duration of one month best matches the actual duration
+range of spotless episodes. Monthly means are thus best for recording the lowest possible mean
+radio fluxes of the fully quiet Sun.
+Finally, we note that this chromospheric background interpretation agrees with the identification
+of two types of emission sources for F10.7 (Tapping, 1987; Tapping and Detracey, 1990; Tapping
+and Zwaan, 2001; Tapping and Morton, 2013; Schonfeld et al. , 2015). While the gyroresonance
+emission is closely associated with strong magnetic fields in sunspots (> 300 G), a so-called ”dif-
+30
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+fuse” component by free-free thermal emission is attributed to plages and the overall chromospheric
+network. The latter is the best candidate for the variable background flux diagnosed here. In this
+respect, we note that while the sunspot component of F10.7 will track instantaneously the evolution
+of active regions (fully linear relation), the plage component will be extended and delayed in time
+relative to the associated sunspots, as it corresponds to the progressive decay and dispersal of flux
+emerging in active regions.
+Therefore, the dependency between the mean background flux and the duration of the spotless
+interval is consistent with this interpretation, and offers an independent indicator of this dual-source
+nature of the F10.7cm radio flux. It also implies that the disagreements between F10.7 and the SN are
+probably due for a significant part to the time delay intrinsic to the free-free emission from plages,
+rather than simply to a non-proportionality with the underlying emerging magnetic fluxes, and thus
+with the SN. Indeed, the latter is only sensitive to strong fluxes freshly emerged in sunspots, without
+mixing with a second magnetic-decay component. The fact that disagreements between those two
+indices increase for short time scales, below one solar rotation and thus below the average plage
+lifetime, also concurs with the prominent role of this temporally-smeared weak-field component.
+7. A simple model for the the F10.7/SN non-linearity
+In the above analyses, we found that daily values indicate that the relation between F10.7 and S N
+is linear almost down to the lowest values of S N = 11 (single isolated spot). Only for smaller S N
+values close to 0, F10.7 stops decreasing and reaches its background level, in the range 68-70 sfu as
+found in Section 6. There is a break from the linear relation, with the last points near S N = 0 located
+several solar flux units above the linear relation, which intercepts the axis at S N = 0 at a value of 58
+to 62 sfu (see tables of polynomial coefficients above).
+When deriving monthly or yearly means, the averaging interval inevitably includes periods of
+different activity levels, including some inactive days, when F10.7 is at its lowest level and thus
+shows an excess relative to a purely linear relation. As the mean activity during the averaging
+interval decreases, the proportion of spotless days increases, and thus also the fraction of points
+bringing an excess above the linear relation. We can thus expect a progressive upwards deviation
+from linearity, like we observe in the monthly and yearly mean curves.
+In order to simulate this scheme, we took the observed SN time series, and synthesized a F10.7
+time series, by converting each S N value via a two-component model:
+1. For most of of the range, we used the linear fit to the linear part of the data (from Table 8).
+2. For the lowest S N values, when the linear relation falls below the base F10.7 background, chosen
+at 70.5 sfu, the output value is set at the constant value of 70.5 sfu. An alternate model for this
+background flux can take into account the fact that the mean background is not constant but
+increases even when the Sun is spotless, up to 74 sfu, as demonstrated in Section 6.
+We then applied the usual monthly and yearly averaging to this synthetic series.
+In Figure 18, the model for daily values assumes a constant background at the mean F10.7 flux for
+a spotless Sun (pink crosses). In Figure 19, the model assumes a progressive rise of the F10.7 flux
+over the range found in our above analysis of the base flux according to the duration of the spotless
+period. It starts from 69 sfu, the low value for a fully spotless Sun and rises to 75 sfu, the upper
+31
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 18. Model of the monthly temporal averaging of daily data built from two components: linear component
+and constant lower background at 70.5 sfu. The pink crosses are the synthesized daily values. The green dots
+are the corresponding monthly mean values, and the blue line is the non-parametric local mean of those
+values. As a comparison, the black line is the 4th-order polynomial fitted to the real monthly mean data
+(Table 3), with uncertainties (black dashes lines).
+value found for isolated spotless days. This can be considered as representative of the flux when
+just one isolated sunspot group is present on the Sun. Here, this ramp connects with the main linear
+relation at about S N = 24.
+Our degree-4 polynomial based on the true data (black line, with uncertainties as dashed lines)
+nicely falls in the middle of the simulated monthly means for both options. The agreement with the
+mean of data values (blue curve) is best for the second model, where the agreement is very tight.
+The first model with a higher but uniform background gives a higher curvature below S N = 24.
+32
+
+105
+Model daily
+Model monthly
+100
+Mean
+Polynomial deg 4
++/- 1 sigma
+95
+90
+F10.7
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+70
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 19. Model of the monthly temporal averaging of daily data built from two components: linear compo-
+nent, and here, a slightly rising background with increasing S N. The elements of the plots are the same as in
+Figure 18.
+We also made the simulations using yearly means (Fig. 20 and 21). They also give a good agree-
+ment, but given the lower number of points and slightly more linear relation, the monthly simu-
+lations shown here illustrate more clearly how temporal averaging is producing a curvature of the
+relation.
+Overall, those two very simple simulations match strikingly well the actual data. They thus
+confirm the mechanism by which time-averaging of the raw linear daily values can produce the
+non-linear proxy relation, thus also indicating that this non-linearity is dependent on the temporal-
+averaging applied to the data before making the regression.
+33
+
+105
+Model daily
+Model monthly
+100
+Mean
+Polynomial deg 4
++/- 1 sigma
+95
+90
+F10.7
+85
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+70
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 20. Model of the yearly temporal averaging of daily data built from two components: linear relation and
+constant lower background at 70.5 sfu. The elements of the plots are the same as in Figure 18, with green
+dots and curve corresponding to yearly averages and the polynomial also to yearly mean data (Table 5).
+8. Temporal variations
+So far, we included the entire duration of the time series, thus making the assumption that both
+the F10.7cm flux and the sunspot number series are homogeneous over the entire 68-year duration
+included here. However, Clette et al. (2016) made a first simple comparison between the newly
+released Version 2 of the sunspot number and F10.7 as a function of time, and found a 12% upward
+jump in the F10.7/S N ratio, occurring between 1979 and 1983.
+Likewise, by a comparison to the sunspot number series (versions 1 and 2) and the total sunspot
+area, Tapping and Vald´es (2011) and Tapping and Morgan (2017) found that the F10.7 time series
+34
+
+105
+Model daily
+Model yearly
+100
+Mean
+Polynomial deg 4
++/- 1 sigma
+95
+90
+F10.7
+85
+80
+×××
+75
+70
++++++
+65
+0
+10
+20
+30
+40
+50
+60
+70
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 21. Model of the yearly temporal averaging of daily data built from two components: linear relation and
+slightly rising background with increasing S N. The elements of the plots are the same as in Figure 18.
+shows an upward deviation in the second half of the series, mostly after 1980, relative to both the
+sunspot number and sunspot areas. Although this trend is stronger when comparing with S Nv1, it is
+still present when S Nv2 is taken as reference. The authors fit a smooth curve as a function of time
+over the whole duration of the series, and they interpret the resulting global trend as a real change
+in solar properties. This evolution would parallel the overall decline of solar cycle amplitudes since
+the mid-20th century, thus invoking a possible genuine change in the properties of the Sun.
+35
+
+105
+Model daily
+Model yearly
+100
+Mean
+Polynomial deg 4
++/- 1 sigma
+95
+90
+F10.7
+85
+XXXXXXxx
+80
+75
+70
+65
+0
+10
+20
+30
+40
+50
+60
+70
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+8.1. A transition between two stable periods
+In order to check for such a change in the relation between the two measurements, we used the
+13-month smoothed monthly means, using the classical Z¨urich smoothing function. This allows to
+reject random fluctuations associated with solar activity at time scales shorted than one year, while
+retaining a better temporal sampling than in yearly means. In our case, in the above F10.7 versus
+S N representation, we checked over which time interval the data follow the same relation in the
+F10.7/S N space.
+In Figures 22 and 23, the resulting curves are shown respectively for the SN version 1 and SN
+version 2. The curves now include the chronology and consist of several narrow loops corresponding
+to each of the solar cycles, varying from very low values at minima to the maxima while staying
+close to a diagonal line, as can be expected given the close proportionality of the two indices, as
+shown in the previous sections.
+We note that two cycles deviate strongly in the case of the SN version 1 (blue loops in Figure
+22). They correspond to cycles 22 and 23, when SN version 1 is know to be affected to drifts in the
+Locarno pilot station (Clette et al., 2014, 2016). On the other hand, the agreement is much tighter
+with SN version 2 (Figure 23), in particular over all cycles after cycle 21 for which the SN was
+entirely re-constructed based on a multi-station reference.
+We will thus now concentrate on this second comparison with SN version 2. The plot immediately
+shows that the curves are grouped along two preferential bands located above (colored blue) and
+below (red) the fit to the entire series (solid black line), with very few points in-between, near the
+global fit. This indicates that the relation was actually very stable during two time periods and
+jumped directly from one relation to the other.
+We then looked for the time sub-intervals following the upper and lower linear relation, and we
+found that the lower relation applies to all data before 1980, while the upper relation is valid for the
+entire period after 1980. The temporal evolution is thus characterized by a single jump separating
+two fully homogeneous periods, during which the F10.7/S N relation is very stable.
+For each homogeneous interval, we could then derive the two corresponding linear relations using
+the same regression methods, as applied before to the whole series (Fig. 23). For the period before
+1980, we find a slope of 0.635 ± 6.6110−3 (dotted line), while after 1980, the slope becomes steeper
+at 0.702±8.8710−3 (dashed line). This corresponds to an upward jump by a factor 1.106±0.017, thus
+of about 10.5%. The slope found for the entire series naturally falls in-between, with an intermediate
+slope of 0.660 ± 6.31310−3 (cf. Table 4). The coefficients for those two linear fits are given together
+with the two 4th-degree polynomial fits in Table 9. Such a jump is highly significant, as Tapping and
+Charrois (1994) and Tapping (2013) give an accuracy of 1% for the flux measurements and of 2%
+for the daily index derived from the ”spot” measurements at 20h00 UT.
+Checking the past published proxies, we note that early proxies that were based only on the first
+part of the series, like Holland and Vaughn (1984) or the IPS formula (Table 1), match the low
+linear fit for the period before 1980 in Table 9, while the NOAA proxy adjusted to the recent SN
+version 2 (1), has a steeper slope matching well the second higher fit for the recent period after
+1980. So, the disagreements between some of the past proxies actually originate from the temporal
+inhomogeneity of the F10.7 index itself, as diagnosed here.
+Another consequence is that when including the entire time series, the least-square regression
+errors will not decrease when using monthly or yearly mean values, exactly as we found in our
+36
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 22. Plot of F10.7 versus S N, using the original SN series (Version 1). The data are smoothed by a 13-
+month running mean. The line connects successive months, and thus illustrate the chronological evolution.
+The curve is colored in blue or red for dates before and after 1981.
+global regressions (sub-Sections 4.1 and 4.2). We can now explain it by the fact that a significant
+part of the deviations of individual monthly or yearly mean values relative to the mean regression
+curve is due to this systematic inhomogeneity in the series, rather than to random errors, and will
+thus not be reduced by temporal averaging.
+8.2. A more precise timing of the jump
+Using the 13-month smoothed monthly values, we can roughly locate the jump in mid-1980, as the
+smoothed monthly means migrate from the low branch to the high branch between January 1980
+and March 1981. However, with such smoothed values, we cannot pinpoint the transition time with
+37
+
+Databefore1980.79
+FCpold4_m
+250
+Dataafter1980.79
+225
+200
+F10.7
+175
+150
+125
+100
+75
+0
+50
+100
+150
+200
+250
+300
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Fig. 23. Plot of F10.7 versus S N, equivalent to figure 22 but using the new re-calibrated version of the SN
+series (Version 2). The data are smoothed by a 13-month running mean. The data curve is in blue or red for
+dates before and after 1981. The black lines correspond to linear fits to the entire series (solid line), the period
+before 1980 (dotted) and the period after 1980 (dashed).
+a better temporal resolution. Still, the fact that the branch-to-branch transition occurs over a duration
+similar to the smoothing duration already indicates that the actual transition must take place over a
+duration much shorter than 13 months, thus very sharply.
+In order to better pinpoint the time of the transition, we checked the (un-smoothed) monthly
+means, and we compared them with the global linear fit to the whole data set (Table 4). In Figure
+24, we plot the resulting monthly ”observed/fit” ratios over the 4-year interval around the suspected
+transition, which is also centered on the maximum of cycle 21. Over this interval, all data values
+are thus in the same high range, and the choice of mean fit has only a small influence on the ratios.
+38
+
+Databefore1980.79
+FCP1_pold1_m
+250
+Dataafter1980.79
+FCP2_pold1_m
+FCRpold1_m
+225
+200
+F10.7
+175
+150
+125
+100
+75
+0
+50
+100
+150
+200
+250
+300
+Sunspot numberClette: F10.7cm proxy relation and temporal homogeneity
+Table 9. Coefficients of order-1 (linear) and order-4 polynomials fitted by the ordinary least-square regres-
+sion on the monthly mean SN for the periods 1947-1980 and 1981-2015, with standard errors σn for each
+coefficient.
+Coefficients
+Period 1947-1980
+Period 1981-2015
+Period 1947-1980
+Period 1981-2015
+Order 1
+Order 1
+Order 4
+Order 4
+C0
+58.09
+58.78
+66.64
+67.85
+σ0
+0.9391
+1.151
+1.476
+1.275
+C1
+0.6345
+0.7020
+0.3658
+0.3845
+σ1
+6.612 10−3
+8.867 10−3
+6.765 10−2
+8.115 10−2
+C2
+2.587 10−3
+2.881 10−3
+σ2
+8.642 10−4
+1.378 10−3
+C3
+−9.906 10−6
+−7.429 10−6
+σ3
+4.029 10−6
+8.344 10−6
+C4
+1.329 10−8
+2.694 10−10
+σ4
+6.152 10−9
+1.645 10−8
+As can be expected, the monthly ratios show rather large month-to-month random fluctuations, but
+19 out of 24 ratios are below 1 before the end of 1980, while 21 out of 24 are above 1 after 1980,
+indicating a clear systematic change. The points are actually grouped respectively around the global
+linear fits calculated for the complete half-series 1947 – 1980 and 1980 – 2015 (dashed lines).
+Moreover, this transition happens very sharply. With the exception of November 1980, there is
+a sudden jump between December 1980 and January 1981, followed by 8 consecutive months in
+1981 all above the mean linear fit. This thus strongly suggests that the jump occurred between
+December 1980 and January 1981, i.e. at the transition between two calendar years. Although a
+slightly earlier transition is not entirely excluded, it cannot be before mid-1980.
+8.3. Validation against multiple independent stations
+Now, it turns out that 1980–1981 also marks a transition for the sunspot number series. This is when
+the production of the sunspot number moved from Zurich to Brussels, with a significant change in
+the production method (Clette et al., 2014, 2016). The former manual processing was computerized
+and the Specola Solare Observatory in Locarno (Z¨urich’s auxiliary station) took over as pilot station
+in replacement for the Z¨urich Observatory. Therefore, as the sunspot number is a synthetic index
+based on a global statistical processing of multiple data sources, we should be careful that the 1980
+jump that we just found above is not entirely due to a sharp change of scale between the Z¨urich and
+Brussels sunspot numbers.
+In order to exclude any processing flaw in the sunspot number, we made direct comparisons with
+a large set of individual stations that provided sunspot counts over a long period extending both
+before and after 1980. In the SILSO database, we found 28 stations fulfilling the requirements. In
+fact, most of those stations were part of the multi-station reference used for the re-calibration of the
+SN version 2 (Clette et al., 2016, Table 1). For each station, we derived the ratio between the raw
+Wolf numbers and the corresponding F10.7-based SN proxy value, as a function of time (single proxy
+relation for all times). As only one station is used in each comparison, this ratio is affected by a larger
+error. Therefore, we averaged the ratios for the whole time intervals before and after 1980, and then
+39
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 24. Monthly ratios between the observed monthly mean F10.7 flux and the proxy value derived from the
+linear fit over the entire interval 1947 – 2015 (Table 4). The dashed lines correspond to the global fits on the
+entire half-series before and after 1980.
+looked at the ratio between the two intervals. This is listed in Table 10. As illustration, Figures
+25 and 26 show the comparisons for two sample stations: the Kislovodsk station (Observatory of
+Pulkovo), a professional observatory, and Thomas Cragg, a dedicated individual observer, who was
+employee of the Mount Wilson Observatory (Howe and Clette, 2015).
+We find that, out of the 28 stations, 24 indicate a higher ratio after 1980, while only one station
+indicates a constant ratio, and only three stations give a decreasing trend. As the curves have larger
+fluctuations, the timing of the transition is less accurate, but all series suggest a short transition
+within at most 3 years around 1980, preceded and followed by periods without systematic trends.
+The simple arithmetic mean of the listed SN ratios is 1.166 (range: 0.812 – 1.731), which is larger
+than the jump amplitude calculated above. However, given the larger uncertainties in individual
+series and the different time intervals covered by each station, this mean value is only indicative. By
+keeping only stations with well-traced reliability, and which were active over most of the F10.7 time
+interval (marked with a * in Table 10), we find a mean value of 1.139 (range: 1.047 – 1.235), which
+is more reliable and comes reasonably close to the value found using the SN series as reference.
+As this comparison involves only raw data from multiple independent observers, this verification
+thus allows us to confirm that the 1980 jump definitely occurs in the F10.7 series and is not due to
+any unsuspected and uncorrected flaw in the SN series.
+40
+
+1.20
+1.15
+d / Mean Fit
+1.10
+1.05
+Observed
+1.00
+0.95
+Ratio
+0.90
+0.85
+0.80
+1979.0
+1979.5
+1980.0
+1980.5
+1981.0
+1981.5
+1982.0
+1982.5
+1983.0
+Time (years)Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 25. Comparison of the F10.7-based SN proxy series with the raw Wolf numbers from the Kislovodsk
+solar Observatory (Russia; SILSO station ID: KS2), cover the period before and after 1980. The upper panel
+shows the two series, smoothed by a 12-month Gaussian kernel to reduce short-term random noise. The
+middle and lower panels show the ratios and differences. No data are available for that station in the blue-
+shaded time interval. An upward jump can been seen around 1980.
+Of course, another independent test could come from a comparison with equivalent radio flux
+measurements, made by another radiotelescope at the same wavelength or at a neighbouring one.
+Nicolet and Bossy (1985) made such a comparison with data from the Toyokawa station (Nagoya,
+41
+
+300
+Y= F10.7
+X= KS2 Sunspot Number
+200
+SN
+100
+0
+2.0
+Ratio smoothed
+Ratio
+1.0
+04B
+Difference smoothed
+Difference Y-X
+20
+0
+-20
+-40
+1950
+1960
+1970
+1980
+1990
+2000
+2010
+Time (years)Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 26. Comparison of the F10.7-based SN proxy series with the raw Wolf numbers from Thomas
+Cragg (SILSO station ID: CRA), an amateur sunspot observer, who was employee of the Mount Wilson
+Observatory. Like in Figure 25, an upward jump in the ratio and differences (middle and lower panels) can
+been seen around 1980.
+Japan), but by then, the data extended only until 1982, thus only two years after the jump, which
+makes the detection of the jump difficult. More recently, another combined study by Yaya et al.
+(2017) also uses F10.7 next to the Toyokawa series. Here also, there is no mention of a jump in
+1980, but this study was focused on short-term predictions and did not look for such long-term
+42
+
+300
+Y= F10.7
+X= CRA Sunspot Number
+200
+100
+0
+2.0
+Ratio smoothed
+Ratio
+1.0
+04B
+Difference smoothed
+Difference Y-X
+20
+0
+-20
+-40
+1950
+1960
+1970
+1980
+1990
+2000
+2010
+Time (years)Clette: F10.7cm proxy relation and temporal homogeneity
+Table 10. k ratios between the raw Wolf number from 28 stations in the SN database (World Data Center
+SILSO) and a fixed F10.7-based proxy of the sunspot number, for observations made before 1980 (column
+3) and after 1980 (column 5). For each station, we also list the corresponding time interval over which the
+data are available (columns 2 and 4). The absolute value of those ratios is different for each station, due to
+the different observing setup (telescope, observing site), but it is not relevant here, as we only trace relative
+differences between different times. The 6th column gives the k2/k1 ratios between the scale ratios for the two
+periods. Stations marked with a * in the first column form a subset of stations with the best reliability and/or
+longest duration before and after the 1980 transition.
+Station
+Period 1
+k1
+Period 2
+k2
+Ratio
+Error
+Comment
+AN *
+1976-1980
+1.435
+1981-1988
+1.608
+1.121
+0.030
+short series
+AT
+1968-1980
+0.878
+1981-1982
+1.096
+1.248
+0.050
+short series
+BN-S
+1965-1980
+0.995
+1981-2014
+0.870
+0.874
+0.016
+decreasing
+BRm
+1974-1980
+0.819
+1981-1998
+1.050
+1.282
+0.015
+short series
+CA
+1949-1980
+1.174
+1981-2015
+1.097
+0.934
+0.013
+drifts before 1980
+CRA *
+1947-1980
+1.267
+1981-2010
+1.521
+1.200
+0.011
+long series
+EB
+1949-1981
+0.746
+1981-2015
+1.291
+1.731
+0.012
+drifts before 1980
+FR-S
+1976-1980
+0.790
+1981-1988
+0.962
+1.218
+0.019
+short series
+FU *
+1968-1980
+1.047
+1981-2015
+1.123
+1.072
+0.012
+long and stable
+GU-S
+1974-1980
+1.617
+1981-1991
+1.936
+1.197
+0.053
+short series
+HD-S
+1967-1980
+1.151
+1981-2013
+1.315
+1.142
+0.024
+unstable
+HU
+1969-1980
+1.323
+1980-2015
+1.164
+0.880
+0.024
+decreasing, unstable
+KH *
+1966-1980
+1.098
+1980-2015
+1.343
+1.223
+0.015
+long series
+KOm
+1947-1980
+1.016
+1981-1996
+1.198
+1.178
+0.016
+slight drift after 1983
+KS2 *
+1954-1980
+1.069
+1981-2015
+1.219
+1.140
+0.009
+long series
+KZm *
+1947-1980
+1.026
+1981-2015
+1.074
+1.047
+0.011
+long series
+LFm *
+1947-1980
+1.136
+1981-1988
+1.403
+1.235
+0.023
+LK *
+1967-1980
+1.321
+1981-1987
+1.457
+1.103
+0.022
+short series
+LO
+1958-1980
+0.822
+1981-2015
+0.945
+1.150
+0.007
+drifts after 1983
+MA *
+1971-1980
+1.039
+1981-1988
+1.183
+1.139
+0.015
+short series
+MD
+1978-1980
+0.988
+1981-1986
+1.532
+1.551
+0.068
+short series
+PO
+1950-1980
+1.122
+1981-2000
+1.106
+0.986
+0.017
+constant ratio
+QU
+1957-1980
+1.411
+1981-2015
+1.820
+1.290
+0.019
+unstable
+SA
+1957-1980
+1.291
+1981-2000
+1.692
+1.311
+0.026
+long series
+SC-S
+1960-1980
+1.396
+1981-2007
+1.133
+0.812
+0.021
+decreasing
+SK
+1950-1980
+1.100
+1981-2012
+1.281
+1.165
+0.015
+unstable
+SM
+1967-1980
+0.794
+1981-2013
+1.031
+1.298
+0.013
+UC *
+1949-1980
+1.113
+1981-2015
+1.236
+1.111
+0.012
+long series
+inhomogeneities. So, a more focused study of long-term radio data series is still needed but goes
+beyond the scope of this article.
+8.4. Possible cause of the 1980 jump: historical elements
+What could be the cause of this transition? Tapping and Morton (2013) and Tapping (2013) mention
+key dates in the history of the F10.7cm radio flux production. Regarding the receiver and facility, the
+43
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+greatest care was devoted to the calibration of the instrument, which was checked against the same
+horn-antenna references (Tapping and Charrois, 1994). The radio-telescope was re-located only
+two times, and this happened in 1962 (Ottawa to Algonquin Park) and 1990 (Algonquin Park to
+Penticton). None of those transfers left a detectable trace in the series, and in particular, those dates
+do not match the 1980 jump indicated by the data.
+However, Tapping (2013) mentions that in the late 1970’s, the original manual processing of the
+daily recordings was replaced by an automated procedure. This happened around the time when
+A.E. Covington, who had initiated the F10.7 standard flux measurements and had directed its pro-
+duction continuously since 1947, went on retirement in 1979. This was thus the very first time a
+new team took over the task. This new team tried to automate the fully manual method that was
+applied until then to empirically eliminate the emission peaks caused by solar flares, but this com-
+puterized method was finally abandoned in 1985 because of insufficient reliability (Tapping 2019,
+private communication).
+The 1979-1985 timing of this change of team and of post-processing method matches quite
+closely the moment when we find this scale jump. Based on similar disruptive events found in
+the history of the Z¨urich sunspot number, we suspect that this transition is the cause of the 1980
+F10.7 scale change. Indeed, only two highly dedicated scientists cared for the production of the F10.7
+index, each one for several decades: A.E. Covington 1947-1979 and K. Tapping 1985-present. Only
+once, in 1979, there was a transition when the experience and practices developed by Covington
+during the first 32 years had to be transmitted to the team that succeeded him. Part of the subtleties
+and habits may get lost in such knowledge transfers, especially as the processing procedure was
+largely manual until then, and proved to be difficult to convert into a computer algorithm.
+Moreover, the extraction of the F10.7cm background flux (the so-called “S” component) requires
+the elimination of flare-associated bursts, of radio-frequency interference, and of the sky back-
+ground emission. Therefore, the final index depends on those post-processing steps, and not only
+on the proper calibration of the receiver and antenna. So, even if the raw fluxes were always accu-
+rately calibrated, this post-processing step may influence the final filtered index. Therefore, we can
+speculate that the changes in the methods that seem to have occurred just after 1979 could perfectly
+have caused this jump in the resulting index, without involving any technical flaw in the instrument
+itself and its calibration.
+Moreover, our results indicate that there was no slow progressive drift of the F10.7 flux relative
+to the sunspot number, but that a scale change occurred abruptly between two constant periods.
+This is in contradiction with the analysis by Tapping and Morgan (2017), who used a temporal
+curve fitting that did not allow to detect such a sharp transition. This abrupt transition occurs at the
+maximum of cycle 21 and is unique over the last 6 solar cycles. Such a step-like jump does not
+match any known solar event in 1980 that would be unique over the last 70 years. Likewise, no
+mechanism generating the solar activity cycle can account for such an abrupt discontinuity, which
+require intrinsic timescales as short as one month.
+Therefore, we consider that it is very unlikely that the Sun itself induced this sudden change
+in the relation between the F10.7cm flux and the sunspot number, while a slow trend, as incorrectly
+diagnosed by Tapping and Morgan (2017), allowed such an hypothesis. By contrast, the production
+process and its historical evolution, retraced above, contain various elements that can induce such a
+sudden jump. So, this past history as recorded in archived documents deserve very careful attention,
+in order to validate or invalidate this processing issue.
+44
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+Fig. 27. Temporal variation of the ratio between the monthly mean F10.7 flux and a constant proxy relation
+(4th-order polynomial) (upper panel), after subtracting a 67 sfu quiet-Sun base flux. The same ratio for the
+yearly mean F10.7 flux is shown in the lower panel. The points near cycle maxima and minima are colored
+in blue and red respectively. The monthly means show a slight solar-cycle modulation, and both curves show
+the 1980 jump between two stable periods.
+8.5. Solar-cycle and other modulations
+As noted for in Section 3, the mean daily F10.7 flux for a given S N shows a significant difference
+between the minima and maxima of the solar cycle. This cycle modulation is largest for raw daily
+data (Figure 7) and decreases for longer time scales. In order to check if this modulation is affecting
+the monthly and yearly means used to build the long-term proxy, we computed the ratio between
+monthly or yearly mean values and a single constant fit: the 4th-order polynomial fitted to the entire
+series, given in Tables 3 and 5. Any deviation from a constant relation will appear as a deviation
+from unity. The result is shown in figure 27, where we marked the periods around maxima and
+minima of the solar cycles by blue and red dots.
+The plot for the monthly means (Figure 27, upper panel) is dominated by random month-to-
+month variations. No annual modulation is present, indicating that the adjusted F10.7 was accurately
+45
+
+1.8
+1.6
+1.4
+tio
+1.2
+Rat
+1.0
+0.8
+0.6
+1950
+1960
+1970
+1980
+1990
+2000
+2010
+2020
+Time (years)1.4
+1.3
+1.2
+.0 1.1
+Ratic
+1.0
+0.9
+0.8
+0.7
+1950
+1960
+1970
+1980
+1990
+2000
+2010
+2020
+Time (years)Clette: F10.7cm proxy relation and temporal homogeneity
+converted to the 1AU distance. The 1980 jump between two stable intervals is also clearly visible.
+Now considering the solar cycle, a slight modulation can be found: although the ranges of monthly
+values around maxima and minima largely overlap, the range near maxima is slightly higher than
+around minima. The difference is subtle and remains smaller than the other deviations mentioned
+above.
+Finally, the ratios for yearly means do not show any cycle modulation (Figure 27, lower panel).
+The only clear systematic variation is the 1980 jump. Virtually all points are below 1 before 1980
+and above 1 after 1980, confirming again clearly the jump and the absence of any progressive trend.
+Overall, we can thus confirm that the solar-cycle modulation is playing a significant role only for
+timescales shorter than a month. We can interpret this effect by invoking the same mechanisms as
+the ones explaining the change of average background during spotless periods (cf. Section 6). As
+activity increases, the plage component can contribute to a flux excess when sunspot activity drops
+momentarily, because the facular and plages associated to all active regions persist for a much
+longer time than the corresponding active regions. The higher activity prevailing around those dips
+in the sunspot number thus prevents the F10.7 to decrease as sharply. Therefore, the net effect must
+always be an excess, which matches the upper tail of the daily distribution in Figure 7. Near the
+minima of cycles, given the small number of active regions, this persisting background is largely
+absent, thus giving a smaller plage excess, leading to the observed solar-cycle variation in Figure 7.
+As this temporal smearing effect corresponds to the lifetime of plages, which ranges from weeks to
+a few months, it should vanish for long timescales, like we find in our analysis (Figure 27).
+9. Conclusions
+Summarizing, our analysis brings the following conclusions regarding the global F10.7/S N proxy
+relation:
+– No previously published F10.7 proxy relation is fully satisfactory. Existing proxies deviate from
+the data points either in the low or high range, though there is a fair agreement in the intermediate
+linear range. Those proxies are also lacking error bars, limiting their applicability.
+– The F10.7/S N relation is fully linear within uncertainties from the lowest to the highest observed
+values, when taking raw daily values without any temporal averaging. The F10.7 flux only deviates
+from the linear relation for S N below 11 (single spot) and even S N = 6. F10.7 reaches a lower base
+background only when the Sun is spotless.
+– When working with monthly and yearly means, the F10.7/S N relation becomes non-linear in the
+low range, for S N below 30 to 50. This non-linearity can be fully explained by the effect of
+temporal averaging on daily data consisting of a fully linear relation, plus a lower background
+(fully inactive Sun). A F10.7 proxy relation is thus only valid for a specific temporal averaging of
+the base daily data.
+– A 4th degree polynomial gives the best fit to the monthly mean data, in particular the non-linear
+section below S N = 50 down to 0. A linear function is sufficient for all S N values above about
+30.
+46
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+– We derived standard errors σp on the polynomial values. As a direct mathematical error-
+propagation calculation does not exist taking into account the inter-dependencies between the
+least-square polynomial coefficient, those errors were derived empirically from the data by de-
+termining the conditional errors for each separate term of different degree in the polynomial. For
+practical applications, we also assembled a simple mathematical formula that closely approxi-
+mates those data-based conditional errors.
+In addition, we derived new properties of the F10.7 quiet-Sun background flux:
+– This background flux for a spotless Sun depends on the duration of the spotless interval. Its mean
+value is 68 sfu for long inactive periods, but rises to 74 sfu for a spotless duration of one day. With
+only a few isolated exceptions, 67 sfu is the lowest flux value, independently from the spotless
+duration, but the mean quiet-Sun background is always higher.
+– Given the actual duration of spotless periods, a temporal averaging over one month is close to
+optimal to reflect the lowest range of F10.7 background flux, with 68 or 69 sfu as the lowest mean
+background over one month.
+– A F10.7 excess flux is present in raw daily data and induces a 12% solar-cycle modulation. The
+latter vanishes at long timescales, and is already barely detectable in monthly averages (≈ 3%).
+This implies that a single standard F10.7 – S N proxy relation, independent of the phase of the solar
+cycle, can only be derived from temporal scales longer than about one month, and will only be
+fully accurate for those timescales.
+Finally, by checking for any temporal variability of the F10.7/S N relation over the entire duration
+of the data series, we found a significant inhomogeneity:
+– The F10.7 series is affected by an upward jump in 1980, separating two stable periods without
+other jumps or trends. Relative to the SN series, the F10.7 index is 10.5% higher after 1980.
+Monthly values suggest that the jump occurred at the transition between two calendar years, i.e.
+between December 1980 and January 1981.
+– In order to exclude a possible flaw on the side of the sunspot number series, which also went
+through a methodological transition in 1980–1981, we compared the F10.7 with raw Wolf numbers
+from a large sets of independent stations. With only a few exceptions, most of them indicate that
+F10.7 becomes higher after 1980. This thus firmly establishes that the scale jump belongs to the
+F10.7 time series.
+– The jump is abrupt and makes any interpretation in terms of a true solar effect difficult. On the
+other hand, this abrupt transition happens close to important changes in the observing team (re-
+tirement of A.E. Covington), and when changes were introduced in the post-processing method
+(computerization of an originally manual processing). Just like the diagnosed jump, this opera-
+tional transition is unique in the history of the F10.7 production.
+This study thus indicates that we can still learn a lot about this fundamental long-term solar-
+activity index. The consistent relation between the F10.7 quiet-Sun background and the duration of
+the quiet period, the role of temporal averaging on the non-linearity of the proxy relation as well
+47
+
+Clette: F10.7cm proxy relation and temporal homogeneity
+as the cycle-dependent excess flux found only in daily data, all invite us to pay more attention to
+temporal scales. Our results confirm the mixed contribution of two components in F10.7, one from
+sunspots and another one from weaker magnetic fields primarily in plages. The latter introduces a
+time-diluted variation relative to the initial magnetic flux emergence in active regions, recorded by
+the SN. As this source-mixing in F10.7 plays a role only at short time-scales, from days to weeks,
+it goes beyond the scope of this long-term proxy study. Still, this aspect calls for more attention in
+futures analyses of F10.7, and it should also be kept in mind in all uses of F10.7 for space-weather or
+space-climate applications.
+Given the inhomogeneity in the F10.7 time series that we diagnose here in detail, it is clear that
+our new global polynomial proxy does not reproduce optimally the actual relation before and after
+1980. Until a correction is adopted for the F10.7 data, it must be considered as the best overall
+sunspot-based proxy. The slope and curvature are largely valid for the whole series, but a different
+scale factor must be applied for each half of the series: proxy values will be about 5% too high
+before 1980 and 5% too low for the more recent years relative to the current version of the F10.7
+series. The polynomials derived for each half of the series in Table 9 can be used in applications
+focusing only on time periods before or after 1980.
+The combined historical evidence indicates that a thorough analysis of the production process of
+the F10.7, in particular in the late 1970’s to early 1980’s, is needed to clarify any possible change,
+and potentially to find a self-consistent correction to restore the homogeneity over the entire series
+since its beginning. If the archived F10.7 data prove to be insufficient, our best second option would
+be to use the long overlap with the sunspot number series. The latter may provide an even better
+reference in the future, as a new re-calibration is in preparation, which could improve in particular
+the S N values from the Z¨urich period before 1981.
+Acknowledgements. This work and the team of the World Data Center SILSO, which produces and dis-
+tributes the international sunspot number used in this study, are supported by Belgian Solar-Terrestrial Center
+of Excellence (STCE) funded by the Belgian Science Policy Office (BelSPo). This work also partly benefited
+from the joint work of the International Team 417 “Recalibration of the Sunspot Number Series”, funded by
+the International Space Science Institute (ISSI, Bern, Switzerland) and chaired by M. Owens and F. Clette.
+We also wish to acknowledge the particularly useful, informative and friendly discussions with Ken Tapping,
+who is the dedicated guardian and the living memory of the F10.7cm radio flux.
+References
+Charbonneau, P., 2010. Dynamo Models of the Solar Cycle, Living Rev. Sol. Phys., 7/3, DOI 10.12942/lrsp-
+2010-3. 1
+Clette, F., Svalgaard, L., Vaquero, J.M. and Cliver, E.W., 2014. Revisiting the Sunspot Number: a 400-year
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf,len=1833
+page_content='submitted to Journal of Space Weather and Space Climate  © The author(s) under the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0 International License (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0)  Is the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm – Sunspot Number relation linear and  stable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Clette  World Data Center SILSO, Royal Observatory of Belgium, 1180 Brussels, Belgium  e-mail: frederic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='clette@oma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='be  ABSTRACT  The F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio flux and the Sunspot Number are the most widely used long-term indices of  solar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They are strongly correlated, which led to the publication of many proxy relations  allowing to convert one index onto the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, those existing proxies show significant  disagreements, in particular at low solar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Moreover, a temporal drift was recently found in  the relative scale of those two solar indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Our aim is to bring a global clarification of those many issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We compute new polynomial  regressions up to degree 4, in order to obtain a more accurate proxy over the whole range of solar  activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We also study the role of temporal averaging on the regression, and we investigate the  issue of the all-quiet F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Finally, we check for any change in the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 – sunspot  number relation over the entire period 1947–2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We find that, with a 4th-degree polynomial, we obtain a more accurate proxy relation than all  previous published ones, and we derive a formula giving standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The relation is different  for daily, monthly and yearly mean values, and it proves to be fully linear for raw non-averaged  daily data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' By a simple two-component model for daily values, we show how temporal averaging  leads to non-linear proxy relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We also show that the quiet–Sun F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background is not abso-  lute and actually depends on the duration of the spotless periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Finally, we find that the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm  time series is inhomogeneous, with an abrupt 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5% upward jump occurring between 1980 and  1981, and splitting the series in two stable intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Our new proxy relations bring a strong improvement and show the importance of temporal  scale for choosing the appropriate proxy and the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 quiet-Sun background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' From historical  evidence, we conclude that the 1981 jump is most likely due to a unique change in the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  scientific team and the data processing, and that the newly re-calibrated sunspot number (version  2) will probably provide the only possible reference to correct this inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Sun – Solar activity – Solar indices – Solar irradiance (radio) – Solar cycle  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Introduction  The sunspot number (hereafter SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' symbol: S N ) (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Clette and Lef`evre, 2016)  and the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio flux (symbol: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7) (Tapping and Morton, 2013) are arguably the most widely  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='02588v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='SR]  6 Jan 2023  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  used solar indices to characterize the long term evolution of the solar activity cycle and of the  underlying dynamo mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In order to be usable over duration of decades to centuries, those  indices must guarantee a long term stability so that levels of activity at two widely spaced epochs  can be compared on exactly the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm and the SN are both absolute indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed,  they do not lean on external references, which anyway are absent over a large part of their temporal  range (410 years for the SN, and 73 years for F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Instead, the index values are only based on  the knowledge of the different steps in their determination, from the raw measurements to the data  processing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For the SN, a full revision process was undertaken in 2011 and led to a first  re-calibration of this multi-century series, with correction reaching up to 20% (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Clette and Lef`evre, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This re-calibration included a full re-construction of the SN from raw  original data for the period 1981 to the present, while correction factors were applied to the original  series built by the Z¨urich Observatory before 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In this article, we will take a closer look at the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio flux, which comes second in duration  after the SN, among the global long-duration solar activity indices based on a single uninterrupted  observing and processing technique, and on a single long-duration standard reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Since the  measurements of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio flux were started in Ottawa in 1947 (Covington, 1948, 1952), this  new solar index proved to be highly correlated with the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This can be explained by the fact that the  background flux, outside flaring events, is associated with the thermal free-free and gyroresonance  emission of electrons trapped in closed loops anchored in the active regions, and thus primarily  in their sunspots (Tapping, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and Detracey, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and Zwaan, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Those  two indices ran in parallel over the last 73 years and they are produced by completely different and  independent processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, as they are supposed to retrace exactly the same evolution of  the last 7 solar activity cycles, studying their mutual relation can give a prime diagnostic of their  long-term stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We will thus focus on the proxy relation between those two indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  It turns out that, given the excellent long-term correlation between the SN and the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio  flux, various proxy relations were derived over past years by different authors, or they were estab-  lished for operational purposes by solar data services like NOAA-SWPC (Space Weather Prediction  Center) in the USA or the IPS (Ionospheric Prediction Service, part of the Bureau of Meteorology)  in Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Those relations are motivated by two kinds of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' One of them is the re-  construction of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 time series before the actual measurements started in 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed, as the  sunspot number extends back over four centuries, it allows to extrapolate this radio index over a  much longer period (Svalgaard, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Another application is producing mid-term predictions of  the future evolution of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As such predictions often need to be calibrated and validated  over many past solar cycles, they are typically based on the SN, and consequently, they produce  their predictions in terms of this SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Moreover, the number of sunspots and sunspot groups give a  direct measure of magnetic flux emergence at the solar surface, and is thus directly related to the  dynamo mechanism at work inside the Sun (Charbonneau, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Hathaway, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Stenflo, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  On the other hand, F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 is a chromospheric/coronal index that combines two kinds of emission:  gyroresonance and free-free, the latter being associated with the magnetic decay of active regions  under the action of the random convection, leading to the chromospheric plage component (Tapping,  1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and Zwaan, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This is why, on timescales shorter than the average lifetime of  individual active regions and their associated plages, daily values of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 are less correlated with  the sunspot number, as first found by Vitinsky and Petrova (1980), Vitinsky (1982), Kopecky  (1982) and Kuklin (1986), and using more modern methods by Dudok de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' (2009) and  2  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Dudok de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' On the other hand, the daily flux offers a better proxy for ultraviolet  (UV) and X-ray fluxes produced in the chromosphere, the transition region and the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  For this reason, F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 is used by preference to the sunspot number for short-term forecasts of solar  irradiance in the UV to X-ray domain and of its influence on the Earth environment (ionosphere,  stratospheric temperatures, chemistry of the upper atmosphere), and for the resulting applications  (radio propagation, atmospheric drag on low-Earth orbiting satellites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This close relation with the  solar UV irradiance also allows to produce backward reconstructions of past UV fluxes for epochs  well before the advent of direct space-based measurements of those fluxes (Svalgaard, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  However, beyond their strong Sun-related similarities on long timescales, the two indices differ  by two base characteristics that play a role primarily at the lowest levels of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Firstly, F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  does not fall to 0 when the Sun is spotless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' A base background flux exists even when the Sun is fully  quiet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This background emission, which corresponds to the spatially diffuse component of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7, is  probably associated with the small magnetic loops rooted in the quiet-Sun chromospheric network  (Tapping and Zwaan, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This lower limit is still a matter of debate, but it is generally estimated  in the range between 64 and 67 solar flux units (sfu) (Tapping and Detracey, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and  Charrois, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Secondly, by its definition (Wolf, 1856;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2014), the SN is quantized at the lowest  values, as each new group (with at least one single spot), adds 11 to the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, for the first  spot, the SN jumps from 0 directly to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This effect quickly decreases for values larger than 22,  as contributions from several groups with multiple sunspots are then combined in the total number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  However, this low jump stretches the SN scale near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As this SN feature is absent in F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7, we can  expect that it will break the proportionality between the two indices in the lowest range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In this article, we first review all F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 – S N proxy relations published in the literature or used  by operational space weather services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Given the mismatches between those existing proxies, we  build more carefully a new least-square polynomial regression, while exploring the effect of tem-  poral averaging of the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We also investigate the issue of the quiet-Sun F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We then check the temporal stability of the relation between F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 and the SN over the entire  duration of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We finally conclude on the new picture emerging from our analysis and on  important aspects to be taken into account for future updates of this relation between those two  most fundamental measures of the long-term solar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In this analysis, we use the sunspot number data provided by the World Data Center SILSO  (Sunspot Index and Long-term Solar Observations) at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='sidc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='be/silso/datafiles  and the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm data series from the Dominion Radio Astrophysical Observatory, available via  the Space Weather Canada service at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='spaceweather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='ca/solarflux/  sx-5-en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='php,  and  also  accessible  through  NOAA  (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='ngdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='gov/  stp/space-weather/solar-data/solar-features/solar-radio/noontime-flux/  penticton/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We use the adjusted F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm flux, which reduces the flux to a fixed distance of 1  Astronomical Unit (AU), and thus eliminates any annual modulation due to the orbital eccentricity  of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  3  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' List of the proxies included in our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For each proxy, we indicate the temporal granularity  of the series used to fit the proxy (day, month, year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' simple means or running means), and the SN version on  which the proxy was based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The label identifies the curves in the associated figures, and the corresponding  formulae are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Source  Temporal base  SN version  Plot label  Kuklin (1984)  Unknown  1  KU1984 V1  Holland and Vaughn (1984)  13-month smoothed  1  HV1984 V1  Xanthakis and Poulakos (1984)  1 day  1  XP1985 V1  Hathaway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' (2002)  24-month Gaussian smoothed  1  HA2002 V1  Zhao and Han (2008) formula 1  1-year mean  1  ZH2008 F1  Zhao and Han (2008) formula 3  1-year mean  1  ZH2008 F3  Svalgaard (2009)  1-month mean  1  SV2009 V1  IPS Australia (Thompson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 2010)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-month mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='IPS2011 V1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Tapping and Vald´es (2011)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-year mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='T2011 V1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Johnson (2011) formula 1 monthly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-month mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='J2011 F1ma  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Johnson (2011) formula 1 yearly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-year mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='J2011 F1ya  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Johnson (2011) formula 2 monthly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-month mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='J2011 F2ma  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Johnson (2011) formula 2 yearly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-year mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='J2011 F2y  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='NOAA-SWPC (2016)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Unknown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='NOAA V2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Tapping and Morgan (2017) S N version 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='10-month smoothed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='T2017 V1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Tapping and Morgan (2017) S N version 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='10-month smoothed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='T2017 V2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Tiwari and Kumar (2018) formula 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-month mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='TK2018 F1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='Tiwari and Kumar (2018) formula 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1-month mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='TK2018 F2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Past F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm – SN proxy relations  Quite a number of proxy relations were proposed in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Here, we first compile all relations  accessible in the literature or documented with associated data products at data centers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They are  listed in Table 1, and the corresponding formulae are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In Figure 1, we plot all those  proxies together, superimposed on the monthly mean values of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 versus S N version 2, the most  recent re-calibration of this series (Clette and Lef`evre, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  1 The Svalgaard (2009) proxy was found in the Web source: https://wattsupwiththat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='com/  2009/05/18/why-the-swpc-10-7-radio-flux-graph-is-wrong/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The  NOAA-SWPC  (2016)  proxy is used for solar cycle predictions provided at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='swpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='gov/products/  predicted-sunspot-number-and-radio-flux  and  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='swpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='gov/products/  solar-cycle-progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This proxy formula was formerly mentioned in an on-line document  (ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='swpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='gov/pub/weekly/Predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='txt), which is not accessible anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Some  related information can presently be found in https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='swpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='gov/sites/default/files/  images/u2/Usr_guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  4  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' List of the proxies (labels from Table 1) and the corresponding formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Plot label  Formula  KU1984 V1  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='74 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2970 S N + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='005146 S 2  N for S N < 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5,  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='63 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3037 S N + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='001817 S 2  N for S N > 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  HV1984 V1  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='97 S N + 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6 (e−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='035 S N − 1)  XP1985 V1  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='15 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='65 S N  HA2002 V1  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='52 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='926 S N  ZH2008 F1  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='932 S N  ZH2008 F3  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='633 S N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='76 10−3S 2  N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='28 10−5S 3  N  SV2009 V1  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='29 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='316 S N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='084 10−2S 2  N + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='813 10−5S 3  N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='314 10−7S 4  N  IPS2011 V1  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='572 S N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='31 10−3S 2  N–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='13 10−6S 3  N  T2011 V1  66 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='446 S N(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' − e−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='027 S N)  J2011 F1ma  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='72 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='900 S N + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0002 S 2  N  J2011 F1ya  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='87 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='835 S N + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0005 S 2  N  J2011 F2ma  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='72 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='686 S N)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0642  J2011 F2y  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='98 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='582 S N)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0970  NOAA V2  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='00 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='4903 S N for S N ≤ 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='06 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7092 S N for S N > 50  T2017 V1  67 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='44 S N [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' − e−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='031 S N]  T2017 V2  67 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='31 S N [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' − e−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='019 S N]  TK2018 F1  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='51 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6422 S N  TH2018 F2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6605 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='500687 S N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='21647 10−3S 2  N + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='71853 10−6S 3  N  So far, only a few recent proxy relations were calculated for this new version of the SN series,  S Nv2 released in July 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, for converting older proxies based on S Nv1 to the S Nv2 scale,  we used the following re-scaling relation:  S Nv2 = S Nv1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='177  (1)  The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6 factor is associated with a change of reference observer, making the former 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6 conven-  tional Z¨urich factor obsolete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The second factor corresponds to a correction for an artificial inflation  of the original SN values due to the use of a weighting according to the sunspot size artificially  introduced in Z¨urich (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This was affecting all the S Nv1 values after 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As the  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 series starts in 1947, this relation is thus valid for the entire time interval considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='177 factor is in fact an asymptotic values reached for medium to high levels of solar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It is  thus variable at low solar activity, dropping to almost 1 near cycle minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, Equation 1  is not fully accurate for low-activity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Still, in our analysis presented below, we did not find  any significant deviation between version 1 and version 2 proxies due to this effect, at the level of  precision associated with the data themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This can be seen in the closeup view (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Overall, we observe that some proxies are very crude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They are simple linear fits, ignoring the  visible deviation from linearity at the lower end of the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Strong deviations also appear at  the high values, in particular for non-linear fits (polynomial or exponential models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This can be  explained by the limited number of such high values in the past solar activity record, which thus  leads to large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We also note that in the low range below S N = 20, virtually all proxies  fall below the observed values, and thus lead to systematic underestimates of the average F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux  at low activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  5  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Combined plot of past published proxy relations giving F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 as a function of the sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The curves are labeled according to the identification in column 4 of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves are superimposed  on the observed monthly mean values (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Moreover, some proxies were derived using monthly means or yearly means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In that case, the  upper range of values is more limited, and the fits should not be trusted beyond their calibration  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Unfortunately, while a few estimates of the error of individual daily F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux values were  published (Nicolet and Bossy , 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and Charrois, 1994), we must note that most of the  available proxy relations are given without any estimate of their uncertainties, and often without  clear indication of the calibration range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Here, we conservatively derived the mean and standard  deviation of all proxy models shown in the plot (black line and shaded band in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 3), to get a  rough first idea of their actual uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  6  350  Data  J2011_F1ya  HV1984_V1  J2011_F2ma  KU1984 V1  J2011_F2ya  300-  HA2002_V1  T2011_V1  ZH2008_F1  T2017_V1  ZH2008_F3  NOAA_V2  SV2009_V1  TK2018_F1  IPS2011_V1  250  TK2018F2  J2011F1ma  T2017_V2  200  150  100  50  0  50  100  150  200  250  300  350  400  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Combined plot of past published proxy relations giving F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 as a function of the sunspot number:  close-up view of the low-activity range of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves are labeled according to the identification in  column 4 of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves are superimposed on the observed monthly mean values (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The proxies giving the best fit to the non-linear section at low SN are those published by Tapping  and Vald´es (2011) (based on S Nv1) and Tapping and Morgan (2017) (based on S Nv1 and S Nv2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The  authors mention that those proxies were defined purely empirically, and they do not explain how  they were adjusted on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Both proxies are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' One can see that the S Nv1  and S Nv2 proxies are almost identical, indicating that the conversion in Equation[1] is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  However, we note that all those proxies reach a value of 67 sfu for S N = 0, while almost all  observed F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values are above this lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In fact, below in Section 6, we find that the most  probable F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 value for a spotless Sun is 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This mismatch indicates that this part of the  curve was not derived by least-squares but was adjusted empirically to reach exactly a tie-point at  7  100  Data  J2011_F1ya  HV1984_V1  J2011_F2ma  KU1984 V1  J2011_F2ya  95  HA2002_V1  T2011_V1  ZH2008F1  T2017_V1  ZH2008 F3  NOAA_V2  06  SV2009_V1  TK2018_F1  IPS2011_V1  TK2018_F2  J2011_F1ma  T2017_V2  85  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  80  75  70  65  0  10  20  30  40  50  60  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean (purple dash-dotted curve) and standard deviation of all proxies in Figure 1 (gray shading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Three recent proxies by Tapping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Table 1) are also included, as well as the observed monthly mean  values (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lower plot is a zoomed-in view of the upper plot for low activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  8  T2011V1  Proxymean  T2017V1  Data  300  T2017 V2  3 Stdv  250  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  200  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  150  100  0  50  100  150  200  250  300  350  Sunspot number100  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  T2011 V1  Proxy mean  T2017 V1  Data  T2017 V2  3 Stdv  95  90  85  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  80  75  70  65  60  0  10  20  30  40  50  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  67 sfu, chosen as base quiet-Sun background when S N=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Given the mismatch with the actual data,  this choice seems questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed, for real applications, users need the most probable F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  flux, and not the lowest possible value, which is rarely reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In Section 6, we will consider more  closely the properties of this quiet-Sun F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Still, the other published proxies are underestimating even more the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux at low activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Therefore, overall, none of the proxies proposed so far are providing a satisfactory representation  of the relation at low solar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Moreover, we note that all past proxies used classical least-  square fits, which assume that errors are present only in the fitted measurement (here F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7), while  the other quantity (S N) is considered as a parameter (without error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As S N is also affected by errors,  this fitting model may thus lead to systematic biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We also checked this aspect as explained in  sub-Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean profiles  In order to extract the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N relation without any parametric model, we first derive the mean of  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values and σm, the standard error of the mean (SEM), for a given value of S N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As the temporal  averaging of raw daily values will influence the relation between the two quantities, we repeated this  calculation for raw daily value pairs, for monthly means, for 13-month smoothed monthly values  and for yearly means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In order to include a sufficiently large sample of values and to reduce random  noise effects, we derived the statistics over a limited S N range centered on each given S N value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The bin width was 3, 20, 20, 60 respectively for daily, monthly, smoothed and yearly values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As our  analysis immediately showed that results for yearly means and 13-month smoothed data are almost  identical, we will not further discuss the 13-month smoothed results here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Figures 4, 5 and 6 show the resulting mean curves and 3-σm band for the daily, monthly, and  yearly calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' While the standard deviation of the base daily data is quite large, in particular  for raw daily values (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 sfu overall), the SEM value is rather small in the low and medium range  (< 1 sfu for daily values), thanks to the fact that each mean value is based on a large number of  data points within each S N slice (about 400 points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It strongly increases in the upper range, above  S N = 200, as data points become sparse, indicating that the fits will become much less precise in  that upper range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Now considering the raw daily values, we observe that the distribution of individual F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values  around the mean is asymmetrical with a more extended wing towards high values (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='This  upper wing may result from different contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This excess flux may come from the incomplete  elimination of eruptive events and from the temporal under-sampling of the 20h00UT ”spot” mea-  surements, as indicated by Tapping and Charrois (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' If any flaring emission was present over  time intervals when the S-component background emission was extracted, it inevitably contributed  to an overestimate of the background flux and thus to a net positive excess, contrary to simple ran-  dom measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This thus leads to an upwards asymmetry of the random deviations, like  we find here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Another consequence of this deviation from the assumed symmetrical Gaussian noise distribution  should be a small upward bias in the estimate of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Moreover, when considering temporal  variations (see Section 8), we find that the distribution is also slightly higher around the maxima  of solar cycles (time of maximum -2 years to +3 years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' blue curve in Figure 7) than around solar  9  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean non-parametric profile (red line) with 3 σm range (gray shading), obtained by averaging all  daily F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values for S N within a narrow band centered on the S N in the bottom axis (See details in the main  text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Gray dots are the daily observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The number of data points strongly decreases above S N = 300  and F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 = 250 sfu, leading to a much larger SEM σm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  minima (red curve), by about 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This thus suggests a significant change in the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 – S N relation  over each solar cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We will further examine this interesting property in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  However, this min – max shift, as well as the asymmetry of the distribution strongly decreases  when considering longer time scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' for monthly and yearly means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The distribution becomes  Gaussian and constant over time over those longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We can observe this by comparing  daily data with the plots for the monthly and yearly means (Figures 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We find that all tempo-  rally averaged values lead to very consistent means, and that those means are all slightly lower than  the mean for daily values (though still marginally consistent within the SEM σm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This indicates  10  400  Mean  3 Stdv  Data  350  300  250  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  200  150  100  0  100  200  300  400  500  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean non-parametric profile (red line) with 3 σm range (gray shading), like in Figure 4 but obtained  by averaging monthly mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values over narrow S N bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Gray dots are the monthly mean observed  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  that the upward bias affecting raw daily values largely vanishes for longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Moreover,  σm values are lower, and similar for the 13-month smoothed and yearly means, as expected: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  sfu (monthly), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2 sfu (smoothed), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3 sfu (yearly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This lower dispersion and higher correlation  between F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 and the SN indicates that for long duration, beyond a single solar rotation, both in-  dices record the level of solar activity (total magnetic flux emergence) essentially in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The global emergence rate dominates the statistics, and the skewed randomness of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux on  short times scales only plays a minor role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We will deepen this interpretation in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In all cases, we can see that over most of the observed range, the means trace a largely linear  proportionality between the two indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Only above S N = 250, F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 tends to fall slightly below the  11  300  Mean  3 Stdv  Data  250  200  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  150  100  0  50  100  150  200  250  300  350  400  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean non-parametric profile (red line) with 3 σm range (grey shading), like in Figure 4 but obtained  by averaging yearly mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values over narrow S N bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Gray dots are the yearly mean observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This slight curvature would suggest that for short periods of extreme activity during  cycle maxima, F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 does not grow as fast relative to the sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, taking into  account the large uncertainty, this is only marginally significant, and a fully linear relation remains  valid up to the highest observed F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  When zooming in on the low values (Figures 8, 9, 10), we find that the means for monthly,  smoothed and yearly data start to deviate from the main linear part for S N values below an inflection  point at about S N = 35 (monthly values) or 50 (smoothed and yearly values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As there are almost  no monthly or yearly periods with a 0 mean S N, the ordinate at S N = 0 can only be extrapolated,  12  250  Mean  3 Stdv  Data  225  200  175  150  125  100  75  0  50  100  150  200  250  300  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Histogram of the ratio between daily F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 data and the mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 value (after subtraction of a 67 sfu  base quiet-Sun flux), for all data with S N > 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The black curve is for the entire data series, while the blue  and red histograms are respectively for the maxima of the solar cycles (time of maximum -2 to +3 years)  and the minima (the rest of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The distributions are slightly asymmetrical with a longer upper wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The means of the distributions are indicated by thick vertical lines, with the matching colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The standard  deviation of the distributions equals 35%, while the SEM equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5% (based on more than 10000 daily ratios  in each distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The distributions for cycle maxima and minima are significantly shifted by 12% above  and below the global mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  and cannot be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Only for monthly means, we find that the means tend towards 68 sfu for SN  below 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  On the other hand, for raw daily values, we find that the means continue to follow linear propor-  tionality down to very low values, around S N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' There are only a very few points between 11  and 0, but at S N = 0 , the mean value is well defined at 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We find that the means reach this  values within the uncertainties for S N = 8, which is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  13  700-  Min  Max  600-  All data  500  Count  400  300  200  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean non-parametric profile (red line) with 3 σm range (grey shading) for daily values: enlarged view  of the low activity part of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We can thus draw three important conclusions:  – F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 and the SN are fully proportional over the full range of observed values, and this propor-  tionality continues down to an almost spotless Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This suggests that the variability of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 is  entirely determined by the level of magnetic activity also controlling the number of sunspots,  without any other contribution to the radio flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The excess associated with the presence of  spots becomes negligible relative to the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux distribution for a fully spotless Sun only below  S N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In other words, the distribution of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 is largely the same for a fully spotless Sun  and when a single isolated and short-lived spot is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' On the other hand, once the number of  spots grows beyond one (one group with a single spot), F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 increases fully proportionally with  the sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, in that sense, the quantization of the SN at low activity (the 0 to 11 jump)  14  120  Mean  3 Stdv  Data  110  100  7  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  90  80  70  60  0  10  20  30  40  50  60  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean non-parametric profile (red line) with 3 σm range (grey shading) for monthly means: enlarged  view of the low activity part of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  does not lead to a significant non-linearity between the two quantities, except for the values at  S N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – As in temporal means, solar activity varies during the chosen time interval, the linear relation  will be changed near the origin, essentially because of this single deviating point at S N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As  the latter is above the overall linear trend, the means will be pushed upwards, and this effect will  increase as the mean SN decreases towards 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed, the time interval used for each mean will  contain a growing proportion of spotless days, dominated by the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  This is exactly what we find in monthly, smoothed and yearly means, with the non-linearity ex-  tending progressively to higher minimum SN as the duration of the temporal averaging increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In Section 7, we will build a simulation to validate this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  15  95  Mean  3 Stdv  Data  90  85  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  80  75  70  65  0  10  20  30  40  50  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mean non-parametric profile (red line) with 3 σm range (grey shading) for yearly means: enlarged  view of the low activity part of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – The proxy relation is consistent with Gaussian statistics and is stable only for timescales equal  to or longer than one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Daily data and short timescales include an excess of high fluxes,  which varies with the solar cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Those data are thus inappropriate for building a reliable proxy  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' New high-degree polynomial fits  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Ordinary least-square polynomial fit: monthly means  In order to obtain a better fit to the data than the earlier, sometimes very crude, fits shown in Section  2, we fitted polynomials with degrees up to 4 by least-square regression of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 versus S N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed,  16  95  Mean  3 Stdv  Data  90  85  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  80  75  70  65  0  10  20  30  40  50  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Coefficients of polynomials of order 1 to 4 fitted by the ordinary least-square regression on the  monthly mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The coefficients Cn correspond to Equation 2, with their standard error σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Coefficients  Order 1  Order 2  Order 3  Order 4  (FCpol1 m)  (FCpol2 m)  (FCpol3 m)  (FCpol4 m)  C0  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='31  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='87  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='64  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='73  σ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5743  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7692  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='9457  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='134  C1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6432  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6279  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='4918  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3368  σ1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='528 10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='478 10−2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='132 10−2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='649 10−2  C2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='141 10−5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='304 10−3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='690 10−3  σ2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='637 10−5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='592 10−4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='699 10−4  C3  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='919 10−6  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='517 10−5  σ3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='946 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='773 10−6  C4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='974 10−8  σ4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='003 10−9  the non-parametric mean curves shown in the previous section indicate that the actual relation is  largely linear over a wide range, with a rather sharp bifurcation towards a constant background in  the low range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The polynomials are of the form (here for a 4th degree polynomial):  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 = C0 + C1 S N + C2 S 2  N + C3 S 3  N + C4 S 4  N  (2)  In Figure 11, we show the fits to the monthly mean values for degrees 1 (linear) to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As we know  that the relation becomes strongly non-linear below S N = 25 , the linear fit (degree 1) was applied  to a restricted range without the interval S N = 0 to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, this fit gives a good model for the main  linear section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  From S N = 30 to 250, all fits are almost identical and remain within the uncertainty range of  the mean values (gray shaded band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Only above 250, there is a slight deviation, with the higher  degrees falling below the linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' But this is hardly significant, given the low number of data points  in this upper range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This is confirmed by the fact that coefficients of the high-degree terms are only  marginally significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In particular for the degree-2 polynomial, only the linear term (degree 1) is  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This curve indeed gives the worst fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  This can be seen in the close-up of the low part (second plot), which shows that the fitted curve  match progressively better the curved lowest part of as the polynomial degree increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Only the  4th-degree polynomial closely reproduces the low part and remains within the uncertainty of the  mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The coefficients for fitted polynomials up to degree 4 are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  As the relation between the two indices is fully linear over a wide range, in this case above  S N = 25, we also derived the linear fits to this linear section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The coefficients are given in Table  4 (also for the orthogonal regression method described below in sub-Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The linearity is  confirmed by the fact that polynomials fits above degree 1 do not give stable solutions, once the  lowest range is excluded (terms of degrees above 1 are not significant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  17  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Polynomial of order 1 (linear fit) to 4 fitted to the monthly mean data by ordinary least-square  regression (OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves are superimposed on the corresponding non-parametric mean (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 5) to  show the agreement within 3 σm, and on the base data (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lower plot is a close-up view of the  low range of the upper plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  18  300  Mean  FCRpold1 m  FCpold2 m  Data  FCpold3 m  3 Stdv  FCpold4 m  250  200  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  150  100  50  100  150  200  250  300  350  Sunspot number105  Mean  FCRpold1 m  FCpold2 m  Data  FCpold3 m  3 Stdv  100  FCpold4 m  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  80  75  70  65  0  10  20  30  40  50  60  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Coefficients of the linear fits to the monthly mean data in the restricted linear range S N = 25 − 290  by ordinary least-squares and by orthogonal distance regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The two fits match closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Coefficients  Order 1  Order 1 (ODR)  (FCRpol1 m)  C0  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='66  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='21  σ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='8801  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='8831  C1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6601  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6720  σ1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='313 10−3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='338 10−3  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Coefficients of polynomials of order 1 to 4 fitted by the ordinary least-square regression on the  yearly mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For the order 2 and 4 polynomials, coefficients for degree 2 and above are not significant  (marked in italics), indicating that the proxy relation is essentially linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Coefficients  Order 1  Order 2  Order 3  Order 4  (FCpol1 y)  (FCpol2 y)  (FCpol3 y)  (FCpol4 y)  C0  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='07  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='64  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='56  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='85  σ0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='323  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='873  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='404  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='220  C1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6555  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6114  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='4163  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3942  σ1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='062 10−2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='893 10−2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='761 10−2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='816 10−1  C2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='862 10−4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='129 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='510 10−3  σ2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='583 10−4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='031 10−4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='849 10−3  C3  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='081 10−6  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='327 10−6  σ3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='062 10−6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='623 10−5  C4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='216 10−9  σ4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='021 10−8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Polynomial fits to yearly means  We repeated the analysis on yearly values and found largely the same conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves are  shown in Figure 12, and the polynomial coefficients for the fits to yearly means are given in Tables  5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In this case, the fit is also not significant at polynomial order 2, and the order-4 polynomial gives  roughly the same quality of fit as order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Although the fits on yearly means are slightly different  from the fits derived from monthly mean values, both are compatible within the uncertainties in  yearly means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This difference is due to a lower non-linearity and the slightly wider range over  which the relation is non-linear for yearly means, but is hardly significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Polynomial fits to daily values  Based on Section 3, we may expect slightly different results for daily values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The polynomial and  linear regressions are shown in Figure 13 and the coefficients are listed in Tables 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We find  that for the main linear range up to S N = 250, the different fits match closely, like for yearly and  monthly means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They then diverge from each other at higher values, which is again due to the  steeply decreasing number of data points in this upper range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The non-linear fits at order 2 to 4 tend  to fall below the linear fit, aligning better with the mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, the linear fit still falls  19  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Polynomial of order 1 (linear fit) to 4 fitted to the yearly mean data by ordinary least-square regres-  sion (OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves are superimposed on the corresponding non-parametric mean (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 6) to show the  agreement within 3 σm, and on the base data (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lower plot is a close-up view of the low range  of the upper plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  20  250  Mean  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  FCRpold1_y  FCpold2_y  Data  FCpold3_y  3 Stdv  225  FCpold4_y  200  175  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  150  125  100  75  0  50  100  150  200  250  300  Sunspot number105  Mean  FCRpold1_y  FCpold2_y  Data  FCpold3_y  3 Stdv  100  FCpold4_y  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  80  75  70  65  0  10  20  30  40  50  60  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Coefficients of the linear fits to the yearly mean data in the restricted linear range S N = 30 - 220 by  ordinary least-squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Coefficients  Order 1  (FCRpol1 y)  C0  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='24  σ0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='259  C1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6936  σ1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='739 10−2  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Coefficients of polynomials of order 1 to 4 fitted by the ordinary least-square regression on the  daily values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The coefficients Cn correspond to Equation 2, with their standard error σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Coefficients  Order 1  Order 2  Order 3  Order 4  (FCpol1 d)  (FCpol2 d)  (FCpol3 d)  (FCpol4 d)  C0  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='72  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='97  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='52  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='41  σ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1654  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2083  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2430  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2711  C1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6198  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5432  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3938  σ1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='253 10−3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='572 10−3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='185 10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='204 10−2  C2  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='068 10−5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='561 10−4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='613 10−3  σ2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='207 10−5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='246 10−5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='432 10−4  C3  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='260 10−6  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='033 10−5  σ3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='027 10−7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='965 10−7  C4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='225 10−8  σ4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='937 10−10  within the uncertainty range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, the differences brought by higher degrees are not fully significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  This is again confirmed by the low level of significance of polynomial coefficients with degrees  higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The low range is where the situation differs markedly from the monthly and yearly mean analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Here (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 13), the linear fit (over the range above S N = 25) remains close to the mean values down  to the lowest values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In the low range, the order-4 curve again gives the best fit, although it fails  for S N values below 6, as the mean then deviates abruptly over a very small range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Ignoring this  section, the order-3 polynomial gives the best fit overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It reaches the mean background value for  S N = 0 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu) at S N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This suggests that for S N below 6, this background value can be used  instead of the polynomial fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Therefore, the fit on daily values helps to confirm the higher linearity of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 – S N relation  down to very low levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, as expected, the fitted curves are significantly higher than the fits  on monthly and yearly means, due to the upward bias characterizing raw daily values (see Section  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Beyond the linearity check, they should thus not be considered for a proxy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Comparison with the Tapping and Morgan (2017) proxy  In order to check if indeed the new polynomial fits bring an improvement on past relations, in Figure  14, we compare the order-4 polynomial with the best curve identified among the past published  proxies, namely the curve by Tapping and Morgan (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Here, we consider the fit on monthly  21  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Polynomial of order 1 (linear fit) to 4 fitted to the daily data by ordinary least-square regression  (OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves are superimposed on the corresponding non-parametric mean (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 4) to show the  agreement within 3 σm, and on the base data (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lower plot is a close-up view of the low range  of the upper plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  22  400  Mean  FCRpold1 d  FCpold2 d  Data  FCpold3 d  3 Stdv  350  FCpold4 d  300  250  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  200  150  100  0  100  200  300  400  500  Sunspot number105  Mean  FCRpold1 d  FCpold2 d  Data  FCpold3 d  3 Stdv  100  FCpold4 d  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  80  75  70  65  0  10  20  30  40  50  60  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Coefficients of the linear fits to the daily data in the restricted linear range S N = 5 - 290 by ordinary  least-squares and by orthogonal distance regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The two fits match closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Coefficients  Order 1  Order 1 (ODR)  (FCRpol1 d)  C0  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='79  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='23  σ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2030  C1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6171  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6419  σ1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='592 10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='610 10−3  means, as most of the fits are based on temporally averaged numbers, in order to smooth out the  random variations due to short-timescale solar variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  One can see that both fits match very closely, within 4 sfu over the whole linear range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The slope  is slightly lower than the slope of a purely linear fit on the range S N = 25 to 290 (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This  may be due to the influence of the upward deviation for S N below 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Now, looking at the lowest range, we find that the 4th-degree polynomial tracks the data slightly  better, and at least remain within the uncertainty range of the means, contrary to the relation by  Tapping and Morgan (2017), which is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It reaches 68 sfu at S N = 0, instead of the 67 sfu  background value used by Tapping and Morgan (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This is in agreement with our analysis of the  background flux for a spotless Sun in Section 6 below: our polynomial fits the most probable flux  instead of the assumed F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background value at S N = 0, which is a lower boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Finally, we  point out that our 4th-order polynomial is entirely defined by a least-square fit to the data, and is not  attached to a predefined tie-point, like the Tapping and Morgan (2017) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It thus allows classical  statistical tests on fitted polynomials, including the estimate of errors on polynomial coefficients and  on the resulting proxy values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Polynomial error determination  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Uncertainties on polynomial values  Regression methods allow to determine the standard errors σn on each polynomial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  However, an exact derivation of the standard error σp on polynomial values themselves, based on  those standard errors σn, does not exist in the literature, due to the mathematical complexity of this  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed, the errors on the coefficients for the different terms are actually inter-correlated,  as they are determined together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, the total variance of the polynomial values is not the  simple naive sum of the individual variances of all terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, a proper estimate of σp can be  derived, based on the fact that the actual error for each term (each degree) is the conditional error  on that coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' the uncertainty of that polynomial term given the values of the coefficients  for all other terms (in the solution of the least-square regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In order to estimate this condi-  tional error, we can make a regression for only one term (one polynomial degree) at a time, after  subtracting all other terms from the original observed F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values, with the other coefficients set at  the values given by the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In order to simplify this calculation, we considered that, as we go to higher degree terms, their  contribution becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, we derived the conditional error for each degree n by re-  gressing for each degree separately (one-term model), after subtracting successively all polynomial  23  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Comparison of our 4th order polynomial with the proxy relation from Tapping and Morgan (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The curves are superimposed on the corresponding non-parametric mean (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 5) to show the agreement  within 3 σm, and on the base data (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lower plot is a close-up view of the low range of the upper  plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  24  105  Mean  FCRpold1 m  T2017V2  Data  FCpold4 m  3 Stdv  100  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  80  75  70  65  0  10  20  30  40  50  60  Sunspot number300  Mean  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  FCRpold1 m  T2017 V2  Data  FCpold4 m  3 Stdv  250  200  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  150  100  0  50  100  150  200  250  300  350  400  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  contributions of lower degrees (< n) from the original F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 data, starting from the lowest degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Thus, the single-term model of degree n is:  Fn = Cn S n  N  n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' , d  (3)  where Cn is the coefficient to be determined (with its error) and d is the degree of the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  This model is fitted to the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 data series, minus all fitted terms of lower degree:  Fcorr = F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 −  n−1  �  k=0  Ck S k  n  (4)  The fact that we did not subtract the terms of higher degrees leads to a slight overestimate of the  residual error for each degree, as it also includes the residual uncertainties of all degrees above n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Therefore, the conditional errors calculated in this way for each separate degree give an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Deriving the above errors from the data requires a statistical processing and multiple regressions  on the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, for practical applications, this approach would be too heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, based  on the data-based errors obtained by this procedure, we found a simple mathematical representation  that gives a good approximation of the σp errors from the full determination described above, and  that can be calculated directly for a polynomial value calculated at any given S N:  σTot =  �  �  � d  �  n=0  �������  S n  N − S N  n  2 n−1 (d + 1 − n)2 σn  �������  2  (5)  where S N is the mean of all S N values in the data set (≈ 120 with the actual data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  It consists in the sum of squared errors for each term, using the (non-conditional) standard error  on each coefficient given by the least-square regression procedure, σn, but with a weight factor that  decreases for increasing degree n (powers of 2), and also decreases for each term of a given degree  n, as the degree d of the polynomial increases (and thus the number of degrees of freedom in the  regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We observe that those rather simple expressions already give a very good agreement with the real  data-based errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This rather simple empirical weighting thus probably reflects the dominant cor-  rections associated with the inter-dependency of the least-square polynomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' A math-  ematical demonstration goes well beyond the scope of this study, but the good match with the  data-based errors indicates that we obtain here a reliable estimate of this polynomial error (Figure  15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This marks a big improvement on all previous proxy relations, where the error was missing and  thus entirely undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The formula in Equation 5 conveniently allows a direct calculation of  the error, without requiring to re-do the above extraction of conditional errors from the data them-  selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Finally, we point out that σTot gives the uncertainty on the proxy values, which combines  both the errors in F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 and S N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This must be distinguished from the standard error of a single daily  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 measurement, which is globally estimated at about 2% by (Tapping and Charrois, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As  expected for such a regression, the smallest errors (±2 sfu) are found in the vicinity of the mean of  all values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' S N = 120 and F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 = 135 sfu, and the error grows in both directions away from this  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  25  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Order-4 polynomial with uncertainty range (1 standard error σp) on polynomial values, superim-  posed on the corresponding non-parametric mean profile and SEM σm obtained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lower plot  is a close-up view of the low-activity range in the upper plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  26  Data  300  Mean  Polynomial deg 4  +/- 1 sigma  3 Stdv  250  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  200  150  100  50  100  150  200  250  300  350  400  Sunspot number105  Data  Mean  100  Polynomial deg 4  +/- 1 sigma  3 Stdv  95  90  85  80  75  70  65  0  10  20  30  40  50  60  70  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Orthogonal-distance regression versus ordinary least-square regression  In the ordinary least-square (OLS) regression, the model assumes that all errors are in the dependent  variable (“response”, here F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7) and not in the independent variable (“explanatory”, here S N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As  we know that in our case, both quantities are actually affected by errors, we repeated the regression,  but using instead the orthogonal distance regression (ODR) technique, which takes into account the  uncertainties in both regressed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We find that the differences between the coefficients derived from the ordinary and ODR re-  gressions are within the computed uncertainties, and are thus not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Likewise, Figure 16  illustrates this close agreement for the 4th-degree polynomials derived by both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore,  we conclude that the ordinary regression gives valid fits in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This can be explained by the  very high level of correlation between the two indices over long timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Background flux for a spotless Sun  In the above curves, we noted that past relations found by Tapping and Vald´es (2011) and Tapping  and Morgan (2017) assumed a base radio flux of 67 sfu at S N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' By contrast, our mean curves  based on daily values indicate a higher mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 for all spotless days in the series, at 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  However, the monthly means tend to converge towards lower values near S N = 0, though still above  67 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We can thus wonder how to reconcile those apparently contradictory determinations of the  same parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As the underlying temporal resolution is different in each case, we suspected that  the temporal scale plays a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Dependency on spotless duration  In order to investigate such a temporal effect, we extracted all spotless days in the SN series, and  the corresponding daily F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Then, we also grouped uninterrupted sequences of contiguous  spotless days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Finally, we computed the distribution of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values for all spotless sequences of  the same length in the observed series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Figure 17 shows the mean values, standard deviations and  extreme values of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm flux for all sequence lengths found in the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In the lower panel, we  also plotted the number of days included in each category, and how many sequences were found for  each duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The longest sequence lasted 42 days, but most spotless days sequences last less than  10 days, with many isolated spotless days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Our analysis shows that the mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux systematically increases as the duration of a spotless-  day sequence decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For duration above 15 days, the most likely F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 value is near 68 or 69 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  On the other hand, it increases to 74 sfu for single spotless days immediately surrounded by active  days with one or more sunspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lowest mean daily value is 67 sfu and is reached only for 6  sequences with duration of 22, 27 and 28 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The red dashed line at 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu corresponds to the  mean flux for all spotless days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Quite logically, it corresponds to the mean levels for duration 5 to  10 days, which is near the mean duration of spotless intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Now, considering the extreme values, we also find a steady increase of the upper values (blue  dots) with decreasing duration, up to as high as 95 sfu for a single spotless day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Based on the above  regressions, such fluxes usually correspond to S N values of 50, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' to moderate levels of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' On  the other hand, the lowest values (green dots) do not show any dependency on duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They stay  around the 67 sfu level, with rare extremely low values down to 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6 sfu (November 3, 1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This  27  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Comparison of 4th-degree polynomials obtained by the ordinary least-square regression (OLS) and  by the orthogonal-distance regression (ODR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The differences are small in comparison with the 1-σp standard  error of the fit, and are thus not significant over the whole range of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  28  Data  300  Polynomial deg 4  +/- 1 sigma  Poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='ODR deg 4  +/- 1 sigma ODR  250  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  200  150  100  50  100  150  200  250  300  350  400  Sunspot number105  Data  Polynomial deg 4  100  +/-1 sigma  Poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' ODR deg 4  +/- 1 sigma ODR  95  90  85  80  75  70  65  0  10  20  30  40  50  60  70  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Plot of the mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background flux for spotless days (red dots) as a function of the duration of the  sequence of contiguous spotless days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The errors bars correspond to one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The upper curve  (blue dots) and lower curve (green dots) are the lowest and largest F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 values for each spotless duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The red dashed line marks the overall mean spotless flux (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu), while the lower black dashed line is the  mean minimum flux (67 sfu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The lower panel gives the number of intervals for each spotless duration (red  bar) and the number of days included in each duration (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  thus validates the choice of 67 sfu as the all-quiet base flux, shown in Figure 17 as the horizontal  black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Actually, there are only 33 values below 66 sfu in the whole series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Moreover, all 8 values below  65 sfu and 14 values out of 25 between 65 and 66 sfu appear exclusively in 1953 and 1954, some  29  95  Meanflux  Lowestvalues  Highest values  Mean Flux & std error  90  85  80  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  山  75  70  65  60  200  Nb days  Nb intervals  Count  100  0  0  5  10  15  20  25  30  35  40  NbzerosdaysClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  of them even on days when the Sun was not spotless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' By contrast, the lowest values recorded after  1954 are always above 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu, and almost all are occurring quite logically during the longest  minimum recorded in the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 series, between cycle 23 – 24 in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' By comparison, the cycle 18  – 19 minimum in 1954 was not particularly low and protracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This suggests that the record-low  values in 1953 – 1954 are either spurious or suffer from a calibration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As indicated by  Tapping (2013) and Tapping and Morgan (2017), those early data may indeed suffer from larger  errors, as the calibration method was not fully standardized, and the location in Ottawa until 1962  caused larger radio interferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, a base background flux as low as 64 sfu, as suggested  by Tapping and Detracey (1990) and Tapping and Charrois (1994), seems doubtful and too low for  the real fully-quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Interpretation  We can explain this dependency of the background levels on the duration of spotless intervals by  the presence of other sources of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 emission, even when there are no associated sunspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This  includes various features in the chromosphere and lower corona associated with closed magnetic  fields weaker than those concentrated in sunspots, like bright chromospheric plages, filaments, coro-  nal condensations (Shimojo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Schonfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' , 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Pevtsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' , 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Ermolli et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In particular, when the Sun is very quiet (few isolated spots) or entirely spotless, the  associated plages have a small extent but still contribute a significant excess in F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed, Figure  17 shows that most of the values are below 80 sfu, although there are a few as high as 80 sfu or  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' On the other hand, there are almost no values below 67 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Such chromospheric plages are typically present just before the emergence of a first spot, or they  remain after the decay of a last sunspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, short spotless intervals surrounded by more ac-  tive periods are most likely to show a significant excess in radio emission above the lowest all-quiet  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Conversely, only very long spotless periods can include days without any activity features on  the Earth-facing solar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The only small active regions present just before the long spotless inter-  val have enough time to decay entirely, well before new bright chromospheric structures develop,  heralding the appearance of the first spots marking the end of the protracted spotless period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Consequently, the mean background level does not have an absolute value, as the lowest level  of 67 sfu is almost never reached, even when there are no sunspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We can thus expect to see a  dependency of the asymptotic F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 value near S N = 0 when applying a temporal averaging to the  raw daily data series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Without any averaging or when the averaging duration is short, days in long  spotless intervals are mixed with those in short intervals, leading to a higher mean, rising to 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  sfu for 1-day timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' At the other extreme, averaging over very long duration, like one year,  inevitably mixes spotless periods with active periods, as virtually no spotless periods have duration  above about 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, the lowest F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 yearly mean values are expected to be also higher  than the 67 sfu minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It turns out that a duration of one month best matches the actual duration  range of spotless episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Monthly means are thus best for recording the lowest possible mean  radio fluxes of the fully quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Finally, we note that this chromospheric background interpretation agrees with the identification  of two types of emission sources for F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 (Tapping, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and Detracey, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping  and Zwaan, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and Morton, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Schonfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' , 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' While the gyroresonance  emission is closely associated with strong magnetic fields in sunspots (> 300 G), a so-called ”dif-  30  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  fuse” component by free-free thermal emission is attributed to plages and the overall chromospheric  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The latter is the best candidate for the variable background flux diagnosed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In this  respect, we note that while the sunspot component of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 will track instantaneously the evolution  of active regions (fully linear relation), the plage component will be extended and delayed in time  relative to the associated sunspots, as it corresponds to the progressive decay and dispersal of flux  emerging in active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Therefore, the dependency between the mean background flux and the duration of the spotless  interval is consistent with this interpretation, and offers an independent indicator of this dual-source  nature of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It also implies that the disagreements between F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 and the SN are  probably due for a significant part to the time delay intrinsic to the free-free emission from plages,  rather than simply to a non-proportionality with the underlying emerging magnetic fluxes, and thus  with the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed, the latter is only sensitive to strong fluxes freshly emerged in sunspots, without  mixing with a second magnetic-decay component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The fact that disagreements between those two  indices increase for short time scales, below one solar rotation and thus below the average plage  lifetime, also concurs with the prominent role of this temporally-smeared weak-field component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' A simple model for the the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/SN non-linearity  In the above analyses, we found that daily values indicate that the relation between F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 and S N  is linear almost down to the lowest values of S N = 11 (single isolated spot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Only for smaller S N  values close to 0, F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 stops decreasing and reaches its background level, in the range 68-70 sfu as  found in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' There is a break from the linear relation, with the last points near S N = 0 located  several solar flux units above the linear relation, which intercepts the axis at S N = 0 at a value of 58  to 62 sfu (see tables of polynomial coefficients above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  When deriving monthly or yearly means, the averaging interval inevitably includes periods of  different activity levels, including some inactive days, when F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 is at its lowest level and thus  shows an excess relative to a purely linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As the mean activity during the averaging  interval decreases, the proportion of spotless days increases, and thus also the fraction of points  bringing an excess above the linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We can thus expect a progressive upwards deviation  from linearity, like we observe in the monthly and yearly mean curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In order to simulate this scheme, we took the observed SN time series, and synthesized a F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  time series, by converting each S N value via a two-component model:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For most of of the range, we used the linear fit to the linear part of the data (from Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For the lowest S N values, when the linear relation falls below the base F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background, chosen  at 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu, the output value is set at the constant value of 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' An alternate model for this  background flux can take into account the fact that the mean background is not constant but  increases even when the Sun is spotless, up to 74 sfu, as demonstrated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We then applied the usual monthly and yearly averaging to this synthetic series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In Figure 18, the model for daily values assumes a constant background at the mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux for  a spotless Sun (pink crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In Figure 19, the model assumes a progressive rise of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux  over the range found in our above analysis of the base flux according to the duration of the spotless  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' It starts from 69 sfu, the low value for a fully spotless Sun and rises to 75 sfu, the upper  31  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Model of the monthly temporal averaging of daily data built from two components: linear component  and constant lower background at 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The pink crosses are the synthesized daily values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The green dots  are the corresponding monthly mean values, and the blue line is the non-parametric local mean of those  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As a comparison, the black line is the 4th-order polynomial fitted to the real monthly mean data  (Table 3), with uncertainties (black dashes lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  value found for isolated spotless days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This can be considered as representative of the flux when  just one isolated sunspot group is present on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Here, this ramp connects with the main linear  relation at about S N = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Our degree-4 polynomial based on the true data (black line, with uncertainties as dashed lines)  nicely falls in the middle of the simulated monthly means for both options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The agreement with the  mean of data values (blue curve) is best for the second model, where the agreement is very tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The first model with a higher but uniform background gives a higher curvature below S N = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  32  105  Model daily  Model monthly  100  Mean  Polynomial deg 4  +/- 1 sigma  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  80  75  70  65  0  10  20  30  40  50  60  70  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Model of the monthly temporal averaging of daily data built from two components: linear compo-  nent, and here, a slightly rising background with increasing S N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The elements of the plots are the same as in  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We also made the simulations using yearly means (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 20 and 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They also give a good agree-  ment, but given the lower number of points and slightly more linear relation, the monthly simu-  lations shown here illustrate more clearly how temporal averaging is producing a curvature of the  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Overall, those two very simple simulations match strikingly well the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They thus  confirm the mechanism by which time-averaging of the raw linear daily values can produce the  non-linear proxy relation, thus also indicating that this non-linearity is dependent on the temporal-  averaging applied to the data before making the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  33  105  Model daily  Model monthly  100  Mean  Polynomial deg 4  +/- 1 sigma  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  80  75  70  65  0  10  20  30  40  50  60  70  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Model of the yearly temporal averaging of daily data built from two components: linear relation and  constant lower background at 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5 sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The elements of the plots are the same as in Figure 18, with green  dots and curve corresponding to yearly averages and the polynomial also to yearly mean data (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Temporal variations  So far, we included the entire duration of the time series, thus making the assumption that both  the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm flux and the sunspot number series are homogeneous over the entire 68-year duration  included here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' (2016) made a first simple comparison between the newly  released Version 2 of the sunspot number and F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 as a function of time, and found a 12% upward  jump in the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N ratio, occurring between 1979 and 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Likewise, by a comparison to the sunspot number series (versions 1 and 2) and the total sunspot  area, Tapping and Vald´es (2011) and Tapping and Morgan (2017) found that the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 time series  34  105  Model daily  Model yearly  100  Mean  Polynomial deg 4  +/- 1 sigma  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  80  ×××  75  70  +++++  65  0  10  20  30  40  50  60  70  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Model of the yearly temporal averaging of daily data built from two components: linear relation and  slightly rising background with increasing S N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The elements of the plots are the same as in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  shows an upward deviation in the second half of the series, mostly after 1980, relative to both the  sunspot number and sunspot areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Although this trend is stronger when comparing with S Nv1, it is  still present when S Nv2 is taken as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The authors fit a smooth curve as a function of time  over the whole duration of the series, and they interpret the resulting global trend as a real change  in solar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This evolution would parallel the overall decline of solar cycle amplitudes since  the mid-20th century, thus invoking a possible genuine change in the properties of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  35  105  Model daily  Model yearly  100  Mean  Polynomial deg 4  +/- 1 sigma  95  90  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  85  XXXXXXxx  80  75  70  65  0  10  20  30  40  50  60  70  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' A transition between two stable periods  In order to check for such a change in the relation between the two measurements, we used the  13-month smoothed monthly means, using the classical Z¨urich smoothing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This allows to  reject random fluctuations associated with solar activity at time scales shorted than one year, while  retaining a better temporal sampling than in yearly means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In our case, in the above F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 versus  S N representation, we checked over which time interval the data follow the same relation in the  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In Figures 22 and 23, the resulting curves are shown respectively for the SN version 1 and SN  version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The curves now include the chronology and consist of several narrow loops corresponding  to each of the solar cycles, varying from very low values at minima to the maxima while staying  close to a diagonal line, as can be expected given the close proportionality of the two indices, as  shown in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We note that two cycles deviate strongly in the case of the SN version 1 (blue loops in Figure  22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' They correspond to cycles 22 and 23, when SN version 1 is know to be affected to drifts in the  Locarno pilot station (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2014, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' On the other hand, the agreement is much tighter  with SN version 2 (Figure 23), in particular over all cycles after cycle 21 for which the SN was  entirely re-constructed based on a multi-station reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We will thus now concentrate on this second comparison with SN version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The plot immediately  shows that the curves are grouped along two preferential bands located above (colored blue) and  below (red) the fit to the entire series (solid black line), with very few points in-between, near the  global fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This indicates that the relation was actually very stable during two time periods and  jumped directly from one relation to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We then looked for the time sub-intervals following the upper and lower linear relation, and we  found that the lower relation applies to all data before 1980, while the upper relation is valid for the  entire period after 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The temporal evolution is thus characterized by a single jump separating  two fully homogeneous periods, during which the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N relation is very stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  For each homogeneous interval, we could then derive the two corresponding linear relations using  the same regression methods, as applied before to the whole series (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For the period before  1980, we find a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='635 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6110−3 (dotted line), while after 1980, the slope becomes steeper  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='702±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='8710−3 (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This corresponds to an upward jump by a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='106±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='017, thus  of about 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The slope found for the entire series naturally falls in-between, with an intermediate  slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='660 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='31310−3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The coefficients for those two linear fits are given together  with the two 4th-degree polynomial fits in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Such a jump is highly significant, as Tapping and  Charrois (1994) and Tapping (2013) give an accuracy of 1% for the flux measurements and of 2%  for the daily index derived from the ”spot” measurements at 20h00 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Checking the past published proxies, we note that early proxies that were based only on the first  part of the series, like Holland and Vaughn (1984) or the IPS formula (Table 1), match the low  linear fit for the period before 1980 in Table 9, while the NOAA proxy adjusted to the recent SN  version 2 (1), has a steeper slope matching well the second higher fit for the recent period after  1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, the disagreements between some of the past proxies actually originate from the temporal  inhomogeneity of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 index itself, as diagnosed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Another consequence is that when including the entire time series, the least-square regression  errors will not decrease when using monthly or yearly mean values, exactly as we found in our  36  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Plot of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 versus S N, using the original SN series (Version 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The data are smoothed by a 13-  month running mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The line connects successive months, and thus illustrate the chronological evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The curve is colored in blue or red for dates before and after 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  global regressions (sub-Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We can now explain it by the fact that a significant  part of the deviations of individual monthly or yearly mean values relative to the mean regression  curve is due to this systematic inhomogeneity in the series, rather than to random errors, and will  thus not be reduced by temporal averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' A more precise timing of the jump  Using the 13-month smoothed monthly values, we can roughly locate the jump in mid-1980, as the  smoothed monthly means migrate from the low branch to the high branch between January 1980  and March 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, with such smoothed values, we cannot pinpoint the transition time with  37  Databefore1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='79  FCpold4_m  250  Dataafter1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='79  225  200  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  175  150  125  100  75  0  50  100  150  200  250  300  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Plot of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 versus S N, equivalent to figure 22 but using the new re-calibrated version of the SN  series (Version 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The data are smoothed by a 13-month running mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The data curve is in blue or red for  dates before and after 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The black lines correspond to linear fits to the entire series (solid line), the period  before 1980 (dotted) and the period after 1980 (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  a better temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Still, the fact that the branch-to-branch transition occurs over a duration  similar to the smoothing duration already indicates that the actual transition must take place over a  duration much shorter than 13 months, thus very sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In order to better pinpoint the time of the transition, we checked the (un-smoothed) monthly  means, and we compared them with the global linear fit to the whole data set (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In Figure  24, we plot the resulting monthly ”observed/fit” ratios over the 4-year interval around the suspected  transition, which is also centered on the maximum of cycle 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Over this interval, all data values  are thus in the same high range, and the choice of mean fit has only a small influence on the ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  38  Databefore1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='79  FCP1_pold1_m  250  Dataafter1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='79  FCP2_pold1_m  FCRpold1_m  225  200  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  175  150  125  100  75  0  50  100  150  200  250  300  Sunspot numberClette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Coefficients of order-1 (linear) and order-4 polynomials fitted by the ordinary least-square regres-  sion on the monthly mean SN for the periods 1947-1980 and 1981-2015, with standard errors σn for each  coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Coefficients  Period 1947-1980  Period 1981-2015  Period 1947-1980  Period 1981-2015  Order 1  Order 1  Order 4  Order 4  C0  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='09  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='78  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='64  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='85  σ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='9391  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='151  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='476  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='275  C1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6345  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3658  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3845  σ1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='612 10−3  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='867 10−3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='765 10−2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='115 10−2  C2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='587 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='881 10−3  σ2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='642 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='378 10−3  C3  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='906 10−6  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='429 10−6  σ3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='029 10−6  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='344 10−6  C4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='329 10−8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='694 10−10  σ4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='152 10−9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='645 10−8  As can be expected, the monthly ratios show rather large month-to-month random fluctuations, but  19 out of 24 ratios are below 1 before the end of 1980, while 21 out of 24 are above 1 after 1980,  indicating a clear systematic change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The points are actually grouped respectively around the global  linear fits calculated for the complete half-series 1947 – 1980 and 1980 – 2015 (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Moreover, this transition happens very sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' With the exception of November 1980, there is  a sudden jump between December 1980 and January 1981, followed by 8 consecutive months in  1981 all above the mean linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This thus strongly suggests that the jump occurred between  December 1980 and January 1981, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' at the transition between two calendar years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Although a  slightly earlier transition is not entirely excluded, it cannot be before mid-1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Validation against multiple independent stations  Now, it turns out that 1980–1981 also marks a transition for the sunspot number series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This is when  the production of the sunspot number moved from Zurich to Brussels, with a significant change in  the production method (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2014, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The former manual processing was computerized  and the Specola Solare Observatory in Locarno (Z¨urich’s auxiliary station) took over as pilot station  in replacement for the Z¨urich Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, as the sunspot number is a synthetic index  based on a global statistical processing of multiple data sources, we should be careful that the 1980  jump that we just found above is not entirely due to a sharp change of scale between the Z¨urich and  Brussels sunspot numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In order to exclude any processing flaw in the sunspot number, we made direct comparisons with  a large set of individual stations that provided sunspot counts over a long period extending both  before and after 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In the SILSO database, we found 28 stations fulfilling the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In  fact, most of those stations were part of the multi-station reference used for the re-calibration of the  SN version 2 (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2016, Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For each station, we derived the ratio between the raw  Wolf numbers and the corresponding F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7-based SN proxy value, as a function of time (single proxy  relation for all times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As only one station is used in each comparison, this ratio is affected by a larger  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, we averaged the ratios for the whole time intervals before and after 1980, and then  39  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Monthly ratios between the observed monthly mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux and the proxy value derived from the  linear fit over the entire interval 1947 – 2015 (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The dashed lines correspond to the global fits on the  entire half-series before and after 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  looked at the ratio between the two intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This is listed in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As illustration, Figures  25 and 26 show the comparisons for two sample stations: the Kislovodsk station (Observatory of  Pulkovo), a professional observatory, and Thomas Cragg, a dedicated individual observer, who was  employee of the Mount Wilson Observatory (Howe and Clette, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We find that, out of the 28 stations, 24 indicate a higher ratio after 1980, while only one station  indicates a constant ratio, and only three stations give a decreasing trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As the curves have larger  fluctuations, the timing of the transition is less accurate, but all series suggest a short transition  within at most 3 years around 1980, preceded and followed by periods without systematic trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The simple arithmetic mean of the listed SN ratios is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='166 (range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='812 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='731), which is larger  than the jump amplitude calculated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' However, given the larger uncertainties in individual  series and the different time intervals covered by each station, this mean value is only indicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' By  keeping only stations with well-traced reliability, and which were active over most of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 time  interval (marked with a * in Table 10), we find a mean value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='139 (range: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='047 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='235), which  is more reliable and comes reasonably close to the value found using the SN series as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  As this comparison involves only raw data from multiple independent observers, this verification  thus allows us to confirm that the 1980 jump definitely occurs in the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 series and is not due to  any unsuspected and uncorrected flaw in the SN series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='15  d / Mean Fit  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='05  Observed  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='95  Ratio  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='80  1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5  1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  Time (years)Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Comparison of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7-based SN proxy series with the raw Wolf numbers from the Kislovodsk  solar Observatory (Russia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' SILSO station ID: KS2), cover the period before and after 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The upper panel  shows the two series, smoothed by a 12-month Gaussian kernel to reduce short-term random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The  middle and lower panels show the ratios and differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' No data are available for that station in the blue-  shaded time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' An upward jump can been seen around 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Of course, another independent test could come from a comparison with equivalent radio flux  measurements, made by another radiotelescope at the same wavelength or at a neighbouring one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Nicolet and Bossy (1985) made such a comparison with data from the Toyokawa station (Nagoya,  41  300  Y= F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  X= KS2 Sunspot Number  200  SN  100  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  Ratio smoothed  Ratio  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  04B  Difference smoothed  Difference Y-X  20  0  20  40  1950  1960  1970  1980  1990  2000  2010  Time (years)Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Comparison of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7-based SN proxy series with the raw Wolf numbers from Thomas  Cragg (SILSO station ID: CRA), an amateur sunspot observer, who was employee of the Mount Wilson  Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Like in Figure 25, an upward jump in the ratio and differences (middle and lower panels) can  been seen around 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Japan), but by then, the data extended only until 1982, thus only two years after the jump, which  makes the detection of the jump difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' More recently, another combined study by Yaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  (2017) also uses F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 next to the Toyokawa series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Here also, there is no mention of a jump in  1980, but this study was focused on short-term predictions and did not look for such long-term  42  300  Y= F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  X= CRA Sunspot Number  200  100  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  Ratio smoothed  Ratio  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  04B  Difference smoothed  Difference Y-X  20  0  20  40  1950  1960  1970  1980  1990  2000  2010  Time (years)Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' k ratios between the raw Wolf number from 28 stations in the SN database (World Data Center  SILSO) and a fixed F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7-based proxy of the sunspot number, for observations made before 1980 (column  3) and after 1980 (column 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For each station, we also list the corresponding time interval over which the  data are available (columns 2 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The absolute value of those ratios is different for each station, due to  the different observing setup (telescope, observing site), but it is not relevant here, as we only trace relative  differences between different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The 6th column gives the k2/k1 ratios between the scale ratios for the two  periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Stations marked with a * in the first column form a subset of stations with the best reliability and/or  longest duration before and after the 1980 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Station  Period 1  k1  Period 2  k2  Ratio  Error  Comment  AN *  1976-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='435  1981-1988  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='608  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='121  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='030  short series  AT  1968-1980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='878  1981-1982  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='096  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='248  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='050  short series  BN-S  1965-1980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='995  1981-2014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='870  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='874  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='016  decreasing  BRm  1974-1980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='819  1981-1998  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='282  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='015  short series  CA  1949-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='174  1981-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='097  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='934  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='013  drifts before 1980  CRA *  1947-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='267  1981-2010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='521  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='011  long series  EB  1949-1981  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='746  1981-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='291  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='731  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='012  drifts before 1980  FR-S  1976-1980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='790  1981-1988  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='962  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='218  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='019  short series  FU *  1968-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='047  1981-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='123  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='072  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='012  long and stable  GU-S  1974-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='617  1981-1991  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='936  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='197  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='053  short series  HD-S  1967-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='151  1981-2013  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='315  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='142  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='024  unstable  HU  1969-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='323  1980-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='164  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='880  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='024  decreasing, unstable  KH *  1966-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='098  1980-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='343  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='223  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='015  long series  KOm  1947-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='016  1981-1996  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='198  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='178  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='016  slight drift after 1983  KS2 *  1954-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='069  1981-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='219  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='140  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='009  long series  KZm *  1947-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='026  1981-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='074  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='047  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='011  long series  LFm *  1947-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='136  1981-1988  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='403  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='235  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='023  LK *  1967-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='321  1981-1987  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='457  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='022  short series  LO  1958-1980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='822  1981-2015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='945  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='007  drifts after 1983  MA *  1971-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='039  1981-1988  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='183  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='139  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='015  short series  MD  1978-1980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='988  1981-1986  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='532  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='551  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='068  short series  PO  1950-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='122  1981-2000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='986  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='017  constant ratio  QU  1957-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='411  1981-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='820  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='290  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='019  unstable  SA  1957-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='291  1981-2000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='692  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='311  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='026  long series  SC-S  1960-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='396  1981-2007  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='133  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='812  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='021  decreasing  SK  1950-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='100  1981-2012  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='281  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='165  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='015  unstable  SM  1967-1980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='794  1981-2013  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='031  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='298  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='013  UC *  1949-1980  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='113  1981-2015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='236  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='111  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='012  long series  inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, a more focused study of long-term radio data series is still needed but goes  beyond the scope of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Possible cause of the 1980 jump: historical elements  What could be the cause of this transition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping and Morton (2013) and Tapping (2013) mention  key dates in the history of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio flux production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Regarding the receiver and facility, the  43  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  greatest care was devoted to the calibration of the instrument, which was checked against the same  horn-antenna references (Tapping and Charrois, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The radio-telescope was re-located only  two times, and this happened in 1962 (Ottawa to Algonquin Park) and 1990 (Algonquin Park to  Penticton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' None of those transfers left a detectable trace in the series, and in particular, those dates  do not match the 1980 jump indicated by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  However, Tapping (2013) mentions that in the late 1970’s, the original manual processing of the  daily recordings was replaced by an automated procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This happened around the time when  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Covington, who had initiated the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 standard flux measurements and had directed its pro-  duction continuously since 1947, went on retirement in 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This was thus the very first time a  new team took over the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This new team tried to automate the fully manual method that was  applied until then to empirically eliminate the emission peaks caused by solar flares, but this com-  puterized method was finally abandoned in 1985 because of insufficient reliability (Tapping 2019,  private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The 1979-1985 timing of this change of team and of post-processing method matches quite  closely the moment when we find this scale jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Based on similar disruptive events found in  the history of the Z¨urich sunspot number, we suspect that this transition is the cause of the 1980  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 scale change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Indeed, only two highly dedicated scientists cared for the production of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  index, each one for several decades: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Covington 1947-1979 and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Tapping 1985-present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Only  once, in 1979, there was a transition when the experience and practices developed by Covington  during the first 32 years had to be transmitted to the team that succeeded him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Part of the subtleties  and habits may get lost in such knowledge transfers, especially as the processing procedure was  largely manual until then, and proved to be difficult to convert into a computer algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Moreover, the extraction of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm background flux (the so-called “S” component) requires  the elimination of flare-associated bursts, of radio-frequency interference, and of the sky back-  ground emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, the final index depends on those post-processing steps, and not only  on the proper calibration of the receiver and antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, even if the raw fluxes were always accu-  rately calibrated, this post-processing step may influence the final filtered index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, we can  speculate that the changes in the methods that seem to have occurred just after 1979 could perfectly  have caused this jump in the resulting index, without involving any technical flaw in the instrument  itself and its calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Moreover, our results indicate that there was no slow progressive drift of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux relative  to the sunspot number, but that a scale change occurred abruptly between two constant periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  This is in contradiction with the analysis by Tapping and Morgan (2017), who used a temporal  curve fitting that did not allow to detect such a sharp transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This abrupt transition occurs at the  maximum of cycle 21 and is unique over the last 6 solar cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Such a step-like jump does not  match any known solar event in 1980 that would be unique over the last 70 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Likewise, no  mechanism generating the solar activity cycle can account for such an abrupt discontinuity, which  require intrinsic timescales as short as one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Therefore, we consider that it is very unlikely that the Sun itself induced this sudden change  in the relation between the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm flux and the sunspot number, while a slow trend, as incorrectly  diagnosed by Tapping and Morgan (2017), allowed such an hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' By contrast, the production  process and its historical evolution, retraced above, contain various elements that can induce such a  sudden jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' So, this past history as recorded in archived documents deserve very careful attention,  in order to validate or invalidate this processing issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  44  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Temporal variation of the ratio between the monthly mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux and a constant proxy relation  (4th-order polynomial) (upper panel), after subtracting a 67 sfu quiet-Sun base flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The same ratio for the  yearly mean F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux is shown in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The points near cycle maxima and minima are colored  in blue and red respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The monthly means show a slight solar-cycle modulation, and both curves show  the 1980 jump between two stable periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Solar-cycle and other modulations  As noted for in Section 3, the mean daily F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux for a given S N shows a significant difference  between the minima and maxima of the solar cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This cycle modulation is largest for raw daily  data (Figure 7) and decreases for longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' In order to check if this modulation is affecting  the monthly and yearly means used to build the long-term proxy, we computed the ratio between  monthly or yearly mean values and a single constant fit: the 4th-order polynomial fitted to the entire  series, given in Tables 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Any deviation from a constant relation will appear as a deviation  from unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The result is shown in figure 27, where we marked the periods around maxima and  minima of the solar cycles by blue and red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The plot for the monthly means (Figure 27, upper panel) is dominated by random month-to-  month variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' No annual modulation is present, indicating that the adjusted F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 was accurately  45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='4  tio  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2  Rat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='6  1950  1960  1970  1980  1990  2000  2010  2020  Time (years)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1  Ratic  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  1950  1960  1970  1980  1990  2000  2010  2020  Time (years)Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  converted to the 1AU distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The 1980 jump between two stable intervals is also clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Now considering the solar cycle, a slight modulation can be found: although the ranges of monthly  values around maxima and minima largely overlap, the range near maxima is slightly higher than  around minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The difference is subtle and remains smaller than the other deviations mentioned  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Finally, the ratios for yearly means do not show any cycle modulation (Figure 27, lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The only clear systematic variation is the 1980 jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Virtually all points are below 1 before 1980  and above 1 after 1980, confirming again clearly the jump and the absence of any progressive trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Overall, we can thus confirm that the solar-cycle modulation is playing a significant role only for  timescales shorter than a month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' We can interpret this effect by invoking the same mechanisms as  the ones explaining the change of average background during spotless periods (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As  activity increases, the plage component can contribute to a flux excess when sunspot activity drops  momentarily, because the facular and plages associated to all active regions persist for a much  longer time than the corresponding active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The higher activity prevailing around those dips  in the sunspot number thus prevents the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 to decrease as sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Therefore, the net effect must  always be an excess, which matches the upper tail of the daily distribution in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Near the  minima of cycles, given the small number of active regions, this persisting background is largely  absent, thus giving a smaller plage excess, leading to the observed solar-cycle variation in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  As this temporal smearing effect corresponds to the lifetime of plages, which ranges from weeks to  a few months, it should vanish for long timescales, like we find in our analysis (Figure 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Conclusions  Summarizing, our analysis brings the following conclusions regarding the global F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N proxy  relation:  – No previously published F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 proxy relation is fully satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Existing proxies deviate from  the data points either in the low or high range, though there is a fair agreement in the intermediate  linear range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Those proxies are also lacking error bars, limiting their applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – The F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N relation is fully linear within uncertainties from the lowest to the highest observed  values, when taking raw daily values without any temporal averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 flux only deviates  from the linear relation for S N below 11 (single spot) and even S N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 reaches a lower base  background only when the Sun is spotless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – When working with monthly and yearly means, the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N relation becomes non-linear in the  low range, for S N below 30 to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This non-linearity can be fully explained by the effect of  temporal averaging on daily data consisting of a fully linear relation, plus a lower background  (fully inactive Sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' A F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 proxy relation is thus only valid for a specific temporal averaging of  the base daily data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – A 4th degree polynomial gives the best fit to the monthly mean data, in particular the non-linear  section below S N = 50 down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' A linear function is sufficient for all S N values above about  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  46  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  – We derived standard errors σp on the polynomial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As a direct mathematical error-  propagation calculation does not exist taking into account the inter-dependencies between the  least-square polynomial coefficient, those errors were derived empirically from the data by de-  termining the conditional errors for each separate term of different degree in the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' For  practical applications, we also assembled a simple mathematical formula that closely approxi-  mates those data-based conditional errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  In addition, we derived new properties of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 quiet-Sun background flux:  – This background flux for a spotless Sun depends on the duration of the spotless interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Its mean  value is 68 sfu for long inactive periods, but rises to 74 sfu for a spotless duration of one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' With  only a few isolated exceptions, 67 sfu is the lowest flux value, independently from the spotless  duration, but the mean quiet-Sun background is always higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – Given the actual duration of spotless periods, a temporal averaging over one month is close to  optimal to reflect the lowest range of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 background flux, with 68 or 69 sfu as the lowest mean  background over one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – A F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 excess flux is present in raw daily data and induces a 12% solar-cycle modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The  latter vanishes at long timescales, and is already barely detectable in monthly averages (≈ 3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  This implies that a single standard F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 – S N proxy relation, independent of the phase of the solar  cycle, can only be derived from temporal scales longer than about one month, and will only be  fully accurate for those timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Finally, by checking for any temporal variability of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7/S N relation over the entire duration  of the data series, we found a significant inhomogeneity:  – The F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 series is affected by an upward jump in 1980, separating two stable periods without  other jumps or trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Relative to the SN series, the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 index is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='5% higher after 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Monthly values suggest that the jump occurred at the transition between two calendar years, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  between December 1980 and January 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – In order to exclude a possible flaw on the side of the sunspot number series, which also went  through a methodological transition in 1980–1981, we compared the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 with raw Wolf numbers  from a large sets of independent stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' With only a few exceptions, most of them indicate that  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 becomes higher after 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This thus firmly establishes that the scale jump belongs to the  F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  – The jump is abrupt and makes any interpretation in terms of a true solar effect difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' On the  other hand, this abrupt transition happens close to important changes in the observing team (re-  tirement of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Covington), and when changes were introduced in the post-processing method  (computerization of an originally manual processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Just like the diagnosed jump, this opera-  tional transition is unique in the history of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  This study thus indicates that we can still learn a lot about this fundamental long-term solar-  activity index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The consistent relation between the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 quiet-Sun background and the duration of  the quiet period, the role of temporal averaging on the non-linearity of the proxy relation as well  47  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  as the cycle-dependent excess flux found only in daily data, all invite us to pay more attention to  temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Our results confirm the mixed contribution of two components in F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7, one from  sunspots and another one from weaker magnetic fields primarily in plages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The latter introduces a  time-diluted variation relative to the initial magnetic flux emergence in active regions, recorded by  the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' As this source-mixing in F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 plays a role only at short time-scales, from days to weeks,  it goes beyond the scope of this long-term proxy study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Still, this aspect calls for more attention in  futures analyses of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7, and it should also be kept in mind in all uses of F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 for space-weather or  space-climate applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Given the inhomogeneity in the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 time series that we diagnose here in detail, it is clear that  our new global polynomial proxy does not reproduce optimally the actual relation before and after  1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Until a correction is adopted for the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 data, it must be considered as the best overall  sunspot-based proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The slope and curvature are largely valid for the whole series, but a different  scale factor must be applied for each half of the series: proxy values will be about 5% too high  before 1980 and 5% too low for the more recent years relative to the current version of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The polynomials derived for each half of the series in Table 9 can be used in applications  focusing only on time periods before or after 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  The combined historical evidence indicates that a thorough analysis of the production process of  the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7, in particular in the late 1970’s to early 1980’s, is needed to clarify any possible change,  and potentially to find a self-consistent correction to restore the homogeneity over the entire series  since its beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' If the archived F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7 data prove to be insufficient, our best second option would  be to use the long overlap with the sunspot number series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The latter may provide an even better  reference in the future, as a new re-calibration is in preparation, which could improve in particular  the S N values from the Z¨urich period before 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This work and the team of the World Data Center SILSO, which produces and dis-  tributes the international sunspot number used in this study, are supported by Belgian Solar-Terrestrial Center  of Excellence (STCE) funded by the Belgian Science Policy Office (BelSPo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' This work also partly benefited  from the joint work of the International Team 417 “Recalibration of the Sunspot Number Series”, funded by  the International Space Science Institute (ISSI, Bern, Switzerland) and chaired by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Owens and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Clette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  We also wish to acknowledge the particularly useful, informative and friendly discussions with Ken Tapping,  who is the dedicated guardian and the living memory of the F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm radio flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
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+page_content=' A Comparison of Variations in the Indices of Sunspots and Solar  Radio Emission in the 19th and 20th Solar-Cycles, Solnechnye Dannye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Byul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Glav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
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+page_content=' 1  50  Clette: F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm proxy relation and temporal homogeneity  Wolf, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 1856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mitteilungen ¨uber die Sonnenflecken I, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Mitteil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Eidgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Sterw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Z¨urich, 1, 3-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 1  Xanthakis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' and Poulakos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Long and short term variation of the 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm solar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' The photo-  spheric granules and the Z¨urich numbers, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' and Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 111, 179-188, DOI 0004-640X/85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='  1  Yaya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
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+page_content=', Le F`evre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' and Bruinsma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Solar radio proxies for improved  satellite orbit prediction, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Space Weather Space Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 7, A35, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='1051/swsc/2017032 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='3  Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' and Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=', 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' Historical dataset reconstruction and a prediction method of solar 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='7cm  radio flux, Chinese J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' of Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' and Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
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+page_content=' http://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content='org/1009-9271/  8/472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
+page_content=' 1  51' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf'}
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+Cuscuta-Galileon Braneworlds
+D. Bazeia,1 A.S. Lob˜ao Jr.,2 M.A. Marques,3 and R. Menezes4
+1Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970 Jo˜ao Pessoa, PB, Brazil
+2Escola T´ecnica de Sa´ude de Cajazeiras, Universidade Federal de Campina Grande, 58900-000 Cajazeiras, PB, Brazil
+3Departamento de Biotecnologia, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, PB, Brazil
+4Departamento de Ciˆencias Exatas, Universidade Federal da Para´ıba, 58297-000 Rio Tinto, PB, Brazil
+We investigate braneworlds modeled by topological solutions that arise from the so-called Cuscuta-
+Galileon model. We develop a first order framework and illustrate our procedure with the scalar
+field having the well-known hyperbolic tangent profile. We find conditions that must be imposed to
+the parameters of the model in order to have solutions connecting minima of the potential, with the
+brane constrained to interpolate Minkowski and anti de Sitter geometries. We also find solutions
+where the brane only interpolates anti de Sitter geometry. In both cases, the gravity sector of the
+brane is stable against small fluctuations of the metric.
+Braneworld models arise in theories of gravity in (4, 1)
+spacetime dimensions. It was conceived in 1999 as a ten-
+tative to explain the hierarchy problem [1]. The original
+model supports the so-called thin brane, with the deriva-
+tive of the warp factor having a discontinuity at its center.
+By including scalar fields in the action, it was shown in
+Ref. [2–5] that a kinklike solution can give rise to thick
+branes. In the presence of scalar fields, it is known that
+modifications in the dynamics may generate interesting
+changes in the brane profile. In this, direction, several
+studies have addressed this issue over the years; see, e.g.,
+Refs. [6–11] and references therein. Among the many
+possibilities in the study of braneworlds, one may find
+the presence of asymmetric structures; see, for instance,
+Refs. [12–23].
+The asymmetry can appear in different
+ways, for instance, one may have it as a consequence of
+interpolation of distinct geometries (see Ref. [12]).
+It
+may also be originated from the internal structure of the
+scalar field [22] or due to the asymptotic behavior of the
+solutions [23].
+Braneworlds may also be investigated in noncanon-
+ical models, in which the Lagrange density associated
+to the scalar field is a general function of the field and
+the kinetic term depending on its first derivative [24–
+27]. A particular model that has been gaining attention
+is the Cuscuton one, introduced in Ref. [28, 29] in the
+context of cosmology. Since then, several papers deal-
+ing with the Cuscuton term have appeared in the litera-
+ture; see Refs. [30–37]. We can also modify the dynam-
+ics by including second order derivatives in the fields in
+the form ∇a∇bφ. This prescription is generally known
+as Horndeski theories or generalized Galileon theories
+[38–43].
+In general, these theories obey the symmetry
+φ → φ + bµxµ + c, where bµ is the constant vector and
+c is a real number. Horndeski theories have been much
+investigated as they have led to new and distinguishable
+inflationary predictions; see, for example [44–48].
+In recent studies, it was also considered the inclusion of
+both kinematic modifications presented above, with ki-
+netic terms depending on the first and the second deriva-
+tives of the field. This is the basis behind the so-called
+Cuscuta-Galileon model, where both the Cuscuton and
+Galileon-like terms are included in the action simultane-
+ously [49–52].
+Horndeski theories have been relatively
+successful in describing aspects of the accelerated ex-
+pansion of the Universe, and the recent results of Refs.
+[51, 52] which nicely provide interesting sequence of the
+thermal cosmological history, have motivated us to in-
+vestigate the possibility to construct braneworld model
+based on the Cuscuta-Galileon dynamics.
+We start the present investigation by considering an
+action that describes a thick brane model in five dimen-
+sions of the spacetime sourced by a single real scalar field
+in the form
+S =
+�
+d5x
+�
+|g|
+�
+−1
+4R + Ls(φ, X)
+�
+,
+(1)
+where g is the determinant of the metric, R is the
+Ricci scalar, Ls(φ, X) is the Lagrange density and X ≡
+(1/2)∇aφ∇aφ represents the dynamical term associated
+to scalar field φ. In this paper, Latin indexes a, b, c run
+from 0 to 4 and Greek indexes µ, ν run from 0 to 3.
+The simplest Lagrange density that support stable
+braneworld configuration is Ls = X −V (φ). In this case,
+the brane can be modeled by a kinklike solution which
+connects the minima of the potential V (φ) [4]. As we
+have commented before, one may consider the inclusion
+of the Cuscuton term [28, 29], with
+Ls = X + f(φ)
+2X
+�
+|2X|
+− V (φ).
+(2)
+This was first investigated in Ref. [36]. There, to keep
+the minima of the potential connected by the solution, it
+was considered a function f(φ) that goes to zero in the
+asymptotic limits of the solution. In this paper, we take a
+novel approach: inspired by the Cuscuta-Galileon model
+investigated in Ref. [51, 52], we consider the inclusion
+of a Horndeski-like term in Eq. (2) and take f(φ) = β,
+where β is constant. The new Lagrange density has the
+form
+Ls = X + β
+2X
+�
+|2X|
++ α ln(|X|)□φ − V (φ).
+(3)
+arXiv:2301.11750v1  [hep-th]  26 Jan 2023
+
+2
+In this expression □ ≡ ∇a∇a, so our model has now the
+dynamics depending on a second derivative of the field.
+By varying the action with respect to the metric we get
+the equation
+Rab − 1
+2gabR = 2Tab,
+(4)
+where the energy momentum tensor is given by
+Tab =
+�
+1 +
+β
+�
+|2X|
+�
+∇aφ∇bφ + α
+X
+�
+□φ∇aφ∇bφ
+− ∇aX∇bφ − ∇bX∇aφ
+�
+− gab
+�
+X + β
+2X
+�
+|2X|
+− V − α
+X ∇cX∇cφ
+�
+.
+(5)
+The equation of motion that arises from the variation of
+the action with respect to the scalar field is
+− ∇a
+� �
+1 +
+β
+�
+|2X|
+�
+∇aφ
+�
+− ∇a
+� α
+X □φ∇aφ
+�
++ ∇a
+� α
+X ∇aX
+�
+= Vφ.
+(6)
+By using the above equation, one can show that the en-
+ergy momentum tensor is conserved, i.e., ∇aT ab = 0.
+Since we are interested to study braneworld scenario, we
+consider the line element in the form
+ds2 = e2Aηµνdxµdxν − dy2 ,
+(7)
+where A is the warp function, ηµν is the four-dimensional
+Minkowski metric with signature (+, −, −, −) and y is the
+extra dimension. In order to obtain localized solutions
+we consider static configurations assuming that the warp
+function and the scalar field only depend on the extra
+dimension y, so that A = A(y) and φ = φ(y).
+This
+makes X = −φ′2/2 such that Eq. (6) become
+φ′′ + 4A′φ′ + 4βA′sgn(φ′) + 8α
+�
+A′′ + 4A′2�
+= Vφ.
+(8)
+Here, Vφ = dV/dφ and the prime stands for the deriva-
+tive with respect to the extra dimension, i.e., φ′ = dφ/dy,
+A′ = dA/dy, etc.
+It is interesting to note that the
+above equation is of second-order, as in the usual Galileon
+model. Regarding the Einstein equation (4), only two
+components survive, leading to
+3A′′ = −2φ′2 − 2β|φ′| + 4α (φ′′ − 4A′φ′) ,
+(9a)
+1
+2φ′2 − 3A′2 + 8αA′φ′ = V.
+(9b)
+It is possible to show that from the three equations (8)
+and (9), only two are independent.
+We can then deal
+with the system of equations (9a) and (9b), noticing that
+these equations are invariant under the change y → −y.
+For simplicity, we consider only monotonically increasing
+solutions for φ, having a kinklike profile.
+We then investigate how the brane behaves for solu-
+tions with exponential tails. To do so, we consider that,
+at y → ±∞, the scalar field obeys
+φ − v± = κ± e−m±|y|,
+(10)
+where v± denotes the asymptotic values of the solution,
+φ(y), at y → ±∞, and κ± and m± > 0 are constants
+which depend on the specific model under investigation.
+In the standard case (α = β = 0), Eqs. (9) read
+3A′′ = −2φ′2,
+(11a)
+V = 1
+2φ′2 − 3A′2.
+(11b)
+Substituting Eq. (10) in Eq. (11a), we get
+A′ = A′
+± ± m±κ2
+±
+3
+e−2m±|y|,
+(12)
+in which A′+ and A′− are both constants of integration
+that represent the asymptotic values of the derivative of
+the warp function at y → ±∞. So, if A′+ = 0 (A′− = 0)
+the brane engender a Minkowski geometry at y → +∞
+(y → −∞). On the other hand, if A′+ < 0 (A′− > 0), we
+have an anti de Sitter (AdS) geometry. To find how the
+potential behaves, we can use Eq. (11b) to get V (v±) =
+−3A′2
+±, Vφ(v±) = 0 and
+Vφφ(v±) = m±(m± ∓ 4A′
+±).
+(13)
+These expressions ensures that the solution connects crit-
+ical points of the potential. Moreover, these points are
+minima of the potential whose values are negative for
+AdS and null for Minkowski geometries. With the in-
+clusion of the Cuscuton term in the Lagrange density, as
+in Eq. (2), the condition Vφ(v±) = 0 is not ensured for
+constant f(φ) and A′± ̸= 0. To remedy this possibility,
+one takes advantage of the Galileon-like term in Eq. (3),
+as we shall see below.
+For general α and β, considering the exponential falloff
+in Eq. (10), we get from Eq. (9a) that the warp function
+behaves asymptotically (y → ±∞) as
+A′ = A′
+± ∓ 2κ±
+3
+(2αm± ± (8αA′
+± + β)) e−m±|y|
+± κ2
+±
+9
+��
+3+32α2�
+m±±16α (8αA′
+±+β)
+�
+e−2m±|y|.
+(14)
+In contrast to the standard case, we now have a contri-
+bution of terms that engender exponential decay. The
+behavior of the potential can be found through Eq. (9b),
+which leads us to the following expressions
+V (v±) = −3A′2
+±,
+(15a)
+Vφ(v±) = 4A′
+±(8αA′
+± + β),
+(15b)
+Vφφ(v±) = 1
+3
+�
+3 + 32α2�
+m± (m± ∓ 4A′
+±)
+− 8
+3 (16αA′
+± + β) (8αA′
+± + β) .
+(15c)
+
+3
+Let us first suppose that v± are critical points of the
+potential. This implies that Vφ(v+) = 0 and Vφ(v−) = 0,
+requiring the need of restrictions on the asymptotic be-
+havior of the warp function, which can be found from
+Eq. (15b). This also implies that one cannot obtain a
+solution connecting AdS geometries. Inevitably, one of
+the tails of the solution has to connect a Minkowski ge-
+ometry (A′+ = 0 or A′− = 0). We consider the situation
+where A′− = 0, and A′+ = −β/8α which gives us a M-
+AdS brane, requiring the parameters α and β to have the
+same sign.
+We then look into each side of the brane, asymptoti-
+cally. At the left tail, in which we have taken A′− = 0,
+we have from Eq. (15) that V (v−) = 0, Vφ(v−) = 0 and
+Vφφ(v−) = 1
+3
+�
+3 + 32α2�
+m2
+− − 8
+3β2.
+(16)
+On the other hand, at the right tail, where we consid-
+ered A′+ = −β/8α, we have the potential behaving as
+V (v+) = −3β2/64α2, Vφ(v+) = 0, and
+Vφφ(v+) = 1
+3
+�
+3 + 32α2�
+m+
+�
+m+ + β
+2α
+�
+.
+(17)
+From the above expression, as α and β must have
+the same sign, v+ defines a minimum of the potential.
+Notwithstanding that, the same cannot be stated about
+v−, which may not lead to a minimum of the potential,
+depending on the values of α, m− and β. To ensure that
+it is a minimum, one must choose the parameters care-
+fully.
+As we have found the conditions to make the solution
+connect minima of the potential, we now investigate a
+procedure to reduce Eq. (9a) to first order. To do so, we
+introduce an auxiliary function h = h(φ) that obeys the
+equation
+φ′ = hφ e−16αφ/3.
+(18)
+The above expression can be used in Eq. (9a), which can
+be integrated to give
+A′ =− β
+8α + 4
+3αφ′− 2
+9
+�
+3+32α2� �
+h+˜h0
+�
+e−16αφ/3,
+(19)
+where ˜h0 is an integration constant. This, combined with
+Eq. (9b), allows us to find the explicit form of the poten-
+tial as a function of the scalar field, V (φ).
+To illustrate our procedure, we work with the model
+described by
+φ′ = λ(1 − φ2),
+(20)
+where λ is a positive real parameter. This equation is
+solved by the function
+φ(y) = tanh(λy),
+(21)
+which connects the values v± = ±1 that define local
+minima of the potential.
+This solution engenders the
+same asymptotic behavior of Eq. (10), with κ± = 2
+and m± = 2λ.
+To calculate the auxiliary function
+h(φ) in this case, we can use Eqs. (18) and (20) to get
+hφ = λe16αφ/3(1 − φ2), which can be integrated to give
+h(φ)=
+3λ
+2048α3
+�
+128α2�
+1 − φ2�
++ 48αφ − 9
+�
+e16αφ/3.
+(22)
+By using this in Eq. (19), one can show that the warp
+function becomes
+A′ = 9λ−32α2(4β+λ)
+1024α3
+−
+λ
+64α2
+�
+3 + 32α2�
+φ
++ λφ2
+8α − h0 e−16αφ/3,
+(23)
+where we redefined the integration constant in a conve-
+nient form, as h0 = 2
+�
+3+32α2� ˜h0/9. The above expres-
+sion can be combined with Eq. (20) in Eq. (9b) to calcu-
+late the potential, V (φ). Its expression is cumbersome,
+so we omit it here.
+Asymptotically for y → ±∞, the derivative of the warp
+function given above behaves as
+A′
+± = − β
+8α + λ
+�
+3 + 32α2�
+(3 ∓ 16α)
+1024α3
+−h0 e∓16α/3. (24)
+Let us first consider the case with h0 = 0.
+For this
+choice, if we want to get a solution that goes from a
+Minkowski geometry at y → −∞ to an AdS one at
+y → +∞, we need to impose that A′
+−
+=
+0, which
+leads us to λ = 128α2β/
+�
+(3 + 16α)
+�
+3 + 32α2��
+, and
+also A′
++ = −4β/ (3 + 16α).
+These conditions makes
+Vφ(v−) = 0, as we have shown before, and
+Vφ(v+) = 16(16α − 3)β2
+(16α + 3)2
+.
+(25)
+Note that, in order to have Vφ(v+) = 0, we must choose
+α = 3/16. However, this restriction leads to
+Vφφ(v−) = 50β2
+33
+and
+Vφφ(v+) = −82β2
+33 .
+(26)
+These expressions shows that φ = v− is a minimum point
+and φ = v+ is a maximum point of the potential. There-
+fore, the case h0 = 0 does not allow for the point φ = v+
+being a minimum of the potential.
+To circumvent this issue, let us now consider the gen-
+eral case in which h0 ̸= 0. We take A′
+− = 0 in Eq. (24)
+and obtain that the integration constant must be
+h0 = e−16α/3�
+512α3λ−128α2β+96α2λ+48αλ+9λ
+�
+1024α3
+.(27)
+As expected from the discussion above Eq. (16), it leads
+to V (v−) = Vφ(v−) = 0. To get a brane with an AdS
+geometry at its right tail (y → +∞), we consider A′
++ < 0
+and Vφ(v+) = 0, which can be attained by taking
+λ = −
+64e−16α/3α2β
+(3 + 32α2)
+�
+3 sinh
+� 16α
+3
+�
+− 16α cosh
+� 16α
+3
+��. (28)
+
+4
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+FIG. 1: The potential for the M-AdS brane, for some values
+of β and α, with h0 and λ given by Eqs.
+(27) and (28).
+In the top panel, we show the potential for β = −0.1 and
+α = −0.025, −0.02 and −0.015.
+In the bottom panel, we
+display the potential for β = 0.1 and α = 0.015, 0.02 and
+0.025. In both panels, the width of the lines increases with α.
+At φ = v+, the potential has the value V (v+) =
+−3β2/64α2, as expected from the expression right above
+Eq. (17). This is always negative, as it usually occurs for
+AdS geometries. In this situation, from Eq. (17) we also
+have
+Vφφ(v+) = 2
+3
+�
+3 + 32α2�
+λ
+�
+2λ + β
+2α
+�
+,
+(29)
+with λ as in Eq. (28).
+From this expression, one can
+show that v+ = 1 is a minimum of the potential, match-
+ing with the discussion below Eq. (17) for solutions with
+exponential tails. On the other hand, we have
+Vφφ(v−) = 4
+3
+�
+3 + 32α2�
+λ2 − 8
+3β2,
+(30)
+with λ as in Eq. (28), again. So, to ensure that v− = −1
+is a minimum of the potential, we impose the conditions
+α < 0 and 0 < α < α∗, where α∗ = 0.096 is obtained nu-
+merically. When choosing α and β, one must remember
+that these parameters must have the same sign.
+Considering the conditions obtained above, we display,
+in Fig. 1, the potential V (φ) in the interval φ ∈ [−2, 2]
+for some values of β and α. Notice that the minimum at
+v− = −1 is also a zero whilst the other one, at v+ = 1,
+is not, attaining the value V (v+) = −3β2/64α2 as given
+above Eq. (17). One can show that V (±∞) → −∞, so it
+has maxima outside the range considered in this figure.
+To calculate the warp function, we combine the solu-
+-4
+-2
+0
+2
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+-4
+-2
+0
+2
+4
+0.0
+0.5
+1.0
+1.5
+FIG. 2: The warp factor e2A(y) associated to the warp func-
+tion in Eq. (32), depicted with the same parameters of Fig.
+1.
+tion in Eq. (21) with Eq. (23) to get
+A′ = 9λ−32α2(4β+λ)
+1024α3
+−
+λ
+64α2
+�
+3+32α2�
+tanh(λy)
++ λ
+8α tanh2(λy) − h0 e−16α tanh(λy)/3,
+(31)
+where h0 is given by Eq. (27) and λ is as in Eq. (28). By
+integrating it, we get the expression
+A(y)= 3+32α2
+64α2
+ln(sech(λy))+
+�
+32α2(3λ−4β)+9λ
+�
+y
+1024α3
+− tanh(λy)
+8α
++ h0
+2λ
+�
+e− 16α
+3 Ei(ξ−)−e
+16α
+3 Ei(ξ+)
+�
+− h0
+2λ
+�
+e− 16α
+3 Ei(16α/3)−e
+16α
+3 Ei(−16α/3)
+�
+,
+(32)
+where ξ± = −(16α/3)(tanh(λy) ± 1) and Ei denotes the
+Exponential Integral function. Here we have used that
+A(0) = 0.
+In Fig. 2, we display the warp factor e2A(y) for the
+warp function in Eq. (32) for some values of β and α.
+We see that e2A(−∞) > 0 and e2A(∞) = 0, showing that
+our solution describes a M-AdS brane.
+We can further study the system, to investigate the
+situation where the warp factor connects two AdS ge-
+ometries, with e2A(±∞) = 0. This is the AdS-AdS brane,
+and it can be implemented with the same Eq. (32), but
+now the constraints in the parameters h0 and λ and in
+Eqs. (27) and (28) have to be discarded. The results are
+depicted in Fig. 3, showing the potential and warp factor
+for some values of α, β, λ and h0. In this case, the scalar
+field solution does not connect minima of the potential
+
+5
+-1
+0
+1
+2
+3
+4
+5
+-20
+-10
+0
+10
+20
+30
+40
+-4
+-2
+0
+2
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 3: In the top panel we display the potential V (φ) for
+the AdS-AdS brane, with β = λ = 1, h0 = −0.1 and α =
+0.2, 0.3 and 0.4.
+In the bottom panel, we depict its warp
+factor associated to the warp function in Eq. (32) for the
+same values of parameters. In both panels, the width of the
+lines increases with α.
+and the brane is asymmetric, due to the presence of the
+Cuscuta-Galileon dynamics.
+Let us now investigate linear stability of the gravity
+sector of the brane considering
+ds2 = e2A(y)�
+ηµν + hµν(xµ, y)
+�
+dxµdxν − dy2,
+(33)
+in which hµν denotes the small fluctuations, where we
+have followed the lines of Ref. [53] to take the axial gauge,
+ha4 = 0, that is convenient for the study and depends on
+the extra dimension and on the 4-dimensional vector xµ.
+We also take small perturbations around the scalar field
+solution, in the form φ → φ(y) + ξ(xµ, y), where φ(y)
+denotes the static solution of Eqs. (9). These fluctuations
+can be inserted into Einstein’s equation (4), which leads
+us to three non null components labeled by G(1)
+µ4 = 2T (1)
+µ4 ,
+G(1)
+44 = 2T (1)
+44 and G(1)
+µν = 2T (1)
+µν , where G(1)
+ab is the term of
+the Einstein tensor that contains the fluctuations at first
+order. In particular, the latter component leads us to
+h′′
+µν + 4A′h′
+µν − e−2A□hµν
++ e−2A�
+∂µ∂γhγν + ∂ν∂γhγµ − ∂µ∂νh
+�
+= ηµν
+�
+h′′ + e−2A�
+∂γ∂σhγσ − □h
+�
++ 4A′h′ + 4Vφξ + 4φ′ξ′ − 8αξ′′�
+,
+(34)
+where we have defined h = ηµνhµν. We then proceed
+as usual and take the transverse and traceless (TT) con-
+dition for hµν, i.e. ∂µhµν = 0 and h = 0. This makes
+-3
+-2
+-1
+0
+1
+2
+3
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-3
+-2
+-1
+0
+1
+2
+3
+-1.0
+-0.5
+0.0
+0.5
+1.0
+FIG. 4: The stability potential for the warp factor of the
+M-AdS brane, depicted with the same parameters of Fig. 1.
+-3
+-2
+-1
+0
+1
+2
+3
+-2
+-1
+0
+1
+2
+FIG. 5: The stability potential for the warp factor of the
+AdS-AdS brane, depicted with the same parameters of Fig.
+3.
+all the components vanish, such that only the one in the
+above equation survives. Also, we take dy = eAdz and
+hµν(xµ, z) = eiωµxµe−3A(z)/2Hµν(z) to get the following
+Schr¨odinger-like eigenvalue equation
+�
+− d2
+dz2 + U(z)
+�
+Hµν = ω2Hµν,
+(35)
+where the stability potential is given exclusively in terms
+of derivatives of the warp function, in the form
+U(z) = 9
+4A2
+z + 3
+2Azz.
+(36)
+In Fig. 4, we display the above stability potential when
+the warp factor connects M and AdS geometries, for
+some values of α and β. In Fig. 5, we display the sta-
+bility potential for the case where the warp factor con-
+nects two AdS geometries, using β = λ = 1, h0 = −0.1
+and α = 0.2, 0.3 and 0.4.
+One can show that the
+operator in the left hand side of Eq. (35) can be writ-
+ten as the product operators, S†S Hµν = ω2 Hµν, where
+
+6
+S = −d/dz−3Az/2 and S† = d/dz−3Az/2. This factor-
+ization ensures the linear stability of the gravity sector
+of the brane, as the operator associated to the eigenvalue
+equation (35) is non negative, implying that ω2 ≥ 0.
+The localization of gravity in the braneworld scenario was
+previously investigated in several works, in particular, in
+Refs. [1, 20, 54]. For solutions of the AdS-AdS brane,
+the 4-dimensional Newtonian potential is given by the
+sum of contributions of the zero mode and the correction
+term associated to the exchange of Kaluza-Klein modes
+[1].
+In the case in which one has the interpolation of
+M-AdS geometries, the Newtonian potential arises due
+to the correction term alone. In both cases, the correct
+4-dimensional Newtonian behavior can be recovered at
+short distances [20, 54].
+In summary,
+we have investigated the Cuscuta-
+Galileon model described by the Lagrange density in
+Eq. (3). The Cuscuton term in Eq. (2) engender a func-
+tion which only depends on the scalar field, f(φ).
+If
+f(φ) is constant, the model (2) cannot support scalar
+field solutions connecting minima of the potential V (φ).
+However, after adding the Galileon term in Eq. (3), we
+have shown that it conspires against the Cuscuton term
+in the Einstein’s equations, leaving room to construct
+interesting solutions that connect minima of the poten-
+tial. Among the results, we recall that we were interested
+in having the minima connected by the solutions, so we
+have first investigated the behavior of the brane asymp-
+totically, with the scalar field having exponential tails,
+as in Eq. (10).
+This led us to find conditions for the
+parameters α, β and m−, which must be chosen care-
+fully. In particular, α and β must have the same sign.
+Moreover, since the investigation involves nonlinear dif-
+ferential equations of second order, we have introduced
+an auxiliary function h(φ) that allows for the presence
+of first order equations compatible with the Einstein’s
+ones.
+To illustrate our procedure, we have taken the
+well-known hyperbolic tangent profile in Eq. (21) and
+found the warp factor associated to the brane that inter-
+polates a Minkowski (M) and an Anti de-Sitter (AdS)
+geometry. We have also investigated the stability of the
+gravity sector of the brane and shown that the operator
+that governs the eigenvalue equation can be factorized,
+ensuring linear stability.
+There are several perspectives, in particular, the case
+concerning asymmetry of the brane and the cosmic ac-
+celeration [55, 56]. The Cuscuta-Galileon model (3) can
+also be investigated in other scenarios of generalized the-
+ories of gravity [57–62].
+The study of time-dependent
+configurations in the five-dimensional spacetime is also
+of interest, as it can shed light on the number of degrees
+of freedom that propagates in the theory [63, 64]. An-
+other possibility concerns absence of gravity, in the case
+of flat spacetime, in the study of kinks [65], vortices [66]
+and monopoles [67, 68]. Some of these issues are cur-
+rently being considered and shall be reported elsewhere.
+This work is supported by the Brazilian agencies Con-
+selho Nacional de Desenvolvimento Cient´ıfico e Tec-
+nol´ogico (CNPq), grants No. 303469/2019-6 (DB) and
+No. 310994/2021-7 (RM), Paraiba State Research Foun-
+dation (FAPESQ-PB), grants No. 0003/2019 (RM) and
+No.
+0015/2019 (DB and MAM), and Federal Univer-
+sity of Para´ıba (UFPB/PROPESQ/PRPG) project code
+PII13363-2020.
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+[66] H.B. Nielsen, P. Olesen, Nucl. Phys. B 61, 45 (1973).
+[67] G.’t Hooft, Nucl. Phys. B 79, 276 (1974).
+[68] A.M. Polyakov, JETP Lett. 20, 194 (1974).
+
diff --git a/y9FKT4oBgHgl3EQfMS25/content/tmp_files/load_file.txt b/y9FKT4oBgHgl3EQfMS25/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='Cuscuta-Galileon Braneworlds  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Bazeia,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Lob˜ao Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=',2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Marques,3 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Menezes4  1Departamento de F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Universidade Federal da Para´ıba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 58051-970 Jo˜ao Pessoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' PB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Brazil  2Escola T´ecnica de Sa´ude de Cajazeiras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Universidade Federal de Campina Grande,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 58900-000 Cajazeiras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' PB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Brazil  3Departamento de Biotecnologia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Universidade Federal da Para´ıba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 58051-900 Jo˜ao Pessoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' PB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Brazil  4Departamento de Ciˆencias Exatas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Universidade Federal da Para´ıba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 58297-000 Rio Tinto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' PB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Brazil  We investigate braneworlds modeled by topological solutions that arise from the so-called Cuscuta-  Galileon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We develop a first order framework and illustrate our procedure with the scalar  field having the well-known hyperbolic tangent profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We find conditions that must be imposed to  the parameters of the model in order to have solutions connecting minima of the potential, with the  brane constrained to interpolate Minkowski and anti de Sitter geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We also find solutions  where the brane only interpolates anti de Sitter geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In both cases, the gravity sector of the  brane is stable against small fluctuations of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Braneworld models arise in theories of gravity in (4, 1)  spacetime dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' It was conceived in 1999 as a ten-  tative to explain the hierarchy problem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' The original  model supports the so-called thin brane, with the deriva-  tive of the warp factor having a discontinuity at its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  By including scalar fields in the action, it was shown in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [2–5] that a kinklike solution can give rise to thick  branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In the presence of scalar fields, it is known that  modifications in the dynamics may generate interesting  changes in the brane profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In this, direction, several  studies have addressed this issue over the years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=',  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [6–11] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Among the many  possibilities in the study of braneworlds, one may find  the presence of asymmetric structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' see, for instance,  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [12–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  The asymmetry can appear in different  ways, for instance, one may have it as a consequence of  interpolation of distinct geometries (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  It  may also be originated from the internal structure of the  scalar field [22] or due to the asymptotic behavior of the  solutions [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Braneworlds may also be investigated in noncanon-  ical models, in which the Lagrange density associated  to the scalar field is a general function of the field and  the kinetic term depending on its first derivative [24–  27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' A particular model that has been gaining attention  is the Cuscuton one, introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [28, 29] in the  context of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Since then, several papers deal-  ing with the Cuscuton term have appeared in the litera-  ture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [30–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We can also modify the dynam-  ics by including second order derivatives in the fields in  the form ∇a∇bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This prescription is generally known  as Horndeski theories or generalized Galileon theories  [38–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In general, these theories obey the symmetry  φ → φ + bµxµ + c, where bµ is the constant vector and  c is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Horndeski theories have been much  investigated as they have led to new and distinguishable  inflationary predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' see, for example [44–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In recent studies, it was also considered the inclusion of  both kinematic modifications presented above, with ki-  netic terms depending on the first and the second deriva-  tives of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This is the basis behind the so-called  Cuscuta-Galileon model, where both the Cuscuton and  Galileon-like terms are included in the action simultane-  ously [49–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Horndeski theories have been relatively  successful in describing aspects of the accelerated ex-  pansion of the Universe, and the recent results of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [51, 52] which nicely provide interesting sequence of the  thermal cosmological history, have motivated us to in-  vestigate the possibility to construct braneworld model  based on the Cuscuta-Galileon dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  We start the present investigation by considering an  action that describes a thick brane model in five dimen-  sions of the spacetime sourced by a single real scalar field  in the form  S =  �  d5x  �  |g|  �  −1  4R + Ls(φ, X)  �  ,  (1)  where g is the determinant of the metric, R is the  Ricci scalar, Ls(φ, X) is the Lagrange density and X ≡  (1/2)∇aφ∇aφ represents the dynamical term associated  to scalar field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In this paper, Latin indexes a, b, c run  from 0 to 4 and Greek indexes µ, ν run from 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  The simplest Lagrange density that support stable  braneworld configuration is Ls = X −V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In this case,  the brane can be modeled by a kinklike solution which  connects the minima of the potential V (φ) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' As we  have commented before, one may consider the inclusion  of the Cuscuton term [28, 29], with  Ls = X + f(φ)  2X  �  |2X|  − V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (2)  This was first investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' There, to keep  the minima of the potential connected by the solution, it  was considered a function f(φ) that goes to zero in the  asymptotic limits of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In this paper, we take a  novel approach: inspired by the Cuscuta-Galileon model  investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [51, 52], we consider the inclusion  of a Horndeski-like term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (2) and take f(φ) = β,  where β is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' The new Lagrange density has the  form  Ls = X + β  2X  �  |2X|  + α ln(|X|)□φ − V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (3)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='11750v1  [hep-th]  26 Jan 2023  2  In this expression □ ≡ ∇a∇a, so our model has now the  dynamics depending on a second derivative of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  By varying the action with respect to the metric we get  the equation  Rab − 1  2gabR = 2Tab,  (4)  where the energy momentum tensor is given by  Tab =  �  1 +  β  �  |2X|  �  ∇aφ∇bφ + α  X  �  □φ∇aφ∇bφ  − ∇aX∇bφ − ∇bX∇aφ  �  − gab  �  X + β  2X  �  |2X|  − V − α  X ∇cX∇cφ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (5)  The equation of motion that arises from the variation of  the action with respect to the scalar field is  − ∇a  � �  1 +  β  �  |2X|  �  ∇aφ  �  − ∇a  � α  X □φ∇aφ  �  + ∇a  � α  X ∇aX  �  = Vφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (6)  By using the above equation, one can show that the en-  ergy momentum tensor is conserved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=', ∇aT ab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Since we are interested to study braneworld scenario, we  consider the line element in the form  ds2 = e2Aηµνdxµdxν − dy2 ,  (7)  where A is the warp function, ηµν is the four-dimensional  Minkowski metric with signature (+, −, −, −) and y is the  extra dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In order to obtain localized solutions  we consider static configurations assuming that the warp  function and the scalar field only depend on the extra  dimension y, so that A = A(y) and φ = φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  This  makes X = −φ′2/2 such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (6) become  φ′′ + 4A′φ′ + 4βA′sgn(φ′) + 8α  �  A′′ + 4A′2�  = Vφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (8)  Here, Vφ = dV/dφ and the prime stands for the deriva-  tive with respect to the extra dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=', φ′ = dφ/dy,  A′ = dA/dy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  It is interesting to note that the  above equation is of second-order, as in the usual Galileon  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Regarding the Einstein equation (4), only two  components survive, leading to  3A′′ = −2φ′2 − 2β|φ′| + 4α (φ′′ − 4A′φ′) ,  (9a)  1  2φ′2 − 3A′2 + 8αA′φ′ = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (9b)  It is possible to show that from the three equations (8)  and (9), only two are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  We can then deal  with the system of equations (9a) and (9b), noticing that  these equations are invariant under the change y → −y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  For simplicity, we consider only monotonically increasing  solutions for φ, having a kinklike profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  We then investigate how the brane behaves for solu-  tions with exponential tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' To do so, we consider that,  at y → ±∞, the scalar field obeys  φ − v± = κ± e−m±|y|,  (10)  where v± denotes the asymptotic values of the solution,  φ(y), at y → ±∞, and κ± and m± > 0 are constants  which depend on the specific model under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In the standard case (α = β = 0), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9) read  3A′′ = −2φ′2,  (11a)  V = 1  2φ′2 − 3A′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (11b)  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (10) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (11a), we get  A′ = A′  ± ± m±κ2  ±  3  e−2m±|y|,  (12)  in which A′+ and A′− are both constants of integration  that represent the asymptotic values of the derivative of  the warp function at y → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' So, if A′+ = 0 (A′− = 0)  the brane engender a Minkowski geometry at y → +∞  (y → −∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' On the other hand, if A′+ < 0 (A′− > 0), we  have an anti de Sitter (AdS) geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' To find how the  potential behaves, we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (11b) to get V (v±) =  −3A′2  ±, Vφ(v±) = 0 and  Vφφ(v±) = m±(m± ∓ 4A′  ±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (13)  These expressions ensures that the solution connects crit-  ical points of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Moreover, these points are  minima of the potential whose values are negative for  AdS and null for Minkowski geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' With the in-  clusion of the Cuscuton term in the Lagrange density, as  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (2), the condition Vφ(v±) = 0 is not ensured for  constant f(φ) and A′± ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' To remedy this possibility,  one takes advantage of the Galileon-like term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (3),  as we shall see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  For general α and β, considering the exponential falloff  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (10), we get from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9a) that the warp function  behaves asymptotically (y → ±∞) as  A′ = A′  ± ∓ 2κ±  3  (2αm± ± (8αA′  ± + β)) e−m±|y|  ± κ2  ±  9  ��  3+32α2�  m±±16α (8αA′  ±+β)  �  e−2m±|y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (14)  In contrast to the standard case, we now have a contri-  bution of terms that engender exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' The  behavior of the potential can be found through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9b),  which leads us to the following expressions  V (v±) = −3A′2  ±,  (15a)  Vφ(v±) = 4A′  ±(8αA′  ± + β),  (15b)  Vφφ(v±) = 1  3  �  3 + 32α2�  m± (m± ∓ 4A′  ±)  − 8  3 (16αA′  ± + β) (8αA′  ± + β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (15c)  3  Let us first suppose that v± are critical points of the  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This implies that Vφ(v+) = 0 and Vφ(v−) = 0,  requiring the need of restrictions on the asymptotic be-  havior of the warp function, which can be found from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (15b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This also implies that one cannot obtain a  solution connecting AdS geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Inevitably, one of  the tails of the solution has to connect a Minkowski ge-  ometry (A′+ = 0 or A′− = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We consider the situation  where A′− = 0, and A′+ = −β/8α which gives us a M-  AdS brane, requiring the parameters α and β to have the  same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  We then look into each side of the brane, asymptoti-  cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' At the left tail, in which we have taken A′− = 0,  we have from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (15) that V (v−) = 0, Vφ(v−) = 0 and  Vφφ(v−) = 1  3  �  3 + 32α2�  m2  − − 8  3β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (16)  On the other hand, at the right tail, where we consid-  ered A′+ = −β/8α, we have the potential behaving as  V (v+) = −3β2/64α2, Vφ(v+) = 0, and  Vφφ(v+) = 1  3  �  3 + 32α2�  m+  �  m+ + β  2α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (17)  From the above expression, as α and β must have  the same sign, v+ defines a minimum of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Notwithstanding that, the same cannot be stated about  v−, which may not lead to a minimum of the potential,  depending on the values of α, m− and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' To ensure that  it is a minimum, one must choose the parameters care-  fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  As we have found the conditions to make the solution  connect minima of the potential, we now investigate a  procedure to reduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9a) to first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' To do so, we  introduce an auxiliary function h = h(φ) that obeys the  equation  φ′ = hφ e−16αφ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (18)  The above expression can be used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9a), which can  be integrated to give  A′ =− β  8α + 4  3αφ′− 2  9  �  3+32α2� �  h+˜h0  �  e−16αφ/3,  (19)  where ˜h0 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This, combined with  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9b), allows us to find the explicit form of the poten-  tial as a function of the scalar field, V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  To illustrate our procedure, we work with the model  described by  φ′ = λ(1 − φ2),  (20)  where λ is a positive real parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This equation is  solved by the function  φ(y) = tanh(λy),  (21)  which connects the values v± = ±1 that define local  minima of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  This solution engenders the  same asymptotic behavior of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (10), with κ± = 2  and m± = 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  To calculate the auxiliary function  h(φ) in this case, we can use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (18) and (20) to get  hφ = λe16αφ/3(1 − φ2), which can be integrated to give  h(φ)=  3λ  2048α3  �  128α2�  1 − φ2�  + 48αφ − 9  �  e16αφ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (22)  By using this in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (19), one can show that the warp  function becomes  A′ = 9λ−32α2(4β+λ)  1024α3  −  λ  64α2  �  3 + 32α2�  φ  + λφ2  8α − h0 e−16αφ/3,  (23)  where we redefined the integration constant in a conve-  nient form, as h0 = 2  �  3+32α2� ˜h0/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' The above expres-  sion can be combined with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (20) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9b) to calcu-  late the potential, V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Its expression is cumbersome,  so we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Asymptotically for y → ±∞, the derivative of the warp  function given above behaves as  A′  ± = − β  8α + λ  �  3 + 32α2�  (3 ∓ 16α)  1024α3  −h0 e∓16α/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (24)  Let us first consider the case with h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  For this  choice, if we want to get a solution that goes from a  Minkowski geometry at y → −∞ to an AdS one at  y → +∞, we need to impose that A′  −  =  0, which  leads us to λ = 128α2β/  �  (3 + 16α)  �  3 + 32α2��  , and  also A′  + = −4β/ (3 + 16α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  These conditions makes  Vφ(v−) = 0, as we have shown before, and  Vφ(v+) = 16(16α − 3)β2  (16α + 3)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (25)  Note that, in order to have Vφ(v+) = 0, we must choose  α = 3/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' However, this restriction leads to  Vφφ(v−) = 50β2  33  and  Vφφ(v+) = −82β2  33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (26)  These expressions shows that φ = v− is a minimum point  and φ = v+ is a maximum point of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' There-  fore, the case h0 = 0 does not allow for the point φ = v+  being a minimum of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  To circumvent this issue, let us now consider the gen-  eral case in which h0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We take A′  − = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (24)  and obtain that the integration constant must be  h0 = e−16α/3�  512α3λ−128α2β+96α2λ+48αλ+9λ  �  1024α3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (27)  As expected from the discussion above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (16), it leads  to V (v−) = Vφ(v−) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' To get a brane with an AdS  geometry at its right tail (y → +∞), we consider A′  + < 0  and Vφ(v+) = 0, which can be attained by taking  λ = −  64e−16α/3α2β  (3 + 32α2)  �  3 sinh  � 16α  3  �  − 16α cosh  � 16α  3  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (28)  4  2  1  0  1  2  2  1  0  1  2  2  1  0  1  2  2  1  0  1  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 1: The potential for the M-AdS brane, for some values  of β and α, with h0 and λ given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In the top panel, we show the potential for β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='1 and  α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='025, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='02 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In the bottom panel, we  display the potential for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='1 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='02 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In both panels, the width of the lines increases with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  At φ = v+, the potential has the value V (v+) =  −3β2/64α2, as expected from the expression right above  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This is always negative, as it usually occurs for  AdS geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In this situation, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (17) we also  have  Vφφ(v+) = 2  3  �  3 + 32α2�  λ  �  2λ + β  2α  �  ,  (29)  with λ as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  From this expression, one can  show that v+ = 1 is a minimum of the potential, match-  ing with the discussion below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (17) for solutions with  exponential tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' On the other hand, we have  Vφφ(v−) = 4  3  �  3 + 32α2�  λ2 − 8  3β2,  (30)  with λ as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (28), again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' So, to ensure that v− = −1  is a minimum of the potential, we impose the conditions  α < 0 and 0 < α < α∗, where α∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='096 is obtained nu-  merically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' When choosing α and β, one must remember  that these parameters must have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Considering the conditions obtained above, we display,  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 1, the potential V (φ) in the interval φ ∈ [−2, 2]  for some values of β and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Notice that the minimum at  v− = −1 is also a zero whilst the other one, at v+ = 1,  is not, attaining the value V (v+) = −3β2/64α2 as given  above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' One can show that V (±∞) → −∞, so it  has maxima outside the range considered in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  To calculate the warp function, we combine the solu-  4  2  0  2  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='4  4  2  0  2  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 2: The warp factor e2A(y) associated to the warp func-  tion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (32), depicted with the same parameters of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  tion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (21) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (23) to get  A′ = 9λ−32α2(4β+λ)  1024α3  −  λ  64α2  �  3+32α2�  tanh(λy)  + λ  8α tanh2(λy) − h0 e−16α tanh(λy)/3,  (31)  where h0 is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (27) and λ is as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' By  integrating it, we get the expression  A(y)= 3+32α2  64α2  ln(sech(λy))+  �  32α2(3λ−4β)+9λ  �  y  1024α3  − tanh(λy)  8α  + h0  2λ  �  e− 16α  3 Ei(ξ−)−e  16α  3 Ei(ξ+)  �  − h0  2λ  �  e− 16α  3 Ei(16α/3)−e  16α  3 Ei(−16α/3)  �  ,  (32)  where ξ± = −(16α/3)(tanh(λy) ± 1) and Ei denotes the  Exponential Integral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Here we have used that  A(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 2, we display the warp factor e2A(y) for the  warp function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (32) for some values of β and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  We see that e2A(−∞) > 0 and e2A(∞) = 0, showing that  our solution describes a M-AdS brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  We can further study the system, to investigate the  situation where the warp factor connects two AdS ge-  ometries, with e2A(±∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This is the AdS-AdS brane,  and it can be implemented with the same Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (32), but  now the constraints in the parameters h0 and λ and in  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (27) and (28) have to be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' The results are  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 3, showing the potential and warp factor  for some values of α, β, λ and h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In this case, the scalar  field solution does not connect minima of the potential  5  1  0  1  2  3  4  5  20  10  0  10  20  30  40  4  2  0  2  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 3: In the top panel we display the potential V (φ) for  the AdS-AdS brane, with β = λ = 1, h0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='1 and α =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In the bottom panel, we depict its warp  factor associated to the warp function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (32) for the  same values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In both panels, the width of the  lines increases with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  and the brane is asymmetric, due to the presence of the  Cuscuta-Galileon dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Let us now investigate linear stability of the gravity  sector of the brane considering  ds2 = e2A(y)�  ηµν + hµν(xµ, y)  �  dxµdxν − dy2,  (33)  in which hµν denotes the small fluctuations, where we  have followed the lines of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [53] to take the axial gauge,  ha4 = 0, that is convenient for the study and depends on  the extra dimension and on the 4-dimensional vector xµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  We also take small perturbations around the scalar field  solution, in the form φ → φ(y) + ξ(xµ, y), where φ(y)  denotes the static solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' These fluctuations  can be inserted into Einstein’s equation (4), which leads  us to three non null components labeled by G(1)  µ4 = 2T (1)  µ4 ,  G(1)  44 = 2T (1)  44 and G(1)  µν = 2T (1)  µν , where G(1)  ab is the term of  the Einstein tensor that contains the fluctuations at first  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In particular, the latter component leads us to  h′′  µν + 4A′h′  µν − e−2A□hµν  + e−2A�  ∂µ∂γhγν + ∂ν∂γhγµ − ∂µ∂νh  �  = ηµν  �  h′′ + e−2A�  ∂γ∂σhγσ − □h  �  + 4A′h′ + 4Vφξ + 4φ′ξ′ − 8αξ′′�  ,  (34)  where we have defined h = ηµνhµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We then proceed  as usual and take the transverse and traceless (TT) con-  dition for hµν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' ∂µhµν = 0 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This makes  3  2  1  0  1  2  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  3  2  1  0  1  2  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 4: The stability potential for the warp factor of the  M-AdS brane, depicted with the same parameters of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  3  2  1  0  1  2  3  2  1  0  1  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 5: The stability potential for the warp factor of the  AdS-AdS brane, depicted with the same parameters of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  all the components vanish, such that only the one in the  above equation survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Also, we take dy = eAdz and  hµν(xµ, z) = eiωµxµe−3A(z)/2Hµν(z) to get the following  Schr¨odinger-like eigenvalue equation  �  − d2  dz2 + U(z)  �  Hµν = ω2Hµν,  (35)  where the stability potential is given exclusively in terms  of derivatives of the warp function, in the form  U(z) = 9  4A2  z + 3  2Azz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  (36)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 4, we display the above stability potential when  the warp factor connects M and AdS geometries, for  some values of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 5, we display the sta-  bility potential for the case where the warp factor con-  nects two AdS geometries, using β = λ = 1, h0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='1  and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  One can show that the  operator in the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (35) can be writ-  ten as the product operators, S†S Hµν = ω2 Hµν, where  6  S = −d/dz−3Az/2 and S† = d/dz−3Az/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' This factor-  ization ensures the linear stability of the gravity sector  of the brane, as the operator associated to the eigenvalue  equation (35) is non negative, implying that ω2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  The localization of gravity in the braneworld scenario was  previously investigated in several works, in particular, in  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' [1, 20, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' For solutions of the AdS-AdS brane,  the 4-dimensional Newtonian potential is given by the  sum of contributions of the zero mode and the correction  term associated to the exchange of Kaluza-Klein modes  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In the case in which one has the interpolation of  M-AdS geometries, the Newtonian potential arises due  to the correction term alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In both cases, the correct  4-dimensional Newtonian behavior can be recovered at  short distances [20, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  In summary,  we have investigated the Cuscuta-  Galileon model described by the Lagrange density in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' The Cuscuton term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (2) engender a func-  tion which only depends on the scalar field, f(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  If  f(φ) is constant, the model (2) cannot support scalar  field solutions connecting minima of the potential V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  However, after adding the Galileon term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (3), we  have shown that it conspires against the Cuscuton term  in the Einstein’s equations, leaving room to construct  interesting solutions that connect minima of the poten-  tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Among the results, we recall that we were interested  in having the minima connected by the solutions, so we  have first investigated the behavior of the brane asymp-  totically, with the scalar field having exponential tails,  as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  This led us to find conditions for the  parameters α, β and m−, which must be chosen care-  fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' In particular, α and β must have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Moreover, since the investigation involves nonlinear dif-  ferential equations of second order, we have introduced  an auxiliary function h(φ) that allows for the presence  of first order equations compatible with the Einstein’s  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  To illustrate our procedure, we have taken the  well-known hyperbolic tangent profile in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' (21) and  found the warp factor associated to the brane that inter-  polates a Minkowski (M) and an Anti de-Sitter (AdS)  geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' We have also investigated the stability of the  gravity sector of the brane and shown that the operator  that governs the eigenvalue equation can be factorized,  ensuring linear stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  There are several perspectives, in particular, the case  concerning asymmetry of the brane and the cosmic ac-  celeration [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' The Cuscuta-Galileon model (3) can  also be investigated in other scenarios of generalized the-  ories of gravity [57–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  The study of time-dependent  configurations in the five-dimensional spacetime is also  of interest, as it can shed light on the number of degrees  of freedom that propagates in the theory [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' An-  other possibility concerns absence of gravity, in the case  of flat spacetime, in the study of kinks [65], vortices [66]  and monopoles [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Some of these issues are cur-  rently being considered and shall be reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  This work is supported by the Brazilian agencies Con-  selho Nacional de Desenvolvimento Cient´ıfico e Tec-  nol´ogico (CNPq), grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 303469/2019-6 (DB) and  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 310994/2021-7 (RM), Paraiba State Research Foun-  dation (FAPESQ-PB), grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 0003/2019 (RM) and  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  0015/2019 (DB and MAM), and Federal Univer-  sity of Para´ıba (UFPB/PROPESQ/PRPG) project code  PII13363-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Menezes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Eto and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Sakai, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 68, 125001 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Melfo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Pantoja and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Skirzewski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D  67, 105003 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [15] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Castillo-Felisola,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Melfo,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Pantoja  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Ramirez, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 70, 104029 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Bazeia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Brito and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Losano, JHEP 11, 064  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Gabadadze, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Grisa and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Shang, JHEP 08, 033  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Takamizu and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Maeda, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 73, 103508  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Wei and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Duan, JCAP  02, 003 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Fonseca, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Brito and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Losano, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' B  697, 493 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Du, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Zhong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Fu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Guo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Menezes and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' da Rocha, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' High En-  ergy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 2014, 276729 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Bazeia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Marques and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Menezes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 74, 085004 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Grandi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Klimas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Sanchez-Guillen and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Wereszczynski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Gomes, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Losano and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Menezes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Dzhunushaliev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Folomeev and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Minamitsuji,  Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Afshordi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Geshnizjani, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Afshordi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Chung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Doran, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Geshnizjani,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 75, 123509 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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+page_content=' Afshordi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 80, 081502 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Wang, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 56, 525 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [33] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Bazeia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Brito, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Costa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 87,  065007 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Gomes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Guariento, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 95, 104049  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Boruah, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Geshnizjani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Cosmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' As-  tropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 1707, 022 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [36] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Andrade, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Marques and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Menezes, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  B 942, 188 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Bazeia and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Lob˜ao, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' C 82, 725  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [38] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Horndeski, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 10, 363 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Nicolis and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rattazzi, JHEP 06, 059 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Nicolis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rattazzi and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Trincherini, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D  79, 064036 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Deffayet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Esposito-Farese and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Vikman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 79, 084003 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [42] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Deffayet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Pujolas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Sawicki and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Vikman, JCAP  10, 026 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' De Felice and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Tsujikawa, JCAP 03, 025 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [44] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Gao and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Steer, JCAP 12, 019 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' De Felice and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Tsujikawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 84, 083504  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [46] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Kobayashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Yamaguchi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Yokoyama, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 126, 511 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [47] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Clifton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Ferreira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Padilla and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Skordis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' 513, 1 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [48] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Zhao and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Liu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D  100, 124057 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [49] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' de Rham and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Motohashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' D 95, 064008  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content='  [50] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Baker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Bellini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Lagos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Noller  and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
+page_content=' Sawicki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FKT4oBgHgl3EQfMS25/content/2301.11750v1.pdf'}
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diff --git a/z9AyT4oBgHgl3EQfPPZM/content/tmp_files/2301.00020v1.pdf.txt b/z9AyT4oBgHgl3EQfPPZM/content/tmp_files/2301.00020v1.pdf.txt
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+Subharmonic Fidelity Revival in a Driven PXP model
+HaRu K. Park1, ∗ and SungBin Lee1, †
+1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea
+(Dated: January 3, 2023)
+The PXP model hosts a special set of nonergodic states, referred to as quantum many-body scars. One of the
+consequences of quantum scarring is the periodic revival of the wave function fidelity. It has been reported that
+quantum fidelity revival occurs in the PXP model for certain product states, and periodic driving of chemical
+potential can enhance the magnitude of quantum revival, and can even change the frequencies of revival showing
+the subharmonic response. Although the effect of the periodic driving in the PXP model has been studied in
+the limit of certain perturbative regimes, the general mechanism of such enhanced revival and frequency change
+has been barely studied. In this work, we investigate how periodic driving in the PXP model can systematically
+control the fidelity revival. Particularly, focusing on the product state so called a N´eel state, we analyze the
+condition of driving to enhance the magnitude of revival or change the frequencies of revival. To clarify the
+reason of such control, we consider the similarities between the PXP model and the free spin-1/2 model in
+graph theoretical analysis, and show that the quantum fidelity feature in the PXP model is well explained by the
+free spin-1/2 model. In addition, under certain limit of the driving parameters, analytic approach to explain the
+main features of the fidelity revival is also performed. Our results give an insight of the scarring nature of the
+periodically driven PXP model and pave the way to understand their (sub-)harmonic responses and controls.
+I.
+INTRODUCTION
+Eigenstate Thermalization Hypothesis(ETH)1–4 is a key
+concept explaining the thermalization of the quantum many-
+body system. Recently, beyond the quantum thermalization,
+the system which strongly violates the ETH has been actively
+studied, such as integrable systems and many-body localiza-
+tion.
+Here, the ”strong violation” of the ETH means ev-
+ery eigenstate breaks the ergodicity and never get thermal-
+ized. There are also the systems which ”weakly violate” the
+ETH, implying only a small portion of the eigenstates violates
+the ergodicity while all the other states get thermalized. In
+particular, quantum many-body scarring(QMBS) systems are
+the examples which weakly violates ETH, containing a small
+number of highly excited non-thermal eigenstates, called scar
+eigenstates5,6. Such scar eigenstates show exotic physical be-
+havior compared to the thermal Gibbs state. For example,
+while the thermal Gibbs state predicts the entanglement en-
+tropy proportional to the volume of subsystem in the middle
+of the spectrum, the entanglement entropy of the scar eigen-
+state scales proportionally to the area of the subsystem. Be-
+cause the QMBS often appear in a tower of scar states where
+a set of states with equidistant energy spacing exists7, if the
+initial product state is a superposition of the scar states, then
+the system exhibits the perfect revival. Conversely, it has been
+also shown that the persistent revival of product state implies
+QMBS8. Hence, it is important to observe the fidelity revival
+as evidence for the QMBS in experiments.
+In recent experiments of a Rydberg atom simulator, it has
+been observed that certain product states show persistent,
+though imperfect revival under van der Waals interaction.
+This indicates the presence of a QMBS in the system9–11. Tak-
+ing the extreme limit of strong van der Waals interaction in
+the Rydberg blockade of the Rydberg atom chain gives rise
+to the so-called PXP model, which has a Hilbert space pro-
+jected onto the states with no neighboring excited states. The
+QMBS structure of the PXP model has been actively stud-
+ied theoretically, including the report that it also shows the
+imperfect revival12. There exist many attempts to reach high
+fidelity revival, for instance, by enhancing its weakly broken
+SU(2) symmetry13,14. There is another way which enhances
+the quantum revival of the states in the PXP model: periodic
+driving. In most of the cases, periodic driving induces ther-
+malization of the scar states, and thus destructs the fidelity re-
+vival. However, recent experimental and theoretical studies
+show10,15 that periodic driving with certain amplitudes sur-
+prisingly enhances the fidelity revival. Furthermore, it has
+also been observed that the subharmonic response of fidelity
+revival exists with doubled period compared to the driving
+mode. This kind of subharmonic response breaks the discrete
+time translational symmetry of the driving mode, and hence
+get attention as a time-version of crystalline order, called a
+discrete time-crystal (DTC) which is also a recently studied
+subject16,17. Although earlier research had demonstrated this
+subharmonic fidelity revival in the limit of large driving am-
+plitude and high frequency18, the general mechanism of such
+subharmonic revival has not been explored well.
+In this paper, we study the periodically driven PXP model,
+focusing on the subharmonic fidelity revivals. In Section II,
+we introduce the PXP model with square pulse driving modes.
+By calculating the average fidelity and the Fourier compo-
+nents of the fidelity signal, we study the conditions under
+which the fidelity is enhanced and subharmonic response oc-
+curs. Then, based on the similarity of the Hamiltonian ad-
+jacent graph between PXP model and free spin model19, we
+show that they can be explained by the free spin-1/2 model
+with the same driving which is exactly solvable. In Section III,
+we introduce an analytic approach for the driven PXP model.
+Within perturbative analysis, we derive the driving conditions
+for subharmonic responses in the driven PXP model and dis-
+cuss their applications.
+In Section IV, we summarize our
+works and suggest interesting future directions.
+arXiv:2301.00020v1  [cond-mat.quant-gas]  30 Dec 2022
+
+2
+II.
+PXP MODEL WITH SQUARE PULSE DRIVE
+In this section, we first introduce the static PXP model and
+then represent how the fidelity revival is controlled under peri-
+odic driving. Particularly, based on the graph theoretical simi-
+larity between PXP model and free spin-1/2 model, we argue
+that various phenomena in the driven PXP model, such as the
+revival enhancement and (sub-) harmonic response, can be ex-
+plained by the free spin-1/2 model.
+The PXP model, describing Rydberg atoms with strong in-
+teraction, is represented as following,
+HPXP = Ω
+�
+j
+Pj−1XjPj+1.
+(1)
+Here, the spin at each site consists of two states, |0⟩ and
+|1⟩, which represent a ground state and an excited state, re-
+spectively.
+Xj = |0j⟩⟨1j| + |1j⟩⟨0j| is the Pauli x ma-
+trix at site j, and Pj = |0j⟩⟨0j| is the projection opera-
+tor for ground state at site j.
+This Hamiltonian describes
+the system which prohibits the spin-flip, unless the neigh-
+boring sites are in the ground state, i.e. only the transition,
+| · · · 010 · · · ⟩ · · · ↔ | · · · 000 · · · ⟩, is allowed. Since this tran-
+sition does not generate or annihilate excited states in any two
+neighboring sites, one can exclude the states where the con-
+secutive neighbors are excited.
+Although the PXP model is a non-integrable chaotic
+system11, there are certain product states which show non-
+ergodicity and decent revival under time evolution, such as
+|Z2⟩ ≡ |0101 · · · 01⟩ called a N´eel state. The non-ergodic
+property of such product states is unique, in a sense that
+the number of them increases linearly with the system size,
+whereas, other ergodic product states show exponential in-
+crease with the system size. It has been understood that their
+fidelity revival is originated from the quantum scarring, i.e.,
+the product state with short-time revival is a linear combina-
+tion of scar eigenstates with equivalent energy spacing7. The
+|Z2⟩ state is mainly composed of the quantum scar states with
+almost equal energy spacing. However, because their energy
+spacing is not perfectly even, it has been pointed out that the
+fidelity revival of the |Z2⟩ state is also imperfect12.
+There have been many suggestions to adjust the system en-
+hancing this imperfect revivals. As one promising way, it has
+been studied in both theoretically and experimentally to the
+addition of cosine modulation. This modulation plays a role
+of controlling chemical potential to the PXP model, and can
+enhance the fidelity revival of the |Z2⟩ state or even induce
+subharmonic responses in certain driving condition10,15. How-
+ever, the systematic ways to find such driving have not been
+studied in detail, which is the focus of this study.
+The periodically driven PXP model is represented as fol-
+lowing.
+H(t) = HPXP + ∆sq(t)
+�
+j
+nj,
+(2)
+where HPXP is defined in Equation 1 and the second term rep-
+resents the periodic driving, where nj = |1j⟩⟨1j| counts the
+number of excited states on each site. In terms of the periodic
+driving, we adopt the square pulse driving protocol as also in-
+troduced in earlier studies for analysis. It shares a very similar
+fidelity profile with the cosine driving case and thus well ex-
+plains the experiments15. The square pulse driving protocol
+∆sq(t) within the period T is defined as,
+∆sq(t) =
+�
+�
+�
+�
+�
+∆0 + ∆m,
+0 ≤ t ≤ T/4
+∆0 − ∆m,
+T/4 < t ≤ 3T/4
+∆0 + ∆m,
+3T/4 < t < T.
+(3)
+This corresponds to the periodic driving with frequency ω0 =
+2π/T, average chemical potential ∆0, and driving amplitude
+∆m.
+Now, we introduce the wave function fidelity F(t) ≡
+|⟨ψ(t)|ψ(0)⟩|2 to measure how the revival of the initial prod-
+uct state, |ψ(0)⟩ = |Z2⟩, changes as the driving parameters
+(∆0, ∆m) are tuned. As a tool to measure the subharmonic re-
+sponse, we also introduce the Fourier component of the wave
+function fidelity defined as,
+F(ω) =
+�����
+1
+nT
+� nT
+0
+F(t)e−iωtdt
+����� .
+(4)
+Later, we will discuss the three main quantities, F(0), F(ω0)
+and F(ω0/2), as functions of (∆0, ∆m): F(0) introduces
+how the fidelity revival gets enhanced, and F(ω0) and
+F(ω0/2) are responsible for harmonic and subharmonic re-
+sponses, respectively. Note that if |ψ(t)⟩ is an eigenstate of
+H(t) with driving parameters (∆0, ∆m), then �
+j Zj|ψ(t)⟩
+with Pauli z matrix Zj = |1j⟩⟨1j| − |0j⟩⟨0j| is also an eigen-
+state of H(t) with the same driving parameters. Hence, with-
+out loss of generality, we only plot the region, ∆m ≥ 0.
+Before investigating the fidelity profile of the |Z2⟩ state in
+the PXP model, let us introduce another model which shows
+very similar feature: the free spin-1/2 chain model with the
+same driving ∆sq(t),
+Hfree(t) =
+�
+j
+Xj + ∆sq(t)
+�
+j
+nj.
+(5)
+For the |Z2⟩ state, the free spin-1/2 model and the PXP model
+share common features in terms of graph theoretical point of
+view. To understand it, notice that the PXP model is noth-
+ing but the free spin-1/2 model with constraints. Hence, the
+graph of length L PXP model, for instance, is a subgraph of
+the free spin-1/2 model with the same length L. Conversely,
+consider the length L PXP model with even L. If we give the
+stronger constraint to the PXP model and only allow the states
+with |0⟩ states at odd sites, then each states |0x0y · · · 0z⟩ can
+be mapped to the state |xy · · · z⟩ in length L/2 free spin-1/2
+model, where x, y, · · · , z are either 0 or 1. This shows that
+the graph of length L/2 free spin-1/2 model is a subgraph of
+length L PXP model.
+Figure 1(a) shows the graph of the PXP Hamiltonian with
+L = 8, and Figure 1(b) shows the graph of the free spin-1/2
+Hamiltonian for L = 4. The blue square in Figure 1(a) marks
+the N´eel state |01010101⟩ in PXP Hamiltonian, which corre-
+sponds to the |1111⟩ state in free spin-1/2 model also marked
+
+3
+FIG. 1. (Top) The Hamiltonian adjacency graph of (a) PXP model for L = 8 and (b) free spin 1/2 model for L = 4. The blue square and
+yellow triangle in Figure (a) represent the states |01010101⟩ and |00000000⟩ respectively, and the blue square and yellow triangle in Figure (b)
+represent the states |1111⟩ and |0000⟩ respectively. (Bottom) Average fidelity, F(0), with initial state |Z2⟩ through 10T time domain. Figure
+(c) shows F(0) of PXP model with the driving period T = 4.788 for system size L = 12; Figure (d) shows F(0), of free spin-1/2 model
+with driving period T = π for system size L = 6. The ”butterfly” peaks are encircled by red line, ”bridge” peaks are encircled by magenta
+line, ”local” (in Figure (c)) or ”steep bridge” (in Figure (d)) peaks are encircled by cyan line, and ”separator” peaks are encircled by blue line.
+by the blue square in Figure 1(b). The yellow triangles in
+Figure 1(a) and (b) represent the polarized state |00000000⟩
+state and |0000⟩ state, respectively. The vertices and edges
+colored in red show the difference between the two graphs
+and show that the graph in Figure 1(b) is indeed a subgraph
+of Figure 1(a). Despite their difference, they share a com-
+mon feature particularly for the |Z2⟩ state, marked by the
+blue square. This common feature is generally applicable for
+the graphs of the length L PXP model and length L/2 free
+spin-1/2 model. To argue it in detail, let’s consider the ex-
+pansion, ⟨Z2(t)|Z2⟩ = ⟨Z2|eiHt|Z2⟩ = �
+n
+(it)n
+n! ⟨Hn⟩ with
+⟨Hn⟩ = ⟨Z2|Hn|Z2⟩. In the graph theoretical point of view,
+⟨Hn⟩ counts the number of walks with length n which starts
+and ends at |Z2⟩ vertex. Because the difference between PXP
+graph and free spin graph mostly occurs for the states with
+high Hamming distance, their difference only affects on the
+long walks, i.e. ⟨Hn⟩ with large n. Hence, we may expect the
+similar behavior in F(t) for a short time scale t in between
+PXP model and free spin model. Later, we will show that it is
+indeed the case by comparing calculation of the fidelity revival
+on both PXP model and free spin model. For completeness we
+note that this argue is not applicable for ergodic initial states:
+for example the polarized state |0p⟩ ≡ |0000 · · · 00⟩, marked
+by the yellow triangle, shows a large difference in graphs even
+for the nearest neighbors, indicating the different fidelity pro-
+file between PXP model and free spin model.
+In the presence of driving, one may also suggest the simi-
+larities between PXP model and free spin-1/2 model, with re-
+spect to the graph theoretical approach. Figure 1(c) and 1(d)
+show the values of F(0) for PXP model and free spin-1/2
+model respectively, as functions of driving parameters ∆0 and
+∆m. As discussed earlier, F(0) indicates the enhancement of
+the fidelity revival. Indeed, Figures 1(c) and 1(d) show very
+similar features up to scale. For calculation, we choose the
+periodicity T0 = 4.788 for the PXP model and Tf = π for the
+free spin model respectively, which are optimized values for
+the fidelity revival observed in the static cases. The system
+size L = 12 is chosen with the time range [0, 10T0] for the
+PXP model, and L = 6 with the time range [0, 10Tf] for the
+free spin model.
+In Figure 1(c) and 1(d), we point out several common fea-
+tures as following. We first note the ”butterfly”-shaped peaks
+on top of each figure, marked by a red circle, and high average
+fidelity region on the lower left and right side. Next, there is a
+wide V -shaped region having relatively small values of F(0)
+in-between, with the following substructures: On the left side
+in both Figures 1(c) and (d), the butterfly peaks are connected
+to the lower left region by some ”bridges”, marked by ma-
+
+16
+14
+0.8
+12
+10
+0.6
+E
+8
+0.4
+6
+4
+0.2
+2
+0
+-10
+-5
+0
+5
+1016
+888
+14
+0.8
+12
+10
+0.6
+m
+8
+0.4
+6
+4
+0.2
+2
+0
+-10
+104
+(a)
+(b)
+FIG. 2. Frequency profile of the fidelity F(t), (a) F(ω0) and (b)
+F(ω0/2) with initial state |Z2⟩, for the PXP model with driving pe-
+riod T = 4.788 and system size L = 12. The ”butterfly” peaks are
+encircled by red line, ”bridges” peaks are encircled by magenta line,
+”local” peaks are encircled by cyan line, and ”separator” peaks are
+encircled by blue line.
+(a)
+(b)
+FIG. 3. Frequency profile of the fidelity F(t), (a) F(ω0) and (b)
+F(ω0/2) with initial state |Z2⟩, for the free spin model with driv-
+ing period T = π and system size L = 12. The ”butterfly” peaks
+are encircled by red line, ”bridges” peaks are encircled by magenta
+line, ”steep bridge” peaks are encircled by cyan line, and ”separator”
+peaks are encircled by blue line.
+genta lines. In addition, there are ”local” peaks between the
+bridges marked by cyan line in Figure 1(c), but instead there
+are ”steep bridge” peaks in Figure 1(d). Later, we will explain
+that they indicate the same phenomena. On the right side in
+both Figures 1(c) and (d), there are long and thin ”separator”
+peak marked by green line, which separates the butterfly peaks
+and the lower right region.
+To determine the origin of these peaks, the frequency pro-
+files of the fidelity are investigated. Figures 2a and 2b plot
+the Fourier component values of the fidelity for the |Z2⟩ state
+in the PXP model, F(ω0) and F(ω0/2), with ω0 = 2π/T0.
+By comparing them with Figure 1(c), one can conclude that
+the butterfly peaks (marked by red line) and the local peaks
+(cyan line) represent harmonic revivals, whereas, the bridges
+(magenta line) and separators (green line) represent the sub-
+harmonic revivals. Notice that the bridges and the separators
+also appear in Figure 2a. However, this does not imply that
+they are harmonic responses, since the subharmonic response
+with nonzero F(ω0/2) also has finite values of F(ω0). There-
+fore, Figure 2b is a direct evidence, showing the subharmonic
+response indeed occurs due to the driving.
+For comparison, we also investigate the free spin-1/2
+model case. Figures 3a and 3b plot the values of F(ωf) and
+F(ωf/2) for the free-spin model, showing harmonic and sub-
+harmonic responses respectively, with ωf = 2π/Tf = 2. The
+butterfly peaks (marked by red line) and the steep bridge peaks
+(cyan line) in Figure 3a again shows the harmonic response,
+while the bridge peaks (magenta line) and the separator peaks
+on the right side (green line) in Figure 3b shows the subhar-
+monic response. Indeed, these features are consistent with the
+case of the PXP model which is explained earlier (see Figure
+2).
+III.
+PERTURBATIVE AND EXACT CALCULATIONS ON
+THE MODELS
+Until now, we have shown that the periodic driving of the
+PXP model can induce the subharmonic responses of the |Z2⟩
+state fidelity and have interpreted them based on the graph the-
+oretical similarities with the free spin-1/2 model. In the fol-
+lowing, for more rigorous argument, alternative analytic ap-
+proaches are presented to understand subharmonic responses
+and to determine the optimal driving conditions. Since our fo-
+cus lies on the subharmonic response of driven PXP model,
+we perform the appropriate perturbation limit which repre-
+sents the V -shaped region in Figure 1(c), where every sub-
+harmonic peaks lies on. We note that perturbation approach
+in another limit has been already performed in earlier work18.
+Its perturbative limit explains the ”butterfly” peaks, marked
+by the red circle in Figure 1(c). In contrast, our perturbative
+limit explains every peaks on the V -shaped region, including
+”bridge”, ”local”, and ”separator” peaks.
+Before moving on, we redefine some notations for simplic-
+ity. Take ∆± ≡ (∆0 ± ∆m)/2, Xj ≡ Pj−1XjPj+1, and
+H± ≡ �
+j H±
+j
+with H±
+j
+≡ ΩXj + ∆±Zj. Notice that
+H± = HPXP + 2∆±
+�
+j nj − ∆±
+�
+j Ij, hence the evolu-
+tion operator in the presence of square pulse driving,
+U = eiH+T0/4eiH−T0/2eiH+T0/4,
+(6)
+is equivalent to the evolution operator of H(t) in one period
+up to phase.
+Consider the limit ∆+ ≫ Ω ≫ ∆−, which is the right
+side of the V -shaped region with low values of F(0) in Fig-
+ure 1.
+We will show that this limit always gives subhar-
+monic response. In this limit, we can approximate H+ ≃
+∆+
+�
+j Zj and H− ≃ ΩHPXP taking the leading terms. Be-
+cause our product state is an eigenstate of H+, if we calcu-
+late ⟨Z′
+2|U|Z2⟩, where Z′
+2 ≡ |1010 · · · 10⟩ is a translated N´eel
+state, one can easily show ⟨Z′
+2|U|Z2⟩ ≃ ⟨Z′
+2|eiHPXPT0/2|Z2⟩
+up to phase.
+For L = 12 case, this value is quite large
+≃ 0.9658. This results in the 2T-periodic revival with high
+lower bound of |⟨Z2|U 2|Z2⟩|, satisfying,
+|⟨Z2|U 2|Z2⟩| ≥ |⟨Z2|eiHPXPT0/2|Z′
+2⟩|2,
+∆+ ≫ Ω ≫ ∆−.
+(7)
+See Appendix A for detailed proof. Thus, one can claim the
+persistent subharmonic revival indeed occurs in ∆+ ≫ Ω ≫
+∆−. We also show that this revival is robust even with O(Ω)
+order terms in Appendix A.
+
+16
+0.2
+14
+12
+10
+8
+6
+4
+2
+0
+-10
+0
+1016
+0.2
+14
+12
+10
+m
+8
+6
+4
+2
+0
+-10
+10
+0
+316
+0.2
+14
+12
+10
+8
+6
+4
+2
+0
+-10
+5
+0
+5
+1016
+0.2
+14
+12
+10
+8
+6
+4
+2
+0
+-10
+0
+105
+Now consider the limit ∆− ≫ Ω ≫ ∆+, which is the
+left side of the V -shaped region with low values of F(0) in
+Figure 1. Similar analysis with the case ∆+ ≫ Ω ≫ ∆−
+leads H+ ≃ ΩHPXP and H− ≃ ∆−
+�
+j Zj. We focus on the
+driving conditions for the parameters ∆− = nπ
+T0 = nω0
+2 , and
+our aim is to show that for even n case the fidelity presents
+subharmonic response and for odd n case the fidelity presents
+harmonic response. First, let n = 2k be even. In this case,
+eiH−T/2 = 1, and thus one again achieve ⟨Z′
+2|U|Z2⟩ ≃
+⟨Z′
+2|eHPXPT0/2|Z2⟩. Thus, we get the very similar result with
+Equation 7,
+|⟨Z2|U 2|Z2⟩| ≥ |⟨Z2|eiHPXPT0/2|Z′
+2⟩|2,
+∆− = kω0 ≫ Ω ≫ ∆+,
+(8)
+showing the subharmonic response mainly occurs at ∆− =
+kω0 for integer k’s. On the other hand, for odd n = 2k + 1’s,
+eiH−T0/2 = �
+j Zj. Using the anti-commutation relation be-
+tween �
+j Zj and HPXP, the evolution operator is represented
+as,
+U ≃ eiHPXPT0/4
+�
+��
+j
+Zj
+�
+� eiHPXPT0/4
+=
+�
+j
+Zje−iHPXPT0/4eiHPXPT0/4 =
+�
+j
+Zj,
+(9)
+and this results in,
+|⟨Z2|U|Z2⟩| ≃ 1,
+∆− =
+�
+k + 1
+2
+�
+ω0 ≫ Ω ≫ ∆+. (10)
+Thus, the harmonic response mainly occurs for ∆− = (k +
+1/2)ω0 region. We again show that this revival is robust up to
+O(Ω) order, see Appendix A.
+In summary of this section, our perturbative analysis pro-
+vides reasonable explanation why the several peaks in V -
+shaped region are emerging in Figure 1. Specifically, Equation
+7 explains the long diagonal ”separator” peaks on the right
+side of V -shaped region, Equation 8 explains the ”bridge”
+peaks on the left side, and Equation 10 explains the ”local”
+peaks between bridge peaks. It is important to note that the
+derivation for Equations 7, 8 and 10 can be generally applica-
+ble.
+IV.
+DISCUSSION AND CONCLUSION
+In this work, we study the wave function fidelity revival on
+the periodically driven PXP model. First, we show that the
+driving on PXP model induces various interesting responses,
+including subharmonic responses. Based on the graph theoret-
+ical similarities between PXP model and free spin-1/2 model,
+we have claimed and numerically confirmed that the driving
+condition which induces the subharmonic response in the PXP
+model can be captured by the free spin-1/2 model. Then, con-
+sidering perturbative analysis, the generic driving conditions
+for subharmonic responses in the PXP model are derived. Our
+work will shed a light on the Rydberg atom simulator, study-
+ing subharmonic responses of the driven quantum many-body
+scarring systems.
+As an interesting future work, one may extend our studies
+with finite van der Waals interaction, and explore the condi-
+tions of the subharmonic revival as interaction changes. Since
+the strength of the van der Waals interaction is determined
+by the distance between the two Rydberg atoms r as ∼
+1
+r6 ,9,
+one could control the atom distance to tune their interaction
+strength and track the revival property of the |Z2⟩ state. One
+can also consider the effect of further neighbor van der Waals
+interactions, and explore how the fidelity revival condition
+changes, which we will leave as a future work.
+ACKNOWLEDGMENTS
+Acknowledgments.— We thank Junmo Jeon for valuable
+discussions.
+This work is supported by National Re-
+search Foundation Grant (No.
+2020R1A4A3079707, No.
+2021R1A2C1093060),).
+Appendix A: Bound of the (sub)harmonic revival on PXP model
+In this section, we show the harmonic and subharmonic re-
+vival is stable under small Ω values up to first order, which is
+discussed in Section III. We show the Equations 7, 8 and 10
+still holds if we include the O(Ω) terms.
+We first consider the condition ∆+ ≫ Ω ≫ ∆−. In this
+case, because for different sites j ̸= k, we have
+[H+
+j , H+
+k ] ∼ O(Ω2),
+(A1)
+hence we may write
+eiH+T0/4 ≃
+�
+j
+eiH+
+j T0/4 =
+�
+j
+ei(ΩXj+
+∆+
+2 Zj)T0/4
+(A2)
+up to O(Ω) order. This can be expanded to cosine and sine
+functions in O(Ω) order, achieving
+eiH+
+j T0/4 ≃ cos ∆+T0
+4
++ i sin ∆+T0
+4
+� Ω
+∆+
+Xj + Zj
+�
+.
+(A3)
+If we product all the eiH+
+j T0/4 terms and left only the O(Ω)
+order terms, then we finally get
+eiH+T0/4
+≃ ei∆+
+�
+j ZjT0/4
+�
+�1 + i sin ∆+T0
+4
+�
+j
+e−i∆+ZjT0/4 Ω
+∆+
+Xj
+�
+�
+=
+�
+�1 + i sin ∆+T0
+4
+�
+j
+Ω
+∆+
+Xje−i∆+ZjT0/4
+�
+� ei∆+
+�
+j ZjT0/4.
+(A4)
+To calculate the subharmonic response, we consider the
+value ⟨Z′
+2|U|Z2⟩:
+if this value is large enough then it
+
+6
+guarantees the 2T-periodic revival with |⟨Z2|U 2|Z2⟩|
+≥
+|⟨Z′
+2|U|Z2⟩|2, because
+⟨Z2|U 2|Z2⟩ = |⟨Z2|U|Z′
+2⟩|2 +
+�
+i
+|⟨Z2|U|ψi⟩|2
+≥ |⟨Z2|U|Z′
+2⟩|2,
+(A5)
+where {ψi} are the basis of the Hilbert space orthogonal to
+|Z′
+2⟩. Now because we are considering Ω ≫ ∆− limit, we
+ignore ∆−, giving eiH−T0/2 ≃ eiHPXPT0/2, then we have
+⟨Z′
+2|U|Z2⟩ ≃ ⟨Z′
+2|eiHPXPT0/2|Z2⟩
++ i Ω
+∆+
+sin ∆+T0
+4
+�
+�⟨Z′
+2|eiHPXPT0/2 �
+j
+Xje−i∆+ZjT0/4|Z2⟩
+�
+�
++ i Ω
+∆+
+sin ∆+T0
+4
+�
+�⟨Z′
+2|
+�
+j
+e−i∆+ZjT0/4XjeiHPXPT0/2|Z2⟩
+�
+� .
+(A6)
+For the second term, observe that
+�
+j
+Xje−i∆+ZjT0/4|Z2⟩ = e−i∆+ZjT0/4 �
+j
+Xj|Z2⟩,
+(A7)
+because there are always excited states between two ground
+states. Hence we get
+⟨Z′
+2|
+�
+j
+e−i∆+ZjT0/4XjeiHPXPT0/2|Z2⟩
+= ⟨Z′
+2|eiHPXPT0/2HPXP|Z2⟩
+= ∂t ⟨Z′
+2|eiHPXPt|Z2⟩
+��
+t=T0/2
+We numerically check that ⟨Z′
+2|eiHPXP|Z2⟩ becomes maxi-
+mized at t = T0/2, and hence conclude this term vanishes.
+Arguing similar for the third term, we get
+⟨Z′
+2|U|Z2⟩ ≃ ⟨Z′
+2|eiHPXPT0/2|Z2⟩,
+(A8)
+showing
+the
+persistent
+subharmonic
+revival
+because
+|⟨Z′
+2|eiHPXPT0/2|Z2⟩| can be taken high enough:
+for ex-
+ample, |⟨Z′
+2|eiHPXPT0/2|Z2⟩| ≃ 0.9658 for L = 12.
+Now we consider the condition ∆− ≫ Ω ≫ ∆+ region,
+which is the left side of the V -shaped low F(0) region. In this
+case, by the similar way achieving A3 we achieve
+eiH−
+j T0/2 ≃ cos ∆−T0
+2
++ i sin ∆−T0
+2
+� Ω
+∆−
+Xj + Zj
+�
+.
+(A9)
+Here, we specifically focus on the area where ∆− = nπ+2η
+T0
+for integer n’s, with small η ≪ L−1.
+We start with even n = 2k, giving
+eiH−
+j T0/2 ≃ ±1 ± iη
+� Ω
+∆−
+Xj + Zj
+�
+,
+(A10)
+and
+eiH−T0/2 ≃ 1 ± iη
+�
+j
+� Ω
+∆−
+Xj + Zj
+�
+.
+(A11)
+Now we calculate
+⟨Z′
+2|U|Z2⟩
+= ⟨Z′
+2|eiHPXPT0/2|Z2⟩
++ ±iη⟨Z′
+2|eiHPXPT0/4
+�
+��
+j
+� Ω
+∆−
+Xj + Zj
+��
+� eiHPXPT0/4|Z2⟩.
+(A12)
+Because the second term can be squeezed by L(1 + Ω/∆−)
+again, we get
+|⟨Z′
+2|U|Z2⟩| ≥ |⟨Z′
+2|eiHPXPT0/2|Z2⟩| − ηL
+�
+1 + Ω
+∆−
+�
+,
+(A13)
+and since the first term is large enough, it shows that the sub-
+harmonic response mainly occurs near ∆− = 2kπ
+T0 = kω0.
+Finally, we take odd n = 2k + 1. In this case, we get
+eiH−
+j T0/2 ≃ ±η ± i
+� Ω
+∆−
+Xj + Zj
+�
+(A14)
+and thus
+eiH−T0/2 ≃ ±
+�
+j
+Zj
+�
+�1 +
+�
+j
+Zj
+�
+η ± i Ω
+∆−
+Xj
+��
+� .
+(A15)
+Now by using the fact that HP XP and �
+j Zj anticommutes,
+we get
+U ≃ ±e−iHPXPT0/4
+�
+�1 +
+�
+j
+Zj
+�
+η ± i Ω
+∆−
+Xj
+��
+� eiHPXPT0/4.
+(A16)
+Calculating ⟨Z2|U|Z2⟩, the first term gives ±1. For the sec-
+ond term, notice that the η-dependent term squeezes �
+j Zj
+operator, which gives its value at most L, and hence squeezed
+by the value ηL. For the
+Ω
+∆− term squeezing �
+j ZjXj =
+i �
+j Y j where Y j = Pj−1YjPj+1 with Yj a Pauli y matrix
+Yj = i|0j⟩⟨1j|−i|1j⟩⟨0j|, we can numerically check that this
+value squeezes below δ ≃ 0.2. Therefore,
+|⟨Z2|U|Z2⟩| ≤ 1 − ηL − Ω
+∆−
+δ
+(A17)
+and because ηL and δ are small enough, this represents the
+persistent harmonic revival.
+
+7
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+
diff --git a/z9AyT4oBgHgl3EQfPPZM/content/tmp_files/load_file.txt b/z9AyT4oBgHgl3EQfPPZM/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4257ea528d8e85508e71a9abd8a27b9b205122ec
--- /dev/null
+++ b/z9AyT4oBgHgl3EQfPPZM/content/tmp_files/load_file.txt
@@ -0,0 +1,409 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf,len=408
+page_content='Subharmonic Fidelity Revival in a Driven PXP model  HaRu K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Park1, ∗ and SungBin Lee1, †  1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea  (Dated: January 3, 2023)  The PXP model hosts a special set of nonergodic states, referred to as quantum many-body scars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' One of the  consequences of quantum scarring is the periodic revival of the wave function fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' It has been reported that  quantum fidelity revival occurs in the PXP model for certain product states, and periodic driving of chemical  potential can enhance the magnitude of quantum revival, and can even change the frequencies of revival showing  the subharmonic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Although the effect of the periodic driving in the PXP model has been studied in  the limit of certain perturbative regimes, the general mechanism of such enhanced revival and frequency change  has been barely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In this work, we investigate how periodic driving in the PXP model can systematically  control the fidelity revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Particularly, focusing on the product state so called a N´eel state, we analyze the  condition of driving to enhance the magnitude of revival or change the frequencies of revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' To clarify the  reason of such control, we consider the similarities between the PXP model and the free spin-1/2 model in  graph theoretical analysis, and show that the quantum fidelity feature in the PXP model is well explained by the  free spin-1/2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In addition, under certain limit of the driving parameters, analytic approach to explain the  main features of the fidelity revival is also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Our results give an insight of the scarring nature of the  periodically driven PXP model and pave the way to understand their (sub-)harmonic responses and controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  INTRODUCTION  Eigenstate Thermalization Hypothesis(ETH)1–4 is a key  concept explaining the thermalization of the quantum many-  body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Recently, beyond the quantum thermalization,  the system which strongly violates the ETH has been actively  studied, such as integrable systems and many-body localiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Here, the ”strong violation” of the ETH means ev-  ery eigenstate breaks the ergodicity and never get thermal-  ized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' There are also the systems which ”weakly violate” the  ETH, implying only a small portion of the eigenstates violates  the ergodicity while all the other states get thermalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In  particular, quantum many-body scarring(QMBS) systems are  the examples which weakly violates ETH, containing a small  number of highly excited non-thermal eigenstates, called scar  eigenstates5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Such scar eigenstates show exotic physical be-  havior compared to the thermal Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' For example,  while the thermal Gibbs state predicts the entanglement en-  tropy proportional to the volume of subsystem in the middle  of the spectrum, the entanglement entropy of the scar eigen-  state scales proportionally to the area of the subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Be-  cause the QMBS often appear in a tower of scar states where  a set of states with equidistant energy spacing exists7, if the  initial product state is a superposition of the scar states, then  the system exhibits the perfect revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Conversely, it has been  also shown that the persistent revival of product state implies  QMBS8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Hence, it is important to observe the fidelity revival  as evidence for the QMBS in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  In recent experiments of a Rydberg atom simulator, it has  been observed that certain product states show persistent,  though imperfect revival under van der Waals interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  This indicates the presence of a QMBS in the system9–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Tak-  ing the extreme limit of strong van der Waals interaction in  the Rydberg blockade of the Rydberg atom chain gives rise  to the so-called PXP model, which has a Hilbert space pro-  jected onto the states with no neighboring excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The  QMBS structure of the PXP model has been actively stud-  ied theoretically, including the report that it also shows the  imperfect revival12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' There exist many attempts to reach high  fidelity revival, for instance, by enhancing its weakly broken  SU(2) symmetry13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' There is another way which enhances  the quantum revival of the states in the PXP model: periodic  driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In most of the cases, periodic driving induces ther-  malization of the scar states, and thus destructs the fidelity re-  vival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' However, recent experimental and theoretical studies  show10,15 that periodic driving with certain amplitudes sur-  prisingly enhances the fidelity revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Furthermore, it has  also been observed that the subharmonic response of fidelity  revival exists with doubled period compared to the driving  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' This kind of subharmonic response breaks the discrete  time translational symmetry of the driving mode, and hence  get attention as a time-version of crystalline order, called a  discrete time-crystal (DTC) which is also a recently studied  subject16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Although earlier research had demonstrated this  subharmonic fidelity revival in the limit of large driving am-  plitude and high frequency18, the general mechanism of such  subharmonic revival has not been explored well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  In this paper, we study the periodically driven PXP model,  focusing on the subharmonic fidelity revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In Section II,  we introduce the PXP model with square pulse driving modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  By calculating the average fidelity and the Fourier compo-  nents of the fidelity signal, we study the conditions under  which the fidelity is enhanced and subharmonic response oc-  curs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Then, based on the similarity of the Hamiltonian ad-  jacent graph between PXP model and free spin model19, we  show that they can be explained by the free spin-1/2 model  with the same driving which is exactly solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In Section III,  we introduce an analytic approach for the driven PXP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Within perturbative analysis, we derive the driving conditions  for subharmonic responses in the driven PXP model and dis-  cuss their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  In Section IV, we summarize our  works and suggest interesting future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='00020v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='quant-gas]  30 Dec 2022  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  PXP MODEL WITH SQUARE PULSE DRIVE  In this section, we first introduce the static PXP model and  then represent how the fidelity revival is controlled under peri-  odic driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Particularly, based on the graph theoretical simi-  larity between PXP model and free spin-1/2 model, we argue  that various phenomena in the driven PXP model, such as the  revival enhancement and (sub-) harmonic response, can be ex-  plained by the free spin-1/2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  The PXP model, describing Rydberg atoms with strong in-  teraction, is represented as following,  HPXP = Ω  �  j  Pj−1XjPj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (1)  Here, the spin at each site consists of two states, |0⟩ and  |1⟩, which represent a ground state and an excited state, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Xj = |0j⟩⟨1j| + |1j⟩⟨0j| is the Pauli x ma-  trix at site j, and Pj = |0j⟩⟨0j| is the projection opera-  tor for ground state at site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  This Hamiltonian describes  the system which prohibits the spin-flip, unless the neigh-  boring sites are in the ground state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' only the transition,  | · · · 010 · · · ⟩ · · · ↔ | · · · 000 · · · ⟩, is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Since this tran-  sition does not generate or annihilate excited states in any two  neighboring sites, one can exclude the states where the con-  secutive neighbors are excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Although the PXP model is a non-integrable chaotic  system11, there are certain product states which show non-  ergodicity and decent revival under time evolution, such as  |Z2⟩ ≡ |0101 · · · 01⟩ called a N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The non-ergodic  property of such product states is unique, in a sense that  the number of them increases linearly with the system size,  whereas, other ergodic product states show exponential in-  crease with the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' It has been understood that their  fidelity revival is originated from the quantum scarring, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=',  the product state with short-time revival is a linear combina-  tion of scar eigenstates with equivalent energy spacing7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The  |Z2⟩ state is mainly composed of the quantum scar states with  almost equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' However, because their energy  spacing is not perfectly even, it has been pointed out that the  fidelity revival of the |Z2⟩ state is also imperfect12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  There have been many suggestions to adjust the system en-  hancing this imperfect revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' As one promising way, it has  been studied in both theoretically and experimentally to the  addition of cosine modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' This modulation plays a role  of controlling chemical potential to the PXP model, and can  enhance the fidelity revival of the |Z2⟩ state or even induce  subharmonic responses in certain driving condition10,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' How-  ever, the systematic ways to find such driving have not been  studied in detail, which is the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  The periodically driven PXP model is represented as fol-  lowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  H(t) = HPXP + ∆sq(t)  �  j  nj,  (2)  where HPXP is defined in Equation 1 and the second term rep-  resents the periodic driving, where nj = |1j⟩⟨1j| counts the  number of excited states on each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In terms of the periodic  driving, we adopt the square pulse driving protocol as also in-  troduced in earlier studies for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' It shares a very similar  fidelity profile with the cosine driving case and thus well ex-  plains the experiments15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The square pulse driving protocol  ∆sq(t) within the period T is defined as,  ∆sq(t) =  �  �  �  �  �  ∆0 + ∆m,  0 ≤ t ≤ T/4  ∆0 − ∆m,  T/4 < t ≤ 3T/4  ∆0 + ∆m,  3T/4 < t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (3)  This corresponds to the periodic driving with frequency ω0 =  2π/T, average chemical potential ∆0, and driving amplitude  ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Now, we introduce the wave function fidelity F(t) ≡  |⟨ψ(t)|ψ(0)⟩|2 to measure how the revival of the initial prod-  uct state, |ψ(0)⟩ = |Z2⟩, changes as the driving parameters  (∆0, ∆m) are tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' As a tool to measure the subharmonic re-  sponse, we also introduce the Fourier component of the wave  function fidelity defined as,  F(ω) =  �����  1  nT  � nT  0  F(t)e−iωtdt  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (4)  Later, we will discuss the three main quantities, F(0), F(ω0)  and F(ω0/2), as functions of (∆0, ∆m): F(0) introduces  how the fidelity revival gets enhanced, and F(ω0) and  F(ω0/2) are responsible for harmonic and subharmonic re-  sponses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Note that if |ψ(t)⟩ is an eigenstate of  H(t) with driving parameters (∆0, ∆m), then �  j Zj|ψ(t)⟩  with Pauli z matrix Zj = |1j⟩⟨1j| − |0j⟩⟨0j| is also an eigen-  state of H(t) with the same driving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Hence, with-  out loss of generality, we only plot the region, ∆m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Before investigating the fidelity profile of the |Z2⟩ state in  the PXP model, let us introduce another model which shows  very similar feature: the free spin-1/2 chain model with the  same driving ∆sq(t),  Hfree(t) =  �  j  Xj + ∆sq(t)  �  j  nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (5)  For the |Z2⟩ state, the free spin-1/2 model and the PXP model  share common features in terms of graph theoretical point of  view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' To understand it, notice that the PXP model is noth-  ing but the free spin-1/2 model with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Hence, the  graph of length L PXP model, for instance, is a subgraph of  the free spin-1/2 model with the same length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Conversely,  consider the length L PXP model with even L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' If we give the  stronger constraint to the PXP model and only allow the states  with |0⟩ states at odd sites, then each states |0x0y · · · 0z⟩ can  be mapped to the state |xy · · · z⟩ in length L/2 free spin-1/2  model, where x, y, · · · , z are either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' This shows that  the graph of length L/2 free spin-1/2 model is a subgraph of  length L PXP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Figure 1(a) shows the graph of the PXP Hamiltonian with  L = 8, and Figure 1(b) shows the graph of the free spin-1/2  Hamiltonian for L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The blue square in Figure 1(a) marks  the N´eel state |01010101⟩ in PXP Hamiltonian, which corre-  sponds to the |1111⟩ state in free spin-1/2 model also marked  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' (Top) The Hamiltonian adjacency graph of (a) PXP model for L = 8 and (b) free spin 1/2 model for L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The blue square and  yellow triangle in Figure (a) represent the states |01010101⟩ and |00000000⟩ respectively, and the blue square and yellow triangle in Figure (b)  represent the states |1111⟩ and |0000⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' (Bottom) Average fidelity, F(0), with initial state |Z2⟩ through 10T time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Figure  (c) shows F(0) of PXP model with the driving period T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='788 for system size L = 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Figure (d) shows F(0), of free spin-1/2 model  with driving period T = π for system size L = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The ”butterfly” peaks are encircled by red line, ”bridge” peaks are encircled by magenta  line, ”local” (in Figure (c)) or ”steep bridge” (in Figure (d)) peaks are encircled by cyan line, and ”separator” peaks are encircled by blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  by the blue square in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The yellow triangles in  Figure 1(a) and (b) represent the polarized state |00000000⟩  state and |0000⟩ state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The vertices and edges  colored in red show the difference between the two graphs  and show that the graph in Figure 1(b) is indeed a subgraph  of Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Despite their difference, they share a com-  mon feature particularly for the |Z2⟩ state, marked by the  blue square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' This common feature is generally applicable for  the graphs of the length L PXP model and length L/2 free  spin-1/2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' To argue it in detail, let’s consider the ex-  pansion, ⟨Z2(t)|Z2⟩ = ⟨Z2|eiHt|Z2⟩ = �  n  (it)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' ⟨Hn⟩ with  ⟨Hn⟩ = ⟨Z2|Hn|Z2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In the graph theoretical point of view,  ⟨Hn⟩ counts the number of walks with length n which starts  and ends at |Z2⟩ vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Because the difference between PXP  graph and free spin graph mostly occurs for the states with  high Hamming distance, their difference only affects on the  long walks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' ⟨Hn⟩ with large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Hence, we may expect the  similar behavior in F(t) for a short time scale t in between  PXP model and free spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Later, we will show that it is  indeed the case by comparing calculation of the fidelity revival  on both PXP model and free spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' For completeness we  note that this argue is not applicable for ergodic initial states:  for example the polarized state |0p⟩ ≡ |0000 · · · 00⟩, marked  by the yellow triangle, shows a large difference in graphs even  for the nearest neighbors, indicating the different fidelity pro-  file between PXP model and free spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  In the presence of driving, one may also suggest the simi-  larities between PXP model and free spin-1/2 model, with re-  spect to the graph theoretical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Figure 1(c) and 1(d)  show the values of F(0) for PXP model and free spin-1/2  model respectively, as functions of driving parameters ∆0 and  ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' As discussed earlier, F(0) indicates the enhancement of  the fidelity revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Indeed, Figures 1(c) and 1(d) show very  similar features up to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' For calculation, we choose the  periodicity T0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='788 for the PXP model and Tf = π for the  free spin model respectively, which are optimized values for  the fidelity revival observed in the static cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The system  size L = 12 is chosen with the time range [0, 10T0] for the  PXP model, and L = 6 with the time range [0, 10Tf] for the  free spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  In Figure 1(c) and 1(d), we point out several common fea-  tures as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' We first note the ”butterfly”-shaped peaks  on top of each figure, marked by a red circle, and high average  fidelity region on the lower left and right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Next, there is a  wide V -shaped region having relatively small values of F(0)  in-between, with the following substructures: On the left side  in both Figures 1(c) and (d), the butterfly peaks are connected  to the lower left region by some ”bridges”, marked by ma-  16  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='8  12  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='6  E  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='4  6  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='2  2  0  10  5  0  5  1016  888  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='8  12  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='6  m  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='4  6  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='2  2  0  10  104  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Frequency profile of the fidelity F(t), (a) F(ω0) and (b)  F(ω0/2) with initial state |Z2⟩, for the PXP model with driving pe-  riod T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='788 and system size L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The ”butterfly” peaks are  encircled by red line, ”bridges” peaks are encircled by magenta line,  ”local” peaks are encircled by cyan line, and ”separator” peaks are  encircled by blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Frequency profile of the fidelity F(t), (a) F(ω0) and (b)  F(ω0/2) with initial state |Z2⟩, for the free spin model with driv-  ing period T = π and system size L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The ”butterfly” peaks  are encircled by red line, ”bridges” peaks are encircled by magenta  line, ”steep bridge” peaks are encircled by cyan line, and ”separator”  peaks are encircled by blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  genta lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In addition, there are ”local” peaks between the  bridges marked by cyan line in Figure 1(c), but instead there  are ”steep bridge” peaks in Figure 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Later, we will explain  that they indicate the same phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' On the right side in  both Figures 1(c) and (d), there are long and thin ”separator”  peak marked by green line, which separates the butterfly peaks  and the lower right region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  To determine the origin of these peaks, the frequency pro-  files of the fidelity are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Figures 2a and 2b plot  the Fourier component values of the fidelity for the |Z2⟩ state  in the PXP model, F(ω0) and F(ω0/2), with ω0 = 2π/T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  By comparing them with Figure 1(c), one can conclude that  the butterfly peaks (marked by red line) and the local peaks  (cyan line) represent harmonic revivals, whereas, the bridges  (magenta line) and separators (green line) represent the sub-  harmonic revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Notice that the bridges and the separators  also appear in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' However, this does not imply that  they are harmonic responses, since the subharmonic response  with nonzero F(ω0/2) also has finite values of F(ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' There-  fore, Figure 2b is a direct evidence, showing the subharmonic  response indeed occurs due to the driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  For comparison, we also investigate the free spin-1/2  model case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Figures 3a and 3b plot the values of F(ωf) and  F(ωf/2) for the free-spin model, showing harmonic and sub-  harmonic responses respectively, with ωf = 2π/Tf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' The  butterfly peaks (marked by red line) and the steep bridge peaks  (cyan line) in Figure 3a again shows the harmonic response,  while the bridge peaks (magenta line) and the separator peaks  on the right side (green line) in Figure 3b shows the subhar-  monic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Indeed, these features are consistent with the  case of the PXP model which is explained earlier (see Figure  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  PERTURBATIVE AND EXACT CALCULATIONS ON  THE MODELS  Until now, we have shown that the periodic driving of the  PXP model can induce the subharmonic responses of the |Z2⟩  state fidelity and have interpreted them based on the graph the-  oretical similarities with the free spin-1/2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In the fol-  lowing, for more rigorous argument, alternative analytic ap-  proaches are presented to understand subharmonic responses  and to determine the optimal driving conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Since our fo-  cus lies on the subharmonic response of driven PXP model,  we perform the appropriate perturbation limit which repre-  sents the V -shaped region in Figure 1(c), where every sub-  harmonic peaks lies on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' We note that perturbation approach  in another limit has been already performed in earlier work18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Its perturbative limit explains the ”butterfly” peaks, marked  by the red circle in Figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In contrast, our perturbative  limit explains every peaks on the V -shaped region, including  ”bridge”, ”local”, and ”separator” peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Before moving on, we redefine some notations for simplic-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Take ∆± ≡ (∆0 ± ∆m)/2, Xj ≡ Pj−1XjPj+1, and  H± ≡ �  j H±  j  with H±  j  ≡ ΩXj + ∆±Zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Notice that  H± = HPXP + 2∆±  �  j nj − ∆±  �  j Ij, hence the evolu-  tion operator in the presence of square pulse driving,  U = eiH+T0/4eiH−T0/2eiH+T0/4,  (6)  is equivalent to the evolution operator of H(t) in one period  up to phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Consider the limit ∆+ ≫ Ω ≫ ∆−, which is the right  side of the V -shaped region with low values of F(0) in Fig-  ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  We will show that this limit always gives subhar-  monic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In this limit, we can approximate H+ ≃  ∆+  �  j Zj and H− ≃ ΩHPXP taking the leading terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Be-  cause our product state is an eigenstate of H+, if we calcu-  late ⟨Z′  2|U|Z2⟩, where Z′  2 ≡ |1010 · · · 10⟩ is a translated N´eel  state, one can easily show ⟨Z′  2|U|Z2⟩ ≃ ⟨Z′  2|eiHPXPT0/2|Z2⟩  up to phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  For L = 12 case, this value is quite large  ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='9658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' This results in the 2T-periodic revival with high  lower bound of |⟨Z2|U 2|Z2⟩|, satisfying,  |⟨Z2|U 2|Z2⟩| ≥ |⟨Z2|eiHPXPT0/2|Z′  2⟩|2,  ∆+ ≫ Ω ≫ ∆−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (7)  See Appendix A for detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Thus, one can claim the  persistent subharmonic revival indeed occurs in ∆+ ≫ Ω ≫  ∆−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' We also show that this revival is robust even with O(Ω)  order terms in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='2  14  12  10  8  6  4  2  0  10  0  1016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='2  14  12  10  m  8  6  4  2  0  10  10  0  316  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='2  14  12  10  8  6  4  2  0  10  5  0  5  1016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='2  14  12  10  8  6  4  2  0  10  0  105  Now consider the limit ∆− ≫ Ω ≫ ∆+, which is the  left side of the V -shaped region with low values of F(0) in  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Similar analysis with the case ∆+ ≫ Ω ≫ ∆−  leads H+ ≃ ΩHPXP and H− ≃ ∆−  �  j Zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' We focus on the  driving conditions for the parameters ∆− = nπ  T0 = nω0  2 , and  our aim is to show that for even n case the fidelity presents  subharmonic response and for odd n case the fidelity presents  harmonic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' First, let n = 2k be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In this case,  eiH−T/2 = 1, and thus one again achieve ⟨Z′  2|U|Z2⟩ ≃  ⟨Z′  2|eHPXPT0/2|Z2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Thus, we get the very similar result with  Equation 7,  |⟨Z2|U 2|Z2⟩| ≥ |⟨Z2|eiHPXPT0/2|Z′  2⟩|2,  ∆− = kω0 ≫ Ω ≫ ∆+,  (8)  showing the subharmonic response mainly occurs at ∆− =  kω0 for integer k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' On the other hand, for odd n = 2k + 1’s,  eiH−T0/2 = �  j Zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Using the anti-commutation relation be-  tween �  j Zj and HPXP, the evolution operator is represented  as,  U ≃ eiHPXPT0/4  �  ��  j  Zj  �  � eiHPXPT0/4  =  �  j  Zje−iHPXPT0/4eiHPXPT0/4 =  �  j  Zj,  (9)  and this results in,  |⟨Z2|U|Z2⟩| ≃ 1,  ∆− =  �  k + 1  2  �  ω0 ≫ Ω ≫ ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' (10)  Thus, the harmonic response mainly occurs for ∆− = (k +  1/2)ω0 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' We again show that this revival is robust up to  O(Ω) order, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  In summary of this section, our perturbative analysis pro-  vides reasonable explanation why the several peaks in V -  shaped region are emerging in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Specifically, Equation  7 explains the long diagonal ”separator” peaks on the right  side of V -shaped region, Equation 8 explains the ”bridge”  peaks on the left side, and Equation 10 explains the ”local”  peaks between bridge peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' It is important to note that the  derivation for Equations 7, 8 and 10 can be generally applica-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSION  In this work, we study the wave function fidelity revival on  the periodically driven PXP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' First, we show that the  driving on PXP model induces various interesting responses,  including subharmonic responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Based on the graph theoret-  ical similarities between PXP model and free spin-1/2 model,  we have claimed and numerically confirmed that the driving  condition which induces the subharmonic response in the PXP  model can be captured by the free spin-1/2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Then, con-  sidering perturbative analysis, the generic driving conditions  for subharmonic responses in the PXP model are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Our  work will shed a light on the Rydberg atom simulator, study-  ing subharmonic responses of the driven quantum many-body  scarring systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  As an interesting future work, one may extend our studies  with finite van der Waals interaction, and explore the condi-  tions of the subharmonic revival as interaction changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Since  the strength of the van der Waals interaction is determined  by the distance between the two Rydberg atoms r as ∼  1  r6 ,9,  one could control the atom distance to tune their interaction  strength and track the revival property of the |Z2⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' One  can also consider the effect of further neighbor van der Waals  interactions, and explore how the fidelity revival condition  changes, which we will leave as a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='— We thank Junmo Jeon for valuable  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  This work is supported by National Re-  search Foundation Grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  2020R1A4A3079707, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  2021R1A2C1093060),).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Appendix A: Bound of the (sub)harmonic revival on PXP model  In this section, we show the harmonic and subharmonic re-  vival is stable under small Ω values up to first order, which is  discussed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' We show the Equations 7, 8 and 10  still holds if we include the O(Ω) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  We first consider the condition ∆+ ≫ Ω ≫ ∆−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In this  case, because for different sites j ̸= k, we have  [H+  j , H+  k ] ∼ O(Ω2),  (A1)  hence we may write  eiH+T0/4 ≃  �  j  eiH+  j T0/4 =  �  j  ei(ΩXj+  ∆+  2 Zj)T0/4  (A2)  up to O(Ω) order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' This can be expanded to cosine and sine  functions in O(Ω) order, achieving  eiH+  j T0/4 ≃ cos ∆+T0  4  + i sin ∆+T0  4  � Ω  ∆+  Xj + Zj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A3)  If we product all the eiH+  j T0/4 terms and left only the O(Ω)  order terms, then we finally get  eiH+T0/4  ≃ ei∆+  �  j ZjT0/4  �  �1 + i sin ∆+T0  4  �  j  e−i∆+ZjT0/4 Ω  ∆+  Xj  �  �  =  �  �1 + i sin ∆+T0  4  �  j  Ω  ∆+  Xje−i∆+ZjT0/4  �  � ei∆+  �  j ZjT0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A4)  To calculate the subharmonic response, we consider the  value ⟨Z′  2|U|Z2⟩:  if this value is large enough then it  6  guarantees the 2T-periodic revival with |⟨Z2|U 2|Z2⟩|  ≥  |⟨Z′  2|U|Z2⟩|2, because  ⟨Z2|U 2|Z2⟩ = |⟨Z2|U|Z′  2⟩|2 +  �  i  |⟨Z2|U|ψi⟩|2  ≥ |⟨Z2|U|Z′  2⟩|2,  (A5)  where {ψi} are the basis of the Hilbert space orthogonal to  |Z′  2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Now because we are considering Ω ≫ ∆− limit, we  ignore ∆−, giving eiH−T0/2 ≃ eiHPXPT0/2, then we have  ⟨Z′  2|U|Z2⟩ ≃ ⟨Z′  2|eiHPXPT0/2|Z2⟩  + i Ω  ∆+  sin ∆+T0  4  �  �⟨Z′  2|eiHPXPT0/2 �  j  Xje−i∆+ZjT0/4|Z2⟩  �  �  + i Ω  ∆+  sin ∆+T0  4  �  �⟨Z′  2|  �  j  e−i∆+ZjT0/4XjeiHPXPT0/2|Z2⟩  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A6)  For the second term, observe that  �  j  Xje−i∆+ZjT0/4|Z2⟩ = e−i∆+ZjT0/4 �  j  Xj|Z2⟩,  (A7)  because there are always excited states between two ground  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Hence we get  ⟨Z′  2|  �  j  e−i∆+ZjT0/4XjeiHPXPT0/2|Z2⟩  = ⟨Z′  2|eiHPXPT0/2HPXP|Z2⟩  = ∂t ⟨Z′  2|eiHPXPt|Z2⟩  ��  t=T0/2  We numerically check that ⟨Z′  2|eiHPXP|Z2⟩ becomes maxi-  mized at t = T0/2, and hence conclude this term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Arguing similar for the third term, we get  ⟨Z′  2|U|Z2⟩ ≃ ⟨Z′  2|eiHPXPT0/2|Z2⟩,  (A8)  showing  the  persistent  subharmonic  revival  because  |⟨Z′  2|eiHPXPT0/2|Z2⟩| can be taken high enough:  for ex-  ample, |⟨Z′  2|eiHPXPT0/2|Z2⟩| ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='9658 for L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Now we consider the condition ∆− ≫ Ω ≫ ∆+ region,  which is the left side of the V -shaped low F(0) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In this  case, by the similar way achieving A3 we achieve  eiH−  j T0/2 ≃ cos ∆−T0  2  + i sin ∆−T0  2  � Ω  ∆−  Xj + Zj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A9)  Here, we specifically focus on the area where ∆− = nπ+2η  T0  for integer n’s, with small η ≪ L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  We start with even n = 2k, giving  eiH−  j T0/2 ≃ ±1 ± iη  � Ω  ∆−  Xj + Zj  �  ,  (A10)  and  eiH−T0/2 ≃ 1 ± iη  �  j  � Ω  ∆−  Xj + Zj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A11)  Now we calculate  ⟨Z′  2|U|Z2⟩  = ⟨Z′  2|eiHPXPT0/2|Z2⟩  + ±iη⟨Z′  2|eiHPXPT0/4  �  ��  j  � Ω  ∆−  Xj + Zj  ��  � eiHPXPT0/4|Z2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A12)  Because the second term can be squeezed by L(1 + Ω/∆−)  again, we get  |⟨Z′  2|U|Z2⟩| ≥ |⟨Z′  2|eiHPXPT0/2|Z2⟩| − ηL  �  1 + Ω  ∆−  �  ,  (A13)  and since the first term is large enough, it shows that the sub-  harmonic response mainly occurs near ∆− = 2kπ  T0 = kω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  Finally, we take odd n = 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' In this case, we get  eiH−  j T0/2 ≃ ±η ± i  � Ω  ∆−  Xj + Zj  �  (A14)  and thus  eiH−T0/2 ≃ ±  �  j  Zj  �  �1 +  �  j  Zj  �  η ± i Ω  ∆−  Xj  ��  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A15)  Now by using the fact that HP XP and �  j Zj anticommutes,  we get  U ≃ ±e−iHPXPT0/4  �  �1 +  �  j  Zj  �  η ± i Ω  ∆−  Xj  ��  � eiHPXPT0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='  (A16)  Calculating ⟨Z2|U|Z2⟩, the first term gives ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' For the sec-  ond term, notice that the η-dependent term squeezes �  j Zj  operator, which gives its value at most L, and hence squeezed  by the value ηL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' For the  Ω  ∆− term squeezing �  j ZjXj =  i �  j Y j where Y j = Pj−1YjPj+1 with Yj a Pauli y matrix  Yj = i|0j⟩⟨1j|−i|1j⟩⟨0j|, we can numerically check that this  value squeezes below δ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Therefore,  |⟨Z2|U|Z2⟩| ≤ 1 − ηL − Ω  ∆−  δ  (A17)  and because ηL and δ are small enough, this represents the  persistent harmonic revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Desaules, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Mukherjee, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content=' Hal-  imeh, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content=' Potirniche,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content=' Michailidis, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
+page_content=' Lukin,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf'}
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+Systematic study of tunable laser cooling for
+trapped-ion experiments
+A. P. Kulosa1, O. N. Prudnikov3,4, D. Vadlejch1, H. A. F¨urst1,2,
+A. A. Kirpichnikova3, A. V. Taichenachev3,4, V. I. Yudin3,4,5,
+and T.E. Mehlst¨aubler1,2
+1 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,
+Germany
+2 Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, Welfengarten 1, 30167
+Hannover, Germany
+3 Institute of Laser Physics, 630090, Novosibirsk, Russia
+4 Novosibirsk State University, 630090, Novosibirsk, Russia
+5 Novosibirsk State Technical University, 630073, Novosibirsk, Russia
+E-mail: andre.kulosa@ptb.de
+Abstract.
+We report on a comparative analysis of quenched sideband cooling in
+trapped ions.
+We introduce a theoretical approach for time-efficient simulation of
+the temporal cooling characteristics and derive the optimal conditions providing fast
+laser cooling into the ion’s motional ground state. The simulations were experimentally
+benchmarked with a single 172Yb+ ion confined in a linear Paul trap. Sideband cooling
+was carried out on a narrow quadrupole transition, enhanced with an additional clear-
+out laser for controlling the effective linewidth of the cooling transition. Quench cooling
+was thus for the first time studied in the resolved sideband, intermediate and semi-
+classical regime. We discuss the non-thermal distribution of Fock states during laser
+cooling and reveal its impact on time dilation shifts in optical atomic clocks.
+Keywords: Tunable laser cooling, atom kinetics, cooling dynamics, trapped ions
+1. Introduction
+Laser cooling is an essential tool for modern quantum optics experiments with trapped
+ions, such as the study of topological defect formation in ion Coulomb crystals [1, 2, 3, 4],
+quantum simulation [5, 6, 7] and quantum computation [8]. Highly-accurate ion optical
+atomic clocks are traditionally operated at the Doppler cooling limit [9, 10].
+As a
+consequence of their constantly improving frequency uncertainty, the time dilation due
+to the residual ion secular motion at Doppler temperature nowadays poses a limiting
+contribution to the clock’s error budget at the low 10−18 level.
+In order to realize
+ion optical clocks at even lower uncertainties, they will have to operate at ultra-cold
+temperatures [11, 12].
+arXiv:2301.03276v1  [physics.atom-ph]  9 Jan 2023
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+2
+Since the first implementation of laser cooling (see e.g. [13]), several sub-Doppler
+cooling mechanisms have been developed and improved. Electromagnetically induced
+transparency (EIT) is an established tool to rapidly cool trapped ions into the motional
+ground state [14, 15]. Recently it was applied to cool the strong transverse mode in
+a single 171Yb+ ion to the motional ground state within a few 100 µs [16, 17]. Similar
+cooling rates were reported for cooling with polarization gradients (PG) [18] created by
+a standing wave field along the trap axis [19]. PG cooling of all motional modes in a
+single 171Yb+ ion to ¯n ≃ 1 was so far achieved within a few ms [20].
+Besides the aforementioned schemes, sideband cooling in the Lamb-Dicke regime is
+a well established technique for trapped ions [21]. Cooling on a narrow optical transition
+can be further enhanced by optical quenching, i.e. a laser-induced increase of the natural
+linewidth γ/2π of the cooling transition [22, 23, 24]. This method has been used in both
+trapped ion [25, 26, 27] and neutral atom experiments [28, 29]. The effective linewidth
+γeff/2π is steered with the intensity of the quenching laser, which couples the excited
+state of the cooling transition to a short-lived intermediate state (see fig. 1a). This
+allows for laser cooling in a broad range of different regimes ranging from the resolved
+sideband regime (γeff ≪ ωosc) to the semi-classical regime (γeff > ωosc), which can be
+described in terms of sub-Doppler [18, 19] or Doppler (see e.g. [30, 31, 32, 33]) forces
+acting from the resonant light field on the ion. Here, ωosc is the motional frequency the
+ion exhibits in the harmonic pseudopotential of the radio-frequency (rf) Paul trap. So
+far, optimisation studies of quench cooling were dedicated either to the limits of low
+saturation on the cooling transition [23, 34, 35] or to the strong-sideband regime, where
+(γeff/2ωosc)2 ≪ η [36, 37]. η =
+�
+ωrec/ωosc is the Lamb-Dicke parameter relating the
+ion’s kinetic energy change due to photon recoil with frequency ωrec to its confinement
+in the harmonic potential.
+In this work, we go beyond the aforementioned limitations by studying optimised
+cooling in the sideband, semi-classical and intermediate regime (γeff ≃ ωosc).
+We
+particularly study the impact of these regimes onto the distribution of atomic Fock
+states. The intermediate regime is naturally present for e.g. cooling of In+ ions [38]
+and can be used to cool the motional modes of multiple ions in the radial directions
+simultaneously.
+We especially focus on the nonlinear dependence of cooling time
+on cooling field intensity.
+We present a versatile method for the calculation of the
+characteristic cooling time, which does not require solving the dynamical evolution of
+the system’s density matrix. With this, we determine the optimal conditions for fast and
+deep laser cooling with significantly reduced computational efforts when compared to the
+full density matrix approach. We derive general analytical expressions for the optimal
+cooling laser Rabi frequency and the minimum cooling time which can be applied to
+any trapped ion species.
+The simulation results are benchmarked against experimental data acquired with a
+single 172Yb+ ion confined in a high-precision rf Paul trap [39]. We study the temporal
+evolution of the Fock state distribution during quench cooling and discuss its influence
+on the thermal time dilation shift in atomic clocks. Our findings pave the way towards
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+3
+fast and deep cooling of larger ion ensembles arranged ion Coulomb crystals.
+This paper is organized as follows: in Chapter 2 we recall the quantum model
+used for ion-light interaction and its reduction to an effective two-level system in the
+frame of optical quenching. Chapter 3 describes our simulation approach of using the
+“ˆτ-matrix” for faster computation of cooling dynamics, which is used for a systematic
+study of the sideband cooling regime in Chapter 4. In Chapter 5 we apply our “ˆτ-
+matrix method” to the specific case of an 172Yb+ ion confined in a rf Paul trap and
+study cooling ranging from the resolved sideband to the Doppler regime. We finally
+discuss the non-thermal distribution of Fock states during cooling and its impact on
+time dilation shifts in optical atomic clocks in Chapter 6.
+2. Quantum model for ion-light interaction
+The cooling dynamics of an ion confined in a rf Paul trap can be described by the
+quantum kinetic equation for the density matrix in single particle approximation
+∂ˆρ
+∂t = − i
+ℏ
+�
+ˆH, ˆρ
+�
++ ˆΓ{ˆρ},
+(1)
+where ˆH is the Hamiltonian and the term ˆΓ{ˆρ} describes the relaxation of the density
+matrix due to spontaneous decay. The Hamiltonian is composed of ˆH = ˆHext + ˆHint +
+ˆV1 + ˆV2, where
+ˆHext = ˆp2
+z
+2M + Mω2
+oscˆz2
+2
+(2)
+is the motional contribution of a harmonically confined ion with mass M in 1D
+approximation. Operator ˆV1 describes transitions induced by the cooling laser
+E1 = E01
+2 exp(ik1z − iω1t) + c.c.,
+(3)
+being in resonance with the |0⟩ → |1⟩ transition and ˆV2 describes the action of the
+quenching field
+E2 = E02
+2 exp(ik2z − iω2t) + c.c.,
+(4)
+resonant with the |1⟩ → |2⟩ transition, as depicted in fig. 1(a). In the rotating wave
+basis the Hamiltonian of the internal ion states is given by
+ˆHint = −δ2 ˆP2 − δ1 ˆP1,
+(5)
+where ˆP1 and ˆP2 are projection operators to the states |1⟩ and |2⟩. δ1 = ω1 − ω10 and
+δ2 = ω2 − ω21 are the detunings of the corresponding light fields (3) and (4), where
+ω10 and ω21 are the resonance frequencies of the unperturbed |0⟩ → |1⟩ and |1⟩ → |2⟩
+transitions, respectively.
+In the basis of Fock states the interaction operators ˆV1 and ˆV2 contain components
+that determine the amplitudes of transitions between the states with different vibrational
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+4
+Figure 1. (a) In a three-level system, the decay rate γ1 of state |1⟩ can be increased
+by laser-coupling to a higher-lying state |2⟩, which features a fast decay γ3 ≫ γ1 to
+the ground state |0⟩. For low saturation on the |1⟩ → |2⟩ transition, state |2⟩ can
+be adiabatically eliminated resulting in an effective two-level system with decay rate
+γeff of state |1⟩ [23]. (b) Temporal evolution of the mean occupation number during
+quench cooling with γeff/2π = 50 kHz and ¯nini = 20. The secular frequency is set to
+ωosc/2π = 600 kHz and the cooling light detuning is δ = −ωosc. The fastest cooling
+rate is observed for a specific Rabi frequency Ωopt of the cooling light, which is not
+expected from previous theoretical discussions [34].
+numbers. The Rabi frequency induced by the field (i = 1 for the cooling light, and i = 2
+for the quenching light) coupling the states with different vibrational numbers m (n) in
+the ground (excited) states is determined by the expression (see [21, 40] for details):
+Ω(i)
+nm = Ω(i)Cnm(ηi),
+(6)
+with
+Cnm(ηi) = L|n−m|
+n<
+�
+η2
+i
+�
+�
+n<!
+n>! (iηi)|n−m| exp
+�
+−η2
+i
+2
+�
+,
+(7)
+where n< = min{n, m}, n> = max{n, m}, and L|n−m|
+n<
+(x) is the generalized Laguerre
+polynomial.
+A small Lamb-Dicke parameter ηi =
+�
+ℏk2
+i /(2Mωosc) ≪ 1, i.e.
+tight
+harmonic confinement, significantly reduces the probability of induced and spontaneous
+transitions between energy states with different vibrational numbers m ̸= n, where the
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+5
+ki are the wave vectors of the corresponding light fields. The full quantum treatment of
+the three-level system with the dynamical evolution of the density matrix elements is
+discussed in Appendix A.
+As shown in [23], the action of the quenching field E2 onto a three-level system,
+leading to a fast decay to the ground state, allows for the reduction to an effective
+two-level system for the |0⟩ and |1⟩ states (see figure 1(a)) with effective decay rate
+γeff = γ1 + γ3
+ρ22
+ρ11 ≃ γ1
+�
+1 + γ3
+γ1
+S2
+�
+,
+(8)
+where S2 = Ω2
+2/[(γ1+γ2+γ3)2+4δ2
+2] is the quenching field saturation parameter and the
+γi (i = 1..3) are the decay rates of the involved transitions. A quantitative analysis of the
+cooling performance is derived from the time evolution of the mean occupation number
+¯n of the Fock states, which according to [34], can be interpolated by an exponential
+decay
+¯n(t) = (¯nini − ¯n∞) e−a t + ¯n∞,
+(9)
+where ¯n∞ is the mean occupation number in steady-state.
+Based on the dynamic
+equations (A.1), we derive the time evolution of ¯n for various Rabi frequencies of the
+cooling light, assuming an effective decay rate γeff = 2π ×50 kHz. The initial conditions
+correspond to a thermal distribution of Fock states in the electronic ground state with
+¯nini = 20 for an ion secular frequency of ωosc/2π = 600 kHz. As can be seen in figure 1(b),
+we expect the existence of an optimal value of cooling laser intensity corresponding to
+a maximised cooling rate, which is not predicted by [34]. In the following, we thus
+investigate the optimal cooling parameters for fastest cooling into the motional ground
+state in the resolved sideband, intermediate and semi-classical regime.
+3. The “ˆτ-matrix method” for fast simulation of cooling dynamics
+The exponential interpolation in equation (9) requires to numerically solve the dynamic
+equations (A.1), which, taking into account a large number n of Fock states, requires
+significant computational resources to solve for a 2n × 2n density matrix for a two-level
+atom. An alternative approach to derive the cooling time is given by the “ˆτ-matrix
+method”, recently introduced in [41] for cooling of neutral atoms. The ˆτ-matrix is given
+by the time integral of the difference of the atomic density matrix ˆρ(t) and its steady
+state solution ˆρst = ˆρ(t)|t=∞
+ˆτ =
+� ∞
+0
+(ˆρ(t) − ˆρst) dt .
+(10)
+The basic equation for the ˆτ-matrix is given by
+− i
+ℏ
+�
+ˆH, ˆτ
+�
++ ˆΓ {ˆτ} = ˆρst − ˆρini,
+(11)
+where ˆρini = ˆρ(t)|t=0 is the density matrix at initial time. As the density matrix ˆρ
+contains all information on external and internal states of the quantum system, the
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+6
+ˆτ-matrix contains all information on temporal characteristics of the system.
+As an
+example for an observable A characterized by the quantum operator ˆA, the characteristic
+evolution time can be extracted from the ˆτ-matrix by the following expression [41]:
+τA =
+Tr
+�
+ˆA ˆτ
+�
+�
+Tr
+�
+ˆA ˆρini
+�
+− Tr
+�
+ˆA ˆρst
+�� .
+(12)
+The cooling time is determined by the evolution rate of the external degrees of freedom
+determined by operator ˆHext in equation (2), and thus can be defined as
+τeff =
+Tr
+�
+ˆHext ˆτ
+�
+�
+Tr
+�
+ˆHext ˆρini
+�
+− Tr
+�
+ˆHext ˆρst
+�� .
+(13)
+In the case of a trapped ion this expression can be reduced to
+τeff =
+1
+(nini − n∞)
+� ∞
+0
+(n(t) − n∞) dt
+(14)
+which exactly corresponds to the 1/e value of an exponential decay of ¯n(t), as given in
+equation (9) with characteristic time τ = 1/a. As an example, for the results depicted
+in figure 1(b) the “ˆτ-matrix method” gives the following values τeff ≃ (115, 31, 55) γ−1
+eff ,
+with Rabi frequencies Ω/2π = (50, 200, 500) kHz, that are in good agreement with the
+results obtained through a fit of the direct numerical solution ¯n(t) by the exponential
+function (9), τ ≃ (115, 29, 50) γ−1
+eff for corresponding Rabi frequencies.
+Differences
+between τ and τeff should become noticeable if the evolution of ¯n(t) is not governed
+by an exponential decay.
+The “ˆτ-matrix method” allows to reduce the analysis based on solving the
+dynamical equations for the density matrix to a more simple task, i.e. the solution
+of the linear equation (11). As an example, taking into account n = 30 Fock states,
+computation of the effective cooling time with the “ˆτ-matrix method” takes approx. 45 s
+in our case, while the direct solution of eq. (1) takes 430 s. This allows us to perform
+a detailed analysis of the characteristic cooling time of a trapped ion with taking a
+sufficiently large number of vibrational states (nmax ≃ 120) into account.
+4. Dynamics of a trapped ion in the sideband cooling regime
+We now use the “ˆτ-matrix method” for a general study of the cooling dynamics in the
+resolved sideband cooling regime (where γeff ≪ ωosc) which, in principle, can be applied
+to any atomic species featuring a level structure as shown in fig. 1(a).
+In figures 2(a) and (b) we plot the mean occupation number ¯n∞ and the cooling
+time τeff, respectively, as a function of Rabi frequency Ω of the cooling light.
+We
+compare our results to calculations derived from simplified balance-rate equations in
+the low-intensity S1 ≪ 1 (S1 is the saturation parameter of the cooling transition)
+and strong Lamb-Dicke limit η ≪ 1 (dashed lines), without taking into account the
+coherence of the Fock states [34, 35]. Obtaining similar results for the low-intensity limit,
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+7
+Figure 2.
+(a) Steady-state mean occupation number and (b) cooling time τeff as
+function of cooling laser Rabi frequency Ω. A minimum cooling time τmin is observed
+at optimal Rabi frequency Ωτ.
+The dashed lines in (a) and (b) are derived from
+simplified balance-rate equations given in [34, 35]. (c) Cooling time τeff as function
+of initial ¯nini for various Rabi frequencies. Quench cooling parameters in (a)-(c) are
+γeff/ωosc = 0.1, η = 0.1 and detuning δ = −ωosc.
+our calculations indicate a global minimum in cooling time, τmin, at Rabi frequencies
+between Ωτ = 0.3 ωosc and Ωτ = 0.4 ωosc. This point defines the optimal parameters for
+both fast and simultaneously deep laser cooling, as the mean occupation number has not
+significantly changed from its minimum value in the low-intensity limit. Furthermore, we
+observe a linear dependence of cooling time τeff on the initial mean occupation number
+¯nini, as shown in Figure 2(c). As a next step, we study the dependence of the optimal
+parameters on the effective quench rate γeff. As shown in figure 3(a), we observe that
+Ωτ strongly depends on γeff in the sideband cooling regime, but does not much depend
+on the Lamb-Dicke parameter η. The available amount of simulation data for cooling
+in the sideband regime with parameters γeff/ωosc < 0.5, η < 0.3 and ¯nini < 20 allows us
+to deduce analytical expressions from fit results for the optimal Rabi frequency Ωτ
+Ωτ ≃
+�
+1.24 · γeff · ωosc
+(15)
+and the minimum cooling time τmin at Ωτ and detuning δ = −ωosc:
+τmin ≃ 1.2 + 2 ¯nini
+γeff
++
+1.9
+η2ωosc
+.
+(16)
+Note that these are general expressions valid for any two-level ion confined in a harmonic
+trap with a decay rate γeff and Lamb-Dicke parameter η. We use equations (15) and (16)
+to plot the dashed black lines in figure 3 and observe good agreement with the results
+for Ωτ and τmin obtained with our direct simulation. In particular, according to equation
+(15), the optimal Rabi frequency providing fast cooling for parameters used in figure
+1(b) results to Ωτ/2π ≃ 193 kHz, which is in excellent agreement with the result of the
+direct simulation of cooling dynamics shown there.
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+8
+Figure 3. (a) Optimal Rabi frequency Ωτ for reaching τmin as function of γeff for
+various Lamb-Dicke parameters η. The black dashed line has been calculated with the
+analytical expression given by Equation (15). (b) Minimum cooling time at optimal
+Rabi frequency Ωτ as function of effective decay rate for various ¯nini (indicated by
+different symbols) and Lamb-Dicke parameters η (indicated by different colours). The
+detuning of the cooling laser corresponds to the first red sideband resonance δ = −ωosc.
+The black dashed lines have been calculated with the expression for τmin given in
+Equation (16) and plotted in units of γ−1
+eff .
+5. Analysis of various quench cooling regimes in 172Yb+
+Turning from a general treatment to a specific case, we now apply our “ˆτ-matrix
+method” to a 172Yb+ ion confined in a rf Paul trap. Figure 4 shows the relevant atomic
+states involved in the quench cooling process. The |0⟩ → |1⟩ cooling transition is given by
+the 2S1/2 → 2D5/2 quadrupole transition near 411 nm, where the 2D5/2 → 2P3/2 dipole
+transition near 1650 nm is used as |1⟩ → |2⟩ quenching transition. In the following,
+we compare the rates and limits of quench cooling in the resolved sideband regime
+γeff ≪ ωosc, the semi-classical regime γeff > ωosc and in the intermediate cooling regime
+γeff ≃ ωosc.
+For each of these regimes, Figure 5 shows the results for the mean occupation
+number ¯n∞ and the cooling time for different Rabi frequencies induced by the cooling
+field E1. In the resolved sideband regime, we observe a minimum of ¯n ≃ 0.0012 for
+a detuning δ = −ωosc at low intensities (Ω = 1/12 ωosc), which is close to the limit
+¯nmin ≃ 7/48 (γeff/ωosc)2 ≃ 0.001 at low intensity as was obtained similarly to [34, 35],
+or ¯nmin ≃ 5/16 (γeff/ωosc)2 ≃ 0.002 in [21]. Only the linear dependence on light field
+intensity was taken into account for ¯n in the simplified balance-rate equations [34, 35],
+which gives slightly underestimated results for the mean occupation number compared
+to our direct simulation obtained by the “ˆτ-matrix method”. For visibility, we used a
+logarithmic scale in Figure 5(a) to pronounce the differences at low intensities (black
+dashed and solid curves). The results for the cooling time coincide with [34, 35] in
+the vicinity of δ = −ωosc at low laser intensity.
+However, with increasing intensity
+the difference between the simple model and our direct simulation becomes more
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+9
+Figure 4. Relevant atomic levels for quench cooling in 172Yb+ including their decay
+rates. Sideband cooling is carried out on the electronic 2S1/2 →
+2D5/2 quadrupole
+transition near 411 nm. The 2D5/2 state is coupled to the short-lived 2P3/2 state by
+means of laser light near 1650 nm.
+pronounced (green dashed and solid lines for Ω = 1/3 ωosc). We also observe that the
+optimal detuning for minimum ¯n∞ and cooling time shifts to smaller absolute values.
+Additionally, at low intensity, local minima are also visible near δ/ωosc = −2, −3, . . ..
+These effects are not predicted by the simplified model [34, 35].
+Note the different dependencies of temperature and cooling time on cooling laser
+intensity: in the weak-field limit, the ion temperature (or the mean occupation number
+¯n) does not significantly depend on intensity, but cooling time decreases proportional
+to it. For high-intensity laser fields the temperature significantly grows, but does not
+result in essential further decreasing of cooling time. Such a dependence of cooling time
+and temperature on field intensity allows to define an optimal cooling light intensity for
+effective, i.e. fast and simultaneously deep quench cooling. A similar behaviour can be
+observed in the intermediate (figure 5(b)) and semi-classical cooling regime (figure 5(c)).
+To select the optimal cooling parameters we use the following algorithm: for various
+intensities of the cooling light field, we determine the optimal detuning δ∗ providing
+the maximum cooling rate. At these values of δ∗, we then analyse the steady-state
+temperature.
+The results are shown in figure 6.
+The optimal detuning δ∗ shifts
+away from the sideband resonance δ = −ωosc with growing intensity (figures 6(a, d,
+g)).
+In each of the studied cooling regimes, a minimum for the cooling time can
+be observed in a certain range of cooling light intensity (figures 6(b, e, h)), which
+allows to select the optimal intensity for fast cooling. In figures 6(c, f, i) we plot the
+steady-state temperature T and average atomic energy Eav as a result from cooling at
+optimal detuning δ∗. The steady-state temperature, which we plot in units of ℏωosc/kB,
+is determined by fitting an exponential distribution to the steady-state populations
+governed by ˆρst. The average atomic energy results as the mean value Eav = �
+n nPn,
+where Pn is a Boltzmann distribution with steady-state temperature T over the atomic
+states with secular frequency ωosc/2π = 600 kHz. Note that Eav is plotted in units of
+ℏωosc with offset of 0.5 to account for the quantum mechanical ground state energy.
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+10
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+Ω = 1/12 ωosc
+Ω = 1/6 ωosc
+Ω = 1/3 ωosc
+Ω = ωosc
+Ω = 5/3 ωosc
+Ω = 1/12 ωosc
+Ω = 1/6 ωosc
+Ω = 1/3 ωosc
+Ω =
+ωosc
+Ω = 5/3 ωosc
+Ω = 1/3 ωosc
+Ω = 2/3 ωosc
+Ω =
+ωosc
+Ω = 5/3 ωosc
+Ω = 2 ωosc
+Ω = 1/3 ωosc
+Ω = 2/3 ωosc
+Ω =
+ωosc
+Ω = 5/3 ωosc
+Ω = 2 ωosc
+Ω = 1/3 ωosc
+Ω = ωosc
+Ω = 2 ωosc
+Ω = 10/3 ωosc
+Ω = 1/3 ωosc
+Ω = ωosc
+Ω = 2 ωosc
+Ω = 10/3 ωosc
+W
+=
+ 
+1
+/
+1
+2
+ 
+w
+o
+s
+c
+Figure 5.
+Comparison of quench cooling in (a) the sideband cooling regime with
+γeff/2π = 50 kHz, (b) intermediate cooling regime with γeff/2π = 600 kHz and (c) semi-
+classical cooling regime with γeff/2π = 2000 kHz. For each of the regimes, both ¯n∞ and
+the cooling time τeff (in units of the decay rate γ−1
+1
+of the 2S1/2 → 2D5/2 transition)
+are plotted for various Rabi frequencies of the cooling light field E1. All cooling laser
+parameters (Ω and δ) are given in units of ion secular frequency ωosc = 2π × 600 kHz.
+Each cooling simulation started with an initial Boltzmann distribution with ¯nini = 20.
+The dashed black and green curves in (a) correspond to ¯n∞ and cooling time at
+Ω/ωosc = 1/12 and 1/3 obtained from analytical expressions given in [34].
+The absolute minimum cooling temperature is reached in the resolved sideband
+cooling regime γeff ≪ ωosc.
+The corresponding minimum cooling time τ ∗ ≃ 2.4 ×
+10−3 γ−1
+1
+≃ 100 µs at optimal detuning δ∗ ≃ −0.89 ωosc is reached for a Rabi frequency
+Ω∗ ≃ ωosc/2 of the cooling light. Here, the steady-state temperature is only slightly
+larger than the minimum value obtained in the low-intensity limit.
+Both in the
+intermediate and semi-classical cooling regime, the minimum cooling time is found for
+a Rabi frequency of the cooling light
+Ω∗ ≃
+�
+γ2
+eff/2 + 2 ω2
+osc ,
+(17)
+where the ion temperature increases proportionally with increasing Rabi frequency Ω∗.
+Note that the analytical approximations for the optimum Rabi frequency are again
+very general and can be applied to quench cooling in atomic species with similar level
+schemes.
+To draw a first conclusion on our findings, the intermediate regime is very
+promising for fast laser cooling to ¯n < 1, as the observed minimum cooling time
+τ ∗ ≃ 1.2 × 10−3 γ−1
+1
+≃ 50 µs is very similar to the semi-classical cooling regime, but
+at the expense of slightly increased temperature.
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+11
+0
+200
+400
+600
+800
+-1,0
+-0,9
+-0,8
+-0,7
+-0,6
+0
+200
+400
+600
+800
+0,000
+0,002
+0,004
+0,006
+0,008
+0
+200
+400
+600
+800
+0,2
+0,4
+0,6
+0,8
+1,0
+0
+200
+400
+600
+800
+1000
+1200
+-1,0
+-0,9
+-0,8
+0
+200
+400
+600
+800
+1000
+1200
+0,000
+0,002
+0,004
+0,006
+0,008
+0
+200
+400
+600
+800
+1000
+1200
+0,4
+0,6
+0,8
+1,0
+1,2
+0
+400
+800
+1200
+1600
+-1,8
+-1,7
+-1,6
+-1,5
+-1,4
+-1,3
+0
+400
+800
+1200
+1600
+0,000
+0,002
+0,004
+0,006
+0,008
+0
+400
+800
+1200
+1600
+1,0
+1,2
+1,4
+1,6
+(c)
+(b)
+g
+ef f
+/2p = 50 kHz
+d
+*
+ / w
+osc
+ 
+(a)
+t
+*
+eff
+ . g
+-1
+1
+ 
+ E
+av
+ T
+E
+av
+ / hw
+osc
+ , k
+B
+T / hw
+osc
+ 
+W  (kHz)
+(f)
+(e)
+g
+ef f
+/2p = 600 kHz
+(d)
+ E
+av
+ T
+W  (kHz)
+(i)
+(h)
+g
+ef f
+/2p = 2000 kHz
+(g)
+ E
+av
+ T
+W  (kHz)
+Figure 6. Optimal parameters for fastest quench cooling in various regimes: resolved
+sideband regime (a, b, c), intermediate regime (d, e, f), and semi-classical cooling
+regime (g, h, i). In each of the cases, the optimal detuning δ∗ will lead to an optimal
+cooling time τ ∗ (in units of the decay rate γ−1
+1
+of the 2S1/2 → 2D5/2 transition). The
+ion temperature T is obtained by fitting the steady-state population governed by ˆρst
+with an exponential distribution. Using this T in a Boltzmann distribution results to
+the average cooling energy Eav. In each of the cooling regimes shown here, the ion
+secular frequency was ωosc/2π = 600 kHz and the simulations started with an initial
+thermal distribution with ¯nini = 20.
+We benchmark our simulation with the “ˆτ-matrix method” against data acquired
+with a single 172Yb+ ion confined in a high-precision rf Paul trap [39]. The rf drive
+frequency is Ωrf = 2π × 24.4 MHz leading to a typical secular frequency of the strong
+radial mode of ωx = 2π × 565(5) kHz. In a first stage, the ion is Doppler-cooled to
+TD ∼ 0.5 mK on the 2S1/2 →
+2P1/2 transition near 370 nm.
+The resulting thermal
+distribution of Fock states features a mean occupation number ¯n = 18(1), measured
+with Rabi flops recorded on the 2S1/2 → 2D5/2 transition near 411 nm.
+Following the Doppler cooling stage, the ion is initialized in the mJ = −1/2 ground
+state. The cooling laser near 411 nm is set to a Rabi frequency Ω/2π = 50(2) kHz and
+detuning δ where we derive the ion temperature as a function of quench cooling time
+tcool via the amplitude ratio R = IBSB/IRSB of blue and red sidebands [22]. The mean
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+12
+occupation number is then given by
+¯nSB =
+1
+R − 1.
+(18)
+Each individual data point of the sideband scans is repeated 200 times for significant
+statistics.
+The effective cooling time τeff is derived as a decay parameter from an
+exponential fit, as shown in the inset of figure 7. We characterized the cooling rate in a
+frequency interval ranging from −0.75 to −2.25 ωosc spanning at least the first two red
+secular sidebands, i.e. δ = −ωosc and δ = −2ωosc, respectively. The impact of sideband
+cooling is well pronounced for a moderate quench with γeff/2π = 50(2) kHz (black curve)
+and our measurement is in excellent agreement with the direct simulation in close vicinity
+of −ωosc and −2 ωosc. However, we observe a faster cooling in the region between the first
+and second red sideband, which we attribute to off-resonant excitation of the sidebands
+due to laser noise at 350 kHz (with a FWHM of approx. 170 kHz, see also Figure B1
+in Appendix B), leading to additional cooling. As expected, the sideband signature
+vanishes for cooling close to the intermediate regime with γeff/2π = 404(21) kHz (blue
+curve). As the cooling laser Rabi frequency is limited by available laser power, we are
+not able to resolve an even faster cooling rate in this regime, as predicted in figure 6(e).
+Figure 7.
+Effective cooling time τeff for quench cooling in the resolved sideband
+regime with γeff/2π = 48(2) kHz (black) and close to the intermediate regime with
+γeff/2π = 404(21) kHz (blue) as a function of cooling laser detuning in units of ion
+secular frequency ωosc = 2π × 565(5) kHz. The dashed line is derived from analytical
+expressions in [34] and the solid curves correspond to our direct simulation with
+Ω = 2π×50 kHz. The filled squares and circles are experimental data. Each experiment
+was carried out with a laser intensity corresponding to Ω = 2π×50(2) kHz and an initial
+thermal distribution with ¯n ≃ 18(1). The inset exemplarily shows how the effective
+cooling time for γeff/2π = 404(21) kHz and δ = −ωosc was determined.
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+13
+6. Fock state distribution during quenched sideband cooling
+A distribution of Fock states being highly non-thermal after cooling is known to
+cause a significant underestimate of temperature [42, 43, 44]. Here, we simulate the
+distribution of the ion Fock states for various cooling times in our trap using a Monte-
+Carlo simulation, similar to [45], as our previously introduced “ˆτ-matrix method” only
+provides the population distribution in steady-state.
+As we experimentally deduce
+the ion temperature from the sideband amplitude ratio R, we model the red and
+blue sideband strengths based on the simulated Fock state distribution and derive a
+corresponding theory value for the temperature according to equation (18).
+For an
+arbitrary Fock state distribution Pn the sideband amplitude ratio R can be expressed
+as follows:
+R(t) = 1 − �
+n Pn cos (Cn+1,n(η)Ω0t)
+1 − �
+n Pn cos (Cn−1,n(η)Ω0t),
+(19)
+where Cn+1,n(η) and Cn−1,n(η) are the sideband strength coefficients of the blue and
+red sideband, respectively, as given in equation (7). Note that in general the ratio R
+is a function of excitation pulse time which is not the case for Pn corresponding to the
+thermal distribution. For the thermal distribution, the equation (19) directly implies
+the well known relation for ¯n shown in expression (18).
+Figures 8(a) to (c) show the temperature as a function of cooling time for various
+effective linewidths γeff/2π, both plotted for the theoretically obtained ¯nSB (black curves)
+and the mean value of the distribution �
+n nPn (red curves).
+As soon as sideband
+cooling is initiated, the data obtained from the measurements (blue squares) indeed
+reveal a non-thermal distribution of Fock states for small effective linewidths (figs. 8(a)
+and (b)), in good agreement with the calculated temperature ¯nSB. Note that the 1/e
+cooling times of τratio = 0.125(2) ms and τmean = 0.714(9) ms derived for ¯nSB and the
+mean value of the distribution, respectively, significantly differ from each other.
+In
+order to pronounce the underestimation of atomic temperature and cooling times, we
+plot the thermal distributions for ¯n = ¯nSB (orange) and ¯n = �
+n n · Pn (beige) in
+figs. 8(d) to (f) together with the previously simulated Fock state distribution (blue).
+For larger effective linewidths, i.e., γeff/2π = 64 kHz, the distribution of Fock states
+slowly resembles a thermal distribution, as can be seen in figs. 8(c) and (f). However,
+we still observe a slightly stronger non-thermal distribution in the experiment. In any
+case, thermal equilibrium of ¯n ≃ 0.06(2) is always reached for sufficiently long cooling
+times. Our findings suggest that the intermediate regime, where γeff ≃ ωosc, is not only
+well suited for fast and deep quench cooling to ¯n < 1, moreover it is expected to be
+rather immune to non-thermal distributions of Fock states during cooling.
+Being of high importance for optical clocks with trapped ions, we calculate the
+temporal evolution of thermal time dilation as experienced by a 172Yb+ ion confined
+in a linear Paul trap. As a basis for kinetic energy we use the mean �
+n nPn of the
+simulated distribution of Fock states. Figure 9(a) shows the 3D relative time dilation
+shift ∆f/f0 = −5/2Ekin/mc2 [46], as a function of cooling time per motional mode
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+14
+Figure 8. Impact of a non-thermal distribution of atomic Fock states on temperature
+evaluation. A Monte-Carlo simulation was used to calculate the population distribution
+for various cooling times, if quench cooling is carried out with γeff/2π = 8, 16, 64 kHz.
+For each value of tcool this Fock state distribution is used to calculate the strengths of
+red and blue sidebands, if driven with a Rabi frequency of Ω411 = 2π×40 kHz. In (a)-(c)
+the calculated sideband ratio (black curves) and the mean value of the distribution (red
+curves) are plotted as a function of cooling time tcool. The blue squares correspond
+to experimentally deduced sideband ratios acquired with (a) γeff/2π = 8.2(2) kHz,
+(b) γeff/2π = 16.4(7) kHz and (c) γeff/2π = 62(4) kHz.
+The shaded region with
+the black curves account for the uncertainty in our experimental Rabi frequency
+Ω411 = 2π × 40(2) kHz. An exponential decay was fitted to each theory curve and
+experimental data to determine the characteristic cooling time with initial temperature
+¯nini = 18. The fit results are given in (a)-(c). (d)-(f) show - exemplarily at tcool = 1 ms
+- the corresponding simulated Fock state distribution (blue) and thermal distributions
+with ¯n = 1/(R − 1) (orange) and ¯n = �
+n nPn (beige).
+with ωosc = 2π × 600 kHz for various effective quench rates. In each case, the initial
+time dilation shift corresponds to the temperature after Doppler cooling. Unless cooling
+is carried out to thermal equilibrium, we observe a significant dependence of the time
+dilation shift on the effective quench rate.
+If cooling was stopped before a thermal
+equilibrium is reached, as depicted in figures 8(d)-(f), temperature would be falsely
+determined with the sideband ratio and accordingly lead to a wrong estimation of the
+time dilation shift. For example, if cooling is carried out for 1 ms in the sideband regime
+with γeff = 8 kHz (figure 8d), the sideband ratio would suggest a time dilation shift of
+−2.3 × 10−20 (according to ¯n = 0.29), while the real distribution of Fock states reveals
+a shift of −2.2 × 10−19 (according to ¯n = 5.23). In this regime, the error budget of
+an atomic clock would be underestimated by one order of magnitude.
+For reduced
+discrepancies, it is advised to cool in a regime where γeff/ωosc > 0.1, as indicated in
+figure 8(f). Finally, we would like to point out that the time dilation shift seen by
+the ion is highly dependent on the cooling laser intensity. In figure 9(b) we fixed the
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+15
+700
+Figure 9. 3D relative time dilation shift experienced by a 172Yb+ ion confined in a
+linear Paul trap. Here we assume a secular frequency of ωosc = 2π × 600 kHz for each
+motional mode and plot the time dilation as a function of quench cooling time per
+dimension. The simulated mean value of the distribution �
+n nPn amongst the Fock
+states, as shown in blue in figure 8, was used as a basis for the shift calculation. (a)
+The Rabi frequency was set to Ω/2π = 40 kHz whilst the effective quench rate was
+varied. (b) For an effective quench rate γeff = 64 kHz we observe an optimal Rabi
+frequency of Ω/2π = 200 kHz allowing for fast quench cooling to the point of thermal
+equilibrium.
+effective quench rate to γeff = 64 kHz and studied the impact of various cooling laser
+intensities. In accordance with our observation of an optimal Rabi frequency for fastest
+cooling, as shown in figures 1(b) and 6(b) , this value also allows to reach the point of
+thermal equilibrium as fast as possible.
+7. Summary
+To conclude, we studied the mechanism of quenched sideband cooling and presented a
+versatile method for fast calculation of the characteristic cooling time which does not
+require to consider the dynamical evolution of the system’s density matrix elements.
+Our “ˆτ-matrix method” significantly reduces computational efforts and agrees within
+90 − 100% with the full quantum model.
+Based on our powerful simulation tool, we derived universal analytical expressions
+for the optimum Rabi frequency Ωτ and the resulting minimum cooling time τmin for
+cooling in the resolved sideband regime, which can be applied to any atomic species
+with decay rate γeff confined in a rf Paul trap. Applying the simulation to a specific
+case, we investigated quench cooling in a 172Yb+ ion, thereby focussing on three different
+regimes: (I) resolved sideband cooling with γeff/2π = 50 kHz, (II) intermediate cooling
+with γeff/2π = 600 kHz and (III) semi-classical cooling with γeff/2π = 2000 kHz. For
+each of these regimes we derived the steady-state temperature and the effective cooling
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+16
+time for various cooling laser parameters, such as Rabi frequency and detuning. From
+this extensive analysis we extracted the optimal parameters to be applied for fast cooling
+into the motional ground state in each of the aforementioned regimes. We benchmarked
+our simulation code against data taken with a single 172Yb+ ion confined with a secular
+frequency ωosc = 2π × 565(5) kHz and observed an agreement between experiment and
+theory.
+We presented a detailed analysis of Fock state population distributions during the
+cooling process. In particular, we compared the time behaviour of temperature derived
+with the sideband ratio method to the actual mean occupation of Fock states and
+revealed discrepancies of more than one order of magnitude, if cooling is not carried out
+to thermal equilibrium. In addition, we investigated the temporal evolution of thermal
+time dilation in an optical clock with 172Yb+ during the process of quench cooling. We
+conclude that quenching in a regime with γeff/ωosc > 0.1 is necessary to stay close to a
+thermal distribution of Fock states.
+The results presented in this work pave the way to flexible quench cooling of ion
+Coulomb crystals with respect to fast laser cooling into the ion’s motional ground
+state. A crystal consisting of N ions requires 3N motional modes to be cooled, thus
+a clever combination of confinement parameters and quench rate is expected to reduce
+the cooling time compared to cooling of each mode individually.
+Acknowledgments
+We thank Jonas Keller for useful comments on optical spectroscopy in the presence
+of laser noise.
+This work was funded by the Deutsche Forschungsgemeinschaft
+(DFG, German Research Foundation) under German’s Excellence Strategy—EXC-2123
+QuantumFrontiers—390837967 and by DFG Grant No. ME 3648/5-1. H. A. F¨urst was
+supported by the EMPIR project 18SIB05 ”Robust Optical Clocks for International
+Timescales”.
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+Appendix A. Dynamical evolution of density matrix elements
+The equation for the evolution of the components of the density matrix of vibrational
+states n, m (see fig. 1(a)) takes the following form:
+∂ρ22
+nm
+∂t
+= − (γ3 + γ2)ρ22
+nm − iωosc(n − m)ρ22
+nm
+− i
+2
+�
+µ
+�
+Ω(2)
+nµρ12
+µm − ρ21
+nµΩ∗ (2)
+µm
+�
+∂ρ11
+nm
+∂t
+= − γ1ρ11
+nm + ˆγ2{ρ22}nm − iωosc(n − m)ρ11
+nm
+− i
+2
+�
+µ
+�
+Ω∗ (2)
+nµ ρ21
+µm + Ω(1)
+nµρ01
+µm − ρ12
+nµΩ(2)
+µm − ρ10
+nµΩ∗ (1)
+µm
+�
+∂ρ00
+nm
+∂t
+= ˆγ3{ρ22}nm + ˆγ1{ρ11}nm − iωosc(n − m)ρ00
+nm
+− i
+2
+�
+µ
+�
+Ω∗ (1)
+nµ ρ10
+µm − ρ01
+nµΩ(1)
+µm
+�
+∂ρ01
+nm
+∂t
+= − (γ1/2 + iδ1) ρ01
+nm − iωosc(n − m)ρ01
+nm
+− i
+2
+�
+µ
+�
+Ω∗ (1)
+nµ ρ11
+µm − ρ00
+nµΩ∗ (1)
+µm − ρ02
+nµΩ(2)
+µm
+�
+∂ρ10
+nm
+∂t
+= − (γ1/2 − iδ1) ρ10
+nm − iωosc(n − m)ρ10
+nm
+− i
+2
+�
+µ
+�
+Ω(1)
+nµρ00
+µm + Ω∗ (2)
+nµ ρ20
+µm − ρ11
+nµΩ(1)
+µm
+�
+∂ρ12
+nm
+∂t
+= − (γ1/2 + γ2/2 + γ3/2 + iδ2) ρ12
+nm
+− iωosc(n − m)ρ12
+nm
+− i
+2
+�
+µ
+�
+Ω(1)
+nµρ02
+µm + Ω∗ (2)
+nµ ρ22
+µm − ρ11
+nµΩ∗ (2)
+µm
+�
+∂ρ21
+nm
+∂t
+= − (γ1/2 + γ2/2 + γ3/2 − iδ2) ρ21
+nm
+− iωosc(n − m)ρ21
+nm
+− i
+2
+�
+µ
+�
+Ω(2)
+nµρ11
+µm − ρ20
+nµΩ∗ (1)
+µm − ρ22
+nµΩ(2)
+µm
+�
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+19
+∂ρ02
+nm
+∂t
+= − (γ2/2 + γ3/2 + γ + iδ1 + iδ2) ρ02
+nm
+− iωosc(n − m)ρ02
+nm
+− i
+2
+�
+µ
+�
+Ω∗ (1)
+nµ ρ12
+µm − ρ01
+nµΩ∗ (2)
+µm
+�
+∂ρ20
+nm
+∂t
+= − (γ2/2 + γ3/2 + Γ − iδ1 − iδ2) ρ20
+nm
+− iωosc(n − m)ρ20
+nm
+− i
+2
+�
+µ
+�
+Ω(2)
+nµρ10
+µm − ρ21
+nµΩ(1)
+µm
+�
+(A.1)
+where the superscripted indices 0, 1, 2 of the density matrix elements denote the states
+|0⟩, |1⟩ and |2⟩, respectively and the subscripted indices n, m and µ are related to
+vibrational states. Γ determines the decay of the coherence between the |2⟩ and |0⟩
+states as a result of an uncorrelated phase of the laser fields E1 and E2.
+The spontaneous relaxation operator ˆγ{ρ}nm determines the spontaneous decay
+rate γ1, γ2, and γ3 (see fig. 1 (a)), the decay of non-diagonal elements of the density
+matrix, as well as contributions from excited states to the ground state. These terms in
+(A.1) have the form
+ˆγi{ρjj}nm =
+�
+νµ
+Γ(i) νµ
+nm ρjj
+νµ.
+(A.2)
+The decay rates Γ(i) νµ
+nm
+for the dipole transitions |2⟩ → |1⟩ and |2⟩ → |0⟩ can be obtained
+from the general expression for the spontaneous relaxation operator taking into account
+recoil effects (see for example [47])
+Γ(i) νµ
+nm
+= γi
+� +1
+−1
+K(d)(h) (Cνn(ηih))+ Cµm(ηih) dh,
+(A.3)
+with K(d)(h) being the dipole pattern for the decay and ηi are the Lamb-Dicke
+parameters for the corresponding dipole transitions (i = 2, 3 in fig. 1 (a)).
+For
+the quadrupole transition the relaxation operator has a similar form to (A.3) with
+replacement of the dipole with the quadrupole pattern for the decay K(q)(h) and
+corresponding Lamb-Dicke parameter η1 [48].
+Appendix B. Off-resonant excitation via laser noise
+We attribute the faster cooling rate observed in our experiment at cooling laser detunings
+around −1.5 ωosc (see fig. 7) to off-resonant excitation induced by laser noise.
+For
+verification, we studied the frequency spectrum of the 2S1/2 →
+2D5/2 transition near
+411 nm ranging from the first-order red sideband to the first-order blue sideband.
+Figure B1 shows such a frequency spectrum, recorded overdriven with interrogation
+time t ≃ 7 · τπ. The spectral features at ±350 kHz correspond to noise modulation of
+
+Systematic study of tunable laser cooling for trapped-ion experiments
+20
+the laser light, most probably caused by the bandwidth of the locking electronics of
+the second harmonic generation cavity for 411 nm. While the frequency detuning of
+the 411 nm cooling laser was set to −1.5 ωosc, with ωosc = 2π × 565(5) kHz, both the
+first and second-order red sidebands were seperated by 283(5) kHz. With a FWHM of
+approx. 310 kHz of the noise spectral feature, a significant overlap to both red sidebands
+is given and most likely the reason for faster cooling observed in the experiment in this
+frequency range.
+Figure B1.
+Frequency spectrum of the 2S1/2 →
+2D5/2 transition near 411 nm,
+relative to the carrier transition. For this measurement, the 172Yb+ ion was confined
+with secular frequency ωosc = 2π × 654(6) kHz and interrogated with a pulse time of
+t ≃ 7 · τπ to enhance the signature of laser noise between the sideband resonances. We
+fit Lorentzian functions to the carrier and the sidebands as a guide to the eye.
+
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Kulosa1, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Prudnikov3,4, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Vadlejch1, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' F¨urst1,2,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Kirpichnikova3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Taichenachev3,4, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Yudin3,4,5,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Mehlst¨aubler1,2  1 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,  Germany  2 Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, Welfengarten 1, 30167  Hannover, Germany  3 Institute of Laser Physics, 630090, Novosibirsk, Russia  4 Novosibirsk State University, 630090, Novosibirsk, Russia  5 Novosibirsk State Technical University, 630073, Novosibirsk, Russia  E-mail: andre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='kulosa@ptb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='de  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We report on a comparative analysis of quenched sideband cooling in  trapped ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We introduce a theoretical approach for time-efficient simulation of  the temporal cooling characteristics and derive the optimal conditions providing fast  laser cooling into the ion’s motional ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The simulations were experimentally  benchmarked with a single 172Yb+ ion confined in a linear Paul trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Sideband cooling  was carried out on a narrow quadrupole transition, enhanced with an additional clear-  out laser for controlling the effective linewidth of the cooling transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Quench cooling  was thus for the first time studied in the resolved sideband, intermediate and semi-  classical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We discuss the non-thermal distribution of Fock states during laser  cooling and reveal its impact on time dilation shifts in optical atomic clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Keywords: Tunable laser cooling, atom kinetics, cooling dynamics, trapped ions  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Introduction  Laser cooling is an essential tool for modern quantum optics experiments with trapped  ions, such as the study of topological defect formation in ion Coulomb crystals [1, 2, 3, 4],  quantum simulation [5, 6, 7] and quantum computation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Highly-accurate ion optical  atomic clocks are traditionally operated at the Doppler cooling limit [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  As a  consequence of their constantly improving frequency uncertainty, the time dilation due  to the residual ion secular motion at Doppler temperature nowadays poses a limiting  contribution to the clock’s error budget at the low 10−18 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  In order to realize  ion optical clocks at even lower uncertainties, they will have to operate at ultra-cold  temperatures [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='03276v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='atom-ph]  9 Jan 2023  Systematic study of tunable laser cooling for trapped-ion experiments  2  Since the first implementation of laser cooling (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' [13]), several sub-Doppler  cooling mechanisms have been developed and improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Electromagnetically induced  transparency (EIT) is an established tool to rapidly cool trapped ions into the motional  ground state [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Recently it was applied to cool the strong transverse mode in  a single 171Yb+ ion to the motional ground state within a few 100 µs [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Similar  cooling rates were reported for cooling with polarization gradients (PG) [18] created by  a standing wave field along the trap axis [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' PG cooling of all motional modes in a  single 171Yb+ ion to ¯n ≃ 1 was so far achieved within a few ms [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Besides the aforementioned schemes, sideband cooling in the Lamb-Dicke regime is  a well established technique for trapped ions [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Cooling on a narrow optical transition  can be further enhanced by optical quenching, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' a laser-induced increase of the natural  linewidth γ/2π of the cooling transition [22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' This method has been used in both  trapped ion [25, 26, 27] and neutral atom experiments [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The effective linewidth  γeff/2π is steered with the intensity of the quenching laser, which couples the excited  state of the cooling transition to a short-lived intermediate state (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' This  allows for laser cooling in a broad range of different regimes ranging from the resolved  sideband regime (γeff ≪ ωosc) to the semi-classical regime (γeff > ωosc), which can be  described in terms of sub-Doppler [18, 19] or Doppler (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' [30, 31, 32, 33]) forces  acting from the resonant light field on the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Here, ωosc is the motional frequency the  ion exhibits in the harmonic pseudopotential of the radio-frequency (rf) Paul trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' So  far, optimisation studies of quench cooling were dedicated either to the limits of low  saturation on the cooling transition [23, 34, 35] or to the strong-sideband regime, where  (γeff/2ωosc)2 ≪ η [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' η =  �  ωrec/ωosc is the Lamb-Dicke parameter relating the  ion’s kinetic energy change due to photon recoil with frequency ωrec to its confinement  in the harmonic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  In this work, we go beyond the aforementioned limitations by studying optimised  cooling in the sideband, semi-classical and intermediate regime (γeff ≃ ωosc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We  particularly study the impact of these regimes onto the distribution of atomic Fock  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The intermediate regime is naturally present for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' cooling of In+ ions [38]  and can be used to cool the motional modes of multiple ions in the radial directions  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We especially focus on the nonlinear dependence of cooling time  on cooling field intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We present a versatile method for the calculation of the  characteristic cooling time, which does not require solving the dynamical evolution of  the system’s density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' With this, we determine the optimal conditions for fast and  deep laser cooling with significantly reduced computational efforts when compared to the  full density matrix approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We derive general analytical expressions for the optimal  cooling laser Rabi frequency and the minimum cooling time which can be applied to  any trapped ion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The simulation results are benchmarked against experimental data acquired with a  single 172Yb+ ion confined in a high-precision rf Paul trap [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We study the temporal  evolution of the Fock state distribution during quench cooling and discuss its influence  on the thermal time dilation shift in atomic clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Our findings pave the way towards  Systematic study of tunable laser cooling for trapped-ion experiments  3  fast and deep cooling of larger ion ensembles arranged ion Coulomb crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  This paper is organized as follows: in Chapter 2 we recall the quantum model  used for ion-light interaction and its reduction to an effective two-level system in the  frame of optical quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Chapter 3 describes our simulation approach of using the  “ˆτ-matrix” for faster computation of cooling dynamics, which is used for a systematic  study of the sideband cooling regime in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In Chapter 5 we apply our “ˆτ-  matrix method” to the specific case of an 172Yb+ ion confined in a rf Paul trap and  study cooling ranging from the resolved sideband to the Doppler regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We finally  discuss the non-thermal distribution of Fock states during cooling and its impact on  time dilation shifts in optical atomic clocks in Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Quantum model for ion-light interaction  The cooling dynamics of an ion confined in a rf Paul trap can be described by the  quantum kinetic equation for the density matrix in single particle approximation  ∂ˆρ  ∂t = − i  ℏ  �  ˆH, ˆρ  �  + ˆΓ{ˆρ},  (1)  where ˆH is the Hamiltonian and the term ˆΓ{ˆρ} describes the relaxation of the density  matrix due to spontaneous decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The Hamiltonian is composed of ˆH = ˆHext + ˆHint +  ˆV1 + ˆV2, where  ˆHext = ˆp2  z  2M + Mω2  oscˆz2  2  (2)  is the motional contribution of a harmonically confined ion with mass M in 1D  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Operator ˆV1 describes transitions induced by the cooling laser  E1 = E01  2 exp(ik1z − iω1t) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=',  (3)  being in resonance with the |0⟩ → |1⟩ transition and ˆV2 describes the action of the  quenching field  E2 = E02  2 exp(ik2z − iω2t) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=',  (4)  resonant with the |1⟩ → |2⟩ transition, as depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In the rotating wave  basis the Hamiltonian of the internal ion states is given by  ˆHint = −δ2 ˆP2 − δ1 ˆP1,  (5)  where ˆP1 and ˆP2 are projection operators to the states |1⟩ and |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' δ1 = ω1 − ω10 and  δ2 = ω2 − ω21 are the detunings of the corresponding light fields (3) and (4), where  ω10 and ω21 are the resonance frequencies of the unperturbed |0⟩ → |1⟩ and |1⟩ → |2⟩  transitions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  In the basis of Fock states the interaction operators ˆV1 and ˆV2 contain components  that determine the amplitudes of transitions between the states with different vibrational  Systematic study of tunable laser cooling for trapped-ion experiments  4  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (a) In a three-level system, the decay rate γ1 of state |1⟩ can be increased  by laser-coupling to a higher-lying state |2⟩, which features a fast decay γ3 ≫ γ1 to  the ground state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For low saturation on the |1⟩ → |2⟩ transition, state |2⟩ can  be adiabatically eliminated resulting in an effective two-level system with decay rate  γeff of state |1⟩ [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (b) Temporal evolution of the mean occupation number during  quench cooling with γeff/2π = 50 kHz and ¯nini = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The secular frequency is set to  ωosc/2π = 600 kHz and the cooling light detuning is δ = −ωosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The fastest cooling  rate is observed for a specific Rabi frequency Ωopt of the cooling light, which is not  expected from previous theoretical discussions [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The Rabi frequency induced by the field (i = 1 for the cooling light, and i = 2  for the quenching light) coupling the states with different vibrational numbers m (n) in  the ground (excited) states is determined by the expression (see [21, 40] for details):  Ω(i)  nm = Ω(i)Cnm(ηi),  (6)  with  Cnm(ηi) = L|n−m|  n<  �  η2  i  �  �  n<!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  n>!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (iηi)|n−m| exp  �  −η2  i  2  �  ,  (7)  where n< = min{n, m}, n> = max{n, m}, and L|n−m|  n<  (x) is the generalized Laguerre  polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  A small Lamb-Dicke parameter ηi =  �  ℏk2  i /(2Mωosc) ≪ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  tight  harmonic confinement, significantly reduces the probability of induced and spontaneous  transitions between energy states with different vibrational numbers m ̸= n, where the  Systematic study of tunable laser cooling for trapped-ion experiments  5  ki are the wave vectors of the corresponding light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The full quantum treatment of  the three-level system with the dynamical evolution of the density matrix elements is  discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  As shown in [23], the action of the quenching field E2 onto a three-level system,  leading to a fast decay to the ground state, allows for the reduction to an effective  two-level system for the |0⟩ and |1⟩ states (see figure 1(a)) with effective decay rate  γeff = γ1 + γ3  ρ22  ρ11 ≃ γ1  �  1 + γ3  γ1  S2  �  ,  (8)  where S2 = Ω2  2/[(γ1+γ2+γ3)2+4δ2  2] is the quenching field saturation parameter and the  γi (i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='.3) are the decay rates of the involved transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A quantitative analysis of the  cooling performance is derived from the time evolution of the mean occupation number  ¯n of the Fock states, which according to [34], can be interpolated by an exponential  decay  ¯n(t) = (¯nini − ¯n∞) e−a t + ¯n∞,  (9)  where ¯n∞ is the mean occupation number in steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Based on the dynamic  equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1), we derive the time evolution of ¯n for various Rabi frequencies of the  cooling light, assuming an effective decay rate γeff = 2π ×50 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The initial conditions  correspond to a thermal distribution of Fock states in the electronic ground state with  ¯nini = 20 for an ion secular frequency of ωosc/2π = 600 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As can be seen in figure 1(b),  we expect the existence of an optimal value of cooling laser intensity corresponding to  a maximised cooling rate, which is not predicted by [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In the following, we thus  investigate the optimal cooling parameters for fastest cooling into the motional ground  state in the resolved sideband, intermediate and semi-classical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The “ˆτ-matrix method” for fast simulation of cooling dynamics  The exponential interpolation in equation (9) requires to numerically solve the dynamic  equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1), which, taking into account a large number n of Fock states, requires  significant computational resources to solve for a 2n × 2n density matrix for a two-level  atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' An alternative approach to derive the cooling time is given by the “ˆτ-matrix  method”, recently introduced in [41] for cooling of neutral atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The ˆτ-matrix is given  by the time integral of the difference of the atomic density matrix ˆρ(t) and its steady  state solution ˆρst = ˆρ(t)|t=∞  ˆτ =  � ∞  0  (ˆρ(t) − ˆρst) dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  (10)  The basic equation for the ˆτ-matrix is given by  − i  ℏ  �  ˆH, ˆτ  �  + ˆΓ {ˆτ} = ˆρst − ˆρini,  (11)  where ˆρini = ˆρ(t)|t=0 is the density matrix at initial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As the density matrix ˆρ  contains all information on external and internal states of the quantum system, the  Systematic study of tunable laser cooling for trapped-ion experiments  6  ˆτ-matrix contains all information on temporal characteristics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  As an  example for an observable A characterized by the quantum operator ˆA, the characteristic  evolution time can be extracted from the ˆτ-matrix by the following expression [41]:  τA =  Tr  �  ˆA ˆτ  �  �  Tr  �  ˆA ˆρini  �  − Tr  �  ˆA ˆρst  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  (12)  The cooling time is determined by the evolution rate of the external degrees of freedom  determined by operator ˆHext in equation (2), and thus can be defined as  τeff =  Tr  �  ˆHext ˆτ  �  �  Tr  �  ˆHext ˆρini  �  − Tr  �  ˆHext ˆρst  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  (13)  In the case of a trapped ion this expression can be reduced to  τeff =  1  (nini − n∞)  � ∞  0  (n(t) − n∞) dt  (14)  which exactly corresponds to the 1/e value of an exponential decay of ¯n(t), as given in  equation (9) with characteristic time τ = 1/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As an example, for the results depicted  in figure 1(b) the “ˆτ-matrix method” gives the following values τeff ≃ (115, 31, 55) γ−1  eff ,  with Rabi frequencies Ω/2π = (50, 200, 500) kHz, that are in good agreement with the  results obtained through a fit of the direct numerical solution ¯n(t) by the exponential  function (9), τ ≃ (115, 29, 50) γ−1  eff for corresponding Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Differences  between τ and τeff should become noticeable if the evolution of ¯n(t) is not governed  by an exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The “ˆτ-matrix method” allows to reduce the analysis based on solving the  dynamical equations for the density matrix to a more simple task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' the solution  of the linear equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As an example, taking into account n = 30 Fock states,  computation of the effective cooling time with the “ˆτ-matrix method” takes approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 45 s  in our case, while the direct solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (1) takes 430 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' This allows us to perform  a detailed analysis of the characteristic cooling time of a trapped ion with taking a  sufficiently large number of vibrational states (nmax ≃ 120) into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Dynamics of a trapped ion in the sideband cooling regime  We now use the “ˆτ-matrix method” for a general study of the cooling dynamics in the  resolved sideband cooling regime (where γeff ≪ ωosc) which, in principle, can be applied  to any atomic species featuring a level structure as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  In figures 2(a) and (b) we plot the mean occupation number ¯n∞ and the cooling  time τeff, respectively, as a function of Rabi frequency Ω of the cooling light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We  compare our results to calculations derived from simplified balance-rate equations in  the low-intensity S1 ≪ 1 (S1 is the saturation parameter of the cooling transition)  and strong Lamb-Dicke limit η ≪ 1 (dashed lines), without taking into account the  coherence of the Fock states [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Obtaining similar results for the low-intensity limit,  Systematic study of tunable laser cooling for trapped-ion experiments  7  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  (a) Steady-state mean occupation number and (b) cooling time τeff as  function of cooling laser Rabi frequency Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A minimum cooling time τmin is observed  at optimal Rabi frequency Ωτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The dashed lines in (a) and (b) are derived from  simplified balance-rate equations given in [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (c) Cooling time τeff as function  of initial ¯nini for various Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Quench cooling parameters in (a)-(c) are  γeff/ωosc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1 and detuning δ = −ωosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  our calculations indicate a global minimum in cooling time, τmin, at Rabi frequencies  between Ωτ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='3 ωosc and Ωτ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='4 ωosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' This point defines the optimal parameters for  both fast and simultaneously deep laser cooling, as the mean occupation number has not  significantly changed from its minimum value in the low-intensity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Furthermore, we  observe a linear dependence of cooling time τeff on the initial mean occupation number  ¯nini, as shown in Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As a next step, we study the dependence of the optimal  parameters on the effective quench rate γeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As shown in figure 3(a), we observe that  Ωτ strongly depends on γeff in the sideband cooling regime, but does not much depend  on the Lamb-Dicke parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The available amount of simulation data for cooling  in the sideband regime with parameters γeff/ωosc < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='5, η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='3 and ¯nini < 20 allows us  to deduce analytical expressions from fit results for the optimal Rabi frequency Ωτ  Ωτ ≃  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='24 · γeff · ωosc  (15)  and the minimum cooling time τmin at Ωτ and detuning δ = −ωosc:  τmin ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2 + 2 ¯nini  γeff  +  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='9  η2ωosc  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  (16)  Note that these are general expressions valid for any two-level ion confined in a harmonic  trap with a decay rate γeff and Lamb-Dicke parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We use equations (15) and (16)  to plot the dashed black lines in figure 3 and observe good agreement with the results  for Ωτ and τmin obtained with our direct simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In particular, according to equation  (15), the optimal Rabi frequency providing fast cooling for parameters used in figure  1(b) results to Ωτ/2π ≃ 193 kHz, which is in excellent agreement with the result of the  direct simulation of cooling dynamics shown there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Systematic study of tunable laser cooling for trapped-ion experiments  8  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (a) Optimal Rabi frequency Ωτ for reaching τmin as function of γeff for  various Lamb-Dicke parameters η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The black dashed line has been calculated with the  analytical expression given by Equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (b) Minimum cooling time at optimal  Rabi frequency Ωτ as function of effective decay rate for various ¯nini (indicated by  different symbols) and Lamb-Dicke parameters η (indicated by different colours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The  detuning of the cooling laser corresponds to the first red sideband resonance δ = −ωosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The black dashed lines have been calculated with the expression for τmin given in  Equation (16) and plotted in units of γ−1  eff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Analysis of various quench cooling regimes in 172Yb+  Turning from a general treatment to a specific case, we now apply our “ˆτ-matrix  method” to a 172Yb+ ion confined in a rf Paul trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Figure 4 shows the relevant atomic  states involved in the quench cooling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The |0⟩ → |1⟩ cooling transition is given by  the 2S1/2 → 2D5/2 quadrupole transition near 411 nm, where the 2D5/2 → 2P3/2 dipole  transition near 1650 nm is used as |1⟩ → |2⟩ quenching transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In the following,  we compare the rates and limits of quench cooling in the resolved sideband regime  γeff ≪ ωosc, the semi-classical regime γeff > ωosc and in the intermediate cooling regime  γeff ≃ ωosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  For each of these regimes, Figure 5 shows the results for the mean occupation  number ¯n∞ and the cooling time for different Rabi frequencies induced by the cooling  field E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In the resolved sideband regime, we observe a minimum of ¯n ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='0012 for  a detuning δ = −ωosc at low intensities (Ω = 1/12 ωosc), which is close to the limit  ¯nmin ≃ 7/48 (γeff/ωosc)2 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='001 at low intensity as was obtained similarly to [34, 35],  or ¯nmin ≃ 5/16 (γeff/ωosc)2 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='002 in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Only the linear dependence on light field  intensity was taken into account for ¯n in the simplified balance-rate equations [34, 35],  which gives slightly underestimated results for the mean occupation number compared  to our direct simulation obtained by the “ˆτ-matrix method”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For visibility, we used a  logarithmic scale in Figure 5(a) to pronounce the differences at low intensities (black  dashed and solid curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The results for the cooling time coincide with [34, 35] in  the vicinity of δ = −ωosc at low laser intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  However, with increasing intensity  the difference between the simple model and our direct simulation becomes more  Systematic study of tunable laser cooling for trapped-ion experiments  9  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Relevant atomic levels for quench cooling in 172Yb+ including their decay  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Sideband cooling is carried out on the electronic 2S1/2 →  2D5/2 quadrupole  transition near 411 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The 2D5/2 state is coupled to the short-lived 2P3/2 state by  means of laser light near 1650 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  pronounced (green dashed and solid lines for Ω = 1/3 ωosc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We also observe that the  optimal detuning for minimum ¯n∞ and cooling time shifts to smaller absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Additionally, at low intensity, local minima are also visible near δ/ωosc = −2, −3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='.  These effects are not predicted by the simplified model [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Note the different dependencies of temperature and cooling time on cooling laser  intensity: in the weak-field limit, the ion temperature (or the mean occupation number  ¯n) does not significantly depend on intensity, but cooling time decreases proportional  to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For high-intensity laser fields the temperature significantly grows, but does not  result in essential further decreasing of cooling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Such a dependence of cooling time  and temperature on field intensity allows to define an optimal cooling light intensity for  effective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' fast and simultaneously deep quench cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A similar behaviour can be  observed in the intermediate (figure 5(b)) and semi-classical cooling regime (figure 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  To select the optimal cooling parameters we use the following algorithm: for various  intensities of the cooling light field, we determine the optimal detuning δ∗ providing  the maximum cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' At these values of δ∗, we then analyse the steady-state  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The results are shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The optimal detuning δ∗ shifts  away from the sideband resonance δ = −ωosc with growing intensity (figures 6(a, d,  g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  In each of the studied cooling regimes, a minimum for the cooling time can  be observed in a certain range of cooling light intensity (figures 6(b, e, h)), which  allows to select the optimal intensity for fast cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In figures 6(c, f, i) we plot the  steady-state temperature T and average atomic energy Eav as a result from cooling at  optimal detuning δ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The steady-state temperature, which we plot in units of ℏωosc/kB,  is determined by fitting an exponential distribution to the steady-state populations  governed by ˆρst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The average atomic energy results as the mean value Eav = �  n nPn,  where Pn is a Boltzmann distribution with steady-state temperature T over the atomic  states with secular frequency ωosc/2π = 600 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Note that Eav is plotted in units of  ℏωosc with offset of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='5 to account for the quantum mechanical ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Systematic study of tunable laser cooling for trapped-ion experiments  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
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+page_content='5  Ω = 1/12 ωosc  Ω = 1/6 ωosc  Ω = 1/3 ωosc  Ω = ωosc  Ω = 5/3 ωosc  Ω = 1/12 ωosc  Ω = 1/6 ωosc  Ω = 1/3 ωosc  Ω =  ωosc  Ω = 5/3 ωosc  Ω = 1/3 ωosc  Ω = 2/3 ωosc  Ω =  ωosc  Ω = 5/3 ωosc  Ω = 2 ωosc  Ω = 1/3 ωosc  Ω = 2/3 ωosc  Ω =  ωosc  Ω = 5/3 ωosc  Ω = 2 ωosc  Ω = 1/3 ωosc  Ω = ωosc  Ω = 2 ωosc  Ω = 10/3 ωosc  Ω = 1/3 ωosc  Ω = ωosc  Ω = 2 ωosc  Ω = 10/3 ωosc  W  =  1  /  1  2  w  o  s  c  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Comparison of quench cooling in (a) the sideband cooling regime with  γeff/2π = 50 kHz, (b) intermediate cooling regime with γeff/2π = 600 kHz and (c) semi-  classical cooling regime with γeff/2π = 2000 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For each of the regimes, both ¯n∞ and  the cooling time τeff (in units of the decay rate γ−1  1  of the 2S1/2 → 2D5/2 transition)  are plotted for various Rabi frequencies of the cooling light field E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' All cooling laser  parameters (Ω and δ) are given in units of ion secular frequency ωosc = 2π × 600 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Each cooling simulation started with an initial Boltzmann distribution with ¯nini = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The dashed black and green curves in (a) correspond to ¯n∞ and cooling time at  Ω/ωosc = 1/12 and 1/3 obtained from analytical expressions given in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The absolute minimum cooling temperature is reached in the resolved sideband  cooling regime γeff ≪ ωosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The corresponding minimum cooling time τ ∗ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='4 ×  10−3 γ−1  1  ≃ 100 µs at optimal detuning δ∗ ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='89 ωosc is reached for a Rabi frequency  Ω∗ ≃ ωosc/2 of the cooling light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Here, the steady-state temperature is only slightly  larger than the minimum value obtained in the low-intensity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Both in the  intermediate and semi-classical cooling regime, the minimum cooling time is found for  a Rabi frequency of the cooling light  Ω∗ ≃  �  γ2  eff/2 + 2 ω2  osc ,  (17)  where the ion temperature increases proportionally with increasing Rabi frequency Ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Note that the analytical approximations for the optimum Rabi frequency are again  very general and can be applied to quench cooling in atomic species with similar level  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  To draw a first conclusion on our findings, the intermediate regime is very  promising for fast laser cooling to ¯n < 1, as the observed minimum cooling time  τ ∗ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2 × 10−3 γ−1  1  ≃ 50 µs is very similar to the semi-classical cooling regime, but  at the expense of slightly increased temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
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+page_content='008  0  400  800  1200  1600  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='0  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='4  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='6  (c)  (b)  g  ef f  /2p = 50 kHz  d / w  osc  (a)  t eff  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' g  1  1  E  av  T  E  av  / hw  osc  , k  B  T / hw  osc  W  (kHz)  (f)  (e)  g  ef f  /2p = 600 kHz  (d)  E  av  T  W  (kHz)  (i)  (h)  g  ef f  /2p = 2000 kHz  (g)  E  av  T  W  (kHz)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Optimal parameters for fastest quench cooling in various regimes: resolved  sideband regime (a, b, c), intermediate regime (d, e, f), and semi-classical cooling  regime (g, h, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In each of the cases, the optimal detuning δ∗ will lead to an optimal  cooling time τ ∗ (in units of the decay rate γ−1  1  of the 2S1/2 → 2D5/2 transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The  ion temperature T is obtained by fitting the steady-state population governed by ˆρst  with an exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Using this T in a Boltzmann distribution results to  the average cooling energy Eav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In each of the cooling regimes shown here, the ion  secular frequency was ωosc/2π = 600 kHz and the simulations started with an initial  thermal distribution with ¯nini = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We benchmark our simulation with the “ˆτ-matrix method” against data acquired  with a single 172Yb+ ion confined in a high-precision rf Paul trap [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The rf drive  frequency is Ωrf = 2π × 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='4 MHz leading to a typical secular frequency of the strong  radial mode of ωx = 2π × 565(5) kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In a first stage, the ion is Doppler-cooled to  TD ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='5 mK on the 2S1/2 →  2P1/2 transition near 370 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The resulting thermal  distribution of Fock states features a mean occupation number ¯n = 18(1), measured  with Rabi flops recorded on the 2S1/2 → 2D5/2 transition near 411 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Following the Doppler cooling stage, the ion is initialized in the mJ = −1/2 ground  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The cooling laser near 411 nm is set to a Rabi frequency Ω/2π = 50(2) kHz and  detuning δ where we derive the ion temperature as a function of quench cooling time  tcool via the amplitude ratio R = IBSB/IRSB of blue and red sidebands [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The mean  Systematic study of tunable laser cooling for trapped-ion experiments  12  occupation number is then given by  ¯nSB =  1  R − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  (18)  Each individual data point of the sideband scans is repeated 200 times for significant  statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The effective cooling time τeff is derived as a decay parameter from an  exponential fit, as shown in the inset of figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We characterized the cooling rate in a  frequency interval ranging from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='75 to −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='25 ωosc spanning at least the first two red  secular sidebands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' δ = −ωosc and δ = −2ωosc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The impact of sideband  cooling is well pronounced for a moderate quench with γeff/2π = 50(2) kHz (black curve)  and our measurement is in excellent agreement with the direct simulation in close vicinity  of −ωosc and −2 ωosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' However, we observe a faster cooling in the region between the first  and second red sideband, which we attribute to off-resonant excitation of the sidebands  due to laser noise at 350 kHz (with a FWHM of approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 170 kHz, see also Figure B1  in Appendix B), leading to additional cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As expected, the sideband signature  vanishes for cooling close to the intermediate regime with γeff/2π = 404(21) kHz (blue  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As the cooling laser Rabi frequency is limited by available laser power, we are  not able to resolve an even faster cooling rate in this regime, as predicted in figure 6(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Effective cooling time τeff for quench cooling in the resolved sideband  regime with γeff/2π = 48(2) kHz (black) and close to the intermediate regime with  γeff/2π = 404(21) kHz (blue) as a function of cooling laser detuning in units of ion  secular frequency ωosc = 2π × 565(5) kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The dashed line is derived from analytical  expressions in [34] and the solid curves correspond to our direct simulation with  Ω = 2π×50 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The filled squares and circles are experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Each experiment  was carried out with a laser intensity corresponding to Ω = 2π×50(2) kHz and an initial  thermal distribution with ¯n ≃ 18(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The inset exemplarily shows how the effective  cooling time for γeff/2π = 404(21) kHz and δ = −ωosc was determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Systematic study of tunable laser cooling for trapped-ion experiments  13  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Fock state distribution during quenched sideband cooling  A distribution of Fock states being highly non-thermal after cooling is known to  cause a significant underestimate of temperature [42, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Here, we simulate the  distribution of the ion Fock states for various cooling times in our trap using a Monte-  Carlo simulation, similar to [45], as our previously introduced “ˆτ-matrix method” only  provides the population distribution in steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  As we experimentally deduce  the ion temperature from the sideband amplitude ratio R, we model the red and  blue sideband strengths based on the simulated Fock state distribution and derive a  corresponding theory value for the temperature according to equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  For an  arbitrary Fock state distribution Pn the sideband amplitude ratio R can be expressed  as follows:  R(t) = 1 − �  n Pn cos (Cn+1,n(η)Ω0t)  1 − �  n Pn cos (Cn−1,n(η)Ω0t),  (19)  where Cn+1,n(η) and Cn−1,n(η) are the sideband strength coefficients of the blue and  red sideband, respectively, as given in equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Note that in general the ratio R  is a function of excitation pulse time which is not the case for Pn corresponding to the  thermal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For the thermal distribution, the equation (19) directly implies  the well known relation for ¯n shown in expression (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Figures 8(a) to (c) show the temperature as a function of cooling time for various  effective linewidths γeff/2π, both plotted for the theoretically obtained ¯nSB (black curves)  and the mean value of the distribution �  n nPn (red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  As soon as sideband  cooling is initiated, the data obtained from the measurements (blue squares) indeed  reveal a non-thermal distribution of Fock states for small effective linewidths (figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 8(a)  and (b)), in good agreement with the calculated temperature ¯nSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Note that the 1/e  cooling times of τratio = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='125(2) ms and τmean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='714(9) ms derived for ¯nSB and the  mean value of the distribution, respectively, significantly differ from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  In  order to pronounce the underestimation of atomic temperature and cooling times, we  plot the thermal distributions for ¯n = ¯nSB (orange) and ¯n = �  n n · Pn (beige) in  figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 8(d) to (f) together with the previously simulated Fock state distribution (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  For larger effective linewidths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=', γeff/2π = 64 kHz, the distribution of Fock states  slowly resembles a thermal distribution, as can be seen in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 8(c) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' However,  we still observe a slightly stronger non-thermal distribution in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In any  case, thermal equilibrium of ¯n ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='06(2) is always reached for sufficiently long cooling  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Our findings suggest that the intermediate regime, where γeff ≃ ωosc, is not only  well suited for fast and deep quench cooling to ¯n < 1, moreover it is expected to be  rather immune to non-thermal distributions of Fock states during cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Being of high importance for optical clocks with trapped ions, we calculate the  temporal evolution of thermal time dilation as experienced by a 172Yb+ ion confined  in a linear Paul trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' As a basis for kinetic energy we use the mean �  n nPn of the  simulated distribution of Fock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Figure 9(a) shows the 3D relative time dilation  shift ∆f/f0 = −5/2Ekin/mc2 [46], as a function of cooling time per motional mode  Systematic study of tunable laser cooling for trapped-ion experiments  14  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Impact of a non-thermal distribution of atomic Fock states on temperature  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A Monte-Carlo simulation was used to calculate the population distribution  for various cooling times, if quench cooling is carried out with γeff/2π = 8, 16, 64 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  For each value of tcool this Fock state distribution is used to calculate the strengths of  red and blue sidebands, if driven with a Rabi frequency of Ω411 = 2π×40 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In (a)-(c)  the calculated sideband ratio (black curves) and the mean value of the distribution (red  curves) are plotted as a function of cooling time tcool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The blue squares correspond  to experimentally deduced sideband ratios acquired with (a) γeff/2π = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2(2) kHz,  (b) γeff/2π = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='4(7) kHz and (c) γeff/2π = 62(4) kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The shaded region with  the black curves account for the uncertainty in our experimental Rabi frequency  Ω411 = 2π × 40(2) kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' An exponential decay was fitted to each theory curve and  experimental data to determine the characteristic cooling time with initial temperature  ¯nini = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The fit results are given in (a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (d)-(f) show - exemplarily at tcool = 1 ms  the corresponding simulated Fock state distribution (blue) and thermal distributions  with ¯n = 1/(R − 1) (orange) and ¯n = �  n nPn (beige).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  with ωosc = 2π × 600 kHz for various effective quench rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In each case, the initial  time dilation shift corresponds to the temperature after Doppler cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Unless cooling  is carried out to thermal equilibrium, we observe a significant dependence of the time  dilation shift on the effective quench rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  If cooling was stopped before a thermal  equilibrium is reached, as depicted in figures 8(d)-(f), temperature would be falsely  determined with the sideband ratio and accordingly lead to a wrong estimation of the  time dilation shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For example, if cooling is carried out for 1 ms in the sideband regime  with γeff = 8 kHz (figure 8d), the sideband ratio would suggest a time dilation shift of  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='3 × 10−20 (according to ¯n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='29), while the real distribution of Fock states reveals  a shift of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2 × 10−19 (according to ¯n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In this regime, the error budget of  an atomic clock would be underestimated by one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  For reduced  discrepancies, it is advised to cool in a regime where γeff/ωosc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1, as indicated in  figure 8(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Finally, we would like to point out that the time dilation shift seen by  the ion is highly dependent on the cooling laser intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In figure 9(b) we fixed the  Systematic study of tunable laser cooling for trapped-ion experiments  15  700  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 3D relative time dilation shift experienced by a 172Yb+ ion confined in a  linear Paul trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Here we assume a secular frequency of ωosc = 2π × 600 kHz for each  motional mode and plot the time dilation as a function of quench cooling time per  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The simulated mean value of the distribution �  n nPn amongst the Fock  states, as shown in blue in figure 8, was used as a basis for the shift calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (a)  The Rabi frequency was set to Ω/2π = 40 kHz whilst the effective quench rate was  varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' (b) For an effective quench rate γeff = 64 kHz we observe an optimal Rabi  frequency of Ω/2π = 200 kHz allowing for fast quench cooling to the point of thermal  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  effective quench rate to γeff = 64 kHz and studied the impact of various cooling laser  intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In accordance with our observation of an optimal Rabi frequency for fastest  cooling, as shown in figures 1(b) and 6(b) , this value also allows to reach the point of  thermal equilibrium as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Summary  To conclude, we studied the mechanism of quenched sideband cooling and presented a  versatile method for fast calculation of the characteristic cooling time which does not  require to consider the dynamical evolution of the system’s density matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Our “ˆτ-matrix method” significantly reduces computational efforts and agrees within  90 − 100% with the full quantum model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Based on our powerful simulation tool, we derived universal analytical expressions  for the optimum Rabi frequency Ωτ and the resulting minimum cooling time τmin for  cooling in the resolved sideband regime, which can be applied to any atomic species  with decay rate γeff confined in a rf Paul trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Applying the simulation to a specific  case, we investigated quench cooling in a 172Yb+ ion, thereby focussing on three different  regimes: (I) resolved sideband cooling with γeff/2π = 50 kHz, (II) intermediate cooling  with γeff/2π = 600 kHz and (III) semi-classical cooling with γeff/2π = 2000 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For  each of these regimes we derived the steady-state temperature and the effective cooling  Systematic study of tunable laser cooling for trapped-ion experiments  16  time for various cooling laser parameters, such as Rabi frequency and detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' From  this extensive analysis we extracted the optimal parameters to be applied for fast cooling  into the motional ground state in each of the aforementioned regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We benchmarked  our simulation code against data taken with a single 172Yb+ ion confined with a secular  frequency ωosc = 2π × 565(5) kHz and observed an agreement between experiment and  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  We presented a detailed analysis of Fock state population distributions during the  cooling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In particular, we compared the time behaviour of temperature derived  with the sideband ratio method to the actual mean occupation of Fock states and  revealed discrepancies of more than one order of magnitude, if cooling is not carried out  to thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' In addition, we investigated the temporal evolution of thermal  time dilation in an optical clock with 172Yb+ during the process of quench cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We  conclude that quenching in a regime with γeff/ωosc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1 is necessary to stay close to a  thermal distribution of Fock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The results presented in this work pave the way to flexible quench cooling of ion  Coulomb crystals with respect to fast laser cooling into the ion’s motional ground  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A crystal consisting of N ions requires 3N motional modes to be cooled, thus  a clever combination of confinement parameters and quench rate is expected to reduce  the cooling time compared to cooling of each mode individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Acknowledgments  We thank Jonas Keller for useful comments on optical spectroscopy in the presence  of laser noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  This work was funded by the Deutsche Forschungsgemeinschaft  (DFG, German Research Foundation) under German’s Excellence Strategy—EXC-2123  QuantumFrontiers—390837967 and by DFG Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' ME 3648/5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
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+page_content=' F¨urst was  supported by the EMPIR project 18SIB05 ”Robust Optical Clocks for International  Timescales”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 11 011002  [40] Leibfried D, Blatt R, Monroe C and Wineland D 2003 Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 75 281  [41] Il’enkov R Y, Prudnikov O N, Taichenachev A V and Yudin V I 2016 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 123  1–11  [42] Che H, Deng K, Xu Z T, Yuan W H, Zhang J and Lu Z H 2017 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A 96 013417  [43] Chen J S, Brewer S M, Chou C W, Wineland D J, Leibrandt D R and Hume D B 2017 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Systematic study of tunable laser cooling for trapped-ion experiments  18  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 118 053002  [44] Rasmusson A J, D’Onofrio M, Xie Y, Cui J and Richerme P 2021 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A 104 043108  [45] Mølmer K, Castin Y and Dalibard J 1993 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' B 10 524–538  [46] Herschbach N, Pyka K, Keller J and Mehlst¨aubler T E 2012 Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' B 107 891–906  [47] Prudnikov O N, Taichenachev A V, Tumaikin A M and Yudin V I 2007 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' A 75 023413  [48] Kirpichnikova A A, Prudnikov O N and Wilkowski D 2019 Quantum Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 49 443  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Dynamical evolution of density matrix elements  The equation for the evolution of the components of the density matrix of vibrational  states n, m (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 1(a)) takes the following form:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − (γ3 + γ2)ρ22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm − iωosc(n − m)ρ22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµρ12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ∗ (2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − γ1ρ11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm + ˆγ2{ρ22}nm − iωosc(n − m)ρ11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω∗ (2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµ ρ21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm + Ω(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµρ01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ∗ (1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ00  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= ˆγ3{ρ22}nm + ˆγ1{ρ11}nm − iωosc(n − m)ρ00  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω∗ (1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµ ρ10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − (γ1/2 + iδ1) ρ01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm − iωosc(n − m)ρ01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω∗ (1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµ ρ11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ00  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ∗ (1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − (γ1/2 − iδ1) ρ10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm − iωosc(n − m)ρ10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµρ00  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm + Ω∗ (2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµ ρ20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − (γ1/2 + γ2/2 + γ3/2 + iδ2) ρ12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− iωosc(n − m)ρ12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµρ02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm + Ω∗ (2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµ ρ22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ∗ (2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − (γ1/2 + γ2/2 + γ3/2 − iδ2) ρ21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− iωosc(n − m)ρ21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµρ11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ∗ (1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Systematic study of tunable laser cooling for trapped-ion experiments  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − (γ2/2 + γ3/2 + γ + iδ1 + iδ2) ρ02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− iωosc(n − m)ρ02  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω∗ (1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµ ρ12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ01  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ∗ (2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂ρ20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='∂t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='= − (γ2/2 + γ3/2 + Γ − iδ1 − iδ2) ρ20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− iωosc(n − m)ρ20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='− i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='Ω(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµρ10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm − ρ21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='nµΩ(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='µm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1)  where the superscripted indices 0, 1, 2 of the density matrix elements denote the states  |0⟩, |1⟩ and |2⟩, respectively and the subscripted indices n, m and µ are related to  vibrational states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Γ determines the decay of the coherence between the |2⟩ and |0⟩  states as a result of an uncorrelated phase of the laser fields E1 and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  The spontaneous relaxation operator ˆγ{ρ}nm determines the spontaneous decay  rate γ1, γ2, and γ3 (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 1 (a)), the decay of non-diagonal elements of the density  matrix, as well as contributions from excited states to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' These terms in  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='1) have the form  ˆγi{ρjj}nm =  �  νµ  Γ(i) νµ  nm ρjj  νµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='2)  The decay rates Γ(i) νµ  nm  for the dipole transitions |2⟩ → |1⟩ and |2⟩ → |0⟩ can be obtained  from the general expression for the spontaneous relaxation operator taking into account  recoil effects (see for example [47])  Γ(i) νµ  nm  = γi  � +1  −1  K(d)(h) (Cνn(ηih))+ Cµm(ηih) dh,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='3)  with K(d)(h) being the dipole pattern for the decay and ηi are the Lamb-Dicke  parameters for the corresponding dipole transitions (i = 2, 3 in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 1 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  For  the quadrupole transition the relaxation operator has a similar form to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='3) with  replacement of the dipole with the quadrupole pattern for the decay K(q)(h) and  corresponding Lamb-Dicke parameter η1 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' Off-resonant excitation via laser noise  We attribute the faster cooling rate observed in our experiment at cooling laser detunings  around −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='5 ωosc (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 7) to off-resonant excitation induced by laser noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  For  verification, we studied the frequency spectrum of the 2S1/2 →  2D5/2 transition near  411 nm ranging from the first-order red sideband to the first-order blue sideband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Figure B1 shows such a frequency spectrum, recorded overdriven with interrogation  time t ≃ 7 · τπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' The spectral features at ±350 kHz correspond to noise modulation of  Systematic study of tunable laser cooling for trapped-ion experiments  20  the laser light, most probably caused by the bandwidth of the locking electronics of  the second harmonic generation cavity for 411 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' While the frequency detuning of  the 411 nm cooling laser was set to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='5 ωosc, with ωosc = 2π × 565(5) kHz, both the  first and second-order red sidebands were seperated by 283(5) kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' With a FWHM of  approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' 310 kHz of the noise spectral feature, a significant overlap to both red sidebands  is given and most likely the reason for faster cooling observed in the experiment in this  frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content='  Frequency spectrum of the 2S1/2 →  2D5/2 transition near 411 nm,  relative to the carrier transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' For this measurement, the 172Yb+ ion was confined  with secular frequency ωosc = 2π × 654(6) kHz and interrogated with a pulse time of  t ≃ 7 · τπ to enhance the signature of laser noise between the sideband resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}
+page_content=' We  fit Lorentzian functions to the carrier and the sidebands as a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E1T4oBgHgl3EQfkwQJ/content/2301.03276v1.pdf'}